Statistical field theory with constraints: application to critical   Casimir forces in the canonical ensemble

Markus Gross; Andrea Gambassi; S. Dietrich

arXiv:1705.11012·cond-mat.stat-mech·August 17, 2017

Statistical field theory with constraints: application to critical Casimir forces in the canonical ensemble

Markus Gross, Andrea Gambassi, S. Dietrich

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TL;DR

This paper develops a statistical field theory approach to analyze how imposing a fixed total order parameter (canonical ensemble) affects finite-size corrections and Casimir forces in critical systems, revealing significant differences from the grand canonical ensemble.

Contribution

It introduces a perturbative method to systematically compute constraint-induced finite-volume corrections in the Landau-Ginzburg model, specifically for the critical Casimir force in constrained ensembles.

Findings

01

Constraint enhances the attractive Casimir force compared to grand canonical case.

02

For thick films, the canonical Casimir force remains negative across boundary conditions.

03

The force's dependence on temperature and scaling variables differs between ensembles.

Abstract

The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter, is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of $ϵ = 4 - d$ , where $d$ is the spatial dimension of the bulk, the finite-size contribution to the free energy…

Tables1

Table 1. Table 1: Glossary of quantities frequently used in the present study. † A subscript R 𝑅 R on a quantity indicates its renormalized counterpart (see Sec. IV ). Periodic, Dirichlet, and Neumann boundary conditions are indicated by the superscripts (p), (D), and (N), respectively. ‡ The canonical ensemble corresponds to the special case 𝓌 = 1 𝓌 1 \mathpzc{w}=1 .

quantity^†	description	definition in
$ϕ$	order parameter (OP) field	Sec. I
$Φ$	total OP (“total mass”) in the system	Eq. 1
$φ$	mean OP, $φ = Φ / V$	Eq. 48
$𝓌$	weight function^‡	Eqs. 6 and 52
$Σ w$	constrained value of the weighted total OP $Φ$	Eq. 6
$Σ$	constrained value of the total OP, $Σ \equiv Σ_{1}$	Eq. 53
$\overset{˚}{𝒵}$	constrained (canonical) partition function^‡	Eq. 9
$𝒵$	unconstrained (grand canonical) partition function	Eq. 11
$ℋ$	effective Hamiltonian	Eqs. 8 and 11
$ℒ$	effective free energy functional	Eqs. 8 and 49
$h$	bulk field	Eqs. 11 and 51
$μ$	Lagrange multiplier associated with the constraint	Eqs. 32 and 54
$ψ$	mean OP field	Eq. 13
$σ$	fluctuation part of the OP field	Eq. 13
$G$	Green function	Eqs. 21 and 60
$\overset{˚}{G}$	constraint-induced Green function	Eq. 26
$ℱ$	unconstrained (grand canonical) film free energy	Eqs. 35 and 119
$\overset{˚}{ℱ}$	constrained (canonical) film free energy	Eqs. 34, 75, and 106
$d$	spatial dimension of the film	Secs. III and 1
$L$	film thickness	Secs. III and 1
$A$	transverse area	Secs. III and 1
$V$	film volume, $V = A L$	Secs. III and 1
$z$	coordinate along the transverse direction	Secs. III and 1
$𝐫_{∥}$	coordinates along the lateral directions	Secs. III and 1
$τ$	temperature parameter	Eq. 49
$g$	quartic coupling constant	Eq. 49
$t$	reduced (renormalized) temperature	Eqs. 50 and 132
$\hat{τ}$	effective temperature parameter	Eq. 79
$ς (z)$	eigenfunctions	Eq. 65
$ρ$	aspect ratio	Eq. 104
$𝒻_{res}, \overset{˚}{} 𝒻_{res}$	residual finite-size free energy per volume	Eqs. 115 and 126
$𝒮$	scaling function of the regularized mode sum	Eqs. 108 and 189
$u^{*}$	fixed point value of the renormalized quartic coupling constant	Eq. 130
$r$	numerical constant	Eq. 131
$x$	finite-size scaling variable associated with $t$	Eq. 132
$\hat{x}$	scaled effective temperature parameter	Eq. 134
$𝓂$	scaled OP	Eq. 132
$𝒽$	scaled bulk field	Eq. 135
$ξ_{+}^{(0)}$ , $ξ_{φ}^{(0)}$ , $ξ_{h}^{(0)}$	correlation length amplitudes associated with $t$ , $φ$ , and $h$	Eqs. 133 and 137
$𝒦, \overset{˚}{𝒦}$	critical Casimir force (CCF)	Eq. 144
$Θ, \overset{˚}{Θ}$	scaling functions of the residual free energy	Eqs. 132 and 135
$Ξ, \overset{˚}{Ξ}$	scaling functions of the CCF	Eqs. 146 and 149

Equations506

Φ = \int_{V} d^{d} r ϕ (r)

Φ = \int_{V} d^{d} r ϕ (r)

\int_{r} \equiv \int_{V} d^{d} r

\int_{r} \equiv \int_{V} d^{d} r

(u, v) \equiv \int_{r} u (r) v (r),

(u, v) \equiv \int_{r} u (r) v (r),

(G, v)_{r} \equiv \int_{r^{'}} G (r, r^{'}) v (r^{'}) = \int_{r^{'}} G (r^{'}, r) v (r^{'}),

(G, v)_{r} \equiv \int_{r^{'}} G (r, r^{'}) v (r^{'}) = \int_{r^{'}} G (r^{'}, r) v (r^{'}),

(u, G, v) \equiv \int_{r} \int_{r^{'}} u (r) G (r, r^{'}) v (r^{'}) .

(u, G, v) \equiv \int_{r} \int_{r^{'}} u (r) G (r, r^{'}) v (r^{'}) .

(\mathpzc w, ϕ) \equiv \int_{r} \mathpzc w (r) ϕ (r) = Σ_{\mathpzc w},

(\mathpzc w, ϕ) \equiv \int_{r} \mathpzc w (r) ϕ (r) = Σ_{\mathpzc w},

\ring P ([ϕ], Σ w) \equiv \frac{1}{\ring Z} exp (- H [ϕ]) δ ((\mathpzc w, ϕ) - Σ_{\mathpzc w}),

\ring P ([ϕ], Σ w) \equiv \frac{1}{\ring Z} exp (- H [ϕ]) δ ((\mathpzc w, ϕ) - Σ_{\mathpzc w}),

H [ϕ] \equiv \int_{r} L (r; [ϕ])

H [ϕ] \equiv \int_{r} L (r; [ϕ])

\ring Z (Σ_{\mathpzc w}) \equiv \int D ϕ exp (- H [ϕ]) δ ((\mathpzc w, ϕ) - Σ_{\mathpzc w}) = \int_{- \infty}^{\infty} \frac{d J}{2 π a ^{- 1 - d /2}} \int D ϕ exp (- H [ϕ] + i J (\mathpzc w, ϕ) - i J Σ_{\mathpzc w}),

\ring Z (Σ_{\mathpzc w}) \equiv \int D ϕ exp (- H [ϕ]) δ ((\mathpzc w, ϕ) - Σ_{\mathpzc w}) = \int_{- \infty}^{\infty} \frac{d J}{2 π a ^{- 1 - d /2}} \int D ϕ exp (- H [ϕ] + i J (\mathpzc w, ϕ) - i J Σ_{\mathpzc w}),

\int D ϕ = i = 1 \prod N \int_{- \infty}^{\infty} \frac{d ϕ _{1}}{a ^{1 - d /2}} \int_{- \infty}^{\infty} \frac{d ϕ _{2}}{a ^{1 - d /2}} \dots \int_{- \infty}^{\infty} \frac{d ϕ _{N}}{a ^{1 - d /2}},

\int D ϕ = i = 1 \prod N \int_{- \infty}^{\infty} \frac{d ϕ _{1}}{a ^{1 - d /2}} \int_{- \infty}^{\infty} \frac{d ϕ _{2}}{a ^{1 - d /2}} \dots \int_{- \infty}^{\infty} \frac{d ϕ _{N}}{a ^{1 - d /2}},

Z (h) \equiv \int D ϕ exp (- H (h; [ϕ])), with H (h; [ϕ]) \equiv H [ϕ] - h \int_{r} ϕ .

Z (h) \equiv \int D ϕ exp (- H (h; [ϕ])), with H (h; [ϕ]) \equiv H [ϕ] - h \int_{r} ϕ .

Z (h) = \int_{- \infty}^{\infty} d Σ e^{h Σ} \ring Z (Σ) .

Z (h) = \int_{- \infty}^{\infty} d Σ e^{h Σ} \ring Z (Σ) .

ϕ (r) = ψ (r) + σ (r) .

ϕ (r) = ψ (r) + σ (r) .

⟨ σ (r)⟩ = 0,

⟨ σ (r)⟩ = 0,

⟨ Σ w ⟩ = Σ_{\mathpzc w} = \int_{r} \mathpzc w (r) [ψ (r) + ⟨ σ (r)⟩] = \int_{r} \mathpzc w (r) ψ (r) \equiv (\mathpzc w, ψ),

⟨ Σ w ⟩ = Σ_{\mathpzc w} = \int_{r} \mathpzc w (r) [ψ (r) + ⟨ σ (r)⟩] = \int_{r} \mathpzc w (r) ψ (r) \equiv (\mathpzc w, ψ),

\int_{r} \mathpzc w (r) σ (r) = 0.

\int_{r} \mathpzc w (r) σ (r) = 0.

H [ψ + σ] = H [ψ] + \int_{r_{1}} \frac{δ H [ ψ ]}{δ ψ ( r _{1} )} σ (r_{1}) + \frac{1}{2 !} \int_{r_{1}} \int_{r_{2}} \frac{δ ^{2} H [ ψ ]}{δ ψ ( r _{1} ) δ ψ ( r _{2} )} σ (r_{1}) σ (r_{2}) + \int_{r} V (r; [ψ, σ]),

H [ψ + σ] = H [ψ] + \int_{r_{1}} \frac{δ H [ ψ ]}{δ ψ ( r _{1} )} σ (r_{1}) + \frac{1}{2 !} \int_{r_{1}} \int_{r_{2}} \frac{δ ^{2} H [ ψ ]}{δ ψ ( r _{1} ) δ ψ ( r _{2} )} σ (r_{1}) σ (r_{2}) + \int_{r} V (r; [ψ, σ]),

V (r; [ψ, σ]) \equiv \frac{1}{3 !} \int_{r_{1}} \int_{r_{2}} \frac{δ ^{3} H [ ψ ]}{δ ψ ( r _{1} ) δ ψ ( r _{2} ) δ ψ ( r )} σ (r_{1}) σ (r_{2}) σ (r) + \dots .

V (r; [ψ, σ]) \equiv \frac{1}{3 !} \int_{r_{1}} \int_{r_{2}} \frac{δ ^{3} H [ ψ ]}{δ ψ ( r _{1} ) δ ψ ( r _{2} ) δ ψ ( r )} σ (r_{1}) σ (r_{2}) σ (r) + \dots .

H [ϕ] \to H [ϕ] - \int_{r} K (r) σ (r) .

H [ϕ] \to H [ϕ] - \int_{r} K (r) σ (r) .

H^{(2)} (r_{1}, r_{2}; [ψ]) \equiv \frac{δ ^{2} H [ ψ ]}{δ ψ ( r _{1} ) δ ψ ( r _{2} )},

H^{(2)} (r_{1}, r_{2}; [ψ]) \equiv \frac{δ ^{2} H [ ψ ]}{δ ψ ( r _{1} ) δ ψ ( r _{2} )},

\int_{r_{2}} G (r_{1}, r_{2}) H^{(2)} (r_{2}, r_{3}) = \int_{r_{2}} H^{(2)} (r_{1}, r_{2}) G (r_{2}, r_{3}) = δ (r_{1} - r_{3}),

\int_{r_{2}} G (r_{1}, r_{2}) H^{(2)} (r_{2}, r_{3}) = \int_{r_{2}} H^{(2)} (r_{1}, r_{2}) G (r_{2}, r_{3}) = δ (r_{1} - r_{3}),

\int D σ exp (- \frac{1}{2} σ_{i} A_{ij} σ_{j} + K_{i} σ_{i}) = \frac{( 2 π ) ^{N /2}}{( det A ) ^{1/2}} exp (\frac{1}{2} K_{i} A_{ij}^{- 1} K_{j})

\int D σ exp (- \frac{1}{2} σ_{i} A_{ij} σ_{j} + K_{i} σ_{i}) = \frac{( 2 π ) ^{N /2}}{( det A ) ^{1/2}} exp (\frac{1}{2} K_{i} A_{ij}^{- 1} K_{j})

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\int_{-\infty}^{\infty}\frac{\mathrm{d}J}{2\pi}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K-\mathrm{i}J\mathpzc{w},G,\frac{\delta\mathcal{H}}{\delta\psi}-K-\mathrm{i}J\mathpzc{w}\right)+\mathrm{i}J(\mathpzc{w},\psi)-\mathrm{i}J\Sigma_{\mathpzc{w}}\Bigg{\}}.

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\int_{-\infty}^{\infty}\frac{\mathrm{d}J}{2\pi}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K-\mathrm{i}J\mathpzc{w},G,\frac{\delta\mathcal{H}}{\delta\psi}-K-\mathrm{i}J\mathpzc{w}\right)+\mathrm{i}J(\mathpzc{w},\psi)-\mathrm{i}J\Sigma_{\mathpzc{w}}\Bigg{\}}.

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\frac{1}{\sqrt{2\pi}}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\frac{\delta\mathcal{H}}{\delta\psi}-K\right)-\frac{1}{2}\ln(\mathpzc{w},G,\mathpzc{w})-\frac{1}{2}\frac{\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\mathpzc{w}\right)^{2}}{(\mathpzc{w},G,\mathpzc{w})}\\ +\frac{\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\mathpzc{w}\right)\left[(\mathpzc{w},\psi)-\Sigma_{\mathpzc{w}}\right]}{(\mathpzc{w},G,\mathpzc{w})}-\frac{1}{2}\frac{[(\mathpzc{w},\psi)-\Sigma_{\mathpzc{w}}]^{2}}{(\mathpzc{w},G,\mathpzc{w})}\Bigg{\}}.

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\frac{1}{\sqrt{2\pi}}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\frac{\delta\mathcal{H}}{\delta\psi}-K\right)-\frac{1}{2}\ln(\mathpzc{w},G,\mathpzc{w})-\frac{1}{2}\frac{\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\mathpzc{w}\right)^{2}}{(\mathpzc{w},G,\mathpzc{w})}\\ +\frac{\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\mathpzc{w}\right)\left[(\mathpzc{w},\psi)-\Sigma_{\mathpzc{w}}\right]}{(\mathpzc{w},G,\mathpzc{w})}-\frac{1}{2}\frac{[(\mathpzc{w},\psi)-\Sigma_{\mathpzc{w}}]^{2}}{(\mathpzc{w},G,\mathpzc{w})}\Bigg{\}}.

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\frac{1}{\sqrt{2\pi}}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\frac{\delta\mathcal{H}}{\delta\psi}-K\right)-\frac{1}{2}\ln(\mathpzc{w},G,\mathpzc{w})-\frac{1}{2}\frac{\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\mathpzc{w}\right)^{2}}{(\mathpzc{w},G,\mathpzc{w})}\Bigg{\}}.

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\frac{1}{\sqrt{2\pi}}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\frac{\delta\mathcal{H}}{\delta\psi}-K\right)-\frac{1}{2}\ln(\mathpzc{w},G,\mathpzc{w})-\frac{1}{2}\frac{\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\mathpzc{w}\right)^{2}}{(\mathpzc{w},G,\mathpzc{w})}\Bigg{\}}.

\ring G (r_{1}, r_{2}) \equiv G (r_{1}, r_{2}) - \frac{( G , \mathpzc w ) _{r_{1}} ( G , \mathpzc w ) _{r_{2}}}{( \mathpzc w , G , \mathpzc w )},

\ring G (r_{1}, r_{2}) \equiv G (r_{1}, r_{2}) - \frac{( G , \mathpzc w ) _{r_{1}} ( G , \mathpzc w ) _{r_{2}}}{( \mathpzc w , G , \mathpzc w )},

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\frac{1}{\sqrt{2\pi}}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,\ring{G},\frac{\delta\mathcal{H}}{\delta\psi}-K\right)-\frac{1}{2}\ln(\mathpzc{w},G,\mathpzc{w})\Bigg{\}}.

\ring{\mathcal{Z}}(\Sigma_{\mathpzc{w}};[K])=\frac{1}{\sqrt{2\pi}}\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,\ring{G},\frac{\delta\mathcal{H}}{\delta\psi}-K\right)-\frac{1}{2}\ln(\mathpzc{w},G,\mathpzc{w})\Bigg{\}}.

(\ring G, \mathpzc w)_{r} = \int_{r^{'}} \ring G (r, r^{'}) \mathpzc w (r^{'}) = 0

(\ring G, \mathpzc w)_{r} = \int_{r^{'}} \ring G (r, r^{'}) \mathpzc w (r^{'}) = 0

\ring{\mathcal{Z}}_{0}(\Sigma{w};[K])\equiv\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}+\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\frac{\delta\mathcal{H}}{\delta\psi}-K\right)\Bigg{\}}

\ring{\mathcal{Z}}_{0}(\Sigma{w};[K])\equiv\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}[\psi]-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}+\frac{1}{2}\left(\frac{\delta\mathcal{H}}{\delta\psi}-K,G,\frac{\delta\mathcal{H}}{\delta\psi}-K\right)\Bigg{\}}

\mathcal{Z}(h;[K])=\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}(h;[\psi])-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}(h;[\psi])}{\delta\psi}-K,G,\frac{\delta\mathcal{H}(h;[\psi])}{\delta\psi}-K\right)\Bigg{\}}.

\mathcal{Z}(h;[K])=\exp\left\{-\int_{\mathbf{r}}\mathcal{V}\left(\mathbf{r};\left[\psi,\sigma\to\frac{\delta}{\delta K(\mathbf{r})}\right]\right)\right\}\exp\Bigg{\{}-\mathcal{H}(h;[\psi])-\frac{1}{2}\mathrm{tr}\ln\mathcal{H}^{(2)}\\ +\frac{1}{2}\left(\frac{\delta\mathcal{H}(h;[\psi])}{\delta\psi}-K,G,\frac{\delta\mathcal{H}(h;[\psi])}{\delta\psi}-K\right)\Bigg{\}}.

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Full text

Statistical field theory with constraints:

application to critical Casimir forces in the canonical ensemble

Markus Gross

[email protected]

Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany

IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Andrea Gambassi

SISSA – International School for Advanced Studies and INFN, via Bonomea 265, 34136 Trieste, Italy

S. Dietrich

Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, 70569 Stuttgart, Germany

IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Abstract

The effect of imposing a constraint on a fluctuating scalar order parameter field in a system of finite volume is studied within statistical field theory. The canonical ensemble, corresponding to a fixed total integrated order parameter (e.g., the total number of particles), is obtained as a special case of the theory. A perturbative expansion is developed which allows one to systematically determine the constraint-induced finite-volume corrections to the free energy and to correlation functions. In particular, we focus on the Landau-Ginzburg model in a film geometry (i.e., in a rectangular parallelepiped with a small aspect ratio) with periodic, Dirichlet, or Neumann boundary conditions in the transverse direction and periodic boundary conditions in the remaining, lateral directions. Within the expansion in terms of $\epsilon=4-d$ , where $d$ is the spatial dimension of the bulk, the finite-size contribution to the free energy of the confined system and the associated critical Casimir force are calculated to leading order in $\epsilon$ and are compared to the corresponding expressions for an unconstrained (grand canonical) system. The constraint restricts the fluctuations within the system and it accordingly modifies the residual finite-size free energy. The resulting critical Casimir force is shown to depend on whether it is defined by assuming a fixed transverse area or a fixed total volume. In the former case, the constraint is typically found to significantly enhance the attractive character of the force as compared to the grand canonical case. In contrast to the grand canonical Casimir force, which, for supercritical temperatures, vanishes in the limit of thick films, in the canonical case with fixed transverse area the critical Casimir force attains for thick films a negative value for all boundary conditions studied here. Typically, the dependence of the critical Casimir force both on the temperature- and on the field-like scaling variables is different in the two ensembles.

I Introduction

In general, statistical ensembles of systems of finite size are not equivalent Lebowitz et al. (1967); Allen and Tildesley (1989); Roman et al. (2008). The primary reason is that imposing a constraint on an extensive thermodynamic variable restricts the fluctuation spectrum of that quantity. For instance, for a fluid the total number of particles is fixed in the canonical ensemble, whereas it fluctuates in the grand canonical one. While liquids are typically studied in the grand canonical ensemble Hansen and McDonald (2006), there is a number of cases in which the difference between the canonical and the grand canonical ensemble becomes significant: most notably, these are systems composed of relatively few particles, such as fluids confined to nanoscale pores or capillaries Gonzalez et al. (1997, 1998). This issue has prompted the development of canonical density functional methods White et al. (2000); White and Velasco (2001); White and Gonzalez (2002); de las Heras and Schmidt (2014) which explicitly take fluctuation corrections into account. Recently, static and dynamic critical phenomena have been investigated also within molecular dynamics Puri and Frisch (1993); Das et al. (2006a, b); Roy and Das (2011); Das et al. (2012); Roy et al. (2016); Puosi et al. (2016) or lattice Boltzmann simulations Gross and Varnik (2012a, b). These simulation methods typically operate in the canonical ensemble and require finite-size corrections in order to extract physical properties of bulk systems Lebowitz and Percus (1961); Salacuse et al. (1996); Roman et al. (1997, 2008). Ensemble differences have also been studied extensively in the context of Bose-Einstein condensation (see, e.g., Refs. Ziff et al. (1977); Gajda and Rza̧żewski (1997); Glaum et al. (2007)).

In the present study, we consider statistical field theory for an order parameter (OP) field $\phi(\mathbf{r})$ , which represents, for instance, the deviation of the density of a one-component fluid from its critical value or the deviation of the local concentration from the critical composition of a binary liquid mixture. For simplicity, henceforth we adopt the notation pertaining to a one-component fluid. While the field theory discussed here is rather general, explicit results for the residual finite-size free energy and the critical Casimir force (CCF) are obtained for the so-called $\phi^{4}$ -Landau-Ginzburg model in a film geometry. We use the notion film for a finite system of volume $V$ with an aspect ratio smaller than unity, while the thin-film limit refers to the limit of a vanishing aspect ratio. The volume integral

[TABLE]

represents the “total mass” in the system, which can fluctuate in the grand canonical ensemble but is fixed to a certain value in the canonical ensemble. This constraint is mirrored by the fluctuations within the system and, as shown here, it turns out to typically enhance the attractive character of the CCF. For a general introduction to the topic of CCFs, we refer to Refs. Krech (1994); Brankov et al. (2000); Gambassi (2009). There are relatively few theoretical studies which focus on the effect of an OP constraint on critical phenomena under confinement Eisenriegler and Tomaschitz (1987); Brankov and Danchev (1989); Blöte et al. (2000); Caracciolo et al. (2001); Pleimling and Hüller (2001); Gulminelli et al. (2003); Deng et al. (2005). Constraining a non-ordering degree of freedom which is coupled to the OP gives rise to the so-called Fisher renormalization of critical exponents and amplitudes Fisher (1968); Imry et al. (1973); Achiam and Imry (1975); Anisimov et al. (1995); Krech (1999); Mryglod and Folk (2001); Izmailian and Kenna (2014). A discussion of ensemble differences for critical fluid films within mean field theory (MFT) is presented in Ref. Gross et al. (2016) for so-called $(++)$ and $(+-)$ boundary conditions, where $\pm$ denotes surface fields of strength $h_{1}=\pm\infty$ , which express the preference of the confining walls for one or the other coexisting liquid phase.

In the present study, we investigate the effect of the OP constraint on the OP fluctuations, focusing on systems of finite volume with periodic, Dirichlet, or Neumann boundary conditions. Within the framework of boundary critical phenomena, the latter two realize the so-called ordinary and special surface universality class, respectively Diehl (1986). In the case of Dirichlet boundary conditions, we focus on the case of zero total mass $\Phi=0$ [Eq. 1], while, for periodic and Neumann boundary conditions, we consider also nonzero values of $\Phi$ . In Ref. Gross et al. (2016), it has been shown that an OP constraint can induce drastic qualitative changes in the CCF, affecting, inter alia, its sign and its decay behavior upon increasing the film thickness or the associated scaling variables. These changes occur already within MFT, i.e., in the absence of fluctuations. Here it is useful to recall that, within MFT and under the same thermodynamic conditions Gross et al. (2016), the film pressures are identical in both ensembles. Accordingly, in this situation, the differences in the CCF are due to the differences in the bulk pressures. In turn, they arise because in the two ensembles film and bulk are coupled differently: in the grand canonical ensemble, film and bulk experience the same chemical potential, whereas, in the canonical ensemble, it is natural to require that film and bulk have the same density. As it will be shown in the present study, fluctuations induce a further change of the CCF in addition to this mean field effect, since the OP constraint explicitly affects the film pressure itself, rather than only the coupling between film and bulk.

The present study is organized as follows: In Sec. II, the statistical field theory which accounts for an OP constraint is presented and the construction of the associated perturbation theory is described. In Sec. III, this field theory is specialized to the Landau-Ginzburg model in a finite volume, and various boundary conditions are investigated. In particular, perturbative expressions of the residual finite-size contribution to the free energy are derived. In Sec. IV, these results are cast into scaling form, and the corresponding scaling functions for the finite-size free energy and the CCFs are obtained. Our main results are discussed in Sec. V and summarized in Sec. VI. Important details of calculations are presented in Appendices A–E. A glossary of the most frequently used quantities is provided in Table 1.

II Statistical field theory with a global constraint

II.1 Notation and conventions

In order to simplify the presentation of the analytical calculations carried out in the present study, we introduce the shorthand notation

[TABLE]

for the integration over a finite, $d$ -dimensional volume $V$ . Following Ref. Rudnick and Jasnow (1981), we define, for two arbitrary scalar functions $u(\mathbf{r})$ and $v(\mathbf{r})$ as well as for a function $G(\mathbf{r},\mathbf{r}^{\prime})$ which is symmetric with respect to its two arguments, the shorthand notations

[TABLE]

and

[TABLE]

In particular, we have $(G,1)_{\mathbf{r}}\equiv\int_{\mathbf{r}^{\prime}}G(\mathbf{r},\mathbf{r}^{\prime})$ . A ring ( $\ring{\>}$ ) above a quantity indicates that it refers to a constrained system.

II.2 General framework

A method to cope with an OP constraint within a statistical field theory has been described in Ref. Rudnick and Jasnow (1981) and is recalled briefly here. Building upon this approach, we study the free energy and correlation functions, focusing on the corrections induced by the constraint, and develop a systematic perturbation theory in the canonical ensemble. We consider in this section a finite $d$ -dimensional volume $V$ with no additional restriction on its geometry. In Sec. III, the theory developed here will be applied to more specific systems. The fluctuating OP field $\phi(\mathbf{r})$ is required to satisfy a constraint of the form

[TABLE]

where $\Sigma_{\mathpzc{w}}$ is a constant and $\mathpzc{w}(\mathbf{r})$ is a given weight function. The case of total mass conservation corresponds to $\mathpzc{w}=1$ . In fact, our expressions generally represent approximations of the true free energy of a constrained system. (An exception is the Gaussian model, for which exact results can be obtained.) The linear nature of Eq. 6 is sufficiently flexible to encompass constraints which fix the value of $\phi$ or its derivative at a certain point $\mathbf{s}$ in space, corresponding to the choices $\mathpzc{w}(\mathbf{r})=\delta(\mathbf{r}-\mathbf{s})$ and $\mathpzc{w}(\mathbf{r})=\delta^{\prime}(\mathbf{r}-\mathbf{s})$ , respectively. In addition, the present framework can be straightforwardly extended to encompass more than a single constraint.

Under the effect of the constraint in Eq. 6, the statistics of the field $\phi$ is governed by the constrained probability distribution

[TABLE]

where

[TABLE]

is the effective Hamiltonian which controls the statistics of the fluctuations of $\phi$ in the absence of the constraint and $\mathcal{L}$ is its density. Accordingly, the constrained partition function $\ring{\mathcal{Z}}$ is given by

[TABLE]

where in the last equation we have made use of the Fourier representation of the $\delta$ -function. As usual, the functional integration in Eq. 9 is defined as the limit $N\to\infty$ of the multiple integrals over a field $\phi_{i}=\phi(\mathbf{r}_{i})$ , $i=1,\ldots,N$ defined on a lattice of size $N$ Zinn-Justin (2002), i.e.,

[TABLE]

where the quantity $a$ represents the lattice constant, the presence of which in Eqs. 9 and 10 renders the partition function dimensionless. However, in order to simplify the notation and because $a$ formally vanishes in the continuum limit, we shall henceforth not indicate it; $a$ can be re-instantiated straightforwardly into the various expressions on the basis of dimensional analysis and of Eqs. 9 and 10. As a consequence, certain logarithms will seemingly have dimensionful arguments, while, in fact, in the corresponding lattice field theory, these arguments are multiplied by suitable powers of $a$ which renders them dimensionless 111Alternatively, this issue can be dealt with by considering $\ring{\mathcal{Z}}/\ring{\mathcal{Z}}_{\text{ref}}$ , where $\ring{\mathcal{Z}}_{\text{ref}}$ is a chosen reference partition function Zinn-Justin (2002); Eisenriegler and Tomaschitz (1987). However, such a definition induces a shift of the associated free energy, which is undesired for the present purposes Krech (1994).. Concerning an example, we refer to the explicit calculations within a lattice field theory presented in Appendix A. We shall occasionally comment on this issue further [see, e.g., Eq. 75 below]. Returning to Eq. 9, we remark that, although $\mathcal{H}[\phi]$ can in principle depend on external fields, this dependence does not affect the construction of the constrained partition function $\ring{\mathcal{Z}}$ and therefore it will not be considered henceforth. The specific expression of $\mathcal{L}$ is not relevant for the general discussion in this section, which will be put in practice for the Landau-Ginzburg model in Sec. III.

The grand canonical partition function $\mathcal{Z}(h)$ in the presence of a (spatially uniform) external field $h$ is given by

[TABLE]

It immediately follows from the first equation in Eq. 9 that, for $\mathpzc{w}=1$ , $\mathcal{Z}(h)$ is related to the canonical partition function $\ring{\mathcal{Z}}(\Sigma)$ at a fixed order parameter $\Sigma\equiv\Sigma_{1}$ via

[TABLE]

This equation forms the basis of many finite-size studies of the grand canonical free energy and of the CCF Rudnick et al. (1985); Brezin and Zinn-Justin (1985); Schloms and Dohm (1989); Dohm (2008, 2011). In contrast to the perturbative approach developed below, in the grand canonical ensemble Eq. 12 treats fluctuations of the total OP non-perturbatively. This allows one to overcome the well-known artifacts related to the presence of a so-called zero mode. We will return to this aspect in Sec. III.4.

Following standard approaches Brezin et al. (1976); Rudnick and Jasnow (1981), the partition functions in Eqs. 9 and 11 are evaluated by means of a saddle-point approximation. To this end, the OP field $\phi(\mathbf{r})$ is split into its mean part $\psi(\mathbf{r})\equiv\langle\phi(\mathbf{r})\rangle$ and a fluctuation $\sigma(\mathbf{r})$ ,

[TABLE]

Accordingly, the integration measure $\int\mathcal{D}\phi$ in Eqs. 9 and 11 turns into $\int\mathcal{D}\sigma$ and Eq. 13 implies that

[TABLE]

where the average $\langle\ldots\rangle=\int\mathcal{D}\phi\ldots\mathcal{\ring{P}}([\phi],\Sigma_{w})$ is performed over the probability distribution given in Eq. 7. The mean OP $\psi$ is left unspecified at this point, but at a later stage it will be determined self-consistently from Eq. 14, which in fact reduces to the equation of state relating $\psi$ and $h$ in the grand canonical and $\psi$ and $\Sigma$ in the canonical ensemble, respectively [see Eq. 32 below]. In the following, we focus on developing a perturbation theory in the presence of a constraint; we simply state the corresponding and well-known Brezin et al. (1976); Zinn-Justin (2002) results in the absence of it.

Inserting Eq. 13 into the constraint in Eq. 6 yields, after averaging,

[TABLE]

i.e., the constant value $\Sigma{w}$ of the constraint is entirely determined by the nonfluctuating part $\psi$ of the OP alone. As an immediate consequence of Eqs. (15) and (6) one finds that the weighted volume integral of the fluctuations must vanish:

[TABLE]

Returning to the calculation of $\ring{\mathcal{Z}}$ , we expand the action $\mathcal{H}$ in terms of $\sigma$ as Brezin et al. (1976)

[TABLE]

where, extending the analysis presented in Ref. Rudnick and Jasnow (1981), we account also for non-Gaussian contributions in the action via the potential $\mathcal{V}$ :

[TABLE]

In order to facilitate the calculation of correlation functions, we add to $\mathcal{H}$ a source term $K(\mathbf{r})$ which couples to the fluctuation $\sigma(\mathbf{r})$ , i.e., in the generating functional in Eq. 9 we replace $\mathcal{H}[\phi]$ according to

[TABLE]

Denoting the quadratic part of the action by

[TABLE]

the Green function $G(\mathbf{r}_{1},\mathbf{r}_{2})$ is defined as the inverse of $\mathcal{H}^{(2)}$ :

[TABLE]

with $G(\mathbf{r}_{1},\mathbf{r}_{2})=G(\mathbf{r}_{2},\mathbf{r}_{1})$ .

In order to proceed, we recall that, for an $N\times N$ matrix $A_{ij}$ and fields $K_{i}$ , $\sigma_{j}$ , the following fundamental result for multidimensional Gaussian integrals holds Zinn-Justin (2002):

[TABLE]

(with summation over repeated indices), as well as the identity $\ln\det A=\mathrm{tr}\ln A$ . With the aid of these relations, the linear and quadratic parts of the action in Eq. 9 can now be integrated over $\sigma$ , yielding

[TABLE]

In the exponent in Eq. 23 we have neglected the term $(N/2)\ln(2\pi)$ stemming from the prefactor on the right hand side of Eq. 22. This term turns infinite in the continuum limit and leads to an unimportant additive shift of the free energy. If $(\mathpzc{w},G,\mathpzc{w})\neq 0$ , one obtains, after performing the Gaussian integration over $J$ , the constrained generating functional

[TABLE]

Due to the constraint expressed by Eq. 15, the last two terms in $\ring{\mathcal{Z}}[K]$ vanish so that

[TABLE]

The last two terms in Eq. 25 emerge as a direct consequence of the constraint. Introducing a Green function $\ring{G}$ which accounts for the constraint as

[TABLE]

the constrained generating functional in Eq. 25 finally reduces to

[TABLE]

It is useful to note that

[TABLE]

for all $\mathbf{r}$ , which follows immediately from Eq. 26. Returning to Eq. 23, we find that, if $(\mathpzc{w},G,\mathpzc{w})=0$ , the integral over $J$ is readily obtained as

[TABLE]

instead of Eq. 27. The case $(\mathpzc{w},G,\mathpzc{w})=0$ occurs for models where the complete set of fluctuation modes [see Eqs. 68 and 73 below] respect the constraint from the outset. For the specific systems investigated in the present study (see Sec. III) one actually has $(\mathpzc{w},G,\mathpzc{w})\neq 0$ and therefore the constrained partition function is the one in Eq. 27. Aside from occasional comments, we shall therefore no longer consider the case $(\mathpzc{w},G,\mathpzc{w})=0$ . Finally, repeating the above derivation for the grand canonical partition function in Eq. 11, one obtains the well-known generating functional Brezin et al. (1976); Zinn-Justin (2002)

[TABLE]

In the case $\mathpzc{w}=1$ , corresponding to a constraint on the total OP, we observe that $\ring{\mathcal{Z}}(\Sigma;[K])$ in Eq. 27 has, apart from the last term in the curly brackets, the same expression as $\mathcal{Z}(h;[K])$ in Eq. 30 provided one replaces $G$ by $\ring{G}$ and $\mathcal{H}(h;[\psi])$ by $\mathcal{H}[\psi]$ . Accordingly, also the perturbation theory for the constrained case based on Eq. 27 or Eq. 29 leads to expressions formally analogous to those in the unconstrained case based on Eq. 30. Note that, even if the constraint does not explicitly appear in the expression of $\ring{\mathcal{Z}}_{0}$ , it still acts via Eqs. (15) and (16), which have to be fulfilled in the construction of $\psi$ and $\sigma$ (see Sec. II.3 below).

II.3 Gaussian approximation

Here, we investigate the constrained generating functional in Eq. 27 within the Gaussian approximation, i.e., neglecting the non-quadratic interactions collected summarily in the potential $\mathcal{V}$ [Eq. 18]. Within this approximation, the condition in Eq. 14 results in

[TABLE]

Due to the property of $\ring{G}$ expressed in Eq. 28, this condition can be satisfied by requiring Rudnick and Jasnow (1981)

[TABLE]

where the spatially constant $\mu$ can be interpreted as a Lagrange multiplier which must be chosen in order to satisfy the constraint in Eq. 15, which leads to $\mu\Sigma{w}=\left(\delta\mathcal{H}/\delta\psi,\psi\right)$ 222Note that the choice $\delta\mathcal{H}/\delta\psi=0$ , which also satisfies Eq. 31, does not lead to a dependence of $\psi$ on $\mu$ , making it impossible to satisfy Eq. 15 in general.. Owing to the dependence of $\mathcal{H}^{(2)}$ on $\psi$ [see Eq. 20], the constraint also affects the fluctuations described by the theory, which will be discussed further in Sec. III. In the case $\mathpzc{w}=1$ , which corresponds to total OP conservation, Eq. 32 represents the equation of state within mean field approximation and $\mu$ plays the role of a bulk field or of the chemical potential. If $(\mathpzc{w},G,\mathpzc{w})=0$ , Eq. 31 must be evaluated with $\ring{\mathcal{Z}}$ [see Eq. 29] replaced by $\ring{\mathcal{Z}}_{0}$ , which yields $(G,\delta\mathcal{H}/\delta\psi)=0$ . This condition can be fulfilled by Eq. 32 with $\mu=0$ , as in the grand canonical case.

Once the mean OP $\psi$ is fixed according to Eq. 32, the constrained generating functional in Eq. 27 reduces to

[TABLE]

As remarked above, terms involving $\mathpzc{w}$ are absent in the analogous expression for the partition function in the unconstrained case or if $(\mathpzc{w},G,\mathpzc{w})=0$ . From Eq. 33, the constrained free energy $\ring{\mathcal{F}}$ within the Gaussian approximation (i.e., at one-loop order) follows as

[TABLE]

For comparison, we also report here the corresponding expression for the unconstrained free energy $\mathcal{F}$ , which, according to Eq. 30, is given by Amit and Martin-Mayor (2005); Zinn-Justin (2002); Brezin et al. (1976)

[TABLE]

In the expression for the free energy [Eq. 34] we keep numerical constants such as $(1/2)\ln(2\pi)$ , because they are required for a consistent relation between the canonical and the grand canonical ensembles according to Eq. 12 (see also Ref. Dohm (2011)). Explicit expressions of $\ring{\mathcal{F}}$ and $\mathcal{F}$ will be presented below in Sec. III, where also the required regularization is discussed. The constraint induced two-point correlation function $\ring{C}$ of the OP fluctuations $\sigma$ follows from Eq. 33 as

[TABLE]

where, as before, the last term is only present if $(\mathpzc{w},G,\mathpzc{w})\neq 0$ . From Eq. 36 it follows directly that

[TABLE]

for all $\mathbf{r}$ , consistently with Eq. 28. In the unconstrained case, the two-point correlation function $C$ coincides with the Green function, i.e., $C=G$ Zinn-Justin (2002); Amit and Martin-Mayor (2005). In contrast, within the Gaussian approximation, the constraint affects the free energy [Eq. 34] and the correlation function [Eq. 36] in two ways: explicitly, via the generation of correction terms involving $\mathpzc{w}$ and, implicitly, via the dependence of $\psi$ on $\mu\mathpzc{w}$ and $\Sigma_{\mathpzc{w}}$ as required by Eq. 32. The latter dependence is a consequence of the fact that the operator $\mathcal{H}^{(2)}$ and, therefore, also the Green function $G$ , which is its functional inverse [Eq. 21], are affected by the constraint only via their dependence on $\psi$ . However, the analytic form of $\mathcal{H}^{(2)}$ and $G$ , as well as the spectrum and the form of the eigenfunctions of $\mathcal{H}^{(2)}$ , are identical in the constrained and the unconstrained cases (see Sec. III below). The fact that the constraint restricts the allowed modes of a fluctuation [see Eq. 16] is accounted for by additive corrections to the free energy [Eq. 34] and the correlation function [Eq. 36]. The meaning of these terms will be further elucidated in Sec. III, where we apply the present framework to specific systems.

II.4 Perturbation theory

In order to be able to illustrate the perturbative calculation of corrections beyond the Gaussian approximation, an expression for the interaction potential $\mathcal{V}$ in Eq. 27 has to be specified. We assume in the following that the corresponding interaction term in $\mathcal{L}$ [Eq. 8] is of the form $g\phi(\mathbf{r})^{4}/4!$ [see also Eq. 49 below], where $g>0$ is a coupling constant. It is well-known that a model based on such a density $\mathcal{L}$ captures properly the universal features associated with critical phenomena in the Ising universality class Zinn-Justin (2002); Amit and Martin-Mayor (2005). Apart from this interaction, no additional non-quadratic terms in $\phi$ are assumed to appear in $\mathcal{L}$ . (This is in line with the vanishing of the coupling constants of the other higher order terms under renormalization group flow.) For this choice of $\mathcal{L}$ , the potential $\mathcal{V}$ defined in Eq. 18 becomes

[TABLE]

where the last expression serves to reveal the functional form and pointlike interaction character of $\mathcal{V}$ . Note the appearance of a three-point vertex proportional to the mean field OP $\psi$ .

II.4.1 Mean order parameter

As a first application, we calculate the perturbative correction to $\mathcal{O}(g)$ of the mean field expression for $\psi$ . Using Eq. 38, the generating functional in Eq. 27 up to $\mathcal{O}(g)$ becomes

[TABLE]

from which the condition in Eq. 14, which defines $\psi$ , results as

[TABLE]

In obtaining the r.h.s. of the last equation, we used $\ln(1+X)\simeq X$ . Analogously to Eq. 31, Eq. 40 must be solved for $\delta\mathcal{H}/\delta\psi$ up to $\mathcal{O}(g)$ . Importantly, at this stage, no assumption concerning the order in $g$ of $\psi$ should be made, i.e., $\psi$ should be formally assumed to be of $\mathcal{O}(g^{0})$ . It is only after imposing the corresponding equation of state [see Eq. 41 below] that $\psi$ turns into a quantity of $\mathcal{O}(g^{-1/2})$ [compare also Eq. 143 and the associated discussion]. The solution of Eq. 40 can thus be iteratively constructed as a series in $g$ by considering $\delta\mathcal{H}/\delta\psi$ and $\psi$ in Eq. 40 to be formally of $\mathcal{O}(g^{0})$ . This yields a perturbatively corrected version of Eq. 32:

[TABLE]

Making use of Eq. 28, we find that this expression of $\delta\mathcal{H}/\delta\psi$ indeed solves Eq. 40 up to and including $\mathcal{O}(g)$ and therefore it implicitly provides the desired leading-order perturbative correction to the mean OP $\psi$ . Note that Eq. 41 in fact coincides (upon interpreting $\mu$ as an external field) with the corresponding expression in the grand canonical ensemble Wilson (1980); Diehl and Dietrich (1981). As in Eq. 32, the parameter $\mu$ in Eq. 41 has to be chosen such that the constraint on $\psi$ in Eq. 15 is fulfilled. We recall that $\ring{G}$ itself depends on $\psi(\mathbf{r})$ through its definition in Eq. 21 as the inverse of $\mathcal{H}^{(2)}$ . In practice, Eq. 41 must therefore be solved iteratively (see Sec. III.2 for further discussion).

In order to obtain the perturbative corrections at $\mathcal{O}(g)$ to the free energy or to a correlation function, Eq. 41 has to be imposed as an implicit definition of $\psi$ . As a consequence, $\psi$ becomes a quantity of $\mathcal{O}(g^{-1/2})$ . Inserting Eq. 41 for $\delta\mathcal{H}/\delta\psi$ into Eq. 27 and using Eqs. 28 and 38, yields the generating functional valid up to $\mathcal{O}(g)$ :

[TABLE]

where we have used the compact notation introduced in Eq. 5, e.g., $(\psi\ring{G},\ring{G},K)=\int_{\mathbf{r}}\int_{\mathbf{r}^{\prime}}\psi(\mathbf{r})\ring{G}(\mathbf{r},\mathbf{r})\ring{G}(\mathbf{r},\mathbf{r}^{\prime})K(\mathbf{r}^{\prime})$ . The term in curly brackets in Eq. 42 arises from an expansion of the first exponential term in Eq. 27, keeping only those terms which contribute up to $\mathcal{O}(g)$ , taking into account that $\psi\sim\mathcal{O}(g^{-1/2})$ . It is interesting to specialize Eq. 42 to the case $\mathpzc{w}=1$ and a translationally invariant system, e.g., a uniform system with periodic boundary conditions in all directions. In this case, one has a spatially constant $\psi(\mathbf{r})=\varphi$ as well as $\ring{G}(\mathbf{r},\mathbf{r})=\ring{G}(\mathbf{0})$ , i.e., also the Green function evaluated at coinciding arguments does not depend on the spatial location 333The Green function $G(\mathbf{r},\mathbf{r})$ is formally infinite and therefore requires a suitable regularization Zinn-Justin (2002). However, this aspect does not affect the conclusions in the present paragraph.. Accordingly, using Eq. 28 with $\mathpzc{w}=1$ , one obtains $(\psi\ring{G},\ring{G},\ldots)=\varphi\ring{G}(\mathbf{0})(1,\ring{G},\ldots)=0$ , implying that the second and the third term in the second exponential of Eq. 42 vanish in this case.

II.4.2 Free energy

The constrained free energy to $\mathcal{O}(g)$ (recall that $\psi\sim\mathcal{O}(g^{-1/2})$ ) follows from Eq. 42 as

[TABLE]

where, as before, we have approximated $\ln(1+X)\simeq X$ in order to evaluate the contribution to $\mathcal{O}(g)$ from the logarithm in the second equation 444 The term $(1/8)g^{2}\left(\psi\ring{G},\ring{G},\psi\ring{G}\right)$ in Eq. 42 is canceled by the perturbative corrections generated by the last term in curly brackets in Eq. 42.. We remark that the expression in Eq. 43 reduces to the corresponding two-loop result for periodic boundary conditions obtained in a different context in Refs. Guo and Jasnow (1987); Dohm (1989). The fourth term in the last equation of Eq. 43 involving the constraint-induced Green function $\ring{G}$ can be rewritten as

[TABLE]

where have we used Eq. 26 as well as the symmetry of $G$ with respect to an exchange of its arguments. An analogous expression applies also to the last term in Eq. 43. Equation (44) explicitly shows the higher-order contributions to the free energy stemming from the constraint. The two-loop constrained free energy including the required renormalization will be discussed further elsewhere.

II.5 Summary

In this section, a statistical field theory for an OP field subject to the integral constraint given in Eq. 6 has been developed based on the approach introduced in Ref. Rudnick and Jasnow (1981). The special case $\mathpzc{w}=1$ of the weight function, which enters into the definition of the constraint, leads to a theoretical description within the canonical ensemble. In order to estimate the typical magnitude of the constraint induced corrections to the free energy [Eq. 34] and to the correlation function [Eq. 36], we consider, having periodic boundary conditions in mind, a film geometry of volume $V=AL$ , where $A$ is the transverse area and $L$ the film thickness. Since the Green function $G$ in fact represents the correlation function, one finds the estimate

[TABLE]

where $\chi$ denotes the global OP susceptibility. Accordingly, one obtains an estimate for the correction term on the r.h.s. in Eq. 26:

[TABLE]

These relations are confirmed below in Sec. III by means of analytical calculations for the Landau-Ginzburg model [see, e.g., Eq. 81 below]. Based on Eq. 46 we conclude that the constraint correction to the Green function vanishes for a system of infinite volume:

[TABLE]

Extending this analysis to the free energy, we note that (after introducing a suitable regularization, see Sec. III.3) the first two terms on the r.h.s. of Eq. 34 scale $\propto V$ at leading order. According to Eq. 45b, the constraint correction $\propto\ln(1,G,1)$ , instead, scales $\propto\ln V$ and, therefore, the constraint correction becomes irrelevant in the thermodynamic limit ( $AL\to\infty$ ) and ensemble equivalence is recovered. In this case, the canonical and the grand canonical free energies are related via a Legendre transform.

Note that the infinite-volume limit encompasses the case in which fewer than the $d$ dimensions of the system become infinite—in particular also the case $A\to\infty$ at fixed $L$ (thin-film limit). Consider, for instance, for fixed $L$ and $A\to\infty$ , the situation at the critical point: assuming that the correlation length $\xi$ scales with the largest size in the system and that the system exhibits critical behavior of a $(d-1)$ -dimensional system, we have $\chi\propto\xi^{2-\eta}\propto A^{(2-\eta)/(d-1)}$ with the usual critical exponent $\eta$ . Hence, provided $d\geq 3-\eta$ , the result in Eq. 47 is expected to hold also near criticality.

We emphasize that this analysis does not imply that in the thin-film limit the residual finite-size free energy or the CCF are generally equivalent among the various ensembles. Indeed, as has been shown in Ref. Gross et al. (2016), this is not the case for systems with inhomogeneities caused by external bulk or surface fields. The reason for the ensemble inequivalence of the CCF in such systems is that the CCF refers to a bulk system, the coupling of which to the film itself depends on the ensemble Gross et al. (2016). In fact, if one considers the thin-film limit, which is natural for MFT, it is reasonable to define the constraint with respect to the total “mass” $\Phi$ per transverse area $A$ , such that Eq. 6 reduces to $\int_{L}\mathrm{d}z\,\phi(z)=\Sigma/A=\mathrm{const}.,$ with formally $\mathpzc{w}=1/A$ . This definition is motivated by the idea that the thermodynamic limit should in general be performed by keeping the mean OP

[TABLE]

(e.g., the particle density) constant.

In a finite volume, the OP constraint generally modifies the fluctuations, reflecting the fact that those fluctuations which change the total number of particles in the system [or, in general, the value of the integral in Eq. 6] are not permitted within the canonical ensemble. This also means that the canonical free energy [Eq. 34] is no longer the Legendre transform of the grand canonical one [Eq. 35], but it exhibits additive corrections [represented by the last term in Eq. 34] 555Indeed, applying a Legendre transform to the grand canonical free energy [Eq. 35] would yield a canonical free energy without the constraint-induced correction terms, which is incorrect in finite volumes. In order to obtain the correct canonical finite-size free energy, the constraint must be imposed at the level of the partition function [Eq. 9].. While the presence of these finite-volume corrections is in principle known White et al. (2000); Roman et al. (2008); Binney et al. (1992), they have so far not been systematically discussed within a statistical field theory and their significance for critical Casimir forces in the canonical ensemble has not been elucidated.

We close this section by summarizing the essential consequences of the OP constraint.

The constraint causes the presence of a bulk field-like parameter $\mu$ in the equation which determines the mean OP $\psi$ [see Eqs. (32) and (41)] Rudnick and Jasnow (1981). This parameter essentially corresponds to the Lagrange multiplier associated with the constrained minimization of the action within the mean field approximation Gross et al. (2016). 2. 2.

The Green function $G$ , defined as the inverse of the quadratic part of the action [see Eq. 20], is affected by the constraint only implicitly, via its dependence on the mean OP $\psi$ . 3. 3.

Within the Gaussian approximation, the constrained free energy $\ring{\mathcal{F}}$ differs from the unconstrained one $\mathcal{F}$ by an additive correction of the form $(1/2)\ln\left[2\pi(\mathpzc{w},G,\mathpzc{w})\right]$ [Eq. 34]. 4. 4.

By introducing a constrained Green function $\ring{G}$ [see Eq. 26], the generating functional in Eq. 27 assumes the same form as in the unconstrained case with a nonzero mean OP. As a consequence, perturbation theory can be introduced analogously, implying that perturbative results formally carry over from the unconstrained to the constrained theory by simply replacing the usual Green function $G$ with $\ring{G}$ . 5. 5.

Within the Gaussian approximation, the two-point correlation function $\ring{C}$ of the OP fluctuations [Eq. 36] is modified compared to the unconstrained case such that its weighted integral vanishes [Eq. 37]. The constrained Gaussian correlation function is, in fact, identical to the constrained Green function, i.e., $\ring{G}=\ring{C}$ . Contributions from higher loop orders provide further constraint corrections to the free energy and to correlation functions. Specific examples are given in Eq. 41 for the one-loop equation determining the constraint-induced OP and in Eq. 43 for the two-loop free energy. 6. 6.

In the limit of infinite volume $V\to\infty$ , the constraint-induced fluctuation corrections to the free energy and the correlation function vanish and $\ring{G}$ reduces to $G$ [see Eq. 47]. As remarked above, for the idealized case of a thin film with a transverse area $A\to\infty$ at fixed thickness $L$ , mean-field contributions to the model can still be affected by the constraint Gross et al. (2016).

III Application to the Landau-Ginzburg model

III.1 Model and general results

In the remaining part of this study, we consider a $d$ -dimensional film of volume $V=AL$ which is translationally invariant and has periodic boundary conditions along the first $(d-1)$ lateral directions, but which can be inhomogeneous in the remaining direction ( $z$ ) of extent $L$ , as sketched in Fig. 1. The boundaries of the film are taken to be located at $z=0$ and $z=L$ , while we indicate the coordinates along the lateral directions by the subscript $\parallel$ , i.e., we decompose the generic position vector as $\mathbf{r}=(\mathbf{r}_{\parallel},z)$ . We shall interchangeably use the notation $G(\mathbf{r}_{\parallel},\mathbf{r}^{\prime}_{\parallel},z,z^{\prime})$ for the Green function $G(\mathbf{r},\mathbf{r}^{\prime})$ . The subsequent discussion shows how the field-theoretical formalism, which is well-known in the grand canonical ensemble Diehl (1986); Brezin et al. (1976); Krech and Dietrich (1992), carries over to the canonical case.

In the following we focus on the one-loop (Gaussian) approximation of the field theory developed in Sec. II. This approximation already displays the essential effects induced by the constraint. Specifically, we consider the scalar Landau-Ginzburg form of the effective free energy density [Eq. 8], i.e.,

[TABLE]

where the second equation defines the effective bulk and surface free energy densities $\mathcal{L}_{b}$ and $\mathcal{L}_{s}$ , respectively. The parameter $\tau$ is proportional to the reduced temperature

[TABLE]

where $T_{c}$ is the bulk critical temperature; $g>0$ is a coupling constant, $h_{1}$ is a surface field, and $c$ is the so-called surface enhancement Diehl (1986). The interaction potential $\mathcal{V}$ [see Eq. 18] which pertains to the action in Eq. 49 has already been reported in Eq. 38. In the grand canonical ensemble, we additionally consider a bulk field $h$ and define

[TABLE]

In order to simplify the notation, we occasionally suppress the dependence of $\mathcal{L}$ , $\mathcal{L}_{b}$ , and $\mathcal{L}_{s}$ on the parameters $\tau$ , $g$ , $h_{1}$ , and $c$ , and write $\mathcal{L}[\phi(\mathbf{r})]\equiv\mathcal{L}(\mathbf{r};[\phi])$ (analogously for $\mathcal{L}_{b}$ and $\mathcal{L}_{s}$ ). Henceforth, in the notation of Eq. 6 we set

[TABLE]

i.e., as it is the case for the canonical ensemble, a constraint is imposed on the spatial integral of the OP [see Eqs. 15 and 16]:

[TABLE]

where $\Sigma$ is the imposed total mass in the system. Since we assume translational invariance in the lateral directions, the mean profile $\psi(\mathbf{r})=\psi(z)$ is a function of $z$ only.

At the leading order, which corresponds to the mean field approximation, in the canonical ensemble $\psi(z)$ is determined by Eq. 32, which yields, for $\mathcal{L}$ given in Eq. 49,

[TABLE]

This expression implies the Euler-Lagrange equation

[TABLE]

and the boundary conditions

[TABLE]

The parameter $\mu$ is the Lagrange multiplier required to satisfy the OP constraint in Eq. 53. Dirichlet boundary conditions [ $\psi(0)=\psi(L)=0$ ] are realized for $|h_{1}|<\infty$ and $c\to\infty$ , while (within MFT) Neumann boundary conditions hold [ $\partial_{z}\psi(0)=\partial_{z}\psi(L)=0$ ] for $h_{1}=0$ and $c=0$ . Upon accounting for the one-loop corrections, Eq. 55 is modified as in Eq. 41 and it turns into 666Concerning methods to solve Eq. 57 we refer to Refs. Diehl and Dietrich (1981); Diehl and Smock (1993), which discuss the analogous grand canonical case.

[TABLE]

Here, for simplicity, we use the notation $\ring{G}(z,z)\equiv\ring{G}(\mathbf{r}_{\parallel},\mathbf{r}_{\parallel},z,z)$ which, due to translation invariance, does actually not depend on $\mathbf{r}_{\parallel}$ . We anticipate that consistency with the $\epsilon$ -expansion of the one-loop free energy [which includes terms up to $\mathcal{O}(\epsilon^{0})$ ] requires to use the mean-field Euler-Lagrange equation in Eq. 55 instead of Eq. 57 in order to obtain $\psi(z)$ . The reason is that $\psi$ itself is a quantity of $\mathcal{O}(g^{-1/2})$ , implying that the last term on the r.h.s. in Eq. 57 becomes formally of $\mathcal{O}(\epsilon^{1/2})$ (see, e.g., Ref. Amit and Martin-Mayor (2005) and the discussion in Sec. IV.1.2). Accordingly, in the grand canonical ensemble one has, analogously to Eq. 55, the mean-field equation of state

[TABLE]

Equation (56) continues to hold for the boundary conditions in the grand canonical case.

Unless specified otherwise, the following expressions apply to both the canonical and the grand canonical ensemble, because neither the constraint-induced field $\mu$ nor the bulk field $h$ appear explicitly in them. Instead, the information about the constraint or the external field is implicitly contained in the mean-field contribution $\psi$ (see also the discussion in Sec. II.5). The expression of $\mathcal{H}^{(2)}$ [Eq. 20] follows from a second functional differentiation of the r.h.s. of Eq. 54 with respect to $\psi(\mathbf{r}^{\prime})$ :

[TABLE]

Accordingly, the definition in Eq. 21 yields the following differential equation for the (unconstrained) Green function $G$ :

[TABLE]

together with the boundary conditions

[TABLE]

where $z_{s}\in\{0,L\}$ denotes the position of one of the surfaces, $z^{\prime}$ is off the surface and the minus (plus) sign applies to the case $z_{s}=L$ ( $z_{s}=0$ ). In the case of Dirichlet boundary conditions, Eq. 61 reduces to $G(\mathbf{r}_{\parallel},\mathbf{r}_{\parallel}^{\prime},z_{s},z^{\prime})=G(\mathbf{r}_{\parallel},\mathbf{r}_{\parallel}^{\prime},z,z_{s}^{\prime})=0$ , while for Neumann boundary conditions, one has $\partial_{z}G(\mathbf{r}_{\parallel},\mathbf{r}_{\parallel}^{\prime},z,z^{\prime})\big{|}_{z=z_{s}}$ = 0. In the case of periodic boundary conditions in the transverse direction, instead of Eq. 61, one has $G(\mathbf{r}_{\parallel},\mathbf{r}_{\parallel}^{\prime},z+L,z^{\prime}+L)=G(\mathbf{r}_{\parallel},\mathbf{r}_{\parallel}^{\prime},z,z^{\prime})$ for all $z$ and $z^{\prime}$ .

In order to proceed, we introduce a complete set of orthonormal eigenfunctions $\sigma_{\mathbf{k}_{\parallel},n}$ of the operator contained in the curly brackets in $\mathcal{H}^{(2)}$ [Eq. 59]:

[TABLE]

where and $A=\prod_{\alpha=1}^{d-1}L_{\alpha}$ and $\mathbf{k}_{\parallel}$ is determined by the periodic boundary conditions in all lateral directions $\alpha=1,\ldots,d-1$ of extent $L_{\alpha}$ as

[TABLE]

The eigenfunctions $\varsigma_{n}$ and the corresponding index $n$ pertain to the transverse direction and the associated boundary conditions. Denoting the eigenvalue of the operator $-\partial_{z}^{2}+(g/2)\psi^{2}(z)$ as $E_{n}$ , the bulk term in Eq. 59 yields the eigenvalue equation for $\sigma_{\mathbf{k}_{\parallel},n}$ :

[TABLE]

which, using Eq. 62, results in an eigenvalue equation for $\varsigma_{n}$ :

[TABLE]

The boundary terms in Eq. 59 imply the boundary conditions

[TABLE]

where, as before, $z_{s}\in\{0,L\}$ and the minus (plus) sign applies to the case $z_{s}=L$ ( $z_{s}=0$ ). Periodic boundary conditions in the transverse directions imply $\varsigma_{n}(z+L)=\varsigma_{n}(z)$ for all $z$ , replacing Eq. 66. Also the functions $\varsigma_{n}$ fulfill completeness and orthonormality relations, i.e.,

[TABLE]

The formal solution of Eqs. 60 and 61 can now be given in terms of the spectral representation of the Green function:

[TABLE]

Due to the assumed translational invariance along the lateral directions, $G$ in fact depends only on the difference $\mathbf{r}_{\parallel}-\mathbf{r}^{\prime}_{\parallel}$ . It is therefore convenient to introduce its Fourier transform $\hat{G}$ along the lateral coordinates,

[TABLE]

referred to as the $pz$ -representation of $G$ . Here, consistently with the periodicity of $G(\mathbf{r}_{\parallel},\mathbf{0},z,z^{\prime})$ along the lateral directions, the components of $\mathbf{p}$ take the discrete values $p_{\alpha}=2\pi n_{\alpha}/L_{\alpha}$ , with $n_{\alpha}=0,\pm 1,\pm 2,\ldots$ for $\alpha=1,\ldots,d-1$ . In obtaining the last expression in Eq. 69, we have furthermore used Eq. 184. The transverse area $A$ appears as a prefactor because here we consider $\hat{G}$ to be a function of only a single wave vector $\mathbf{p}$ , whereas, in real space, $G$ is defined as a function of two positions, $\mathbf{r}_{\parallel}$ and $\mathbf{r}_{\parallel}^{\prime}$ (see Appendix B). The fluctuating field $\sigma$ [see Eq. 13] can also be expanded in terms of the eigenfunctions in Eq. 62:

[TABLE]

with the coefficients $c_{n}$ given by $c_{n}(\mathbf{k}_{\parallel})=(1/\sqrt{A})\int_{0}^{L}\mathrm{d}z\int_{A}\mathrm{d}^{d-1}r_{\parallel}\exp(-\mathrm{i}\mathbf{k}_{\parallel}\cdot\mathbf{r}_{\parallel})\varsigma_{n}^{*}(z)\sigma(\mathbf{r}_{\parallel},z)$ . Since Eq. 16 constrains the function $\sigma$ as a whole, nothing can be stated at this point about each individual $c_{n}$ , except that

[TABLE]

In particular, we emphasize that it is not justified to include in the expansion in Eq. 70 only those eigenfunctions $\varsigma_{n}$ which satisfy the constraint in Eq. 16 [with $\mathpzc{w}=1$ , see Eq. 53].

In order to be able to calculate the constrained Green function $\ring{G}$ [Eq. 26] and the constrained free energy $\ring{\mathcal{F}}$ [Eq. 34], expressions for the quantities $(G,1)$ and $(1,G,1)$ have to be worked out. Making use of the spectral representation of $G$ [Eq. 68] as well as of Eqs. (184) and (69), we eventually find:

[TABLE]

As a consequence of translational invariance, this expression does not depend on the lateral coordinate $\mathbf{r}_{\parallel}$ . The quantity $(1,G,1)$ follows as

[TABLE]

which, in general, does not vanish (see Sec. III.2 below), so that the generating functional defined in Eq. 27 applies to the constrained case. The $pz$ -representation of the constrained correlation function in Eq. 36 follows, analogously to Eq. 69, as

[TABLE]

The constrained (canonical) free energy $\ring{\mathcal{F}}$ within the Gaussian approximation [see Eq. 34] takes the form

[TABLE]

The r.h.s. of this expression depends on $\Sigma$ via the mean field $\psi$ [Eqs. 55 and 53], the eigenvalues $E_{n}$ [Eq. 65], and the Green function [Eq. 60]. The second term in Eq. 75 requires a suitable regularization in order to render a physically meaningful, finite result. This issue as well as the relevance of the constraint correction will be discussed in Sec. III.3 below. We recall that in Eq. 75 we have suppressed the lattice constant $a$ , the presence of which is implied within the corresponding discrete field theory via the definition of the functional integral in Eq. 10. Accordingly, the arguments of the first and second logarithm in Eq. 75 would have to be multiplied by $a^{2}$ and $a^{2+d}$ , respectively, which renders them dimensionless (see, e.g., Refs. Dohm (2008, 2011) and Appendix A). However, a physical observable with universal features, such as the CCF, is independent of the lattice constant.

III.2 Specialization to various boundary conditions

We now specialize the general expressions derived above to single-phase systems having periodic, Dirichlet, or Neumann boundary conditions at both boundaries $z=0$ and $z=L$ . Within the Gaussian approximation, the latter two boundary conditions are realized by setting $h_{1}=0,c=\infty$ and $h_{1}=0,c=0$ , respectively, in $\mathcal{L}_{s}$ [Eq. 49]. In the case of periodic boundary conditions, instead, one has $\mathcal{L}_{s}=0$ and requires $\phi(\mathbf{r}_{\parallel},0)=\phi(\mathbf{r}_{\parallel},L)$ . In all cases, periodic boundary conditions along the lateral directions are applied (see Fig. 1). The calculation of the regularized free energy [see Eq. 75] is deferred to Sec. III.3.

III.2.1 Periodic boundary conditions

For periodic boundary conditions along the $z$ direction, the system is homogeneous in all directions, with the mean OP [see Eqs. 15 and 48]

[TABLE]

which does not vary spatially. Within the mean-field approximation, $\varphi$ is determined by Eq. 55, which, for periodic boundary conditions, turns into:

[TABLE]

The parameter $\mu$ must be chosen such that the constraint in Eq. 76 is obeyed by the solution of Eq. 77 for $\varphi$ . The orthonormal eigenfunctions $\sigma_{\mathbf{k}}\equiv\sigma_{\mathbf{k}_{\parallel},n}$ [see Eq. 62] are given by

[TABLE]

for $\alpha=1,2,\ldots,d$ and $V=\prod_{\alpha=1}^{d}L_{\alpha}$ . Note that here we have simplified the notation used in Eq. 62. The functions $\sigma_{\mathbf{k}}$ in Eq. 78 fulfill the eigenvalue equation in Eq. 64 with $E_{n}=k_{z}^{2}+(g/2)\varphi^{2}$ and $n\equiv n_{z}$ (denoting by $z$ the last of the $d$ Cartesian coordinates). The temperature parameter $\tau$ enters these expressions in combination with the mean OP density $\varphi$ in the form of an effective, shifted temperature

[TABLE]

Due to Eq. 184, the eigenfunctions in Eq. 78 for periodic boundary conditions include a single zero mode $\sigma_{\mathbf{k}=\mathbf{0}}$ , which is spatially constant. Accordingly, the Green function $G^{\text{(p)}}$ has the spectral representation [see Eq. 68]

[TABLE]

By using this equation together with Eq. 184, one readily finds

[TABLE]

and

[TABLE]

Since $\chi=1/\hat{\tau}$ is the susceptibility within MFT, these results confirm the estimates in Eq. 45.

In order to gain further insight into the effect of the constraint on the free energy, we insert Eq. 82 into Eq. 75 and obtain

[TABLE]

As expected for this particular case, the effect of the constraint consists of, apart from generating an additional term $\propto\ln V$ , the cancellation of the zero mode contribution from the free energy. (Regarding the dimensions of the last two terms in Eq. 83 see the discussions after Eqs. 75 and 10.) Constrained free energies of the type given in Eq. 83 have in fact been studied previously in the context of finite-size criticality within the grand canonical ensemble Rudnick et al. (1985); Brezin and Zinn-Justin (1985); Guo and Jasnow (1987); Dohm (1989); Caracciolo et al. (2001); Dohm (2008, 2011). Here, we have obtained Eq. 83 by explicitly taking into account the OP constraint. In particular, the contribution $\propto\ln V$ in Eq. 83 is relevant for the calculation of the canonical CCF. Finite-size properties of the free energy will be investigated further in Sec. III.3 below.

According to Eq. 36, the correlation function $\ring{C}^{\text{(p)}}$ of a constrained system with periodic boundary conditions is, within the Gaussian approximation, given by

[TABLE]

As expected from Eq. 37, one has $\int_{\mathbf{r}}\ring{C}^{\text{(p)}}(\mathbf{r})=0$ . Since $C^{\text{(p)}}(\mathbf{r})$ typically vanishes exponentially upon increasing $|\mathbf{r}|$ , the fact that $\ring{C}^{\text{(p)}}$ is shifted by the amount $-1/(\hat{\tau}V)$ relative to $C^{\text{(p)}}=G^{\text{(p)}}$ means that the constraint induces anti-correlations of fluctuations at large distances. However, at least within the Gaussian approximation, the constraint does not cause $\ring{C}^{\text{(p)}}(\mathbf{r})$ to approach its limit for large $|\mathbf{r}|$ differently than in the unconstrained system. We finally note that, in the continuum limit, i.e., with $\sum_{\mathbf{k}}\to\frac{V}{(2\pi)^{d}}\int_{\mathbf{k}}$ , Eq. 80 becomes $G^{\text{(p)}}\simeq(2\pi)^{-d}\int_{\mathbf{k}}e^{\mathrm{i}\mathbf{k}\cdot(\mathbf{r}-\mathbf{r}^{\prime})}/(\mathbf{k}^{2}+\hat{\tau})$ , showing that $G^{\text{(p)}}$ is a quantity of $\mathcal{O}(V^{0})$ . Hence, for $V\to\infty$ , which includes the case of a film with $A\to\infty$ at finite $L$ , it follows from Eq. 84 that $\ring{G}^{\text{(p)}}=G^{\text{(p)}}$ , as anticipated in Eq. 47.

III.2.2 Dirichlet boundary conditions

For a system with Dirichlet boundary conditions at $z=0$ and $L$ , the mean OP $\psi$ within MFT is determined by [Eq. 55]

[TABLE]

According to Eqs. 66 and 70, the fluctuating component $\sigma$ of the OP [see Eq. 13] also fulfills Dirichlet boundary conditions, i.e., $\sigma(\mathbf{r}_{\parallel},0)=\sigma(\mathbf{r}_{\parallel},L)=0$ . For non-vanishing $\mu$ , an explicit analytical solution of Eq. 85 is not available 777For $\mu=0$ , instead, see, e.g., the Appendix in Ref. Gambassi and Dietrich (2006). Although in principle Eq. 85 can be solved numerically, this poses additional challenges due to the presence of a spatially varying profile $\psi(z)$ . For the purpose of highlighting the effects of the constraint, in the following we focus on the simpler case $\mu=0$ (and $\tau\geq 0$ ), corresponding to a vanishing total mass $\Sigma=0$ , for which Eq. 85 is solved by

[TABLE]

Consequently, the set of orthonormal eigenfunctions $\varsigma_{n}$ [see Eq. 62] is given by

[TABLE]

with eigenvalues [see Eq. 65]

[TABLE]

as in the grand canonical ensemble Krech and Dietrich (1992). Since

[TABLE]

all eigenfunctions $\varsigma_{n}^{\text{(D)}}$ with odd $n$ contribute to the fluctuation constraint in Eq. 71, in contrast to the case of periodic boundary conditions, in which only the mode [see Eq. 78] with $\mathbf{k}=0$ contributes. The Green function can be straightforwardly obtained by solving the differential equation in Eq. 60, subject to the boundary conditions given in Eq. 61 (with $\psi=0$ and $\mathcal{L}_{s}^{\prime\prime}[\psi]=c=\infty$ ), in the $pz$ -representation [see Eq. 69]. This yields Diehl (1986); Gambassi and Dietrich (2006); Kleinert (2009)

[TABLE]

$\hat{G}^{\text{(D)}}$ is symmetric with respect to $z\leftrightarrow z^{\prime}$ and it has a finite limit for $\kappa\to 0$ :

[TABLE]

The evaluation of the two-point correlation function $\ring{C}$ according to Eq. 36 requires the calculation of the term $(G^{\text{(D)}},1)$ defined in Eq. 72, which is easily inferred from Eq. 90:

[TABLE]

This quantity is finite for all $\tau\geq 0$ and, for $\tau\to 0$ , it turns into $(G^{\text{(D)}},1)_{z}|_{\tau\to 0}=(L-z)z/2$ . Further integrations over $\mathbf{r}_{\parallel}$ and $z$ of Eq. 92 yield [see Eq. 73]

[TABLE]

For $\tau L^{2}\to 0$ , $(1,G^{\text{(D)}},1)$ is finite with the expansion

[TABLE]

where the latter behavior also applies to the case $\tau\to\infty$ at fixed $L$ . In contrast, for $L\to\infty$ at fixed $\tau$ , one has $(1,G^{\text{(D)}},1)\simeq AL/\tau$ .

In the $pz$ -representation, the constraint-induced correlation function $\hat{\ring{C}}^{\text{(D)}}$ [Eq. 74] for Dirichlet boundary conditions is then given by

[TABLE]

Note that both terms on the r.h.s. of Eq. 95 are proportional to $A$ . Figure 2 illustrates the typical behavior of $\hat{\ring{C}}^{\text{(D)}}(\mathbf{p}=\mathbf{0},z,z^{\prime})$ as a function of $z$ for a fixed value of $z^{\prime}$ . In contrast to the corresponding unconstrained correlation function $\hat{C}$ , which takes only positive values and is identical to $\hat{G}^{\text{(D)}}$ , $\hat{\ring{C}}^{\text{(D)}}(\mathbf{p}=\mathbf{0},z,z^{\prime})$ is modified such that, in accordance with Eq. 37, the integral over either of its arguments $z$ and $z^{\prime}$ vanishes.

III.2.3 Neumann boundary conditions

In the case of Neumann boundary conditions the mean OP $\psi$ is determined by Eq. 55. For $\tau\geq 0$ , this equation is solved by a constant OP profile $\psi(z)=\varphi$ , which fulfills

[TABLE]

and satisfies the boundary conditions $\psi^{\prime}(0)=\psi^{\prime}(L)=0$ . Equations (65) and (66) for the eigenfunctions $\varsigma_{n}(z)$ turn into

[TABLE]

As in the case of periodic boundary conditions [see Eq. 79], the temperature parameter $\tau$ enters these expressions in combination with the mean OP $\varphi$ in the form given by Eq. 79. Equation (97) is solved by the eigenfunctions

[TABLE]

with eigenvalues

[TABLE]

Since

[TABLE]

Neumann boundary conditions entail a well defined zero mode, $\sigma_{\mathbf{k}_{\parallel}=0,n=0}$ , similarly to the case of periodic boundary conditions. Equations 72 and 73, upon using Eq. 68, render the expressions

[TABLE]

and

[TABLE]

which coincide with the ones obtained for periodic boundary conditions and reported in Eqs. 81 and 82. The (unconstrained) Green function in the $pz$ -representation [see Eq. 69] is given by Diehl (1986)

[TABLE]

with $\kappa\equiv\sqrt{\mathbf{p}^{2}+\hat{\tau}}$ . According to Eqs. 101, 102, and 26, the presence of the constraint simply gives rise to an overall $\tau$ -dependent shift of the unconstrained correlation function, as it is the case for the periodic boundary conditions discussed above.

III.3 Canonical free energy

Here, we discuss, within the one-loop (Gaussian) approximation, the canonical free energy $\ring{\mathcal{F}}$ [Eq. 75] for finite systems with aspect ratio

[TABLE]

and exhibiting periodic, Dirichlet, or Neumann boundary conditions at both surfaces, located at $z=0$ and $z=L$ . In all three cases periodic boundary conditions are imposed in the remaining lateral directions and phase separation is excluded. Analytical results for the finite-size free energy of constrained systems with periodic boundary conditions have been presented for $\rho=1$ , e.g., in Refs. Brezin (1982); Rudnick et al. (1985); Eisenriegler (1985); Eisenriegler and Tomaschitz (1987); Esser et al. (1995) and, for $\varphi=0$ and arbitrary $\rho$ , in Refs. Dohm (2008, 2011). The finite-size free energy for Dirichlet boundary conditions and cubical volumes (i.e., $\rho=1$ ) has been studied, e.g., in Refs. Dohm (1989, 2014). With the exception of Ref. Eisenriegler and Tomaschitz (1987), these studies aimed, however, for the grand canonical free energy [which, according to Eq. 12, can be constructed from the canonical free energy discussed here]. Instead, here we focus on the canonical ensemble and, extending previous studies, we allow also for a nonzero mean OP $\varphi$ in the case of periodic and Neumann boundary conditions. The details of the corresponding perturbative calculation are deferred to Appendix C. The results reported here are subsequently improved in Sec. IV by means of renormalization-group theory.

Analogously to what is expected for the grand canonical free energy $\mathcal{F}$ Privman (1990); Brankov et al. (2000), the canonical free energy $\ring{\mathcal{F}}$ of a $d$ -dimensional system of volume $V=AL$ decomposes into a bulk ( $\,\ring{\mathpzc}{f}_{b}$ ), a surface ( $\,\ring{\mathpzc}{f}_{s}$ ) and a residual finite size contribution $\ring{\mathcal{F}}_{\text{res}}$ [see also Eq. 121 below]:

[TABLE]

We anticipate that in our case the residual finite-size free energy (per area $A$ ) $\ring{\mathcal{F}}_{\text{res}}$ depends on the area only via the aspect ratio $\rho$ . Explicitly, from Eq. 75, the total, regularized free energies for periodic, Dirichlet, and Neumann boundary conditions turn out to be [see Eqs. 196, 204, and 213]

[TABLE]

where

[TABLE]

and $\hat{\tau}$ is defined in Eq. 79 (recall that, for Dirichlet boundary conditions, $\hat{\tau}=\tau$ because in that case we focus on the choice $\varphi=0$ [see Eq. 86]). The quantities $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}$ , $\mathcal{S}^{\text{(D)}}_{d,\text{reg}}$ , and $\mathcal{S}^{\text{(N)}}_{d,\text{reg}}$ represent the regularized dimensionless expression of the corresponding mode sum, i.e., the second term in Eq. 75. They are given by [see Eqs. 189, 202, and 210]

[TABLE]

for periodic, Dirichlet, and Neumann boundary conditions, respectively. In Eq. 108a, $\vartheta$ is a Jacobi theta function [see Eq. 190]. In Eq. 108, we introduced the notions $x=\tau L^{2}$ and $\hat{x}=\hat{\tau}L^{2}$ . For all these boundary conditions, far from criticality one has

[TABLE]

For periodic and Neumann boundary conditions, $\mathcal{S}_{d,\text{reg}}(\hat{x},\rho)$ diverges logarithmically upon approaching bulk criticality, i.e., for $\hat{x}\to 0$ [see Eq. 191]:

[TABLE]

while $\mathcal{S}^{\text{(D)}}_{d,\text{reg}}$ is finite in that limit. We shall return to this aspect in Sec. III.4. The quantities $\delta F^{(p,D,N)}$ in Eq. 106 represent the correction stemming from the OP constraint in Eq. 53. Within the one-loop approximation, one has [see Eqs. (82), (93), and (102)]

[TABLE]

for the indicated boundary conditions. It is interesting to note that the same form of the constraint correction $\delta F^{\text{(p)}}$ as in Eq. 111a is obtained also for an uncorrelated Gaussian field [which is described by the Hamiltonian $\int_{V}\mathrm{d}^{d}r\,\tau\phi^{2}(\mathbf{r})/2$ instead of the one in Eq. 49] in a finite volume (see Appendix A). Since the perturbation theory in the canonical ensemble is based on the modified Green function $\ring{G}$ [see Eq. 26], higher-order constraint corrections are, however, sensitive to the presence of a finite correlation length in the system [see, e.g., Eq. 43].

In view of the formulation of the scaling theory in Sec. IV below, it is convenient to cast $\delta F$ given by Eq. 111 into the form

[TABLE]

where

[TABLE]

is a “scaling” contribution, which is specific to each boundary condition, while

[TABLE]

is a “non-scaling” contribution, which is common to all boundary conditions considered here. Upon reinstating the lattice spacing $a$ , which we formally disregarded in taking the continuum limit [see Eq. 10], the arguments of the logarithms in Eqs. 111 and 114 are divided by a factor $a^{d+2}$ , which renders them dimensionless [compare Eq. 180].

According to Eq. 105, the residual finite-size free energy per volume, $\ring{\mathpzc}{f}_{\text{res}}$ , is given by

[TABLE]

which, upon using Eqs. 106 and 112 and noting that $AL=L^{d}/\rho^{d-1}$ [see Eq. 104], can be expressed as

[TABLE]

The scaling function $\ring{\Theta}$ introduced in this expression contains the contribution $\delta F_{\text{s}}$ from the constraint correction given in Eq. 113:

[TABLE]

Note that the divergence of $\mathcal{S}_{d,\text{reg}}^{\text{(p,N)}}$ for $\hat{x}\to 0$ [see Eq. 110] is canceled by that of $\delta F_{\text{s}}^{\text{(p,N)}}$ [Eq. 113a], such that $\ring{\Theta}$ remains finite for $\hat{x}=0$ . In the case of Dirichlet boundary conditions, neither $\mathcal{S}^{\text{(D)}}_{d,\text{reg}}$ [Eq. 108b] nor $\delta F_{\text{s}}^{\text{(D)}}$ [Eq. 113b] diverge in the limit $x\to 0$ . Accordingly, at bulk criticality, $\ring{\mathpzc}{f}_{\text{res}}$ [Eq. 116] generally scales $\propto L^{-d}$ while $\ring{\mathcal{F}}_{\text{res}}$ [Eqs. 105 and 115] scales $\propto L^{-(d-1)}$ , as in the grand canonical ensemble. Since $\mathcal{S}_{d,\text{reg}}(\hat{x}\to\infty)\to 0$ , the canonical scaling function $\ring{\Theta}$ turns out to diverge logarithmically for $\hat{x}\gg 1$ , i.e.,

[TABLE]

due to the term $\delta F_{\text{s}}$ in Eq. 113. This behavior applies to all three boundary conditions considered here. In the thin-film limit, one has $\ring{\Theta}(\hat{x}\gg 1,\rho\to 0)\to 0$ . The residual finite-size free energy will be discussed further in Sec. V.

III.4 Grand canonical free energy

For comparison, here we report the corresponding expressions for the one-loop free energy $\mathcal{F}(\tau,h,A,L)$ in the grand canonical ensemble. Field theory yields the well-known perturbative expression for $\mathcal{F}$ in Eq. 35 Brezin (1982); Rudnick et al. (1985); Eisenriegler (1985); Eisenriegler and Tomaschitz (1987); Krech and Dietrich (1992); Esser et al. (1995); Dohm (2008, 2011, 1989, 2014). The corresponding regularized forms of $\mathcal{F}$ coincide with the ones in Eq. 106 for the various boundary conditions, except for the fact that $\mathcal{L}_{b}(\varphi)$ is replaced by $\mathcal{L}_{b}(\varphi,h)\equiv\mathcal{L}_{b}(\varphi)-h\varphi$ [see Eq. 51] and that the constraint induced term $\delta F$ [Eq. 111] is absent:

[TABLE]

The expressions for $\mathcal{S}_{d,\text{reg}}$ are reported in Eq. 108. In the case of periodic and Neumann boundary conditions, the mean OP $\varphi$ is a function of the bulk field $h$ via the equation of state, which, within the presently considered approximation, is given by Eq. 58:

[TABLE]

Equation (120) takes already into account that for periodic and Neumann boundary conditions the system is spatially homogeneous so that $\psi(\mathbf{r})=\varphi$ . For Dirichlet boundary conditions, as explained below Eq. 85, we focus on the simple case $h=0$ , i.e., $\varphi=0$ . Note that Eq. 120 in fact coincides with the equation of state for the corresponding bulk system. Finite-size corrections enter through higher loop orders, analogous to Eq. 57. The renormalized forms of these expressions will be discussed in Sec. IV below.

As it is well known from general finite-size scaling arguments Barber (1983); Privman (1990); Brankov et al. (2000), the grand canonical free energy $\mathcal{F}$ of a confined $d$ -dimensional system of volume $V=AL$ decomposes into a bulk ( $\mathpzc{f}_{b}$ ), a surface ( $\mathpzc{f}_{s}$ ) and a residual finite size $\mathcal{F}_{\text{res}}$ contribution [compare Eq. 105]:

[TABLE]

Crucially, in order to be able to cast Eq. 119 into the form prescribed by Eq. 121, $\mathcal{F}$ must be first expressed as a function of the bulk field $h$ , which is the relevant thermodynamic control parameter in the grand canonical ensemble. In their present form, the expressions in Eq. 119 are still explicit functions of $\varphi$ . To proceed, any $\varphi$ occurring in Eq. 119 must therefore be replaced by the $\varphi(h)$ determined from the equation of state. Before turning to the specific approximation for the latter as given by Eq. 120, for the time being we adopt a generic equation of state of the form $\varphi=\varphi(\tau,h,\rho,L)$ . In this case, bulk and surface free energies can be identified based on their scaling behavior with $L$ according to Eq. 121. In particular, the bulk limit is obtained by taking $L\to\infty$ and by assuming $A$ to be either constant or to scale with a certain positive power of $L$ (the precise formulation does not matter here). Accordingly, from Eq. 119 the bulk free energy follows as

[TABLE]

with the bulk OP given by

[TABLE]

The surface free energy (per area $A$ ) is defined as the $L$ -independent part of the total free energy. Therefore it can be obtained as the dominant contribution to $\mathcal{F}$ in the limit $L\to\infty$ after subtracting the bulk contribution $\mathpzc{f}_{b}$ :

[TABLE]

Note that the limit $L\to\infty$ implies again the use of the bulk OP for the evaluation of $\mathpzc{f}_{s}$ . Neumann boundary conditions are the only case considered in this study for which $\mathpzc{f}_{s}$ does not vanish, and Eq. 119c yields

[TABLE]

The residual finite-size free energy per volume $\mathpzc{f}_{\text{res}}=\mathcal{F}_{\text{res}}/V$ in the grand canonical ensemble follows according to Eq. 121 as

[TABLE]

with $\mathpzc{f}\equiv\mathcal{F}/V$ . Since one generally expects $\varphi\neq\varphi_{b}$ due to finite-size effects (see, e.g., Refs. Chen et al. (2000); Gross et al. (2016)), it follows from Eq. 126 that the residual finite-size free energy in this case does not necessarily coincide with the last term in each of the Eqs. (119a) $-$ (119c), in spite of the fact that they apparently display the appropriate scaling as a function of $L$ , but for fixed $\varphi$ only. However, within the presently considered approximation of $\mathcal{O}(g^{0})$ for the equation of state in Eq. 120, finite-size effects are absent and therefore

[TABLE]

Accordingly, Eqs. 126 and 119 immediately yield

[TABLE]

with the scaling function

[TABLE]

The subscript on the r.h.s. of Eq. 128 indicates that $\hat{\tau}$ [Eq. 79] is to be evaluated by using $\varphi=\varphi(h)$ .

According to Eqs. 129 and 110, the grand canonical residual finite-size free energy for periodic and Neumann boundary conditions diverges logarithmically for $\hat{x}\to 0$ in the case $\rho>0$ , while this divergence is absent in the thin-film limit ( $\rho=0$ ). This behavior is a well-known artifact of perturbation theory and stems from the contribution of the zero mode to the free energy Rudnick et al. (1985); Brezin and Zinn-Justin (1985); Schloms and Dohm (1989); Diehl et al. (2006); Dohm (2008); Grüneberg and Diehl (2008) (see also Eq. 83 and the related discussion). In order to overcome this problem, the zero mode must be treated non-perturbatively, which results in a finite residual finite-size free energy for $\hat{x}=0$ . Since here we are interested in a comparison between the canonical and grand-canonical ensemble, we do not consider such improvements of the theory further. Instead, we note that, for the grand canonical ensemble in the case $\rho>0$ , the perturbative expressions of the residual finite-size free energy and the CCF for periodic and Neumann boundary conditions are reliable only for $\hat{x}\gtrsim 1$ . Since Dirichlet boundary conditions do not involve zero-mode fluctuations, the perturbative results for $\Theta^{\text{(D)}}$ are well behaved for all $x\geq 0$ .

IV Renormalization and scaling

IV.1 Residual finite-size free energy

IV.1.1 Canonical residual finite-size free energy

In order to be applicable in the critical regime, the perturbative results of Sec. III must be renormalized Zinn-Justin (2002); Amit and Martin-Mayor (2005). On general grounds, it is expected that the short-distance singularities of field theory are not affected by the finiteness of the volume of the system Brezin (1982); Guo and Jasnow (1987, 1989); Amit and Martin-Mayor (2005); Zinn-Justin (2002). As it has been shown in Ref. Eisenriegler and Tomaschitz (1987), renormalization based on minimal subtraction of dimensional poles in conjunction with an expansion in $\epsilon=4-d$ is applicable also in the canonical ensemble and it requires the same additive and multiplicative counterterms which are known from the grand canonical case Diehl (1986); Krech and Dietrich (1992). In particular, the findings of Ref. Eisenriegler and Tomaschitz (1987) apply also to the present study, because here we focus on planar surfaces only (compare Ref. Eisenriegler (1985)) and do not consider surface correlation functions. Furthermore, because off the surfaces neither Dirichlet nor Neumann boundary conditions introduce new dimensional poles as $d\nearrow 4$ , the same counterterms as for periodic boundary conditions can be used in these cases as well. The renormalized (grand) canonical free energy can be constructed by following the same steps as in Refs. Diehl (1986); Eisenriegler and Tomaschitz (1987); Krech and Dietrich (1992); for further details we refer to these studies 888Conventionally, the definition of the renormalized total free energy involves the subtraction of the bare free energy and of its first two temperature derivatives taken at a certain reference temperature (see, e.g., Eq. (3.1) in Ref. Krech and Dietrich (1992)). With such a prescription, the non-scaling contribution $\delta F_{\text{ns}}$ [see Eqs. 106 and 114] to the canonical free energy is eliminated. However, such a definition also introduces a shift of the renormalized residual finite-size free energy, which is undesired for our purposes. Consistently with the literature (see, e.g., Refs. Krech and Dietrich (1992); Krech (1994); Grüneberg and Diehl (2008)), we therefore proceed by studying the un-subtracted but renormalized residual finite-size free energy.. Along these lines one obtains the expected scaling laws for the free energy near the infrared renormalization-group (RG) fixed point, at which, within the $\epsilon$ -expansion, the renormalized coupling constant $u$ is given by

[TABLE]

We adopt the same conventions as in Refs. Diehl (1986); Krech and Dietrich (1992) and define $g=\mu^{\epsilon}Z_{u}ru$ , where $\mu$ is the RG momentum scale, $Z_{u}$ is the standard $Z$ -factor for the coupling constant, and

[TABLE]

In the following, we focus directly on the renormalization of the residual finite-size free energy, which turns out to not require any additive renormalization in order to cancel its dimensional poles (see also Ref. Grüneberg and Diehl (2008)). From Eq. 116, one obtains the following finite-size scaling form of the residual finite-size free energy per volume and per $k_{B}T_{c}$ :

[TABLE]

where $t=(T-T_{c})/T_{c}$ is the reduced temperature [Eq. 50]. We recall that $\delta F_{\text{ns}}(L)=(1/2)\ln L^{d+2}$ [see Eq. 114] is a contribution stemming from the constraint which cannot be expressed solely in terms of scaling variables. The subscript $R$ indicates a dimensionless, renormalized quantity. Specifically, $t$ and $\varphi_{R}$ can be related to the correlation length $\xi(t,\varphi_{R})$ via the non-universal critical amplitudes $\xi_{+}^{(0)}$ and $\xi_{\varphi}^{(0)}$ Pelissetto and Vicari (2002):

[TABLE]

The amplitude $\xi_{\varphi}^{(0)}$ can be related to the amplitude $\xi_{h}^{(0)}$ of the correlation length at $T_{c}$ as a function of the bulk field [see Eq. 138 below]. The scaling variable corresponding to $\hat{\tau}$ , which has been introduced in Eq. 79, is defined by

[TABLE]

The expressions of the scaling functions $\ring{\Theta}$ are reported in Eqs. 117, 113, and 108 for the respective boundary conditions. Consistently with the considered one-loop approximation for the free energy, the scaling functions $\ring{\Theta}$ are to be evaluated to $\mathcal{O}(\epsilon^{0})$ , i.e., for $d=4$ . Since the constraint-induced terms $\delta F_{\text{s,ns}}$ turn out to be $\propto\rho^{d-1}$ , they are negligible in the thin-film limit $\rho\to 0$ . They are also negligible, together with $\ring{\mathpzc}{f}_{\text{res}}$ , in the thermodynamic limit obtained for $L\to\infty$ . We note that $\mathpzc{m}^{2}\sim O(u^{-1})$ , such that the last term in Eq. 134 is actually of $\mathcal{O}(\epsilon^{0})$ . This expression and the equation of state [see Eq. 140 below] are the only instances in which within the approximation $\mathcal{O}(\epsilon^{0})$ the renormalized coupling constant $u$ appears in the final expressions of the residual finite-size free energy and of the CCF.

IV.1.2 Grand canonical residual finite-size free energy

Here, we summarize the scaling forms obtained for the grand canonical residual free energy based on the renormalization of the perturbative results in Sec. III.4. In particular, at the fixed point, the RG yields the scaling property of the renormalized grand canonical residual free energy per volume and per $k_{B}T_{c}$ (see, e.g., Refs. Diehl (1986); Krech and Dietrich (1992); Grüneberg and Diehl (2008))

[TABLE]

The scaling function $\widetilde{\Theta}(x,\mathpzc{h},\rho)$ is related to $\Theta(\hat{x}(x,\mathpzc{m}),\rho)$ in Eq. 129 via

[TABLE]

and $\mathpzc{m}(x,\mathpzc{h},\rho)$ is the scaling form of the equation of state (see Eq. 142 below). The renormalized bulk field $h_{R}$ can be introduced on the basis of the correlation length Pelissetto and Vicari (2002):

[TABLE]

which also serves as a definition of the amplitude $\xi_{h}^{(0)}$ . It is useful to recall the relation $\Delta=\delta\beta$ between standard bulk critical exponents. We emphasize that Eq. 133b can be obtained from Eq. 137 and from the relation $\varphi_{R}(t=0,h_{R}\to 0)=\phi_{h}^{(0)}h_{R}^{1/\delta}$ Pelissetto and Vicari (2002), which defines the amplitude $\phi_{h}^{(0)}$ and thereby yields the expression

[TABLE]

for the amplitude $\xi_{\varphi}^{(0)}$ .

The equation of state exhibits the scaling form

[TABLE]

where the scaling function $\mathpzc{h}$ results from Eq. 120 within the approximation $\mathcal{O}(\epsilon^{0})$ as

[TABLE]

which is, in fact, independent of $\rho$ . Within that approximation, this equation of state applies to all boundary conditions and it coincides with the one in the bulk [see Eq. 127]. An alternative form of the equation of state can be obtained from the total grand canonical free energy $\mathpzc{f}_{R}$ via the basic thermodynamic relation $\varphi_{R}=\partial\mathpzc{f}_{R}/\partial h_{R}$ . This leads to the scaling form (see, e.g., Ref. Gross et al. (2016)):

[TABLE]

It can be shown that the scaling function $\mathpzc{m}(x,\mathpzc{h},\rho)\equiv\hat{\mathpzc}{m}(x,\mathpzc{h},\rho)(\xi_{h}^{(0)})^{\beta\delta/\nu}$ is universal Privman and Fisher (1984). In the bulk limit, i.e., for $x\gg 1$ or $\mathpzc{h}\gg 1$ , the scaling function $\mathpzc{m}$ reduces to (see, e.g., Ref. Amit and Martin-Mayor (2005))

[TABLE]

Within the considered approximation $\mathcal{O}(\epsilon^{0})$ , Eq. 142 holds for all $x$ and $\mathpzc{h}$ , where $\mathpzc{m}_{b}$ follows from Eq. 140 as 999It is useful to note that Eq. 142 can expressed alternatively as $\mathpzc{m}(x,\mathpzc{h},\rho)\simeq\hat{\mathpzc{h}}^{1/\delta}\hat{\mathpzc{m}_{b}}(x\hat{\mathpzc{h}}^{-1/\delta\beta},\rho)$ with a scaling function $\hat{\mathpzc{m}_{b}}(y,\rho)=(ru^{*})^{-1/2}\left[2y\left(\sqrt{9+8y^{3}}-3\right)^{1/3}-\left(\sqrt{9+8y^{3}}-3\right)^{1/3}\right]$ and $\hat{\mathpzc{h}}\equiv(ru^{*})^{1/2}\mathpzc{h}$ . This form shows explicitly that $\mathpzc{m}$ and $\mathpzc{h}$ are quantities of $O(u^{-1/2})$ .

[TABLE]

which is, in fact, independent of $\rho$ . Finite-size effects for a certain boundary condition enter the equation of state at $\mathcal{O}(\epsilon)$ . Within field theory, the finite-size scaling function $\mathpzc{m}(x,\mathpzc{h},\rho)$ has been investigated further, e.g., in Ref. Chen et al. (2000).

IV.2 Critical Casimir force

The critical Casimir force $\mathcal{K}$ (per area and per $k_{B}T_{c}$ ) is defined in terms of the residual finite-size free energy $L\mathpzc{f}_{\text{res}}$ per area $A$ and per $k_{B}T_{c}$ Krech (1994); Brankov et al. (2000); Gambassi (2009):

[TABLE]

We emphasize that this derivative is to be calculated by keeping the area as well as the appropriate thermodynamic control parameters of the respective ensemble constant: these are, in the grand canonical ensemble, the reduced temperature $t$ and the bulk field $h_{R}$ , whereas in the canonical ensemble, these are $t$ and the total mass $\Sigma$ [Eq. 53]. Furthermore, in order to obtain the CCF for a system with vanishing aspect ratio, in Eq. 144 the limit $\rho\to 0$ must be taken only at the end of the calculation. Alternatively to Eq. 144, the CCF can be defined as the pressure difference between the film and the surrounding fluid. While these definitions are equivalent in the grand canonical ensemble, this is not necessarily the case in the canonical ensemble Gross et al. (2016). We briefly discuss these aspects in Appendix D, but continue to use the definition in Eq. 144 for the remainder of the present study. The consequences of defining the CCF under the condition of a fixed total volume $V=AL$ instead of a fixed area are discussed in Appendix E.

As alluded to above, in order to evaluate Eq. 144 in the canonical ensemble, we have to take into account the global OP constraint [Eq. 53], $\varphi AL=\Sigma=\mathrm{const}.$ , which immediately implies a dependence of the mean OP $\varphi$ on $L$ according to

[TABLE]

assuming a fixed transverse area $A$ . (We note that, as a consequence of this assumption, $\rho$ varies upon changing $L$ .) From Eqs. 132 and 144 we then obtain the canonical CCF (per area and per $k_{B}T_{c}$ )

[TABLE]

with the universal scaling function

[TABLE]

where $\ring{\hat{\Theta}}(x,\mathpzc{m},\rho)\equiv\ring{\Theta}(\hat{x}(x,\mathpzc{m}),\rho)$ . The contribution

[TABLE]

stems from the non-scaling term $\delta F_{\text{ns}}$ in Eq. 114. Note that, while $\delta F_{\text{ns}}$ is an explicitly $L$ -dependent contribution to the residual finite-size free energy [see Eq. 132], $\delta\ring{\Xi}_{\text{ns}}$ can be expressed fully in terms of the scaling variable $\rho$ and therefore can be considered as a universal contribution to the CCF.

In the grand canonical ensemble, assuming a fixed bulk field $h_{R}$ , one obtains from Eqs. 135 and 144 the CCF (per area and per $k_{B}T_{c}$ )

[TABLE]

with the universal scaling function

[TABLE]

in terms of $\tilde{\Theta}(x,\mathpzc{h},\rho)$ defined in Eq. 135. Furthermore, the r.h.s. of Eq. 150 can be expressed in terms of $\hat{\Theta}(x,\mathpzc{m},\rho)=\Theta(\hat{x}(x,\mathpzc{m}),\rho)$ defined in Eq. 136 as:

[TABLE]

where $\mathpzc{m}$ is determined as a function of $x$ and $\mathpzc{h}$ via the corresponding equation of state [see Eq. 141].

We now focus specifically on the approximation $\mathcal{O}(\epsilon^{0})$ of the CCF. In this case, the scaling form of the equation of state in Eq. 142 applies and can be used to express Eq. 151 as a function of $\mathpzc{m}$ instead of $\mathpzc{h}$ :

[TABLE]

Beyond $\mathcal{O}(\epsilon^{0})$ , Eq. 142 applies in general only in the bulk limit, i.e., for $x,\mathpzc{h}\gg 1$ . We note that Eq. 152 presupposes that both in Eq. 142 and in Eq. 151 the same approximation for the values of the critical exponents is used. Within the mean-field or Gaussian approximation considered here, one has in particular $\beta=\nu$ and, consequently, in Eq. 147 the term proportional to $\partial{m}\ring{\hat{\Theta}}$ vanishes. Furthermore, upon using Eq. 134, the fact that $\beta=1/2$ , and noting that $\partial_{\rho}\mathpzc{m}=0$ for the boundary conditions considered here [see Eq. 143], we can express Eq. 152 in terms of the scaling function $\Theta(\hat{x}(x,\mathpzc{m}),\rho)=\hat{\Theta}(x,\mathpzc{m},\rho)$ [see Eq. 129] as

[TABLE]

In order to analogously simplify the canonical CCF [Eq. 147], we define $\delta\ring{\mathcal{K}}$ as the total contribution to the canonical CCF $\ring{\mathcal{K}}$ [Eq. 146] stemming from the constraint-induced term $\delta F$ [Eq. 111]:

[TABLE]

with the associated universal scaling function [see Eq. 134]

[TABLE]

We note that, in fact, $\delta\ring{\Xi}=\delta\ring{\Xi}_{\text{ns}}+\delta\ring{\Xi}_{\text{s}}$ , where $\delta\ring{\Xi}_{\text{ns}}$ is given in Eq. 148 and the scaling function $\delta\ring{\Xi}_{\text{s}}$ is defined, analogously to Eq. 154, by $L^{-d}\delta\ring{\Xi}_{\text{s}}=-(1/A)\mathrm{d}\delta F_{\text{s}}/\mathrm{d}L$ in terms of $\delta F_{\text{s}}$ in Eq. 113. Using Eqs. 154, 117, and 129, $\ring{\Xi}$ in Eq. 147 can now be expressed in terms of $\Theta(\hat{x},\rho)$ as

[TABLE]

Comparing Eqs. 156 and 153 reveals that the scaling functions of the canonical and grand canonical CCF are related as

[TABLE]

where $\Theta(\hat{x},\rho)=\frac{1}{2}\mathcal{S}_{d,\text{reg}}(\hat{x},\rho)$ with $\mathcal{S}_{d,\text{reg}}$ given in Eq. 108. We also recall that, according to Eq. 134, $ru^{*}\mathpzc{m}^{2}\partial_{\hat{x}}\Theta(\hat{x},\rho)=\mathpzc{m}\partial{m}\Theta(\hat{x}(x,\mathpzc{m}),\rho)$ . In the next section, the implications of Eq. 157 are discussed further. We remark that, if the CCF is defined under the condition of a fixed volume $V$ [see Appendix E], the resulting scaling functions for periodic and Neumann boundary conditions coincide in the two ensembles [see Eq. 242]. In fact, in those cases the constraint-induced correction $\delta\ring{\Xi}$ in Eq. 155 vanishes identically, whereas for Dirichlet boundary conditions [see Eq. 239] it is nonzero and takes a form different from Eq. 155b.

V Discussion

Here we discuss the residual finite-size free energy reported in Eqs. 132 and 135, with the scaling functions defined in Eqs. 117 and 129, respectively, and the associated CCF given in Eqs. 156 and 153 within the one-loop approximation, i.e., to $\mathcal{O}(\epsilon^{0})$ . A meaningful comparison of the two ensembles requires to evaluate the scaling functions for the same value of the scaled mean OP $\mathpzc{m}$ . This can be achieved by relating $\mathpzc{m}$ to $\mathpzc{h}$ via the finite-size equation of state reported in Eqs. 142 and 143. In Sec. III.4 it has been shown that, within the approximation $\mathcal{O}(\epsilon^{0})$ considered here, the grand canonical residual finite-size free energy can be expressed as in Eq. 129 [see also Eq. 136]. This result provides the desired grand canonical scaling function $\Theta(\hat{x}(x,\mathpzc{m}),\rho)$ expressed in terms of $\mathpzc{m}$ . Furthermore, in the discussion of the residual finite-size free energy, we shall omit the non-scaling contribution $\delta F_{\text{ns}}$ in Eq. 114 stemming from the constraint. However, this latter contribution is taken into account for the scaling function of the CCF, because, as shown in Eq. 148, it takes on a scaling form. We recall here that the perturbative results in the present study refer to cubical systems with aspect ratios $0\leq\rho\lesssim 1$ . The description of rod-like geometries with $\rho\gg 1$ would require, inter alia, a different set of scaling variables Dohm (2011); Hucht et al. (2011). In addition, certain features of the (grand canonical) CCF for $\rho\simeq 1$ near bulk criticality ( $x=\mathpzc{h}=0$ ) Hucht et al. (2011) are not captured by our analytical expressions. Instead, they require more refined approaches, such as those described in Ref. Dohm (2011). In the subsequent discussion of the residual finite-size free energy and CCF for the various boundary conditions, we shall therefore focus on the case $0\leq\rho\lesssim 1$ .

V.1 Periodic boundary conditions

V.1.1 Residual finite-size free energy

We recall that, in the grand canonical ensemble, the scaling function of the renormalized residual finite-size free energy is given by Eqs. 129 and 136:

[TABLE]

whereas, in the canonical ensemble, we have [see Eq. 117]

[TABLE]

The function $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}$ is reported in Eq. 108a and $\hat{x}=\hat{x}(x,\mathpzc{m})$ is defined in Eq. 134. In the canonical ensemble, the constraint contributes to the scaling function $\ring{\Theta}^{\text{(p)}}$ the expression [see Eq. 113a]

[TABLE]

Within the considered one-loop approximation, both $\Theta^{\text{(p)}}$ and $\ring{\Theta}^{\text{(p)}}$ have to be evaluated at $\epsilon=0$ , i.e., $d=4$ . For a system with $\rho\neq 0$ and either periodic or Neumann boundary conditions, perturbative results for the grand canonical residual finite-size free energy are applicable only for $\hat{x}\gtrsim 1$ (see, in this respect, the discussion in Sec. III.4). Accordingly, in these cases, the region $\hat{x}\lesssim 1$ will be excluded from the corresponding plots. For $\rho=0$ , our perturbative results for periodic or Neumann boundary conditions are well behaved even for $x,\mathpzc{h}\lesssim 1$ and agree with the ones reported in Ref. Krech and Dietrich (1992) (see also Refs. Diehl et al. (2006); Grüneberg and Diehl (2008) for further discussions).

Since the contribution $\delta\ring{\Theta}_{\text{s}}$ due to the constraint [Eq. 160] vanishes for $\rho\to 0$ , the canonical and grand canonical scaling functions for periodic boundary conditions become identical in the thin-film limit, i.e.,

[TABLE]

This is visualized in Fig. 3, where $\Theta^{\text{(p)}}(\hat{x},\rho=0)$ is plotted as a function of $\hat{x}$ . In panels (b) and (c) of Fig. 3, we compare the dependence on $\hat{x}$ of the scaling functions in the two ensembles for fixed nonzero aspect ratios $\rho$ . The difference between $\ring{\Theta}^{\text{(p)}}$ (solid curve) and $\Theta^{\text{(p)}}$ (dashed curve) stems solely from the constraint-induced term $\delta\ring{\Theta}_{\text{s}}^{\text{(p)}}$ in Eq. 159, because the contribution from the regularized mode sum $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}$ is the same in both ensembles. Consequently, while $\Theta^{\text{(p)}}$ vanishes for $\hat{x}\to\infty$ , $|\ring{\Theta}^{\text{(p)}}|$ grows logarithmically upon increasing $\hat{x}$ [see Eq. 118]. This behavior stems from the absence of the zero-mode fluctuations in the canonical ensemble [see also Eq. 83], which, being spatially homogeneous, affect the residual free energy of a finite system for all values of $L$ . Figure 3(d) illustrates that a change in the aspect ratio $\rho$ has a strong effect on the canonical residual finite-size free energy, inducing, inter alia, a change of sign of $\ring{\Theta}^{\text{(p)}}$ at small $\hat{x}$ . In contrast, in the grand canonical case (not shown), increasing $\rho$ leads, within the considered range of $\hat{x}$ , mainly to an increase in the overall strength of $\Theta^{\text{(p)}}$ .

V.1.2 Critical Casimir force

At $\mathcal{O}(\epsilon^{0})$ , the difference between the canonical and grand canonical CCF is given by Eq. 157. Since $\delta\ring{\Xi}^{\text{(p)}}(\hat{x},\rho=0)=0$ [see Eq. 155a], Eq. 157 in the thin-film limit renders

[TABLE]

where we have used Eq. 134. We recall that the term $\mathpzc{m}\partial{m}\ring{\Theta}$ in Eq. 162 stems from Eq. 145, which is a direct consequence of the OP constraint and of the assumption of a fixed transverse area $A$ . For aspect ratios $\rho>0$ , $\delta\ring{\Xi}^{\text{(p)}}$ [Eq. 155a] is in general nonzero and reduces to the limiting expressions

[TABLE]

In both limits, $\delta\ring{\Xi}^{\text{(p)}}$ is independent of $\mathpzc{m}$ and $x$ . According to Eqs. 157 and 155a, in general the constraint-induced contribution $\delta\ring{\Xi}^{\text{(p)}}$ enhances the attractive character of the CCF compared to the unconstrained case. This is expected intuitively, because the constraint reduces the number of available fluctuation modes and thus the “fluctuation pressure” of the confined system compared to that of the bulk. Interestingly, however, this effect is absent if the CCF is defined under the condition of a fixed total volume $V$ instead of a fixed transverse area (see Appendix E). In this case, the CCF for periodic boundary conditions is identical in the two ensembles. For $\mathpzc{m}=0$ , the canonical and the grand canonical CCFs defined with fixed transverse area are related by a constant shift:

[TABLE]

Note that, beyond the approximation at $\mathcal{O}(\epsilon^{0})$ , $\mathpzc{m}$ is in general expected to acquire a dependence on $\rho$ Chen et al. (2000), such that Eq. 152 has to be used.

The scaling functions $\ring{\Xi}^{\text{(p)}}$ and $\Xi^{\text{(p)}}$ of the CCF at $\mathcal{O}(\epsilon^{0})$ in the two ensembles are shown in Fig. 4 for a vanishing mean OP, i.e., $\mathpzc{m}=0$ . According to Eq. 162, $\ring{\Xi}^{\text{(p)}}$ and $\Xi^{\text{(p)}}$ become identical in the thin-film limit $\rho=0$ , as shown in Fig. 4(a). For $\rho>0$ [Figs. 4(b) $-$ (d)], $\ring{\Xi}^{\text{(p)}}$ approaches the constant in Eq. 163a for large values $x\gg 1$ , while, correspondingly, $\Xi^{\text{(p)}}$ vanishes. We recall that, for $\rho>0$ , the results obtained perturbatively in the grand canonical ensemble are not expected to be reliable near the bulk critical point. Correspondingly, in spite of Eq. 164, we plot the grand canonical CCF in this case only for $\hat{x}\gtrsim 1$ . As Fig. 4(d) illustrates, upon increasing $\rho$ the absolute strength of $\ring{\Xi}^{\text{(p)}}$ for $\mathpzc{m}=0$ increases, while its functional form does not change significantly.

In Fig. 5, the scaling functions of the CCF are shown as functions of the scaled mean OP $\mathpzc{m}$ for $x=0$ . In the thin-film limit $\rho=0$ [Fig. 5(a)], in which the perturbative results at this order in $\epsilon$ are reliable in the whole domain of $\mathpzc{m}$ , the only difference between $\ring{\Xi}^{\text{(p)}}$ and $\Xi^{\text{(p)}}$ is due to Eq. 162. We conclude that, in contrast to $\delta\ring{\Xi}$ [Eq. 155a], the constraint-induced effect expressed in Eq. 145 increases the value of $\ring{\Xi}$ compared to the one of $\Xi$ . For nonzero $\rho$ [Figs. 5(b) and 5(c)], the OP constraint decreases the value of the canonical CCF relative to the grand canonical one by the amount given in Eq. 163b. Accordingly, for nonzero aspect ratios $\rho$ and in the limit $\mathpzc{m}\to\infty$ , the canonical CCF approaches a negative value 101010A similar result has been obtained in Ref. Gross et al. (2016) for $(++)$ boundary conditions.. Figure 5(d) illustrates in more detail the dependence of the canonical scaling function $\ring{\Xi}^{\text{(p)}}$ on $\mathpzc{m}$ for $x=0$ upon changing the aspect ratio. In passing, we mention that in the limit $\mathpzc{m}\to\infty$ the CCF defined under the condition of constant volume (see Appendix E) vanishes in both ensembles.

V.2 Dirichlet boundary conditions

V.2.1 Residual finite-size free energy

For Dirichlet boundary conditions we consider only the case $\mathpzc{m}=\mathpzc{h}=0$ ; hence the scaling functions of the residual finite-size free energy [Eqs. 117 and 129] depend solely on $x$ . The only difference between the residual finite-size free energies in the two ensembles is provided by the constraint-induced term $\delta F^{\text{(D)}}$ [Eq. 111b], which contributes to $\ring{\Theta}^{\text{(D)}}$ with the expression [Eq. 113b]

[TABLE]

In the thin-film limit $(\rho\to 0$ ), $\delta\ring{\Theta}_{\text{s}}^{\text{(D)}}(x,\rho)$ vanishes, so that in this case the canonical and grand canonical scaling functions are identical. In Fig. 6 the canonical ( $\ring{\Theta}^{\text{(D)}}$ ) and grand canonical ( $\Theta^{\text{(D)}}$ ) scaling functions are shown for Dirichlet boundary conditions, for $\mathpzc{m}=0$ , and for various aspect ratios $\rho$ . Due to Eq. 165, $\ring{\Theta}^{\text{(D)}}$ significantly differs from $\Theta^{\text{(D)}}$ upon increasing the aspect ratio $\rho$ . In particular, while $\Theta^{\text{(D)}}$ vanishes exponentially for $x\to\infty$ , $\ring{\Theta}^{\text{(D)}}$ diverges logarithmically in the same limit; this latter behavior is similar to the one discussed above for periodic boundary conditions [see Sec. V.1.1] and is due to the constraint-induced contribution [see Eq. 118].

V.2.2 Critical Casimir force

Since here we are considering $\mathpzc{m}=0$ , according to Eq. 157 the constraint-induced term $\delta\ring{\Xi}^{\text{(D)}}$ in Eq. 155b provides the only difference between the canonical and grand canonical CCFs. Therefore, in the thin-film limit ( $\rho\to 0$ ) the CCFs for Dirichlet boundary conditions and $\mathpzc{m}=0$ are identical in the two ensembles, as are the corresponding residual finite-size free energies. The quantity $\delta\ring{\Xi}^{\text{(D)}}$ attains two distinct $\rho$ -dependent values for $x\to 0$ and $x\to\infty$ :

[TABLE]

which coincide with the corresponding limits of $\delta\ring{\Xi}^{\text{(p)}}$ for periodic boundary conditions [see Eq. 163]. Accordingly, while $\Xi^{\text{(D)}}$ vanishes in the limit $x\to\infty$ , the scaling function $\ring{\Xi}^{\text{(D)}}$ of the canonical CCF does not. This means that the effect of the OP constraint [Eq. 6] on the fluctuations manifests itself in the form of an attractive contribution to the CCF even for arbitrarily thick films. The scaling functions $\Xi^{\text{(D)}}$ and $\ring{\Xi}^{\text{(D)}}$ of the CCF for Dirichlet boundary conditions are illustrated in Figs. 7(a) $-$ (c) for various aspect ratios. In general, the canonical CCF turns out to be attractive for all aspect ratios considered here and its strength is found to be significantly larger than that of the grand canonical CCF. We remark that a similar constraint-induced effect is present also for the CCF defined under the constraint of a fixed volume and for Dirichlet boundary conditions (see Appendix E). As Fig. 7(d) shows, the strength of the canonical CCF significantly grows upon increasing the aspect ratio from thin-film geometry towards a cubical system. In contrast to the canonical CCF, the grand canonical CCF changes its character from attractive to repulsive upon increasing the aspect ratio $\rho$ [see Fig. 7(c)].

V.3 Neumann boundary conditions

V.3.1 Residual finite-size free energy

The scaling functions $\ring{\Theta}^{\text{(N)}}$ and $\Theta^{\text{(N)}}$ of the canonical and the grand canonical residual finite-size free energy for Neumann boundary conditions [Eqs. 117 and 129, respectively] are shown in Fig. 8 as functions of the scaling variable $\hat{x}$ [Eq. 134] for various values of the aspect ratio $\rho$ . Due to the presence of the constraint-induced term $\delta F_{\text{s}}^{\text{(N)}}$ [Eq. 113a], $\ring{\Theta}$ and $\Theta$ are equal only for $\rho=0$ , while they increasingly differ for larger values of $\rho$ . The qualitative behavior of $\Theta^{\text{(N)}}$ is similar to that of the scaling function $\Theta^{\text{(p)}}$ for periodic boundary conditions [see Fig. 3]. However, for $\rho=0$ , $\Theta^{\text{(N)}}$ is about 50 times smaller in strength than $\Theta^{\text{(p)}}$ ; the strengths become comparable only for $\rho\simeq 1$ . As discussed in Sec. III.4, in the grand canonical ensemble and for $\rho>0$ , the perturbative expressions for the residual finite-size free energy and the CCF are reliable only for $\hat{x}\gtrsim 1$ .

V.3.2 Critical Casimir force

The scaling functions $\ring{\Xi}^{\text{(N)}}$ and $\Xi^{\text{(N)}}$ of the canonical and the grand canonical CCF are shown in Fig. 9 as functions of the scaled temperature $x$ (for $\mathpzc{m}=0$ ) and in Fig. 10 as functions of the scaled mean OP $\mathpzc{m}$ (for $x=0$ ) for various aspect ratios $\rho$ . For $\rho=0$ , the contribution $\delta\ring{\Xi}^{\text{(N)}}$ in Eq. 155a vanishes, such that, according to Eqs. 157 and 134, the scaling functions of the canonical and the grand canonical CCF are related as follows:

[TABLE]

Hence, for $\mathpzc{m}=0$ and $\rho=0$ , the CCFs in the two ensembles are identical, as illustrated in Fig. 9(a). For nonzero mean OP $\mathpzc{m}\neq 0$ and $\rho=0$ , the difference between the CCFs stems from the last term in Eq. 167, the presence of which is a direct consequence of Eq. 145. As shown in Fig. 10(a), similarly to the case with periodic boundary conditions, this contribution causes the canonical CCF to be less attractive than the grand canonical one. In contrast, for nonzero aspect ratios $\rho>0$ , the contribution $\delta\ring{\Xi}^{\text{(N)}}$ [Eq. 155a] dominates in Eq. 157 and typically leads to a more attractive canonical CCF compared to the grand canonical one. This is illustrated by the panels (b) and (c) of Figs. 9 and 10. For $\rho>0$ , $\ring{\Xi}^{\text{(N)}}$ approaches a nonzero constant in the limit $\hat{x}\to\infty$ , whereas $\Xi^{\text{(N)}}$ vanishes. Specifically, Eq. 155a implies

[TABLE]

as in the case of periodic and Dirichlet boundary conditions. The canonical CCF remains attractive for all aspect ratios $\rho$ considered here and its strength grows significantly upon increasing $\rho$ [see Figs. 9(d) and 10(d)]. In contrast, the grand canonical CCF changes its character from attractive to repulsive upon increasing $\rho$ [see Figs. 9(c) and 10(c)]. Under the constraint of a fixed total volume, the associated CCF for Neumann boundary conditions (see Appendix E) turns out to be identical in the two ensembles for all values of $x$ , $\mathpzc{m}$ , and $\rho$ considered here.

VI Summary and outlook

In this study, we have investigated the implications of a global constraint on a scalar OP within a field theoretical approach. Generic results, which are independent of the specific form of the field-theoretic action describing the confined system are summarized in Sec. II.5 and will not be repeated here. We have subsequently applied this formalism to a Landau-Ginzburg model for a finite cubical volume $V=AL$ in the supercritical regime ( $T\geq T_{c}$ , where $T_{c}$ is the bulk critical temperature). We have considered periodic, Dirichlet, and Neumann boundary conditions along the transverse direction of extent $L$ and periodic boundary conditions along the remaining lateral directions [see Fig. 1]. Our approach is expected to be applicable for films, i.e., for systems with aspect ratios $\rho$ [Eq. 104 and Fig. 1] fulfilling $0\leq\rho\lesssim 1$ . Perturbative expressions for the residual finite-size free energy and the CCF have been obtained within an $\epsilon$ -expansion to $\mathcal{O}(\epsilon^{0})$ , corresponding to the Gaussian approximation of the free energy and to the mean-field approximation of the equation of state. While most of the present study focuses on the CCF emerging under the condition of fixed transverse area $A$ [see Eq. 144], Appendix E considers the alternative condition of having a fixed volume $V$ , which is also briefly summarized below. Within the Gaussian approximation, the OP constraint can be implemented exactly and one obtains the expressions of the canonical free energy presented in Eq. 106. Non-Gaussian contributions to the field-theoretic action can be accounted for perturbatively, based on a suitably defined Green function [Eq. 26]. Apart from the contribution of the $\phi^{4}$ -term to the effective temperature variable $\hat{\tau}$ [Eq. 79], non-Gaussian effects have not been considered.

The consequences of the constraint on the residual finite-size free energy and on the CCF are summarized as follows:

The canonical residual finite-size free energy $\ring{\mathpzc}{f}_{\text{res}}$ (per volume $AL$ and per $k_{B}T_{c}$ , see Eq. 115) differs at $\mathcal{O}(\epsilon^{0})$ from the grand canonical $\mathpzc{f}_{\text{res}}$ by an extra contribution $\delta F(\hat{\tau},L,A)$ [Eq. 111]. This term is induced by fluctuations only. Its presence can be most easily understood for periodic boundary conditions: in this case, in fact, it simply removes the zero mode from the mode sum in the free energy [see Eq. 83]. $\delta F$ can be decomposed into a scaling and a non-scaling contribution as in Eq. 112, where the latter explicitly depends on the film thickness $L$ . For periodic and Neumann boundary conditions, the constraint-induced contribution cancels the divergence of the grand canonical residual finite-size free energy at criticality caused by the zero-mode Brezin and Zinn-Justin (1985); Rudnick et al. (1985). In the limit $x\to\infty$ or $\mathpzc{m}\to\infty$ , $\delta F$ gives rise to a logarithmic divergence of the scaling function of the residual finite-size free energy in the canonical ensemble, $\ring{\Theta}\propto\rho^{d-1}\ln\hat{x}$ , independently of the choice of the boundary conditions [see Eq. 118 and panels (b), (c), and (d) of Figs. 3, 6, and 8.]. 2. 2.

For a vanishing aspect ratio ( $\rho=0$ ), the constraint-induced contribution to the residual finite-size free energy vanishes [see Eqs. 116 and 117]. This holds for all boundary conditions considered here and to all orders in perturbation theory (see the discussion in Sec. II.5). As a consequence, at $\mathcal{O}(\epsilon^{0})$ the canonical and the grand canonical residual finite-size free energies are identical [see panel (a) in Figs. 3, 6, and 8] and reduce (for $\mathpzc{m}=0$ ) to the one-loop results of Ref. Krech and Dietrich (1992). For the boundary conditions considered here, the equation of state acquires finite-size corrections only beyond the leading order in the $\epsilon$ -expansion 111111For nonzero symmetry breaking surface fields $h_{1}$ , the equation of state acquires finite-size contributions already at the mean-field level Gross et al. (2016). This is expected to hold also for Dirichlet boundary conditions for a nonzero mean OP.. 3. 3.

The CCF depends on whether it is defined under the constraint of a constant transverse area $A$ [Sec. IV.2] or a constant total volume $V=AL$ [Appendix E]. In the latter case, at $\mathcal{O}(\epsilon^{0})$ the OP constraint has no effect on the scaling functions for periodic and Neumann boundary conditions (see Fig. 11), i.e., the canonical and the grand canonical CCF coincide. For Dirichlet boundary conditions, instead, ensemble differences are present for both definitions of the CCF. They vanish, however, in the thin-film limit ( $\rho\to 0$ ). 4. 4.

If the CCF is defined under the condition of a fixed transverse area $A$ , the OP constraint is reflected by two distinct contributions to the canonical CCF: first, the fluctuation-induced term $\delta F$ in the residual finite-size free energy yields, via Eq. 144, a contribution $\delta\ring{\Xi}$ to the scaling function of the CCF [see Eq. 154]. Within the Gaussian approximation, the expressions of $\delta F$ and hence $\delta\ring{\Xi}$ coincide for periodic and Neumann boundary conditions [see Eq. 155a; $\delta\ring{\Xi}^{\text{(D)}}$ for Dirichlet boundary conditions is reported in Eq. 155b]. These contributions vanish in the thin-film limit ( $\rho\to 0$ ). A second difference between the canonical and the grand canonical CCFs arises from the fact that the mean OP $\varphi$ , which enters into the definition of the OP scaling variable $\mathpzc{m}$ [see Eq. 146], is affected, via Eq. 145, by the constraint of having a fixed total OP $\Phi$ and a fixed area $A$ . This effect occurs also within MFT Gross et al. (2016). 5. 5.

For nonzero aspect ratios ( $\rho>0$ ), the canonical CCF is typically more attractive than the grand canonical CCF (see Figs. 4, 5, 7, 9, and 10). Indeed, a restriction on the OP [see Eq. 16] is expected to reduce the fluctuation contribution to the pressure of the confined system and therefore leads to an additional, attractive contribution to the CCF. However, this effect is absent for periodic and Neumann boundary conditions if the CCF is defined with a fixed total volume $V$ (see Appendix E). In the limit $x\to\infty$ or $\mathpzc{m}\to\infty$ , the scaling function $\ring{\Xi}$ of the canonical CCF for fixed transverse area approaches a negative constant $\propto\rho^{d-1}$ [see Eq. 163]. This asymptotic value is the same for all boundary conditions studied here. In contrast, the CCF defined with constant volume vanishes in the limit $x\to\infty$ or $\mathpzc{m}\to\infty$ for all boundary conditions considered. 6. 6.

For $\rho=0$ and vanishing mean OP $\mathpzc{m}=0$ , the canonical and the grand canonical CCF defined for fixed transverse areas coincide [see panels (a) of Figs. 4, 7, and 9] and reduce to the expressions reported in Ref. Krech and Dietrich (1992). In contrast, for $\rho=0$ and nonzero mean OPs $\mathpzc{m}\neq 0$ , the OP constraint yields, via Eq. 145, a repulsive contribution to the canonical CCF [see Figs. 5(a) and 10(a)], although in that case the corresponding residual finite-size free energies are identical. In general, this repulsive contribution is absent for the CCF defined for constant volume, because in that case $\mathrm{d}\varphi/\mathrm{d}L=0$ . For $\rho=0$ , in general the canonical CCF vanishes in the limits $x\to\infty$ or $\mathpzc{m}\to\infty$ .

We mention that the perturbative results obtained here for the grand canonical CCF agree qualitatively with corresponding Monte Carlo simulation data Hucht et al. (2011); Hasenbusch (2011); Vasilyev et al. (2009). If one aims at improving the analytical predictions in the grand canonical ensemble, in particular the issues associated with the presence of a zero mode must be dealt with appropriately (see, e.g., Refs. Esser et al. (1995); Dohm (2011, 2014); Grüneberg and Diehl (2008); Diehl et al. (2006)). The purpose of the present study is, however, not to present quantitatively accurate expressions for the grand canonical CCF, but to provide a self-contained treatment of ensemble differences due to fluctuations in a near-critical, confined fluid. We remark that, for the Ising model at very high temperatures [see Eqs. (131) and (132) in Ref. Gross et al. (2016)], the constraint induced contribution to the free energy and CCF reduces (up to irrelevant constants) to the expressions given in Eqs. 118 and 163.

The present study can be considered as a sequel to Ref. Gross et al. (2016), where we have investigated the ensemble differences within MFT for $(++)$ and $(+-)$ boundary conditions, i.e., for a confined, near-critical fluid which exhibits strong adsorption at the container walls in the transverse direction. For periodic and Neumann boundary conditions, as well as for the disordered phase with Dirichlet boundary conditions, a mean-field contribution to the residual finite-size free energy is in general absent. Periodic boundary conditions, although not experimentally relevant (see, however, Ref. Rancon et al. (2016)), are arguably the simplest case for which the influence of a constraint on the OP fluctuations can be studied analytically. Dirichlet boundary conditions apply at the RG fixed point of the so-called ordinary surface universality class Diehl (1986). Generically, confining surfaces exhibit a preference for one of the two species of a binary liquid mixture, which gives rise to a symmetry breaking surface field and therefore to ( $++$ ) or ( $+-$ ) boundary conditions. If the surface is endowed with a periodically striped pattern of alternating surface fields, for thick films such surfaces behave effectively as if there is a Dirichlet boundary condition (see Sec. IIIB in Ref. Sprenger et al. (2006)). This way, Dirichlet boundary conditions can be realized even for classical binary liquid mixtures. Neumann boundary conditions apply at the fixed-point of the so-called special surface universality class and correspond to weak adsorption. Our predictions lend themselves to be tested by Monte Carlo Vasilyev (2015); Gross et al. (2016) or molecular dynamics Puosi et al. (2016); Roy et al. (2016) simulations. In future studies, the theory developed here could be extended to the sub-critical region, where, so far, predictions for the canonical CCF are not available.

Acknowledgements.

We thank O. Vasilyev for informing us about preliminary simulation results and D. Dantchev and M. Krüger for useful discussions.

Appendix A Gaussian lattice field theory and dimensional considerations

Here, we consider an uncorrelated Gaussian random field on a $d$ -dimensional hypercubic lattice of volume $V=Na^{d}$ , where $N$ is the number of lattice points and $a$ the lattice constant. This is arguably the simplest system for which the effect of the constraint on the OP field [Eq. 1] can be studied exactly. In addition, the finite lattice constant of the model provides a natural regularization. Before proceeding, we recall that the OP field $\phi$ , the reduced temperature $\tau$ , and the bulk field $h$ appearing in the action [see Eq. 49] must have the engineering dimensions

[TABLE]

in order to render $\mathcal{H}$ in Eq. 11, and thus also the free energy $\mathcal{F}$ in units of $k_{B}T_{c}$ , dimensionless Amit and Martin-Mayor (2005).

A.1 Grand canonical ensemble

We consider a dimensionless Hamiltonian of the form

[TABLE]

corresponding to a Gaussian ensemble of uncorrelated random variables $\phi_{i}$ . Using the lattice constant $a$ to render the integration measure dimensionless, the lattice partition function corresponding to Eq. 11 is given by

[TABLE]

We point out that the contribution which diverges in the continuum limit $a\to 0$ (with fixed volume $V$ ) is contained completely in $\mathcal{Z}(0)$ . It is therefore convenient to define the actual grand canonical partition function by dividing $\mathcal{Z}(h)$ by $\mathcal{Z}(0)$ 121212This is, in fact, a commonly adopted definition of a path integral Kleinert (2009).. However, since this term does not interfere with others, we carry it along in our calculations. The bulk field $h$ can be related to the average $\langle\Phi\rangle$ of the total OP

[TABLE]

by noting that, according to Eq. 171, $\left\langle\Phi\right\rangle=\partial\ln\mathcal{Z}/\partial h=Vh/\tau$ and thus

[TABLE]

Accordingly, the free energy in units of $k_{B}T_{c}$ follows as

[TABLE]

Below, we compare these results are compared with the corresponding expressions in the canonical ensemble.

A.2 Canonical ensemble

The counterpart of Eq. 170 in the canonical ensemble is given by

[TABLE]

subject to a constraint of the form

[TABLE]

which is imposed on the field $\{\phi_{i}\}$ such that Eq. 172 is recovered for $\mathpzc{w}_{i}=1$ . Here, we keep the general expression involving $\mathpzc{w}_{i}$ in order to be able to track the influence of the constraint. Using the lattice constant $a$ in order to render the integration measures and the argument of the $\delta$ function dimensionless (note that $[\Phi]=a^{1+d/2}$ ), the constrained lattice partition function corresponding to Eq. 9 is given by

[TABLE]

Because we require the weights $\mathpzc{w}_{i}$ to be dimensionless, the dimension of the auxiliary integration variable $J$ is $[J]=a^{-(1+d/2)}$ . Accordingly, $\ring{\mathcal{Z}}$ in Eq. 177 is dimensionless. In Eq. 177, first performing the Gaussian integrals over $\{\phi_{j}\}$ and then the remaining one over $J$ , we obtain

[TABLE]

This result can also be obtained directly from Eqs. 171 and 177 by noting that

[TABLE]

Accordingly, the free energy of the constrained system, in units of $k_{B}T_{c}$ , is given by

[TABLE]

In the last step of Eq. 180, we have introduced the aspect ratio $\rho=L/A^{1/(d-1)}$ , assuming a lattice of cubical geometry and volume $V=AL$ . Introducing the lattice correlation function $G_{ij}$ for this model of an uncorrelated random field $\{\phi_{i}\}$ in the form $G_{ij}=\delta_{i,j}/(a^{d}\tau)$ (which has an engineering dimension of $a^{2-d}$ ), the second term on the r.h.s. of the second equation in Eq. 180 can be written alternatively as $\frac{1}{2}\ln\left(2\pi a^{d-2}\sum_{i,j}\mathpzc{w}_{i}G_{ij}\mathpzc{w}_{j}\right)$ . In the continuum limit ( $a\to 0$ with fixed $V$ ), this expression reduces, upon recalling that $\sum_{i}a^{d}\to\int\mathrm{d}V$ and by neglecting a divergent factor $a^{-d-2}$ , to the term $\frac{1}{2}\ln\left(2\pi(\mathpzc{w},G,\mathpzc{w})\right)$ in the corresponding expression for the free energy in Eq. 34. The second term on the r.h.s. of the last equation in Eq. 180 is a constraint-induced contribution which appears in the same form also for periodic and Neumann boundary conditions within the corresponding continuum field theory (see Eqs. 111a and 111c, respectively). The last term in Eq. 180 is the usual canonical bulk free energy. Its sign is opposite to that of the analogous term in Eq. 174 because the bulk contributions to $\mathcal{F}(h)$ and $\ring{\mathcal{F}}(\Phi)$ are related via a Legendre transform.

Appendix B Fourier transforms

We consider a function $f(\mathbf{r})$ which is periodic with respect to a $D$ -dimensional macroblock of volume $A=\Pi_{\alpha=1}^{D}L_{\alpha}$ ( $D\leq d$ ) with edges of length $L_{\alpha}$ :

[TABLE]

This function can be expressed in terms of a Fourier series

[TABLE]

Equation 181 implies that $\mathbf{k}$ is discrete, i.e., $\mathbf{k}=2\pi\left(n_{1}/L_{1},\ldots,n_{D}/L_{D}\right)$ with $\mathbf{n}\in\mathbb{Z}^{D}$ . Forming $\int_{A}\mathrm{d}^{D}r\,\exp(-\mathrm{i}\mathbf{k}\cdot\mathbf{r})f(\mathbf{r})$ and inserting Eq. 182 yields the inverse Fourier transform

[TABLE]

by using

[TABLE]

The Fourier transform $\hat{F}(\mathbf{k},\mathbf{k}^{\prime})$ of a function $F(\mathbf{r},\mathbf{r}^{\prime})=f(\mathbf{r}-\mathbf{r}^{\prime})$ , where $f$ is periodic and $k_{\alpha}$ and $k_{\alpha}^{\prime}$ are discrete (as above), is given by

[TABLE]

where Eq. 184 has been used.

Appendix C Canonical finite-size free energy

C.0.1 Periodic boundary conditions

In order to determine the regularized finite-size free energy for a system with periodic boundary conditions in all spatial directions and arbitrary aspect ratio, we follow the approach as taken in Refs. Dohm (2008, 2011). In these studies, only the case $\varphi=0$ was considered. Within the present theory, the generalization of the free energy to nonzero $\varphi$ amounts to replacing the temperature parameter $\tau$ by $\hat{\tau}$ [see Eq. 79]. In order to extract the finite-size part of the mode sum [see Eqs. 75 and 83]

[TABLE]

we introduce as a regularization the subtraction of the corresponding bulk expression:

[TABLE]

As shown in Refs. Dohm (2008, 2011), this expression can be simplified to

[TABLE]

with

[TABLE]

where

[TABLE]

is the elliptic Jacobi theta function $\theta_{3}(z|q)$ Olver et al. (2010). Due to the presence of the theta function, $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}$ is not a homogeneous function of its first argument, i.e., there is no value of $\kappa$ for which, with arbitrary $\hat{x}$ and $b$ , one has $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}(b\hat{x},\rho)\,=\,b^{\kappa}\mathcal{S}^{\text{(p)}}_{d,\text{reg}}(\hat{x},\rho)$ . In order to facilitate the analysis of the scaling behavior (see Sec. IV), $\mathrm{S}^{\text{(p)}}_{d,\text{reg}}$ in Eq. 188 has been brought directly into a suitable scaling form.

It is useful to note the limiting behaviors $\vartheta(y\to\infty)=1$ and $\vartheta(y\to 0)\simeq(\pi/y)^{1/2}[1+2\exp(-\pi^{2}/y)]$ . Accordingly, the integrand in Eq. 189 vanishes in the limit $y\to 0$ for all $\rho$ and $\hat{x}$ 131313For $\rho<1$ , $y\to 0$ , and all $\hat{x}$ , the integrand in Eq. 189 behaves in leading order as $2\pi^{d/2}y^{-d/2-1}\exp(-\pi^{2}/y)$ ., and decays exponentially as a function of $y$ for $y\to\infty$ and $\hat{x}\neq 0$ . Thus $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}$ is finite for all $d$ and $\hat{x}\neq 0$ . In contrast, for $\hat{x}\to 0$ and nonzero $\rho$ , $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}$ diverges asymptotically as

[TABLE]

due to the leading behavior of the integrand in Eq. 189 at the upper limit of integration 141414For a sufficiently large constant $A$ ensuring $\vartheta(A)\simeq 1$ and $B\gg A$ , the r.h.s. of Eq. 189 can be estimated as $C(\hat{x})-D(\hat{x})$ , where $D(\hat{x})\equiv\rho^{d-1}\int_{A}^{B}\mathrm{d}yy^{-1}\exp(-\hat{x}y/(4\pi^{2}))$ and $C(\hat{x})$ captures the remaining contributions of the integrand for small $y$ . According to the preceding discussion in the main text, $C(\hat{x})$ is thus finite for all $\hat{x}$ . In fact, the dominant contribution for $\hat{x}\to 0$ stems from the term $D(\hat{x})$ . This follows from noting that, for small $\hat{x}$ , the integrand in $D(\hat{x})$ contributes only if $y\ll 4\pi^{2}/\hat{x}$ , for which $\exp(-\hat{x}y/(4\pi^{2}))\simeq 1$ . Accordingly, one obtains the estimates $D(\hat{x})\sim-\rho^{d-1}\int_{A}^{4\pi^{2}/\hat{x}}\mathrm{d}yy^{-1}\sim-\rho^{d-1}\ln(4\pi^{2}/(\hat{x}A))$ up to $\hat{x}$ -independent terms, which gives the asymptotic result in Eq. 191.. Below this property will be discussed further [see Eq. 198]. Since $\lim_{\rho\to 0}\rho\vartheta(\rho^{2}y)=\sqrt{\pi/y}$ , Eq. 189 reduces in the thin-film limit $\rho\to 0$ to

[TABLE]

which can be shown Kastening and Dohm (2010) to be identical to the expression derived in Ref. Krech and Dietrich (1992):

[TABLE]

where $\Gamma$ is the Gamma function and

[TABLE]

As implied by Eq. 191, $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}(x,\rho=0)$ [Eq. 192] is finite for all $\hat{x}\geq 0$ .

In order to evaluate the bulk expression appearing in the subtraction in Eq. 187, we note that, for an arbitrary constant $a>0$ , one has in dimensional regularization Rudnick et al. (1985); Amit and Martin-Mayor (2005):

[TABLE]

with

[TABLE]

In summary, for periodic boundary conditions and finite aspect ratio $\rho$ , the total free energy defined in Eq. 75 takes the form

[TABLE]

where $\mathcal{L}_{b}$ is defined in Eq. 49 and the constraint-induced contribution $\delta F^{\text{(p)}}$ is reported in Eq. 111a. The contribution in Eq. 196 involving the term $\int\mathrm{d}^{d}k\ln(\mathbf{k}^{2})$ formally vanishes in dimensional regularization Zinn-Justin (2002) and will be disregarded henceforth. (This term would be canceled also by additive renormalization of the total free energy Krech and Dietrich (1992).)

Following Eq. 115, we extract from Eq. 196 the residual finite-size free energy per volume,

[TABLE]

with the scaling function

[TABLE]

where $\delta F^{\text{(p)}}_{\text{s}}=-\frac{1}{2}\ln(\rho^{d-1}\hat{x}/(2\pi))$ [Eq. 113a] is the scaling contribution to the constraint-induced term and $\delta F_{\text{ns}}$ is the non-scaling contribution [Eq. 114]. The divergence of $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}$ expressed in Eq. 191 is canceled by $\delta F^{\text{(p)}}_{\text{s}}$ in $\ring{\Theta}^{\text{(p)}}$ , which therefore remains finite for all $\hat{x}\geq 0$ and all aspect ratios $\rho$ . In contrast, for an unconstrained system with $\rho\neq 0$ , the corresponding residual finite-size free energy diverges for $\hat{x}\to 0$ as in Eq. 191. This divergence originates from the contribution of the mode with $\mathbf{k}=\mathbf{0}$ in the mode sum in Eq. 186 151515We remark that, also in the thin-film limit $\rho\to 0$ , a problematic infrared divergence of the residual finite-size free energy occurs at higher orders in perturbation theory Diehl et al. (2006); Grüneberg and Diehl (2008), despite the finiteness of $\mathpzc{f}_{\text{res}}$ at the Gaussian level [see Eq. 193]. However, this fact is immaterial for the present discussion.. Since $\mathcal{S}^{\text{(p)}}_{d,\text{reg}}(\hat{x}\to\infty)\to 0$ , the presence of the constraint-induced term $\delta F^{\text{(p)}}_{\text{s}}$ leads to a logarithmic divergence of $\ring{\Theta}^{\text{(p)}}$ for $\hat{x}\gg 1$ :

[TABLE]

C.0.2 Dirichlet boundary conditions

We consider a $d$ -dimensional box with periodic boundary conditions in the $d-1$ lateral directions and Dirichlet boundary conditions at $z=0,L$ (see Fig. 2). In the basic expression for the free energy in Eq. 75, we have $\psi=\varphi=0$ ; $E_{n}$ is defined in Eq. 88 and the quantity $(1/2)\ln(\mathpzc{w},G,\mathpzc{w})$ is reported in Eq. 111b. The expression of the corresponding mode sum can be obtained from the one for periodic boundary conditions by noting that, due to Eqs. (75) and (88), one has

[TABLE]

with $L^{\prime}\equiv 2L$ and where we introduced the wavenumber $p^{\prime}\equiv 2\pi n/L^{\prime}$ with $n\in\mathbb{Z}$ . In the last line in Eq. 200, we have identified the first term as (half of) the mode sum of a $d$ -dimensional system with periodic boundary conditions and aspect ratio $L^{\prime}/L_{\parallel}$ , and the second term as (half of) the mode sum of a $(d-1)$ -dimensional cubic system of volume $L_{\parallel}^{d-1}$ with periodic boundary conditions (see Eq. 187). Using Eq. 188, we can thus express the regularized mode sum for Dirichlet boundary conditions as

[TABLE]

with

[TABLE]

Since, for $x\to\infty$ , $\mathcal{S}^{\text{(p)}}_{d-1,\text{reg}}(x,1)$ vanishes exponentially as a function of $x$ , upon using Eq. 193 one recovers in the thin-film limit ( $\rho\to 0$ ) the expression contained in Ref. Krech and Dietrich (1992):

[TABLE]

Furthermore, due to Eq. 191, the divergences for $x\to 0$ of the two separate terms in Eq. 202 cancel, rendering $\mathcal{S}^{\text{(D)}}_{d,\text{reg}}$ finite for $x=0$ and all aspect ratios. Taking into account Eqs. 187 and 195, the free energy of the constrained system [Eq. 75] for Dirichlet boundary conditions with vanishing mean OP follows as

[TABLE]

The constraint-induced term $\delta F^{\text{(D)}}$ is reported in Eq. 111b. From Eq. 115, one obtains the residual finite-size free energy per volume:

[TABLE]

with the scaling function

[TABLE]

where $\delta F_{\text{s}}$ and $\delta F_{\text{ns}}$ are given in Eqs. 113b and 114, respectively. For $x\gg 1$ , $\mathcal{S}^{\text{(D)}}_{d,\text{reg}}(x,\rho)$ vanishes exponentially so that the asymptotic behavior of $\delta F_{\text{s}}^{\text{(D)}}$ dominates, resulting in a logarithmic divergence of $\ring{\Theta}^{\text{(D)}}$ :

[TABLE]

analogously to $\Theta^{\text{(p)}}$ [Eq. 199]. Since both $\mathcal{S}^{\text{(D)}}_{d,\text{reg}}$ and $\delta F_{\text{s}}^{\text{(D)}}$ are finite for $x\to 0$ [see Eq. 94], also $\ring{\Theta}^{\text{(D)}}$ remains finite in that limit.

C.0.3 Neumann boundary conditions

The mode sum for Neumann boundary conditions can be related to the one for Dirichlet boundary conditions [Eq. 200] by writing

[TABLE]

From Eqs. 201 and 202 we thus obtain the regularized mode sum

[TABLE]

with

[TABLE]

where $\mathrm{S}^{\text{(p)}}_{d,\text{reg}}$ is given by Eq. 189. In the thin-film limit ( $\rho\to 0$ ) the second term in Eq. 210 vanishes so that

[TABLE]

with $\mathcal{S}^{\text{(D)}}_{d,\text{reg}}(\hat{x},\rho=0)$ given by Eq. 203, in agreement with explicit calculations for $\rho=0$ reported in Ref. Krech and Dietrich (1992). However, in contrast to the case of Dirichlet boundary conditions, $\mathcal{S}^{\text{(N)}}_{d,\text{reg}}$ diverges for $\hat{x}\to 0$ and nonzero $\rho$ :

[TABLE]

due to Eq. 191. The free energy of the constrained system [Eq. 75] for Neumann boundary conditions is given by

[TABLE]

where $\mathcal{L}_{b}$ is defined in Eq. 49 and the constraint-induced contribution $\delta F^{\text{(N)}}$ is reported in Eq. 111c. We remark that, for $\varphi=0$ , the grand canonical $\mathcal{F}^{\text{(N)}}$ and $\mathcal{F}^{\text{(D)}}$ are identical [Eq. 204], except that the sign of the surface contribution is reversed. The residual finite-size free energy per volume follows from Eq. 115 as

[TABLE]

with the scaling function

[TABLE]

where $\delta F_{\text{s}}$ and $\delta F_{\text{ns}}$ are given in Eqs. 113a and 114, respectively. In the thin-film limit ( $\rho\to 0$ ) and for a vanishing mean OP $\varphi$ (implying $\hat{x}\to x$ , see Eq. 134), the scaling functions for Dirichlet and Neumann boundary conditions are identical, $\ring{\Theta}^{\text{(D)}}=\ring{\Theta}^{\text{(N)}}$ , as a consequence of Eq. 211. Since $\mathcal{S}^{\text{(N)}}_{d,\text{reg}}(\hat{x},\rho)\to 0$ for $x\to\infty$ , it follows from the presence of $\delta F_{\text{s}}^{\text{(N)}}$ that $\ring{\Theta}^{\text{(N)}}$ diverges logarithmically for $\hat{x}\gg 1$ :

[TABLE]

The divergences for $\hat{x}\to 0$ of $\mathcal{S}^{\text{(N)}}_{d,\text{reg}}$ [Eq. 212] and $\delta F_{\text{s}}^{\text{(N)}}$ [Eq. 113a] cancel in $\ring{\Theta}^{\text{(N)}}$ , rendering the residual finite-size free energy in the canonical case and at bulk criticality finite for all aspect ratios. In contrast, in the grand canonical case, Eq. 212 implies a divergent residual finite-size free energy for $\hat{x}\to 0$ and nonzero $\rho$ . This divergence is due to a zero mode in the fluctuation spectrum, as it is also the case for periodic boundary conditions.

Appendix D Critical Casimir forces obtained from pressure differences

Alternatively to the definition based on the residual finite-size free energy [Eq. 144], the CCF $\mathcal{K}$ can be defined as the difference between the pressure $p$ in the confined system and the pressure $p_{b}$ in the surrounding bulk medium:

[TABLE]

For fixed area $A=V/L$ , these pressures follow from the corresponding free energy densities $\mathpzc{f}$ and $\mathpzc{f}_{b}$ :

[TABLE]

The same relations apply also in the canonical ensemble. The bulk pressure can be obtained from the thermodynamic limit:

[TABLE]

which is to be performed by keeping a fixed mean OP $\varphi$ in the canonical ensemble Gross et al. (2016) and a fixed bulk field $h$ in the grand canonical ensemble.

Turning first to the canonical ensemble, we employ the decomposition property in Eq. 105 to formally write the pressure $\ring{p}$ as consisting of bulk, surface, and residual finite-size contributions:

[TABLE]

where

[TABLE]

is a “surface” pressure and

[TABLE]

is the excess contribution.

In the following, we focus on Neumann boundary conditions, because only in this case the CCF derived from Eq. 217 differs from the one obtained on the basis of Eq. 144. For simplicity, we analyze the regularized (but not yet renormalized) expressions of the free energy, as given in Secs. III.3 and III.4. Renormalization produces (via additive counterterms) contributions to the bulk free energy Eisenriegler and Tomaschitz (1987), but does not change the conclusions of this section regarding the CCF. According to Eq. 106c the bulk free energy density is given by

[TABLE]

where $\mathcal{L}_{b}$ is defined in Eq. 49. Inserting $\ring{\mathpzc}{f}_{b}$ [Eq. 223] into Eq. 218b yields the bulk pressure

[TABLE]

where we made use of Eq. 145. As a manifestation of ensemble equivalence in the thermodynamic limit, the grand canonical bulk free energy density $\mathpzc{f}_{b}$ can be obtained from $\ring{\mathpzc}{f}_{b}$ [Eq. 223] via a Legendre transform:

[TABLE]

with $\varphi=\varphi(h)$ determined from the implicit equation

[TABLE]

In the grand canonical case, $h$ is an external field and Eq. 226 does not introduce any dependence on $L$ . Therefore, by using the above equations, the grand canonical film pressure follows as

[TABLE]

As expected, the canonical and the grand canonical bulk pressures are identical.

From Eq. 106c one infers the canonical surface free energy per area for Neumann boundary conditions:

[TABLE]

Upon using Eq. 145, the “surface” pressure [Eq. 221] follows as

[TABLE]

Due to the OP constraint [Eq. 145], the surface pressure is nonzero in the canonical ensemble. This result can be compared with the corresponding one in the grand canonical ensemble, in which, according to Eq. 125, the surface free energy $\mathpzc{f}_{s}^{\text{(N)}}$ has the same formal expression as $\ring{\mathpzc}{f}_{s}^{\text{(N)}}$ [Eq. 228], except that $\varphi=\varphi_{b}(\tau,h)$ is a function of the external field $h$ via the bulk equation of state. Since the latter is independent of $L$ , we immediately infer, analogously to Eq. 229, that

[TABLE]

as expected. As a direct consequence of Eq. 229, the canonical CCF $\ring{\mathcal{K}}$ defined by Eq. 217 is in general different from the CCF defined by Eq. 144, because the latter simply coincides with $\ring{p}_{\text{res}}$ . Following Sec. IV, Eq. 229 can be cast into scaling form:

[TABLE]

with the scaling function

[TABLE]

which is to be evaluated for $d=4$ . In Eq. 231 $\ring{p}_{s}$ , is to be understood as per $k_{B}T_{c}$ , so that $\ring{\Theta}_{\text{s}}$ is dimensionless.

Appendix E Critical Casimir force for constant volume

In Eq. 144 we have defined the CCF under the condition of a fixed transverse area $A$ , implying a change of the volume of the film upon its action and thereby of the mean OP [see Eq. 145]. In the case of a binary liquid mixture, the near-incompressibility of the liquid (close to demixing) strongly opposes changes of volume and, therefore, of the distance between the plates realizing the confinement. In the grand canonical ensemble the change of volume of the film occurs (easily) via exchange with the reservoir, but not due to compression. Alternatively, one may thus consider the CCF (per area) under the constraint of constant volume $V=AL$ ,

[TABLE]

An analogous definition applies to the corresponding canonical CCF $\ring{\mathcal{K}}_{V}$ , where, as before, additionally to $V$ also the total OP $\Phi$ is held constant. We further note that, for constant volume, Eq. 104 implies $\mathrm{d}\rho/\mathrm{d}L=\rho Ld/(d-1)$ and

[TABLE]

instead of Eq. 145. Using Eqs. 233 and 132, the canonical CCF $\ring{\mathcal{K}}_{V}$ can be shown to fulfill Eq. 146 with the scaling function

[TABLE]

instead of $\ring{\Xi}$ . The expression for $\delta\ring{\Xi}_{\text{ns}}$ is the same as in Eq. 148, and the expressions of the scaling functions $\ring{\Theta}(\hat{x}(x,\mathpzc{m}),\rho)=\ring{\hat{\Theta}}(x,\mathpzc{m},\rho)$ are reported in Eqs. 117, 113, and 108 for the various boundary conditions. Using, analogously, Eqs. 233 and 135, the grand canonical CCF $\mathcal{K}_{V}$ fulfills Eq. 149 with the scaling function

[TABLE]

instead of $\tilde{\Xi}$ . Analogously to Eq. 152, expressing $\tilde{\Theta}(x,\mathpzc{h},\rho)$ in terms of $\hat{\Theta}(x,\mathpzc{m}(x,\mathpzc{h},\rho),\rho)$ and using the scaling form of the equation of state in Eq. 142 (which is valid in the bulk limit as well as at $\mathcal{O}(\epsilon^{0})$ ), yields

[TABLE]

Since, at $\mathcal{O}(\epsilon^{0})$ , $\partial\mathpzc{m}/\partial\rho=0$ for the considered boundary conditions, the scaling functions $\ring{\Xi}_{V}$ and $\Xi_{V}$ in Eqs. 235 and 237 have formally identical expressions in terms of the corresponding scaling functions $\ring{\hat{\Theta}}$ and $\hat{\Theta}$ .

In order to asses the actual difference between the two ensembles, we must take into account that, in the canonical ensemble, the constraint-induced term $\delta F$ [Eq. 111] contributes to $\ring{\hat{\Theta}}$ with a term which is given in Eq. 113. According to Eq. 233, the total constraint-induced contribution to the CCF $\ring{\mathcal{K}}_{V}$ [including the term $\delta\ring{\Xi}_{\text{ns}}$ in Eq. 148] is given by

[TABLE]

with

[TABLE]

These expressions can be contrasted to the corresponding ones for $\delta\ring{\Xi}$ reported in Eq. 155. Notably, the constraint-induced contribution to the CCF defined with fixed volume $V$ vanishes for periodic and Neumann boundary conditions. At $\mathcal{O}(\epsilon^{0})$ , upon using Eq. 134 and $\beta=1/2$ , we can express Eq. 237 in terms of the scaling function $\Theta(\hat{x}(x,\mathpzc{m}),\rho)=\hat{\Theta}(x,\mathpzc{m},\rho)$ as

[TABLE]

Since Eq. 239 contains the contributions to the CCF from both the scaling and non-scaling terms in the residual finite-size free energy, Eq. 235 can, owing to Eqs. 117 and 129, be expressed analogously in terms of $\Theta(\hat{x},\rho)$ as

[TABLE]

Accordingly, at $\mathcal{O}(\epsilon^{0})$ , the canonical and the grand-canonical CCFs are identical for periodic and Neumann boundary conditions:

[TABLE]

For Dirichlet boundary conditions with $\mathpzc{m}=0$ we have, instead,

[TABLE]

where $\delta\ring{\Xi}_{V}$ is negative for all $x$ and vanishes in the limit $x\to\infty$ . We finally note that, for a fully isotropic cube, the CCF for conserved volume is expected to vanish by symmetry. Indeed, using Eqs. 134, 240, and 189, for periodic boundary conditions with $\rho=1$ and $d=4$ one finds

[TABLE]

where the last step follows from the asymptotic behavior of the theta function $\vartheta(y)$ [see Eq. 190 and the associated comments].

Figure 11 shows the numerically evaluated scaling functions $\ring{\Xi}_{V}^{\text{(p,D,N)}}$ of the canonical CCF for conserved volume. Note that, according to Eqs. 239, 240, and 241, $\Xi_{V}$ and $\ring{\Xi}_{V}$ can be considered as functions of the combined scaling variable $\hat{x}$ [Eq. 134]. For periodic boundary conditions [Fig. 11(a)], the CCF at constant volume is identical in the two ensembles and its absolute strength decreases upon increasing the aspect ratio $\rho$ . This trend is opposite to the behavior of the CCF at constant transverse area displayed in Figs. 4 and 5. For Dirichlet boundary conditions, the CCF at constant volume [Fig. 11(b)] is qualitatively similar to that at constant transverse area [Fig. 7], except that, in the latter case, $\ring{\Xi}^{\text{(D)}}$ attains a nonzero value for $x\to\infty$ , whereas $\ring{\Xi}_{V}^{\text{(D)}}$ vanishes in that limit. In the case of Neumann boundary conditions [Fig. 11(c)], the scaling function $\ring{\Xi}_{V}^{\text{(N)}}=\Xi_{V}^{\text{(N)}}$ of the CCF at constant volume shows a behavior distinct from that of $\ring{\Xi}^{\text{(N)}}$ [see Figs. 9 and 10], as it depends non-monotonically on the effective scaled temperature $\hat{x}$ and exhibits a pronounced minimum at intermediate values of $\hat{x}$ .

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