Non-Universal Equation of State of the Two-Dimensional Bose Gas

TL;DR

This paper derives non-universal corrections to the equation of state of a two-dimensional Bose gas by accounting for finite-range effects, revealing complex dependencies on system parameters beyond the universal logarithmic form.

Contribution

It presents a novel analytical approach to include finite-range effects in the equation of state of 2D Bose gases, extending beyond mean-field approximations.

Findings

Pressure depends nonpolynomially on the finite-range parameter.

Quantum fluctuations lead to complex, nontrivial dependencies on chemical potential and temperature.

Results are obtained using dimensional regularization within a finite-temperature functional integral framework.

Abstract

For a dilute two-dimensional Bose gas the universal equation of state has a logarithmic dependence on the s-wave scattering length. Here we derive non-universal corrections to this equation of state taking account finite-range effects of the inter-atomic potential. Our beyond-mean-field analytical results are obtained performing dimensional regularization of divergent zero-point quantum fluctuations within the finite-temperature formalism of functional integration. In particular, we find that in the grand canonical ensemble the pressure has a nonpolynomial dependence on the finite- range parameter and it is a highly nontrivial function of chemical potential and temperature.

Equations

$$\mathcal{L}$$

$$\tilde{V}(q)\simeq g_0+g_2{q}^2={\tilde{V}}_{p,2}(q),$$

$$g_0=\tilde{V}(0)=\int d^2\mathbf{r}V(r)$$

$$g_2=\frac{1}{2}{\tilde{V}}^{\prime \prime }(0)=-\frac{1}{4}\int d^2\mathbf{r}{r}^{2}V(r).$$

$$\mathcal{L}$$

$$\mathcal{Z}=\int \mathcal{D}[\psi ,{\psi }^{\ast }]\exp \left\{-\frac{S[\psi ,{\psi }^{\ast }]}{\hslash }\right\},$$

$$S[\psi ,{\psi }^{\ast }]={\int }_0^{\hslash \beta }d\tau {\int }_{L^2}{d}^2\mathbf{r}\mathcal{L}(\psi ,{\psi }^{\ast })$$

$$\Omega=-{1\over\beta}\ln{\big{(}{\cal Z}\big{)}}\;.$$

$$\psi (\mathbf{r},\tau )={\psi }_0+\eta (\mathbf{r},\tau )$$

$$\Omega (\mu ,T,{\psi }_0)={\Omega }_0(\mu ,{\psi }_0)+{\Omega }_g^{(0)}(\mu ,{\psi }_0)+{\Omega }_g^{(T)}(\mu ,{\psi }_0),$$

$${\Omega }_0(\mu ,{\psi }_0)=\left(-\mu {\psi }_0^2+\frac{1}{2}{g}_0{\psi }_0^4\right)L^2$$

$${\Omega }_g^{(0)}(\mu ,{\psi }_0)=\frac{1}{2}\sum _{\mathbf{q}}{E}_{\mathbf{q}}(\mu ,{\psi }_0)$$

$$E_{\mathbf{q}}(\mu ,{\psi }_0)$$

$${\Omega }_g^{(T)}(\mu ,{\psi }_0)=\frac{1}{\beta }\sum _{\mathbf{q}}\ln \left(1-e^{-\beta E_{\mathbf{q}}(\mu ,{\psi }_0)}\right)$$

$$\frac{\partial {\Omega }_0(\mu ,{\psi }_0)}{\partial {\psi }_0}=0,$$

$${\psi }_0(\mu )=\sqrt{\frac{\mu }{g_0}}$$

$$E_q(\mu )=\sqrt{\frac{{\hslash }^2{q}^2}{2m}\left(\lambda (\mu )\frac{{\hslash }^2{q}^2}{2m}+2\mu \right)},$$

$$\lambda (\mu )=1+\frac{4m}{{\hslash }^2}\frac{g_2}{g_0}\mu$$

$${\Omega }_0(\mu )=-\frac{{\mu }^2}{2g_0}.$$

$${\Omega }_g^{(0)}(\mu )=\frac{1}{2}\sum _{\mathbf{q}}{E}_q(\mu ).$$

$$\frac{{\Omega }_g^{(0)}}{L^D}$$

$$\frac{{\Omega }_g^{(0)}(\mu )}{L^D}=-\frac{A(\mu )}{2\epsilon {\kappa }^{\epsilon }}{\mu }^2.$$

$$\frac{{\Omega }^{(0)}(\mu )}{L^D}=\frac{{\Omega }_0(\mu )}{L^D}+\frac{{\Omega }_g^{(0)}(\mu )}{L^D}=-\frac{{\mu }^2}{2{\xi }_r(\mu ,\kappa ,\epsilon )},$$

$$\frac{1}{{\xi }_r(\mu ,\kappa ,\epsilon )}=\frac{1}{g_0{\kappa }^{\epsilon }}+\frac{A(\mu )}{\epsilon {\kappa }^{\epsilon }}$$

$$\frac{1}{{\xi }_r(\mu ,\kappa ,\epsilon {)}^2}\frac{d{\xi }_r(\mu ,\kappa ,\epsilon )}{d\kappa }=\frac{\epsilon }{g_0{\kappa }^{\epsilon +1}}+\frac{A(\mu )}{{\kappa }^{\epsilon +1}}.$$

$$\frac{1}{{\xi }_r(\mu ,\kappa ,0{)}^2}\frac{d{\xi }_r(\mu ,\kappa ,0)}{d\kappa }=\frac{A(\mu )}{\kappa }.$$

$$\frac{1}{{\xi }_r(\mu ,{\kappa }^{\prime },0)}-\frac{1}{{\xi }_r(\mu ,\kappa ,0)}=-A(\mu )\ln \left(\frac{{\kappa }^{\prime }}{\kappa }\right).$$

$$\frac{{\Omega }^{(0)}(\mu )}{L^2}=-\frac{m}{8\pi {\hslash }^2\lambda (\mu {)}^{3/2}}{\mu }^2\ln \left(\frac{{ϵ}_c}{\mu }\right).$$

$$P^{(0)}(\mu )=\frac{m}{8\pi {\hslash }^2}\frac{{\mu }^2}{{\left(1+\chi \mu \right)}^{3/2}}\ln \left(\frac{{ϵ}_c}{\mu }\right)$$

$$\chi =\frac{4m}{{\hslash }^2}\frac{g_2}{g_0}$$

Full text

Introduction. The equation of state of a uniform weakly-interacting Bose gas has a long history. Universal beyond-mean-field theoretical results, which depend only on the s-wave scattering length $a_{s}$ of the inter-atomic potential, were obtained for the three-dimensional (3D) bosonic system by Bogoliubov bogoliubov and by Lee, Huang, and Yang ly1957 ; lhy1957 . In one dimension (1D), based on a previous investigation of the 1D Bose-Fermi mapping girardeau1960 , Lieb and Liniger lieb1963 obtained the exact equation of state of a Bose gas with contact repulsive interaction. In the case of two spatial dimensions (2D), Schick schick1971 found that the equation of state of a uniform 2D repulsive Bose gas contains a nontrivial logarithmic term. This remarkable result was improved by Popov popov who obtained an equation of state which, at the leading order, reduces to Schick’s one in the dilute limit (see also lozovik ; hines ; fisher ; straley ). More recently, Andersen andersen2d and Mora and Castin mora-castin went one step further than Popov finding a next-next to leading universal equation of state for the two-dimensional weakly-interacting Bose gas. It is important to stress that, in the last years, various experiments with ultracold and dilute atomic gases in 3D papp2008 ; wild2012 and 2D salomon2010 ; dalibard2011 have put in evidence beyond-mean-field effects on the equation of state of repulsive bosons. Moreover, experiments on 1D bosons kinoshita2004 ; paredes2004 have shown that the Lieb-Liniger theory is needed to accurately describe the strong-coupling (i.e. low 1D density) regime.

The universal theory of the 3D weakly-interacting Bose gas has been extended including corrections due to the finite range of the inter-atomic potential sala-old ; braaten ; andersen ; roth ; gao ; boris ; pethick ; ketterle . These corrections give a modified Gross-Pitaevskii equation gao ; boris ; pethick ; ketterle ; zinner ; sgarlata for the nonuniform condensate and non-universal effects for quantum fluctuations at zero temperature braaten ; andersen . For a deeper understanding of the behavior of interacting bosonic systems in lower dimensionality, it is extremely important to analyze and control non-universal effect induced by the finite range in the equation of state also in the case of 2D and 1D Bose gases. In this Letter we investigate finite-range effects on quantum fuctuations of a 2D Bose gas by using the finite-temperature functional integration schakel ; atland on a local effective action. We derive the finite-temperature beyond-mean-field (one-loop, Gaussian) equation of state of the bosonic system performing dimensional regularization thooft of zero-point energy. The final non-universal analytical result, which reduces to the universal Popov equation of state popov in the zero-range case, exhibits a nonpolynomial dependence on the finite-range parameter.

Effective field theory for the 2D Bose gas. In the study of the two-dimensional interacting Bose gas we adopt the path integral formalism, where the atomic bosons are described by a complex field $\psi({\bf r},\tau)$ atland . The Euclidean Lagrangian density of the system with chemical potential $\mu$ is given by

[TABLE]

where $V(|{\bf r}-{\bf r}^{\prime}|)$ is the two-body interaction potential between bosons.

Given the Fourier transform ${\tilde{V}}(q)$ of the interaction potential $V(r)$ one can expand it at the second order in $q$ around $q=0$ finding

[TABLE]

where

[TABLE]

and

[TABLE]

Thus, within this approximation where the true interatomic potential ${\tilde{V}}(q)$ is substituted by the pseudo-potential $\tilde{V}_{p,2}(q)$ of Eq. (2), the effective local Lagrangian density becomes

[TABLE]

The term proportional to $g_{2}$ gives an improvement with respect to the contact (zero-range) approximation usually adopted in the case of ultracold and dilute atoms. In the three-dimensional case, Gaussian (one-loop) results of Eq. (5) have been obtained in Refs. braaten ; andersen ; roth , but only at zero temperature. Here we investigate the two-dimensional case, which is nontrivial also in the absence of finite range corrections, both at zero and finite temperature.

Partition function and grand potential. The partition function ${\cal Z}$ of the system at temperature $T$ can then be written as atland

[TABLE]

where

[TABLE]

is the Euclidean action, $L^{2}$ is the area of the system, and $\beta\equiv 1/(k_{B}T)$ with $k_{B}$ being Boltzmann’s constant. The grand potential $\Omega$ of the system, which is a function of $\mu$ and $T$, is then obtained as atland

[TABLE]

All the thermodynamical properties of the system can be deduced from $\Omega$ but, due to the interaction, some approximated procedure is needed to explicitly calculate $\Omega$.

The mean-field plus Gaussian (one-loop) approximation is obtained setting

[TABLE]

and expanding the action $S[\psi,\psi^{*}]$ of Eq. (7) around the uniform and constant $\psi_{0}$ up to quadratic (Gaussian) order in $\eta({\bf r},\tau)$ and $\eta^{*}({\bf r},\tau)$. In this way, taking into account Eq.(5) we find the grand potential (see also andersen ; schakel ; sala-review )

[TABLE]

where

[TABLE]

is the mean-field contribution (assuming a real $\psi_{0}$),

[TABLE]

is the zero-point energy of bosonic excitations

[TABLE]

i.e. the zero-temperature contribution of quantum Gaussian fluctuations, while

[TABLE]

takes into account thermal Gaussian fluctuations.

Zero-temperature results. Imposing the crucial saddle-point condition

[TABLE]

we get

[TABLE]

and the following spectrum of collective excitations

[TABLE]

where

[TABLE]

takes into account finite range effects of the inter-atomic potential.

By using Eq. (16) the mean-field grand potential (11) becomes

[TABLE]

Instead, the one-loop grand potential reads

[TABLE]

In the continuum limit, where $\sum_{\bf q}\to L^{2}\int d^{2}{\bf q}/(2\pi)^{2}$, $\Omega_{g}^{(0)}(\mu)$ is ultraviolet divergent with $E_{q}(\mu)$ given by Eq. (17). This divergence can be regularized with dimensional regularization, where the space dimension $D$ is analytically continued andersen ; thooft ; sala-review . To this end we extend the two-dimensional integral to a generic complex $D=2-\varepsilon$ dimension, and then take the limit $\varepsilon\to 0$. In this way

[TABLE]

where the regulator $\kappa$ is a crucial scale wavenumber which enters for dimensional reasons: $L^{D}=L^{2}\kappa^{\varepsilon}$. In Eq. $(\ref{om-eps})$ we have defined $A(\mu)=m/(2\pi\hbar^{2}\lambda(\mu)^{3/2})$ and $\Gamma(z)$ is the Euler gamma function, such that $\Gamma(-2+\varepsilon/2)=1/\varepsilon+O(\varepsilon^{0})$ for $\varepsilon\to 0$. Notice that, the strengths $g_{0}$ and $g_{2}$ of the 2D Lagrangian density (5) become $g_{0}\kappa^{\varepsilon}$ and $g_{2}\kappa^{\varepsilon}$ in $D$ dimensions, but the adimensional parameter $\lambda(\mu)$ of Eq. (18) remains unchanged.

It follows that, to leading order in $1/\varepsilon$, the Gaussian grand potential in $D$ dimensions reads

[TABLE]

This expression is still divergent. Nevertheless, comparing $\Omega_{g}(\mu)$ with $\Omega_{0}(\mu)$ in $D=2-\varepsilon$ dimensions we find the total zero-temperature grand potential

[TABLE]

where it appears the “running constant”

[TABLE]

which runs by changing $\kappa$ and depends on the dimension $D$ through $\varepsilon=2-D$ schakel ; thooft ; sala-review .

To remove the divergence $1/\varepsilon$ in Eq. (24) we calculate the derivative of $1/\xi_{r}(\mu,\kappa,\varepsilon)$ with respect to $\kappa$ finding

[TABLE]

Now, in the limit $\varepsilon\to 0$ (i.e. $D\to 2$) we get

[TABLE]

This first order differential equation can be easily solved by separation of variables, and the result is

[TABLE]

We set the Landau pole of Eq. (27) at the high energy scale of the system $\epsilon_{c}$, i.e. we set $1/\xi_{r}(\mu,\kappa^{\prime},0)=0$ at $\kappa^{\prime}$ such that $\hbar^{2}\kappa^{\prime 2}/(2m)=\epsilon_{c}$. Then, when $\kappa$ corresponds to the actual energy of our system, i.e. $\hbar^{2}\kappa^{2}/(2m)=\mu$. It follows that, from Eqs. (23) with $\varepsilon\to 0$ and $A(\mu)=m/(2\pi\hbar^{2}\lambda(\mu)^{3/2})$ we obtain

[TABLE]

Thus, taking into account Eq. (18) and the formula $P=-\Omega/L^{2}$ which relates the pressure $P$ to the grand potential $\Omega$, we finally get the zero-temperature beyond-mean-field pressure

[TABLE]

where

[TABLE]

with $g_{0}$ given by Eq. (3) and $g_{2}$ given by Eq. (4). Moreover, following Mora and Castin mora-castin , we set

[TABLE]

that is the high-energy scale fixed by the 2D s-wave scattering length $a_{s}$, with $\gamma\simeq 0.5772$ is the Euler-Mascheroni constant. Given the inter-atomic potential ${\tilde{V}}(q)$, the corresponding 2D scattering length $a_{s}$ is obtained calculating the s-wave phase shift $\delta_{0}(q)$ that is related to $a_{s}$ by the expression mora-castin ; richard ; werner-castin ; stoof ; stoof2 ; burnett

[TABLE]

In the case of contact interaction, where $\chi=0$, Eq. (29) reduces to the equation of state derived by Popov popov from a 2D hydrodynamic Hamiltonian with $\epsilon_{c}$ an ultraviolet cutoff, which depends on the s-wave scattering length $a_{s}$ nota1 . Moreover, using Eq. (31), one finds exactly the grand potential derived by Mora and Castin expanding the energy in powers of a small parameter mora-castin . Instead, if $\chi\neq 0$ Eq. (29) generalizes the zero-temperature Popov’s equation of state giving a nonpolynomial finite-range correction.

The relative difference of the pressure (29) with and without the finite-range correction is given by $|1/(1+\chi\mu)^{3/2})-1|\simeq(3/2)|\chi\mu|=12\pi nR^{2}/|\ln{(na_{s}^{2})}|$, by using $R=2\sqrt{|g_{2}/g_{0}|}$ as characteristic range of the inter-atomic potential nota2 and $\mu=8\pi\hbar^{2}n/(m|\ln{(na_{s}^{2})}|)$ as leading-order chemical potential in terms of the gas parameter $na_{s}^{2}$ with $n=\partial P^{(0)}(\mu)/\partial\mu$ the 2D number density schick1971 ; popov ; andersen2d ; mora-castin . Choosing, for example, $na_{s}^{2}=10^{-5}$ and $nR^{2}=6\cdot 10^{-2}$ we get a correction to the pressure of about $20\%$ due to finite-range effects, which is much larger than the Mora-Castin next-next-to-leading universal correction mora-castin of about $2\%$ for the same value of the gas parameter $na_{s}^{2}$ mora-castin . This regime can be experimentally achieved with 87Rb atoms, where $R=1.07\cdot 10^{-2}$ micron dalibard , using $n=524$ atoms/micron2 and tuning the 2D scattering length via Feshbach resonance dalibard2 to $a_{s}=1.38\cdot 10^{-4}$ micron. In general, given a quite small gas parameter $na_{s}^{2}$, finite-range effects become relevant for larger values of the non-universal adimensional parameter $nR^{2}$. In other words, sizable non-universal effects without next-next-to-leading universal corrections can be reached experimentally by decreasing the scattering length $a_{s}$ (through Feshbach-resonance techniques) and increasing the 2D number density $n$.

Note that, instead of using Eqs. (3) and (4) which immediately give the parameters $g_{0}$ and $g_{2}$ knowing the inter-atomic potential ${\tilde{V}}(q)$, one can alternatively establish a connection between $g_{0}$ and $g_{2}$ and familiar low-energy scattering quantities such as the s-wave scattering length $a_{s}$ and the s-wave effective range $r_{s}$ (which is not the characteristic range $R$ of the potential). In two spatial dimensions this connection is very cumbersome and highly nonlinear richard ; werner-castin .

Finite-temperature results. The finite-temperature one-loop contribution to the equation of state is obtained from Eq. (12) with Eq. (16), which gives the finite-temperature contribution

[TABLE]

to the total pressure, within our Gaussian scheme. Introducing the variable $x=\beta E_{q}$ we get

[TABLE]

where $q(x)$ is given by

[TABLE]

Expanding this expression at low temperature $T$ we find

[TABLE]

where $\Gamma(x)$ is the Euler gamma function and $\lambda(\mu)$ is given by Eq. (18). Thus, the final grand-canonical equation of state $P(\mu,T)$, that gives the pressure as a function of both the chemical potential $\mu$ and the temperature. Explicitly,

[TABLE]

where $P^{(0)}(\mu)$ is given by Eq. (29) and $P_{g}^{(T)}(\mu)$ is given by Eq. (36). As clearly shown in Eq. (36), at finite temperature $T$ the role of non-universal effects (which are encoded into $\lambda(\mu)$) increases as ratio $k_{B}T/\mu$ grows. This effect is somehow expected since the details of the potential become more relevant when atoms scatter at higher energy.

Conclusions. We have used finite-temperature one-loop functional integration to obtain the non-universal equation of state of a dilute and ultracold gas of bosons. We have adopted an effective field theory which includes a low-energy finite-range contribution of the inter-atomic interaction. The divergent zero-point energy of the system has been regularized by performing dimensional regularization. Our analytical results at zero and finite temperature are highly nontrivial generalizations of old but tricky universal formulas schick1971 ; popov ; mora-castin which depend only on the s-wave scattering length $a_{s}$.

Acknowledgments. The author acknowledges for partial support the 2016 BIRD project ”Superfluid properties of Fermi gases in optical potentials” of the University of Padova. The author thanks Alberto Cappellaro and Flavio Toigo for enlightening discussions and Henk Stoof for useful e-suggestions.