Multiparameter universality and directional nonuniversality of exact anisotropic critical correlation functions of the two-dimensional Ising universality class

TL;DR

This paper proves that the critical correlation functions of 2D anisotropic Ising models follow a multiparameter universality, with directional nonuniversality caused by principal axes depending on microscopic details, and determines the exact anisotropy matrices.

Contribution

It establishes the validity of multiparameter universality for exact critical correlation functions and amplitude relations in anisotropic 2D Ising models, linking microscopic details to macroscopic critical behavior.

Findings

Correlation functions exhibit directional nonuniversality due to principal axes.

Exact anisotropy matrices are determined for bulk and finite-size behavior.

Multiparameter universality is validated for amplitude relations.

Abstract

We prove the validity of multiparameter universality for the exact critical bulk correlation functions of the anisotropic square-lattice and triangular-lattice Ising models on the basis of the exact scaling structure of the correlation function of the two-dimensional anisotropic scalar $\varphi^4$ model with four nonuniversal parameters. The correlation functions exhibit a directional nonuniversality due to principal axes whose orientation depends on microscopic details. In particular we determine the exact anisotropy matrices governing the bulk and finite-size critical behavior of the $\varphi^4$ and Ising models. We also prove the validity of multiparameter universality for an exact critical bulk amplitude relation.

Equations

$$G_{\pm }^{\text{i}so}(\mathbf{x},t)=D_1^{\text{i}so}|\mathbf{x}{|}^{-d+2-\eta }{\Phi }_{\pm }(|\mathbf{x}|/{\xi }_{\pm }^{\text{i}so}),$$

$${\chi }_{\pm }^{\text{i}so}(t)$$

$${\tilde{\Phi }}_{\pm }$$

$$G_{\pm }^{\text{i}so}(\mathbf{x},t)={\Gamma }_{+}^{\text{i}so}({\xi }_{0+}^{\text{i}so}{)}^{-2+\eta }|\mathbf{x}{|}^{-d+2-\eta }{\Psi }_{\pm }(|\mathbf{x}|/{\xi }_{\pm }^{\text{i}so}),$$

$${\Psi }_{+}(y)={\Phi }_{+}(y)/{\tilde{\Phi }}_{+},{\Psi }_{-}(y)={\Phi }_{-}(y)/{\tilde{\Phi }}_{+},$$

$${\Psi }_{+}(0)={\Psi }_{-}(0)={\tilde{Q}}_3=\frac{2^{d-2+\eta }\Gamma [(d-2+\eta )/2]}{(4\pi {)}^{d/2})\Gamma [(2-\eta )/2]}{Q}_3.$$

$$(B^{\text{i}so}{)}^2({\Gamma }^{\text{i}so}{)}^{-1}({\xi }_{0+}^{\text{i}so}{)}^2=Q_c$$

$$\displaystyle H=\tilde{a}^{2}\Bigg{[}\sum_{i=1}^{\widetilde{N}}\left(\frac{r_{0}}{2}\varphi_{i}^{2}+u_{0}\varphi_{i}^{4}\right)+\sum_{i,j=1}^{\widetilde{N}}\frac{K_{i,j}}{2}(\varphi_{i}-\varphi_{j})^{2}\Bigg{]},\;\;\;\;$$

$$G_{\pm }({\mathbf{x}}_{\mathbf{i}}-{\mathbf{x}}_{\mathbf{j}},t)=\underset{\tilde{N}\to \infty }{\lim }[<{\phi }_i{\phi }_j>-{\mathcal{M}}^2],$$

$$\mathbf{A}$$

$$A_{\alpha \beta }$$

$$\displaystyle G_{\pm}({\bf x},t)=\frac{\Gamma_{+}\;(\bar{\xi}_{0+})^{-7/4}}{[{\bf x}\cdot{\bf\bar{A}}^{-1}{\bf x}]^{1/8}}\;\Psi_{\pm}\Big{(}\frac{[{\bf x}\cdot{\bf\bar{A}}^{-1}{\bf x}]^{1/2}}{\bar{\xi}_{\pm}(t)}\Big{)}\;\;\;$$

$$\displaystyle{\bar{\xi}_{\pm}(t)}=\bar{\xi}_{0\pm}|t|^{-1},\;\;\bar{\xi}_{0\pm}=\big{[}\xi_{0\pm}^{(1)}\xi_{0\pm}^{(2)}\big{]}^{1/2}$$

$$B^2{\Gamma }_{+}^{-1}({\bar{\xi }}_{0+}{)}^2=Q_c$$

$$q=({\lambda }_1/{\lambda }_2{)}^{1/2}={\xi }_{0\pm }^{(1)}/{\xi }_{0\pm }^{(2)}$$

$${\mathbf{e}}^{(1)}$$

$${\lambda }_1$$

$$w$$

$$\tan \Omega$$

$${\lambda }_1$$

$$\displaystyle{\bf\bar{A}}(q,\Omega)=\left(\begin{array}[]{ccc}q\;c_{\Omega}^{2}+q^{-1}s_{\Omega}^{2}&\;\;\;(q-q^{-1})\;c_{\Omega}\;s_{\Omega}\\ (q-q^{-1})\;c_{\Omega}\;s_{\Omega}&q\;s_{\Omega}^{2}+q^{-1}\;c_{\Omega}^{2}\end{array}\right)$$

$$[\mathbf{x}\cdot ({\bar{\mathbf{A}}}^{-1}\mathbf{x}){]}^{1/2}/{\bar{\xi }}_{\pm }(t)=r/{\xi }_{\pm }(t,\theta ,q,\Omega )$$

$$G_{\pm }(\mathbf{x},t)$$

$${\xi }_{\pm }(t,\theta ,q,\Omega )$$

$$f(\theta ,q,\Omega )$$

$$\displaystyle f(\theta,q,0)\equiv f_{\text{r}ec}(\theta,q)=\big{(}q^{-1}\;\cos^{2}\theta+\;q\;\sin^{2}\theta\big{)}^{1/2}.$$

$$H^{\text{I}s}=\sum _{j,k}[-E_1{\sigma }_{j,k}{\sigma }_{j,k+1}$$

$$<{\sigma }_{0,0}{\sigma }_{M,N}{>}_{\pm }=R^{-1/4}{\mathcal{F}}_{\pm }(R/{\xi }_{\pm }^{\text{I}s},E_1,E_2,E_3)$$

$$<{\sigma }_{0,0}{\sigma }_{M,N}{>}_{\pm }^{\text{s}ub}=<{\sigma }_{0,0}{\sigma }_{M,N}{>}_{\pm }-({\mathcal{M}}^{\text{I}s}{)}^2$$

$${\xi }_{0+}^{\text{I}s}(E,E,0)=(4{\beta }_{c}E{)}^{-1}=[2\ln (1+2^{1/2}){]}^{-1}\equiv {\xi }_{0+}^{\text{I}s},$$

$${\mathcal{M}}^{\text{I}s}(E,E,0)=2^{5/16}[\ln (1+2^{1/2}){]}^{1/8}|t{|}^{1/8}\equiv B^{\text{I}s}|t{|}^{1/8},$$

$$C_{0+}(E,E,0)=2^{19/8}\pi [{\xi }_{0+}^{\text{I}s}(E,E,0){]}^{7/4}{p}_{+}\equiv C_{0+}^{\text{I}s},$$

$$p_{+}=C_{0+}(E,E,0)/D=\mathrm{0.1592846958...},$$

$$<{\sigma }_{0,0}{\sigma }_{M,N}{>}_{\pm }^{\text{s}ub}=\frac{C_{0+}^{\text{I}s}({\xi }_{0+}^{\text{I}s}{)}^{-7/4}}{R^{1/4}}{\Psi }_{\pm }(R/{\xi }_{\pm }^{\text{I}s}),$$

$${\Psi }_{+}(y_{+})=(2^{19/8}\pi p_{+}{)}^{-1}{\tilde{F}}_{+}(y_{+}),$$

$${\Psi }_{-}(y_{-})=(2^{19/8}\pi p_{+}{)}^{-1}$$

Full text

The concept of bulk universality classes plays a fundamental role in the theory of critical phenomena fish-1 ; priv ; pelissetto . They are characterized by the spatial dimension $d$ and the symmetry of the ordered state which we assume here to be $O(n)$ symmetric with an $n$-component order parameter. Within a given $(d,n)$ universality class, critical exponents and bulk scaling functions are independent of microscopic details, such as the couplings of (short-range) interactions or the lattice structure. It was asserted that, once the universal quantities of a universality class are given, two-scale-factor universality priv ; pelissetto ; stau ; hohenberg1976 implies that the asymptotic (small $t=(T-T_{c})/T_{c}$ ) critical behavior of any particular system of this universality class is known completely provided that only two nonuniversal amplitudes are specified. It has been shown cd2004 ; dohm2006 ; dohm2008 ; kastening-dohm , however, that it is necessary to distinguish subclasses of isotropic and weakly anisotropic systems within a universality class and that two-scale-factor universality is not valid for the subclass of weakly anisotropic systems. In the latter systems there exists no unique bulk correlation-length amplitude but rather $d$ independent nonuniversal amplitudes in the $d$ principal directions. This has a significant effect on the anisotropic bulk order-parameter correlation function $G({\bf x},t)$ but a clear classification of its universality properties was not developed dohm2008 .

Recently dohm2018 the notion of multiparameter universality, originally introduced for critical amplitude relations dohm2008 , was formulated for the scaling structure of $G({\bf x},t)$ within the anisotropic $\varphi^{4}$ theory where $G({\bf x},t)$ depends on up to $d(d+1)/2+1$ independent nonuniversal parameters in $d$ dimensions, i.e., up to four or seven parameters in two or three dimensions, respectively. It was hypothesized that multiparameter universality of $G({\bf x},t)$ is valid not only for ”soft-spin” $\varphi^{4}$ models but also for all weakly anisotropic systems within a given universality class including fixed-length spin models such as Ising $(n=1)$, $XY$ $(n=2)$, and Heisenberg $(n=3)$ models. No general proof was given for this hypothesis, except for an analytic verification for a special example within the two-dimensional anisotropic Ising model at $T=T_{c}$ Wu1966 .

A unique opportunity for a significant test of the validity of multiparameter universality is provided by an analysis of the exact results for the bulk correlation function of the anisotropic ”square-lattice” and ”triangular-lattice” Ising models WuCoy ; Vaidya1976 in the asymptotic scaling region near $T_{c}$. Such an analysis is made possible by deriving the exact scaling structure of $G({\bf x},t)$ of the general anisotropic two-dimensional scalar $\varphi^{4}$ lattice model which belongs to the same universality class as the $d=2$ Ising model Mehlig . We introduce angular-dependent correlation lengths which permits us to determine the principal axes via an extremum criterion and to derive the exact anisotropy matrices. This leads to a proof of multiparameter universality for the Ising models with three or four nonuniversal parameters for the square-lattice or triangular-lattice model, respectively. However, the correlation functions exhibit a directional nonuniversality due to the principal axes whose orientation depends on microscopic details. This dependence is different for Ising and $\varphi^{4}$ models. Our results are expected to make an impact on scaling theories for $G({\bf x},t)$ of real anisotropic systems such as magnetic materials alpha , superconductors schneider2004 , alloys onukiBook , and solids with structural phase transitions bruce-1 , where angular-dependent correlation functions are measurable quantities. Multiparameter universality is relevant also for finite-size effects, e.g., the critical Casimir force dohm2018 .

It is necessary to first reformulate the standard scaling form of $G_{\pm}^{\text{i}so}({\bf x},t)$ for isotropic systems. In the limit of large $|{\bf x}|$ and large $\xi^{\text{i}so}_{\pm}(t)=\xi^{\text{i}so}_{0\pm}|t|^{-\nu}$ at fixed $|{\bf x}|/\xi^{\text{i}so}_{\pm}\geq 0$ the scaling form for $n=1$ and $2\leq d<4$ reads pri ; pelissetto

[TABLE]

with the universal scaling function $\Phi_{\pm}(y)$ above $(+)$ and below ($-$) $T_{c}$ and the nonuniversal amplitudes $D^{\text{i}so}_{1},\xi^{\text{i}so}_{0\pm}$, where $\xi^{\text{i}so}_{0+}/\xi^{\text{i}so}_{0-}$ is universal but $D^{\text{i}so}_{1}$ still contains a universal part. We employ the ”true” (exponential) correlation lengths $\xi^{\text{i}so}_{\pm}$ which are defined by the exponential decay $\sim{\text{e}xp}\;(-|{\bf x}|/\xi^{\text{i}so}_{\pm})$ of $\Phi_{\pm}$ for large $|{\bf x}|/\xi^{\text{i}so}_{\pm}$ and which are universally related to the second-moment correlation lengths pelissetto . The exact sum rule dohm2008 $\chi_{\pm}^{\text{i}so}(t)=\int d^{d}{\bf x}\;G_{\pm}^{\text{i}so}({\bf x},t)$ yields the susceptibility

[TABLE]

with $2-\eta=\gamma/\nu$ and the universal quantities $\widetilde{\Phi}_{+}$, $\widetilde{\Phi}_{-}$, and $\Gamma^{\text{i}so}_{+}/\Gamma^{\text{i}so}_{-}$. This implies $D^{\text{i}so}_{1}=\Gamma^{\text{i}so}_{+}\;(\xi^{\text{i}so}_{0+})^{-2+\eta}/\widetilde{\Phi}_{+}$, thus $G_{\pm}^{\text{i}so}$ can be uniquely divided into universal and nonuniversal parts

[TABLE]

with two nonuniversal amplitudes $\Gamma^{\text{i}so}_{+}$ and $\xi_{0+}^{\text{i}so}$ and the universal scaling function $\Psi_{\pm}(y)$. At $T_{c}$ it is related to the universal constants dohm2008 ; priv ; tarko $\widetilde{Q}_{3}$ and $Q_{3}$ by

[TABLE]

For $d=2$, $\Gamma^{\text{i}so}_{+}$ and $\xi_{0+}^{\text{i}so}$ are related to the amplitude $B^{\text{i}so}$ of the order parameter ${\cal M}^{\text{i}so}=B^{\text{i}so}|t|^{1/8}$ and to the specific-heat amplitude $A^{\text{i}so}$ through pelissetto

[TABLE]

and $A^{\text{i}so}(\xi^{\text{i}so}_{0+})^{2}=(R^{+}_{\xi})^{2}$ where $Q_{c}$ and $R^{+}_{\xi}=(2\pi)^{-1/2}$ are universal constants according to Eqs. (2.50),(3.49), and (6.31) of priv . We present the exact value of $Q_{c}$ in (84) below. Our only assumption is the validity of two-scale-factor universality for isotropic systems which implies that $\Psi_{\pm},Q_{c},R^{+}_{\xi}$ are the same for isotropic Ising and $\varphi^{4}$ models with $\gamma=2-\eta=7/4,\nu=1$, and $\xi^{\text{i}so}_{0+}/\xi^{\text{i}so}_{0-}=2$.

We first consider the anisotropic scalar $\varphi^{4}$ model on $\widetilde{N}$ lattice points ${\bf x}_{i}\equiv(x_{i1},x_{i2})$ of a square lattice with lattice spacing $\tilde{a}$ and finite-range interactions $K_{i,j}$. The Hamiltonian divided by $k_{B}T$ and the bulk correlation function are defined by dohm2008

[TABLE]

where ${\cal M}^{2}=\lim_{|{\bf x_{i}-x_{j}}|\to\infty}<\varphi_{i}\varphi_{j}>$. The large-distance anisotropy is described by the anisotropy matrix

[TABLE]

where weak anisotropy requires ${\det\bf A}>0,a>0,b>0$ which ensures unchanged critical exponents cd2004 . It has been shown recently dohm2018 that $G_{\pm}({\bf x},t)$ has the asymptotic scaling form

[TABLE]

with ${\bf\bar{A}}={\bf A}/({\det\bf A})^{1/2}$ where $\Psi_{\pm}$ is the same scaling function as that in (4) for isotropic systems (${\bf\bar{A}}={\bf 1}$). We have obtained (14) from Eqs. (5.61) and (5.32) of dohm2018 by employing the sum rule for the susceptibility of the anisotropic system $\chi_{\pm}(t)=\int d^{2}{\bf x}\;G_{\pm}({\bf x},t)=\Gamma_{\pm}|t|^{-7/4}$ which yields the nonuniversal constant $D_{1}=\Gamma_{+}\;(\bar{\xi}_{0+})^{-7/4}/\widetilde{\Phi}_{+}$ dohm2018 . Here ${\bar{\xi}_{\pm}(t)}$ is the geometric mean

[TABLE]

of the principal correlation lengths $\xi_{\pm}^{(\alpha)}(t)=\xi^{(\alpha)}_{0\pm}|t|^{-1}$ where the principal axes are defined by the eigenvectors ${\bf e}^{(\alpha)}$ determined by ${\bf Ae}^{(\alpha)}=\lambda_{\alpha}{\bf e}^{(\alpha)},\alpha=1,2$. The eigenvalues $\lambda_{\alpha}>0$ determine the amplitudes $\xi_{0\pm}^{(\alpha)}=\lambda_{\alpha}^{1/2}\xi_{0\pm}^{\prime}$ with $\xi_{0+}^{(\alpha)}/\xi_{0-}^{(\alpha)}=\xi_{0+}^{\prime}/\xi_{0-}^{\prime}=2$ where $\xi_{0\pm}^{\prime}$ is the correlation length of the isotropic system obtained after a shear transformation that consists of a rotation and a rescaling in the ${\bf e}^{(\alpha)}$ directions cd2004 ; dohm2006 ; dohm2008 ; dohm2018 . The amplitudes $\xi_{0+}^{(\alpha)}$ are independent of the amplitude $B$ of the order parameter ${\cal M}=B|t|^{1/8}$ of the anisotropic model. From the shear transformations dohm2006 ; dohm2008 ; dohm2018 $(\xi^{\prime}_{0+})^{2}=({\det\bf A})^{-1/2}(\bar{\xi}_{0+})^{2}$, $(B^{\prime})^{2}=({\det\bf A})^{1/2}B^{2},A^{\prime}=({\det\bf A})^{1/2}A$, and $\Gamma^{\prime}_{+}=\Gamma_{+}$ we find the relations for the anisotropic system

[TABLE]

and $A(\bar{\xi}_{0+})^{2}=(R^{+}_{\xi})^{2}$ where $Q_{c}$ and $R^{+}_{\xi}$ are the same as in the isotropic case. Thus the susceptibility amplitude $\Gamma_{+}$ is determined by three independent nonuniversal parameters $\xi_{0+}^{(1)},\xi_{0+}^{(2)},B$ whereas the specific-heat amplitude $A$ is determined by two parameters $\xi_{0+}^{(1)},\xi_{0+}^{(2)}$, and $B^{2}$ can be expressed as $B^{2}=A\Gamma_{+}Q_{c}/(R^{+}_{\xi})^{2}$. The individual lengths $\xi^{(\alpha)}_{0+}$ cannot be determined from $A,B,$ and $\Gamma_{+}$.

Contours of constant correlations are ellipses determined by ${\bf x}\cdot{\bf\bar{A}}^{-1}{\bf x}={\text{c}onst}$ whose excentricity and orientation are characterized by

[TABLE]

and by the angle $\Omega$ determining the principal axes, i.e.,

[TABLE]

For $a\neq b$ we define

[TABLE]

and for $a=b,c\neq 0$

[TABLE]

From {\bf\bar{A}}={\bf U}^{-1}{\bf\bar{\mbox{\boldmath\lambda}}}{\bf U} with the rotation and rescaling matrices {\bf U}=\left(\begin{array}[]{ccc}c_{\Omega}&s_{\Omega}\\ -s_{\Omega}&c_{\Omega}\\ \end{array}\right) and {\bf\bar{\mbox{\boldmath\lambda}}}=\left(\begin{array}[]{ccc}q&0\\ 0&\;q^{-1}\\ \end{array}\right) we obtain

[TABLE]

with the abbreviations $c_{\Omega}\equiv\cos\Omega,s_{\Omega}\equiv\sin\Omega$. Using polar coordinates ${\bf x}=(x_{1},x_{2})=(r\cos\theta,r\sin\theta)$ (Fig. 1) we define the angular-dependent correlation length $\xi_{\pm}(t,\theta,q,\Omega)$ by

[TABLE]

which yields the exact reformulation of (14)

[TABLE]

where the directional dependence is described by

[TABLE]

In the limit $c\to 0$ at fixed $a-b\neq 0$ a ”rectangular” anisotropy is obtained with $\lambda_{1}=a,\lambda_{2}=b,q=(a/b)^{1/2}=\xi_{0\pm}^{(1)}/\xi_{0\pm}^{(2)}$, $\Omega=0$, and

[TABLE]

For $q\neq 1$ the requirement $\partial\xi_{\pm}(t,\theta,q,\Omega)/\partial\theta=0$ yields $\sin[2(\theta-\Omega)]=0$ implying that $\xi_{\pm}$ has extrema at $\theta^{(1)}=\Omega$ and $\theta^{(2)}=\Omega+\pi/2$ defining the two principal directions.

In contrast to $G^{\text{i}so}_{\pm}({\bf x},t)$, $G_{\pm}({\bf x},t)$ depends on four independent nonuniversal parameters $\Gamma_{+},\bar{\xi}_{0+},q,\Omega$ which violates two-scale-factor universality. Unlike $q$, the angle $\Omega(a,b,c)$ cannot be parameterized in terms of $\xi_{0\pm}^{(\alpha)}$ but depends on the lattice structure and the microscopic couplings $K_{i,j}$ through $a,b,c$. Thus ${\bf\bar{A}}$ depends not only on bulk correlation lengths through $q$ but also on other microscopic details through $\Omega$. The parametrization of (17)-(34) is valid in the unrestricted range $0<q<\infty$ above, at, and below $T_{c}$ eigen . The same matrix ${\bf\bar{A}}$ also enters the finite-size critical behavior dohm2018 . These results derived for a $\varphi^{4}$ model on a square lattice remain valid more generally for a $\varphi^{4}$ model with couplings $K_{i,j}$ on two-dimensional Bravais lattices dohm2008 .

The hypothesis of multiparameter universality dohm2018 predicts that the critical correlation functions of all anisotropic Ising models with short-range interactions can be expressed in the same form as (31)-(33) with the same universal functions $\Psi_{\pm}$ and $f$ and the same critical exponents, but with up to four different nonuniversal parameters. We shall show that this is indeed valid for Ising models with the Hamiltonian WuCoy ; Vaidya1976 ; stephenson

[TABLE]

where $\sigma_{j,k}=\pm 1$ are spin variables on a square lattice (with the lattice spacing $\tilde{a}=1$) with horizontal, vertical, and diagonal couplings $E_{1}>0,E_{2}>0$, $E_{3}$ (Fig.2). The exact correlation function $<\sigma_{0,0}\;\sigma_{M,N}>_{\pm}$ at vanishing external field was calculated for $E_{3}=0$ in WuCoy and for positive and negative $E_{3}\neq 0$ in Vaidya1976 , resulting in the scaling form

[TABLE]

with a nonuniversal scaling function ${\cal F}_{\pm}$, a distance $R(E_{1},E_{2},E_{3})$, and a correlation length $\xi_{\pm}^{\text{I}s}=\xi^{\text{I}s}_{0\pm}(E_{1},E_{2},E_{3})|t|^{-1}$ with $\xi_{+}^{\text{I}s}/\xi_{-}^{\text{I}s}=2$. The exact amplitude $C_{0\pm}(E_{1},E_{2},E_{3})$ of the susceptibility was also calculated. So far the universality properties of (36) have not been analyzed in the literature priv ; pelissetto ; Vaidya1976 ; WuCoy ; CoyWu , and the universal part of the function ${\cal F}_{\pm}$ has not been identified. In particular, the principal axes and principal correlation lengths of the triangular-lattice model ($E_{3}\neq 0$) are as yet unknown, and only a conjecture exists for the correlation lengths in the direction of the bonds Indekeu . For comparison with (9) below $T_{c}$ we need to consider the subtracted correlation function

[TABLE]

where ${\cal M}^{\text{I}s}(E_{1},E_{2},E_{3})$ is the spontaneous magnetization, with ${\cal M}^{\text{I}s}=0$ for $T\geq T_{c}$. We shall analyze three cases.

We start from the isotropic case $E_{1}=E_{2}=E>0$, $E_{3}=0$, where $R(E,E,0)=(M^{2}+N^{2})^{1/2}\equiv R$, and WuCoy

[TABLE]

with $\beta_{c}=(k_{B}T_{c})^{-1},\sinh 2\beta_{c}E=1$. The constant $p_{+}$ is expressed analytically in terms of a Painlevé function of the third kind and its numerical value follows from Eq. (2.52S) of WuCoy . According to (4) we reformulate (37) as

[TABLE]

with $y_{\pm}=R/\xi^{\text{I}s}_{\pm}$, $\xi_{\pm}^{\text{I}s}\equiv\xi^{\text{I}s}_{0\pm}(E,E,0)|t|^{-1}$ where the functions $\widetilde{F}_{+}(y_{+})$ and $\widetilde{F}_{-}(y_{-}/2)$ are given by the right-hand side of Eq. (2.39) of WuCoy with $\sinh 2\beta_{c}E_{1}+\sinh 2\beta_{c}E_{2}$ replaced by $2$ and with the argument $\theta$ replaced by $y_{+}/2$ or $y_{-}/4$, respectively. The unsubtracted correlation function WuCoy $<\sigma_{0,0}\;\sigma_{M,N}>_{\pm}$ is easily obtained by dropping the last term in (Multiparameter universality and directional nonuniversality of exact anisotropic critical correlation functions of the two-dimensional Ising universality class) which comes from $-({\cal M}^{\text{I}s})^{2}$ in (37). According to two-scale-factor universality the functions $\Psi_{\pm}$ identify the exact universal scaling functions above and below $T_{c}$ of all systems in the subclass of isotropic systems in the $(d=2,n=1)$ universality class. In particular, $\Psi_{+}(0)=\Psi_{-}(0)=\widetilde{Q}_{3}$ is a universal amplitude. Its exact value is $\widetilde{Q}_{3}=0.270969...$ where we have used Eq. (5.10S) of WuCoy . This implies $Q_{3}=0.414131...$.

Now we turn to the case of a ”rectangular” anisotropy $E_{1}\neq E_{2}$, $E_{3}=0$ where the condition of criticality is $S_{1}S_{2}=1$ with $S_{\alpha}=\sinh 2\beta^{\text{r}ec}_{c}E_{\alpha},\alpha=1,2$ WuCoy . Using $x_{1}=N=r\cos\theta,x_{2}=M=r\sin\theta$ we derive from Eqs. (2.6), (2.8), (2.10), and (2.44) of WuCoy

[TABLE]

where $\xi_{0\pm}^{(1){\text{r}ec}}$ and $\xi_{0\pm}^{(2){\text{r}ec}}$ are the correlation-length amplitudes in the principal directions $(1,0)$ and $(0,1)$, respectively, corresponding to $\Omega^{\text{r}ec}=0$ and where $f_{\text{r}ec}(\theta,q^{\text{r}ec})$ is the same function as defined in (34), with $q$ replaced by $q^{\text{r}ec}$. From Eq. (2.8) of WuCoy we derive

[TABLE]

where $C_{\alpha}=\coth 2\beta^{\text{r}ec}_{c}E_{\alpha}$. This identifies ”the correlation length $\xi$” in WuCoy as $\bar{\xi}_{+}^{{\text{\;}rec}}(t)$ above $T^{{\text{\;}rec}}_{c}$ and $2\bar{\xi}_{-}^{{\text{\;}rec}}(t)$ below $T^{{\text{\;}rec}}_{c}$ as follows from Eq. (2.31) of WuCoy . From Eqs. (2.46a) and (2.48) of WuCoy we obtain the exact amplitude $C^{\text{r}ec}_{0+}\equiv C_{0+}(E_{1},E_{2},0)$ in the form

[TABLE]

and from Eq. (2.39) of WuCoy we obtain

[TABLE]

Together with (43) and (Multiparameter universality and directional nonuniversality of exact anisotropic critical correlation functions of the two-dimensional Ising universality class) this leads to the exact reformulation of the asymptotic result of WuCoy

[TABLE]

with the angular-dependent correlation lengths $\xi^{\text{r}ec}_{\pm}(t,\theta)$ and with ${\bf\bar{A}}^{\text{r}ec}(q^{\text{r}ec})={\bf\bar{A}}(q^{\text{r}ec},0)$, in exact agreement with (14) and (29)-(34) for $\Omega^{\text{r}ec}=0$, thus confirming multiparameter universality above, at, and below $T^{\text{r}ec}_{c}$ with three nonuniversal parameters $C^{\text{r}ec}_{0+},\bar{\xi}_{0+}^{{\text{\;}rec}},q^{\text{r}ec}$. This is valid for both $E_{1}\geq E_{2}$ and $E_{1}\leq E_{2}$ in the unrestricted range $0<q^{\text{r}ec}<\infty$. For $E_{1}=E_{2}$ the isotropic results are recovered.

We proceed to the case of a ”triangular” anisotropy Vaidya1976 ; stephenson with $E_{3}\neq 0$ and the condition of criticality $\hat{S}_{1}\hat{S}_{2}+\hat{S}_{2}\hat{S}_{3}+\hat{S}_{3}\hat{S}_{1}=1$ with $\hat{S}_{\alpha}=\sinh 2\beta^{\text{t}r}_{c}E_{\alpha},\alpha=1,2,3$ Berker . We first determine the angle $\Omega^{\text{t}r}$ describing the orientation of the principal axes. For $T\approx T^{\text{t}r}_{c}$ the angular dependence of the distance $R(E_{1},E_{2},E_{3})\equiv R_{\text{t}r}(\theta)=rf_{\text{t}r}(\theta)$ of Eq. (11) of Vaidya1976 is given by

[TABLE]

We define the angular-dependent correlation length $\xi^{\text{t}r}_{\pm}(t,\theta)$ by rewriting the scaled variable of Eq. (10) of Vaidya1976 for $T\approx T^{\text{t}r}_{c}$ as

[TABLE]

The requirement $\partial\xi^{\text{t}r}_{\pm}(t,\theta)/\partial\theta=0$ yields $\theta=\Omega^{\text{t}r}$ with

[TABLE]

implying that $\xi^{\text{t}r}_{\pm}(t,\theta)$ has extrema at $\theta^{(1)}=\Omega^{\text{t}r}$ and $\theta^{(2)}=\Omega^{\text{t}r}+\pi/2$ defining the two principal directions. Clearly $\Omega^{\text{t}r}(E_{1},E_{2},E_{3})$ is a nonuniversal quantity that depends on microscopic details and differs from $\Omega$, (25), of the $\varphi^{4}$ model even if the $\varphi^{4}$ and Ising models have the same couplings on the same lattices. This is due to the nonuniversal difference between a fixed-length spin model and a soft-spin model. From (60)-(64) we determine the ratio $q^{\text{t}r}=\xi_{0\pm}^{(1){\text{t}r}}/\xi_{0\pm}^{(2){\text{t}r}}$ of the amplitudes of the principal correlation lengths

[TABLE]

where now the $\pm$ sign in front of the square root term means $E_{1}>E_{2}$ (+) and $E_{1}<E_{2}$ $(-)$, respectively, and

[TABLE]

for $E_{1}=E_{2}=E>0$. From (63)-(66) we derive

[TABLE]

for both $E_{1}\geq E_{2}$ and $E_{1}\leq E_{2}$. Together with (60) these equations prove the validity of the identification $f_{\text{t}r}(\theta)=f(\theta,q^{\text{t}r},\Omega^{\text{t}r})$ in the unrestricted range smalllarge $0<q^{\text{t}r}<\infty$ above, at, and below $T_{c}^{\text{t}r}$ where $f$ is indeed the same function (33) as derived within the $\varphi^{4}$ theory. This completes the determination of the angular dependence of the anisotropy matrix ${\bf\bar{A}}^{\text{t}r}\equiv{\bf\bar{A}}(q^{\text{t}r},\Omega^{\text{t}r})$ for the triangular-lattice Ising model (35) where ${\bf\bar{A}}$ is the same matrix as in (29) for the $\varphi^{4}$ model, in exact agreement with multiparameter universality. Our results for rectangular anisotropy are recovered from (60)-(69) in the limit $E_{3}\to 0$.

We mention two earlier conjectures. (i) From (60) and (62) we derive

[TABLE]

where $\xi_{0\pm}^{({\text{d}iag}){\text{t}r}}$ $,\xi_{0\pm}^{({\text{h}or}){\text{t}r}}$, and $\xi_{0\pm}^{({\text{v}ert}){\text{t}r}}$ denote the correlation lengths in the $(1,1)$ $,(1,0)$, and $(0,1)$ directions and the factor $2^{1/2}$ accounts for the diagonal lattice spacing. This confirms the conjecture in Eq. (2.6) of Indekeu . (ii) The ratio (66) used in Sec. V. C of dohm2018 was based on the conjecture in Eq. (A22) of Indekeu and is derived here directly from the exact result (60).

In the remaining analysis of the triangular case we confine ourselves to $E_{1}=E_{2}=E_{3}=E>0$ where

[TABLE]

By expanding Eqs. (2) and (10) of Vaidya1976 around $T^{\text{t}r}_{c}$ to leading order in $|t|=|T-T^{\text{t}r}_{c}|/T^{\text{t}r}_{c}$ we determine the magnetization, the mean correlation lengths $\bar{\xi}_{\pm}^{{\text{\;}tr}}(t)=\bar{\xi}_{0\pm}^{{\text{\;}tr}}|t|^{-1}$, and the principal correlation lengths $\xi_{0+}^{(\alpha){\text{t}r}}$ as

[TABLE]

From Eqs. (12) and (14)-(16) of Vaidya1976 we obtain

[TABLE]

Together with (43) and (Multiparameter universality and directional nonuniversality of exact anisotropic critical correlation functions of the two-dimensional Ising universality class) this leads to the exact reformulation of the asymptotic result of Vaidya1976

[TABLE]

with the same universal functions $\Psi_{+}$, $\Psi_{-}$, and $f$ as in (31)-(33), (43), and (Multiparameter universality and directional nonuniversality of exact anisotropic critical correlation functions of the two-dimensional Ising universality class) and the same matrix ${\bf\bar{A}}^{\text{t}r}\equiv{\bf\bar{A}}(q^{\text{t}r},\Omega^{\text{t}r})$ as in (29), with the four nonuniversal parameters $C^{\text{t}r}_{0+},\bar{\xi}_{0+}^{{\text{\;}tr}},q^{\text{t}r},\Omega^{\text{t}r}$ given in (78), (76), (73), (72), respectively, thus proving the validity of multiparameter universality for the triangular-lattice Ising model above, at, and below $T_{c}^{\text{t}r}$, and disproving two-scale-factor universality. Our hypothesis of multiparameter universality predicts the structure of (Multiparameter universality and directional nonuniversality of exact anisotropic critical correlation functions of the two-dimensional Ising universality class)-(82) to be valid also in the general case $E_{1}\neq E_{2},E_{3}\neq 0$.

In order to complete our analysis we show that the universal amplitude relations (7) and (16) derived for the $\varphi^{4}$ model remain valid also for the Ising model. We first employ (38)-(41) for the isotropic Ising model to derive

[TABLE]

in structural agreement with (7). Thus our analysis identifies the exact universal constant $Q_{c}$ for $d=2,n=1$ as

[TABLE]

This can be confirmed by means of a different derivation from Eqs. (6.29) and (6.31) of priv which determines $Q_{c}=(R^{+}_{\xi})^{2}/R_{C}$. From the rectangular and triangular results (50)-(55) and (75)-(78), respectively, we derive

[TABLE]

which agrees with (16) for the anisotropic $\varphi^{4}$ model. Thus both the anisotropic Ising and $\varphi^{4}$ models have universal amplitude relations with the same universal constant $Q_{c}$ as for the isotropic models, in agreement with the hypothesis of multiparameter universality. In the anisotropic cases (85) and (86) three independent nonuniversal parameters are involved for the same reasons as given in the context of (16).

Multiparameter universality for other critical bulk amplitude relations within $\varphi^{4}$ theory in $d$ dimensions follows from Sec. III of dohm2008 , e.g., Eqs. (3.32)-(3.36). In particular, multiparameter universality is predicted, for general $n$, for anisotropic systems at $T_{c}$ in the presence of an ordering field $h$ with the amplitude $\Gamma_{c}$ of the susceptibility and the principal correlation lengths $\xi_{c}^{(\alpha)}$ according to Eq. (3.35) of dohm2008 for each $\alpha$, with a universal constant $Q_{2}(d,n)$ that is the same as for the corresponding relation tarko of isotropic systems at $T_{c}$ in the same $(d,n)$ universality class. A verification of such relations within anisotropic fixed-length spin models would be interesting.

To summarize, we have determined the exact anisotropy matrix ${\bf\bar{A}}$ for anisotropic $\varphi^{4}$ and Ising models WuCoy ; Vaidya1976 and have confirmed the validity of multiparameter universality for the exact bulk order-parameter correlation functions of these models above, at, and below $T_{c}$, thereby answering the longstanding question WuCoy as to the universality properties of the Ising models. It is reassuring that the leading scaling part of the detailed expressions for $<\sigma_{0,0}\;\sigma_{M,N}>_{\pm}$ presented in WuCoy ; Vaidya1976 can be condensed into the same compact universal forms (58) and (81) as the exact result (31) for the anisotropic $\varphi^{4}$ model, with three universal functions $\Psi_{+}$, $\Psi_{-}$, and $f$. We have also found agreement with multiparameter universality for the exact critical bulk amplitude relations (16), (85), and (86) with three independent nonuniversal parameters. These results support the validity of multiparameter universality for the large class of weakly anisotropic systems within the $(d,n)$ universality classes which is of relevance for studying the correlation functions in real anisotropic systems alpha ; schneider2004 ; onukiBook ; bruce-1 . The significance of multiparameter universality for finite-size effects, e.g., on the critical Casimir force and the specific heat, has been pointed out in dohm2018 . In all cases the universal critical exponents are not changed by weak anisotropy cd2004 ; dohm2008 , unlike the case of strong anisotropy tonchev . Nonuniversality enters $<\sigma_{0,0}\;\sigma_{M,N}>_{\pm}$ through the anisotropy matrix ${\bf\bar{A}}$, the mean correlation length, and the susceptibility amplitude in the prefactor. ${\bf\bar{A}}$ is temperature-independent and is applicable above, at, and below $T_{c}$ in bulk and confined systems dohm2018 . As an appropriate parametrization of ${\bf\bar{A}}$ we have employed the ratio $q$ of the principal correlation lengths and the angle $\Omega$ determining the principal directions. Both parameters are nonuniversal microscopic quantities. While for $\varphi^{4}$ models $\Omega$ is known explicitly according to (13) and (25), this is not generally the case for Ising models. We agree with the assertion night1983 that, apart from the Ising models WuCoy ; Vaidya1976 analyzed in this paper, the principal directions ”generically depend in an unknown way on the anisotropic interactions.” Since the principal directions enter the angular dependence of correlation functions in a crucial way the unknown dependence of $\Omega$ on microscopic details introduces a significant nonuniversality into the correlation functions of weakly anisotropic systems, in contrast to isotropic systems of the same universality class. This underscores the necessity of distinguishing subclasses of isotropic and anisotropic systems within a given $(d,n)$ universality class. The latter are less universal than the former and require significantly more nonuniversal input in order to achieve quantitative predictions. This statement applies also to finite-size effects in anisotropic systems where up to $d(d+1)/2+1$ nonuniversal parameters enter the finite-size scaling form of the free energy density dohm2008 ; dohm2018 . This sheds new light on the general belief that the critical behavior of systems with short-range interactions is largely independent of microscopic details fish-1 ; priv ; pelissetto .