with a known universal constant $X_{\infty\rm T}$ priv . By means of the
inverse special shear transformation the anisotropic transverse correlation
function $G_{\infty,\rm T}({\bf x},t)$ of the $\varphi^{4}$ model in the
large-$n$ limit is obtained as

$\displaystyle G_{\infty,\rm T}({\bf x},t)$ $\displaystyle=$
$\displaystyle(\det{\bf A})^{-1/2}G^{\prime}_{\infty,\rm T}(|{\bf
x}^{\prime}|,t)$ (5.90) $\displaystyle=$ $\displaystyle{\cal C}_{\rm
T}\big{[}{\cal M}_{\infty}(t)\big{]}^{2}\Big{(}\frac{\bar{\xi}_{\infty,\rm
T}}{[{\bf x}\cdot({\bf\bar{A}}^{-1}{\bf
x})]^{1/2}}\Big{)}^{d-2},\;\;\;\;\;\;\;\;\;\;$ (5.91)
$\displaystyle\big{[}{\cal M}_{\infty}(t)\big{]}^{2}$ $\displaystyle=$
$\displaystyle\frac{r_{\infty,0c}-r_{0}}{4u_{0}n}=(B_{\infty})^{2}|t|,$ (5.92)
$\displaystyle(B_{\infty})^{2}$ $\displaystyle=$ $\displaystyle
a_{0}/(4u_{0}n),$ (5.93) $\displaystyle r_{\infty,0c}$ $\displaystyle=$
$\displaystyle-4u_{0}n\int_{\bf k}{\bf k}^{-2}=r^{\prime}_{\infty,0c},$ (5.94)
$\displaystyle\bar{\xi}_{\infty,\rm T}(t)$ $\displaystyle=$
$\displaystyle\Big{[}\prod^{d}_{\alpha=1}\xi_{\infty,\rm
T}^{(\alpha)}(t)\Big{]}^{1/d},$ (5.95) $\displaystyle\xi_{\infty,\rm
T}^{(\alpha)}(t)$ $\displaystyle=$ $\displaystyle\xi_{\infty,{\rm
T},0}^{(\alpha)}|t|^{-\nu_{\infty}},\;\;$ (5.96)
$\displaystyle\xi_{\infty,{\rm T},0}^{(\alpha)}$ $\displaystyle=$
$\displaystyle\lambda_{\alpha}^{1/2}\xi_{\infty,{\rm
T},0}^{\prime}.\;\;\;\;\;\;$ (5.97)

where ${\bf\bar{A}}$ must be expressed in terms of $\xi_{\infty,{\rm
T},0}^{(\alpha)}$, compare Eqs. (5.41)-(5.44) of dohm2018  for finite $n>1$.

Our proof of multiparameter universality in Sec. IV. C. implies that the
correlation functions of the anisotropic $n$-vector model (2.22) in the
large-$n$ limit have the same structure as those derived for the anisotropic
$\varphi^{4}$ model, with the same universal scaling functions $\Psi_{\infty}$
and $\hat{\Psi}_{\infty}$ and with the reduced anisotropy matrix
${\bf\bar{A}^{\rm sp}}$ and two different nonuniversal parameters $\Gamma^{\rm
sp}_{\infty}$ and $\bar{\xi}^{\rm sp}_{\infty,0}$ that depend on the couplings
$E_{i,j}$. The same statement applies to other models of the spherical
universality class.

###  V.4 Exact anisotropic Gaussian correlation function

The following is based on the continuum version of the anisotropic Gaussian
model (4.58) and the isotropic Gaussian Hamiltonian (4.59) obtained after the
special shear transformation (4.1)-(4.3). The structure of the anisotropic and
isotropic Gaussian correlation functions $G^{\rm G}({\bf x},t)$, $\hat{G}^{\rm
G}({\bf k},t)$ and $G^{\prime\rm G}(|{\bf x}^{\prime}|,t)$,
$\hat{G}^{\prime\rm G}(|{\bf k}^{\prime}|,t)$, respectively, are closely
related to those in the large-$n$ limit with essentially the same scaling
functions but with Gaussian critical exponents

$\displaystyle\nu^{\rm G}=1/2,\;\gamma^{\rm G}=1,\;\eta^{\rm G}=0,$ (5.98)

and different nonuniversal parameters $\Gamma^{\rm G}_{+}$ and $\bar{\xi}^{\rm
G}_{0+}$. These correlation functions are defined for $d\geq 2$ by

$\displaystyle G^{\rm G}({\bf x},t)$ $\displaystyle=$ $\displaystyle
n<\varphi({\bf x})\varphi({\bf 0})>=\int_{\bf k}\;\hat{G}^{\rm G}({\bf
k},t)e^{i{\bf k}\cdot{\bf x}},\;\;\;\;\;\;\;\;\;$ (5.99) $\displaystyle
G^{\prime\rm G}(|{\bf x}^{\prime}|,t)$ $\displaystyle=$
$\displaystyle\int_{\bf k^{\prime}}\hat{G}^{\prime\rm G}(|{\bf
k}^{\prime}|,t)\;e^{i{\bf k^{\prime}}\cdot{\bf x}^{\prime}},$ (5.100)
$\displaystyle\hat{G}^{\prime\rm G}(|{\bf k}^{\prime}|,t)$ $\displaystyle=$
$\displaystyle n\;[r_{0}+({\bf k^{\prime}})^{2}]^{-1}\;\;\;,$ (5.101)

where

$r_{0}=a_{0}t=[\xi_{+}^{\prime\rm G}(t)]^{-2},\;\;\;t\geq 0,$ (5.102)

with the isotropic Gaussian correlation length dohm2008

$\xi_{+}^{\prime\rm G}(t)=r_{0}^{-1/2}=\xi_{0+}^{\prime\rm
G}t^{-1/2},\;\;\xi_{0+}^{\prime\rm G}=a_{0}^{-1/2}.$ (5.103)

The evaluation of the integral (5.100) in the asymptotic region is analogous
to that of (5.60) and leads to

$\displaystyle G^{\prime\rm G}(|{\bf x}^{\prime}|,t)$ $\displaystyle=$
$\displaystyle\frac{1}{|{\bf x}^{\prime}|^{d-2}}\Psi^{\rm G}(|{\bf
x}^{\prime}|/\xi_{+}^{\prime G}(t)),$ (5.104) $\displaystyle\Psi^{\rm G}(y)$
$\displaystyle\equiv$ $\displaystyle n\Psi_{\infty}(y),$ (5.105)
$\displaystyle\widetilde{Q}^{\rm G}_{3}$ $\displaystyle=$
$\displaystyle\Psi^{\rm G}(0)=n\frac{\Gamma[(d-2)/2]}{4\pi^{d/2}},\;\;\;\;$
(5.106)

where the Gaussian scaling function (5.105) divided by $n$ is identical with
the scaling function (5.75) in the large-$n$ limit. There exists agreement
with the structure of (3.45) because of the identity $1=\Gamma^{\prime\rm
G}_{+}(\xi^{\prime\rm G}_{0+})^{-2}$ where $\Gamma^{\prime\rm G}_{+}$ is the
amplitude of the isotropic Gaussian susceptibility

$\displaystyle\chi^{\prime\rm G}(t)$ $\displaystyle=$
$\displaystyle\Gamma^{\prime\rm G}_{+}t^{-1}=r_{0}^{-1}=[\xi_{+}^{\prime\rm
G}(t)]^{2},$ (5.107) $\displaystyle\Gamma^{\prime\rm G}_{+}$ $\displaystyle=$
$\displaystyle(\xi_{0+}^{\prime\rm G})^{2}.$ (5.108)

The universal isotropic scaling form for $t\geq 0$ in ${\bf k^{\prime}}$ space
is analogous to (5.69),

$\displaystyle\hat{G}^{\prime\rm G}(|{\bf k}^{\prime}|,t)$ $\displaystyle=$
$\displaystyle({\bf k^{\prime}})^{-2}\hat{\Psi}^{\rm G}(|{\bf
k^{\prime}}|\xi_{+}^{\prime\rm G}(t)),$ (5.109) $\displaystyle\hat{\Psi}^{\rm
G}(y)$ $\displaystyle=$ $\displaystyle(1+y^{-2})^{-1},$ (5.110) $\displaystyle
Q^{\rm G}_{3}$ $\displaystyle=$ $\displaystyle\hat{\Psi}^{\rm G}(\infty)=1.$
(5.111)

Correspondingly we obtain the exact anisotropic correlation function of the
Gaussian model for $d\geq 2$ and $t\geq 0$

$\displaystyle G^{\rm G}({\bf x},t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{+}^{\rm G}\;(\bar{\xi}^{\rm G}_{0+})^{-2}}{[{\bf
x}\cdot{\bf\bar{A}}^{-1}{\bf x}]^{(d-2)/2}}\Psi^{\rm G}\Big{(}\frac{[{\bf
x}\cdot{\bf\bar{A}}^{-1}{\bf
x}]^{1/2}}{\bar{\xi}^{G}_{+}(t)}\Big{)},\;\;\;\;\;\;\;\;\;\;$ (5.112)
$\displaystyle\hat{G}^{\rm G}({\bf k},t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{+}^{\rm G}\;(\bar{\xi}^{\rm G}_{0+})^{-2}}{[{\bf
k}\cdot\;{\bf\bar{A}}{\bf k}]}\hat{\Psi}^{\rm G}\Big{(}[{\bf
k}\cdot\;{\bf\bar{A}}{\bf k}]^{1/2}\bar{\xi}_{+}^{\rm G}(t)\Big{)},\;$ (5.113)
$\displaystyle\Gamma_{+}^{\rm G}\;(\bar{\xi}^{\rm G}_{0+})^{-2}$
$\displaystyle=$
$\displaystyle\Big{(}\prod^{d}_{\alpha=1}{\mbox{$\lambda$}_{\alpha}}\Big{)}^{-1/d}=(\det{\bf
A})^{-1/d},$ (5.114)

where ${\bf\bar{A}}$ must be expressed in terms of $\xi_{0+}^{{\rm
G}(\alpha)}$. Here $\Gamma^{\rm G}_{+}=\Gamma^{\prime\rm G}_{+}$ and
$\bar{\xi}^{\rm G}_{0+}$ are the amplitudes of the susceptibility per
component and mean correlation length of the anisotropic system above $T_{c}$,

$\displaystyle\chi^{\rm G}_{+}(t)$ $\displaystyle=$ $\displaystyle\Gamma^{\rm
G}_{+}\;t^{-1},$ (5.115) $\displaystyle\bar{\xi}^{\rm G}_{+}(t)$
$\displaystyle=$ $\displaystyle\bar{\xi}^{\rm G}_{0+}t^{-1/2},$ (5.116)
$\displaystyle\bar{\xi}^{\rm G}_{0+}$ $\displaystyle=$
$\displaystyle\big{[}\prod^{d}_{\alpha=1}\xi_{0+}^{{\rm
G}(\alpha)}\big{]}^{1/d}$ (5.117)

where the amplitudes $\xi_{0+}^{{\rm G}(\alpha)}$ of the principal correlation
lengths $\xi^{{\rm
G}(\alpha)}_{0+}={\mbox{$\lambda$}_{\alpha}}^{1/2}\xi^{\prime\rm G}_{0+}$ are
determined by the eigenvalues ${\mbox{$\lambda$}_{\alpha}}$ of the anisotropy
matrix ${\bf A}$ and the amplitude $\xi^{\prime\rm G}_{0+}=a_{0}^{-1/2}$,
(5.103), of the isotropic correlation length above $T_{c}$ of the Gaussian
Hamiltonian (4.59). As in the large-$n$ limit, the single-parameter isotropic
scaling forms (5.104) and (5.109) are due to the special relation (5.108).

Owing to two-scale-factor universality all isotropic systems of the Gaussian
universality class have an asymptotic correlation function that has the same
structure as that of (5.104) and (5.109) with two nonuniversal amplitudes
$\Gamma^{\rm G,iso}_{+}$ and $\xi^{\rm G,iso}_{0+}$ which in general are
independent of each other. This statement applies to the Gaussian version of
the $n$-vector model (2.22) (with spin variables $-\infty\leq
S_{i}^{(\mu)}\leq\infty$). Thus, according to the proof of multiparameter
universality in Sec. IV.C., the same structure of the anisotropic Gaussian
correlation function as in (5.112) and (5.21) is predicted for the Gaussian
version of the isotropic $n$-vector model where the reduced anisotropy matrix
${\bf\bar{A}}^{\rm sp}$ comes into play as well as two nonuniversal parameters
$\Gamma^{\rm sp}_{+}$ and $\bar{\xi}^{\rm sp}_{0+}$ which need to be
determined as functions of $E_{i,j}$.

###  V.5 Application to three dimensions:
Determination of $\hat{\Psi}_{\pm}$ from the FRG

Application of the preceding analysis to three dimensions is of interest in
view of the various anisotropic three-dimensional systems in condensed matter
physics. Thereby one encounters, in general, three problems:

(i) the determination of the three-dimensional universal isotropic scaling
functions $\Psi_{\pm},\hat{\Psi}_{\pm}$ or, equivalently, $\hat{D}_{\pm}$, of
the $O(n)$-symmetric universality classes,

(ii) the determination of up to seven independent nonuniversal parameters
$\Gamma_{+},\xi^{(1)}_{0+},\xi^{(2)}_{0+},\xi^{(3)}_{0+},\Omega_{1},\Omega_{2},\Omega_{3}$
where the principal angles $\Omega_{\alpha}$ describe the orientation of the
three principal directions, and

(iii) the construction of the reduced anisotropy matrix

$\displaystyle{\bf\bar{A}}={\bf\bar{A}}\big{(}\\{\xi_{0\pm}^{(\alpha)},{\bf
e}^{(\alpha)}\\}\big{)}={\bf\bar{A}}\Big{(}\frac{\xi^{(1)}_{0\pm}}{\bar{\xi}_{0\pm}},\frac{\xi^{(2)}_{0\pm}}{\bar{\xi}_{0\pm}},\frac{\xi^{(3)}_{0\pm}}{\bar{\xi}_{0\pm}},\Omega_{1},\Omega_{2},\Omega_{3}\Big{)}$

where ${\bf e}^{(\alpha)}$ are the three principal unit vectors and where
$\bar{\xi}_{0\pm}$ is the amplitude of the mean correlation length. Several
examples of ${\bf\bar{A}}$ for three-dimensional anisotropic $\varphi^{4}$
models have been presented cd2004 ; dohm2008 ; dohm2018 ; DWKS2021 , in
particular with general principal angles $\Omega$ and $\Omega+\pi/2$
describing planar anisotropies in a three-dimensional environment DW2021 .
This matrix appears not only in the bulk correlation function but also in the
finite-size properties of the excess free energy and the Casimir force of
weakly anisotropic systems cd2004 ; dohm2008 ; dohm2018 ; DW2021 . A
determination of the nonuniversal parameters (ii) in terms of the couplings is
a nontrivial problem for three-dimensional anisotropic $n$-vector models and
real systems that we shall not further discuss in this paper.

Here we confine ourselves to a discussion of the problem (i) in the context of
the isotropic order-parameter correlation function $\hat{G}_{\pm}^{\rm
FRG}(k,t)$ derived in the framework of the FRG hassel2007 ; sinner2008 . The
calculation was performed within the isotropic $\varphi^{4}$ theory with a
finite cutoff $\Lambda_{0}$ and a four-point coupling $u_{0}$ in $d<4$
dimensions for $n=1$. On the basis of truncated FRG flow equations the
following approximate result was obtained above $(+)$ and below $(-)$ $T_{c}$

$\displaystyle\hat{G}_{\pm}^{\rm FRG}(k,t)$ $\displaystyle=$ $\displaystyle
k_{c}^{-2}g^{\pm}(k\xi^{\rm FRG}_{\pm},k/k_{c}),$ (5.119) $\displaystyle
g^{\pm}(x_{\pm},y)$ $\displaystyle=$
$\displaystyle[y^{2}+\sigma^{\pm}(x_{\pm},y)]^{-1}$ (5.120)

with $x_{\pm}=k\xi^{\rm FRG}_{\pm}$, $y=k/k_{c}$, $k=|{\bf k}|$ where ${\bf
k}$ is the wave vector, $\xi^{\rm FRG}_{\pm}$ is the correlation length, and
$k_{c}$ is a finite nonuniversal wavenumber that depends on the cutoff and the
four-point coupling. For the definition of the functions $g^{\pm}(x_{\pm},y)$
and $\sigma^{\pm}(x_{\pm},y)$ we refer to hassel2007 ; sinner2008 . This
result is supposed to be applicable to the region $k\ll k_{c}$ and to the
crossover to a nonasymptotic critical region $k_{c}\ll k\ll\Lambda_{0}$ as
well as in the limit $k\to 0$ at fixed $\xi^{\rm FRG}_{\pm}<\infty$. The
implications of the exact sum rule (3.53) were not discussed. It was claimed
hassel2007 , without justification, that the scaling function
$\sigma^{\pm}(x_{\pm},y)$ is universal. Below we shall refute this claim by
showing that $g^{\pm}(x_{\pm},y)$ is not universal.

We first discuss (5.119) in the standard asymptotic scaling region
$k_{c}\xi^{\rm FRG}_{\pm}\gg 1$ and $k/k_{c}\ll 1$ at fixed $k\xi^{\rm
FRG}_{\pm}$, including the case $\xi^{\rm FRG}_{\pm}=\infty$ and the limit
$k\to 0$ at $\xi^{\rm FRG}_{\pm}<\infty$. We shall show that
$\hat{G}_{\pm}^{\rm FRG}(k,t)$ can be written in the two-parameter scaling
forms (3.51), (3.55), and (3.57) and that the universal scaling function
$\hat{\Psi}_{\pm}$ is not determined by $g^{\pm}$ or by $\sigma^{\pm}$ but by
a certain ratio of $g^{\pm}$ and a ratio of its overall amplitudes.

Comparison of (5.119) with (3.51) in the limit $k_{c}\to\infty$ at fixed
$k\xi^{\rm FRG}_{\pm}$ yields the universal ratio

$\displaystyle\lim_{k_{c}\to\infty}\frac{\hat{G}_{\pm}^{\rm
FRG}(k,t)}{\hat{G}_{+}^{\rm FRG}(k,0)}$ $\displaystyle=$
$\displaystyle\lim_{k_{c}\to\infty}\frac{g^{\pm}(k\xi^{\rm
FRG}_{\pm},k/k_{c})}{g^{+}(\infty,k/k_{c})}\;\;\;\;\;\;$ (5.121)
$\displaystyle=$ $\displaystyle\frac{\hat{\Psi}_{\pm}(k\xi^{\rm
FRG}_{\pm})}{\hat{\Psi}_{+}(\infty)}\;.$ (5.122)

Next we determine the universal constant $Q_{3}$. We first consider the case
$T=T_{c}$, $\xi^{\rm FRG}_{\pm}=\infty$,

$\displaystyle\hat{G}_{\pm}^{\rm FRG}(k,0)$ $\displaystyle=$ $\displaystyle
k_{c}^{-2}g^{\pm}(\infty,k/k_{c}).$ (5.123)

Since $\hat{G}_{\pm}^{\rm FRG}(k,0)$ must behave as $\simeq k^{-2+\eta}$ for
small $k$ we obtain for $k/k_{c}\ll 1$

$\displaystyle g^{\pm}(\infty,k/k_{c})$ $\displaystyle=$ $\displaystyle A^{\rm
FRG}\;(k_{c}/k)^{2-\eta},$ (5.124) $\displaystyle\hat{G}_{\pm}^{\rm FRG}(k,0)$
$\displaystyle=$ $\displaystyle A^{\rm FRG}\frac{k_{c}^{-\eta}}{k^{2-\eta}}.$
(5.125)

The $d$-dependent constant $A^{\rm FRG}>0$ is denoted by $A_{D}^{-1}$ in Eq.
(58) of sinner2008 , with $A_{3}\approx 1.075$ in three dimensions, i.e.,

$\displaystyle A^{\rm FRG}\approx 0.930\;,\;\;d=3.$ (5.126)

Furthermore, compatibility of (5.119) with the free-energy density of the
$\varphi^{4}$ model requires that (5.119) satisfies the exact sum rule (3.53)
which yields the susceptibility $\chi^{\rm FRG}_{\pm}$

$\displaystyle\hat{G}_{\pm}^{\rm FRG}(0,t)$ $\displaystyle=$
$\displaystyle\chi^{\rm FRG}_{\pm}$ (5.127) $\displaystyle=$ $\displaystyle
k_{c}^{-2}\lim_{k\to 0}g^{\pm}(k\xi^{\rm
FRG}_{\pm},k/k_{c})\;\;\;\;\;\;\;\;\;\;\;\;$ (5.128) $\displaystyle=$
$\displaystyle k_{c}^{-2}f_{\chi}^{\pm}(k_{c}\xi^{\rm FRG}_{\pm}).$ (5.129)

As discussed in Sec. II. A, Eqs. (5.127)-(5.129) are exactly valid in the
entire range where (5.119) is valid. For $k_{c}\xi^{\rm FRG}_{\pm}\gg 1$,
$\chi^{\rm FRG}_{\pm}$ must behave as

$\displaystyle\chi^{\rm FRG}_{\pm}$ $\displaystyle=$ $\displaystyle\Gamma^{\rm
FRG}_{\pm}t^{-\gamma},$ (5.130)

with $\gamma=\nu(2-\eta)$ and $\xi^{\rm FRG}_{\pm}=\xi^{\rm
FRG}_{0\pm}t^{-\nu}$. This implies

$\displaystyle f_{\chi}^{\pm}(k_{c}\xi^{\rm FRG}_{\pm})$ $\displaystyle=$
$\displaystyle C^{\rm FRG}_{\pm}(k_{c}\xi^{\rm FRG}_{\pm})^{2-\eta}\;,$
(5.131) $\displaystyle\Gamma^{\rm FRG}_{\pm}$ $\displaystyle=$ $\displaystyle
C^{\rm FRG}_{\pm}k_{c}^{-\eta}(\xi^{\rm FRG}_{0\pm})^{2-\eta},$ (5.132)

with the $d$-dependent constant $C^{\rm FRG}_{\pm}$. The factor
$k_{c}^{-\eta}$ in (5.132) and (5.125) can be eliminated. This yields at
$T_{c}$

$\displaystyle\hat{G}_{+}^{\rm FRG}(k,0)$ $\displaystyle=$
$\displaystyle\frac{A^{\rm FRG}}{C^{\rm FRG}}\frac{\Gamma^{\rm
FRG}_{+}}{(k\xi^{\rm FRG}_{0+})^{2-\eta}}.$ (5.133)

From a comparison with the exact asymptotic isotropic scaling form at $T_{c}$,
(3.54) and (3.55), we obtain the identification of the universal quantity
$Q_{3}$ in terms of the ratio

$\displaystyle Q_{3}=\frac{A^{\rm FRG}}{C_{+}^{\rm
FRG}}=\hat{\Psi}_{+}(\infty)=\hat{\Psi}_{-}(\infty)=\;\;\text{universal}.\;\;\;\;\;$
(5.134)

Together with the determination of the ratio (5.122) this completes our
determination of the isotropic two-parameter scaling form (3.51),

$\displaystyle\hat{G}_{\pm}^{\rm FRG}(k,t)=\frac{\Gamma^{\rm
FRG}_{+}}{\big{(}k\;\xi^{\rm
FRG}_{0+}\big{)}^{2-\eta}}\;\hat{\Psi}_{\pm}\Big{(}k\xi^{\rm
FRG}_{\pm}\Big{)}\;$ (5.135)

with the universal scaling function $\hat{\Psi}_{\pm}$ for $d<4$ in the regime
$k/k_{c}\ll 1,k_{c}\xi^{\rm FRG}_{\pm}\gg 1$, $k\xi^{\rm FRG}_{0+}>0$,

$\displaystyle\hat{\Psi}_{\pm}(k\xi^{\rm FRG}_{\pm})=\frac{A^{\rm
FRG}}{C_{+}^{\rm FRG}}\lim_{k_{c}\to\infty}\frac{g^{\pm}(k\xi^{\rm
FRG}_{\pm},k/k_{c})}{g^{+}(\infty,k/k_{c})}.$ (5.136)

We note that in (5.135) $\hat{G}_{\pm}^{\rm FRG}(k,t)$ is uniquely divided
into nonuniversal and universal parts. The corresponding two-parameter Fisher-
Aharony scaling form for $t\neq 0,k\geq 0,$

$\displaystyle\hat{G}_{\pm}^{\rm FRG}(k,t)$ $\displaystyle=$
$\displaystyle\Gamma^{\rm
FRG}_{\pm}|t|^{-\gamma}\;\hat{D}_{\pm}\Big{(}k\xi^{\rm
FRG}_{\pm}(t)\Big{)}\;\;$ (5.137)

follows from (3.57)-(3.60). The structure of (5.135) and (5.137) has been
confirmed by exact analytic results for isotropic Ising and $\varphi^{4}$
models on two-dimensional lattices dohm2019 ; DKW2023 . We see that, unlike
the original formulation (5.119) in terms of the nonuniversal model parameter
$k_{c}$, this parameter has been completely eliminated in (5.135) and (5.137)
in favor of the observable thermodynamic amplitude $\Gamma^{\rm FRG}_{\pm}$ of
the susceptibility, as expected on general grounds. But there still exist two
independent nonuniversal thermodynamic parameters $\Gamma^{\rm FRG}_{+}$ and
$\xi^{\rm FRG}_{0+}$ in the asymptotic scaling forms (5.135) and (5.137)
including $T=T_{c}$. This refutes the existence of a ”one-parameter scaling
picture” and ”one-parameter scaling hypothesis” hassel2007  for systems with
$\eta\neq 0$ for which the principle of two-scale-factor universality with two
independent thermodynamic parameters is well established.

After the identification of $\hat{\Psi}\pm$ in terms of the ratios $A^{\rm
FRG}/C_{+}^{\rm FRG}$ and $g^{\pm}(x_{\pm},y)/g^{+}(\infty,y)$ from hassel2007
; sinner2008  we substitute (5.136) into (5.21) which yields the bulk
correlation function $\hat{G}_{\pm}$ of the anisotropic $\varphi^{4}$ model in
the asymptotic critical region in three dimensions for $n=1$. Due to
multiparameter universality this also yields the prediction of the correlation
function of all other weakly anisotropic systems in the $(d=3,n=1)$
universality class provided that the nonuniversal parameters (ii) specified
above are known.

The value of $C_{+}^{\rm FRG}$ was not calculated in hassel2007 ; sinner2008 .
It would be worthwhile to determine this value in order to obtain an estimate
for $Q_{3}$ which then can be compared with the known estimates $Q_{3}\approx
0.922$ ($\varepsilon$ expansion) or $0.90\pm 0.01$ (numerical studies) for
$d=3,n=1$ priv . (Note that these values are defined in combination with the
”true” (exponential) correlation length $\xi_{0+}$ which differs from the
second-moment correlation length pelissetto ; tarko ; brezin1974 .)

So far our analysis confirms the universality only of the ratios of the
amplitudes $A^{\rm FRG}$ and $C_{+}^{\rm FRG}$ and of the functions
$g^{\pm}(x_{\pm},y)$ and $g^{+}(\infty,y)$ appearing in (5.121), (5.134), and
(5.136) in the asymptotic scaling regime. Thus there is no compelling reason
for the assertion hassel2007 ; sinner2008  that the functions
$\sigma^{\pm}(x_{\pm},y)$ and $g^{\pm}(x_{\pm},y)$ themselves are universal.
No argument in support of this assertion was given in hassel2007 ; sinner2008
. This claim would imply the existence of new universal amplitudes $A^{\rm
FRG}$ or $C_{+}^{\rm FRG}$ that do not exist in the established theory of bulk
critical phenomena in the asymptotic region of isotropic systems priv ;
pelissetto ; zinn2007 ; fish-1 ; bre-1 .

In the following we show that the function $g^{\pm}(x_{\pm},y)$ is not
universal. From (5.125), (5.133), and (5.134) we obtain the relation between
amplitudes

$\displaystyle A^{\rm FRG}k_{c}^{-\eta}=Q_{3}\;\frac{\Gamma^{\rm
FRG}_{+}}{(\xi^{\rm FRG}_{0+})^{2-\eta}}\;\;.$ (5.138)

The right-hand side is uniquely divided into a universal part $Q_{3}$ and a
nonuniversal amplitude ratio of thermodynamic quantities. The nonuniversal
parameter $k_{c}$ on the left-hand side characterizes the nonasymptotic range
$k>k_{c}$ of wave numbers of the correlation function of the $\varphi^{4}$
theory and is determined by non-thermodynamic quantities, i. e., the cutoff
$\Lambda_{0}$ and the four-point coupling $u_{0}$ of the $\varphi^{4}$
Hamiltonian hassel2007 ; sinner2008 ,

$\displaystyle k_{c}$ $\displaystyle=$ $\displaystyle\Lambda_{0}\exp(-{\it
l}_{c})$ (5.139)

where ${\it l}_{c}(u_{0}\Lambda_{0}^{d-4})$ is a dimensionless flow parameter.
Thus $k_{c}^{-\eta}$ is different from the ratio $\Gamma^{\rm
FRG}_{+}/(\xi^{\rm FRG}_{0+})^{2-\eta}$ and is not universally related to this
ratio. We conclude that the dimensionless amplitude $A^{\rm FRG}$ must be
different from $Q_{3}$ and must contain a nonuniversal part, i.e., $A^{\rm
FRG}$ is a nonuniversal quantity. Since $A^{\rm FRG}$ governs the small-$k$
behavior of $g^{\pm}(\infty,k/k_{c})$ according to (5.124) we conclude that
the function $g^{\pm}(\infty,k/k_{c})$ is nonuniversal in the asymptotic
region at $T_{c}$ for small $k/k_{c}$. By continuity and analyticity
requirements this nonuniversal dependence for small $k/k_{c}$ cannot turn
smoothly into a universal dependence for larger $k/k_{c}$ in the nonasymptotic
regime $k>k_{c}$.

Furthermore, because of the relation

$\displaystyle C_{+}^{\rm FRG}$ $\displaystyle=$ $\displaystyle A^{\rm
FRG}/Q_{3}$ (5.140) $\displaystyle=$ $\displaystyle k_{c}^{\eta}\Gamma^{\rm
FRG}_{+}/(\xi^{\rm FRG}_{0+})^{2-\eta}$ (5.141)

we conclude that $C_{+}^{\rm FRG}$ must be a nonuniversal quantity. This
conclusion is supported by exact analytic results for the ratio $\Gamma^{\rm
iso}_{+}/(\xi^{\rm iso}_{0+})^{2-\eta}$ of isotropic Ising and $\varphi^{4}$
models on two-dimensional square and parallelogram lattices dohm2019 ; DKW2023
.

We also consider the case $k=0$ in the asymptotic range $k_{c}\xi^{\rm
FRG}_{+}\gg 1$ where we have from (5.127)-(5.132)

$\displaystyle k_{c}^{2}\hat{G}^{\rm FRG}_{+}(0,t)$ $\displaystyle=$
$\displaystyle\lim_{k\to 0}g^{+}(k\xi^{\rm FRG}_{+},k/k_{c})$ (5.142)
$\displaystyle=$ $\displaystyle C^{\rm FRG}_{+}(k_{c}\xi^{\rm
FRG}_{+})^{2-\eta}.$ (5.143)

The nonuniversality of $C_{+}^{\rm FRG}$ implies that the function
$g^{+}(k\xi^{\rm FRG}_{+},k/k_{c})$ is a nonuniversal quantity for $k\to 0$
and large $k_{c}\xi^{\rm FRG}_{+}$. A similar argumentation holds for
$g^{-}(k\xi^{\rm FRG}_{-},k/k_{c})$ for $k\to 0$. Together with the
nonuniversality of $g^{\pm}(\infty,k/k_{c})$ this means that the function
$g^{\pm}(x_{\pm},y)$ is not universal along the two vertical and horizontal
axes $(\xi^{\rm FRG}_{\pm})^{-1}=0$ and $k=0$ of the $k-(\xi^{\rm
FRG}_{\pm})^{-1}$ plane. By continuity and analyticity requirements this
nonuniversal dependence cannot turn smoothly into a universal dependence in
the plane away from these axes. This implies the nonuniversality of
$g^{\pm}(x_{\pm},y)$ in the region between these axes including both the
asymptotic and nonasymptotic regions, thus the curves for $\Delta\sigma^{\pm}$
shown in Fig. 5 of hassel2007  and in Fig. 3 of sinner2008  are nonuniversal.

We briefly comment on the consequences of the structure of (5.119) for $t\neq
0$. We rewrite (5.127) and (5.129) in the form

$\displaystyle\chi^{\rm FRG}_{\pm}$ $\displaystyle=$ $\displaystyle(\xi^{\rm
FRG}_{\pm})^{2}{\cal F}_{\pm}(k_{c}\xi^{\rm FRG}_{\pm})$ (5.144)

with

$\displaystyle{\cal F}_{\pm}(k_{c}\xi^{\rm FRG}_{\pm})$ $\displaystyle=$
$\displaystyle(k_{c}\xi^{\rm FRG}_{\pm})^{-2}\;f_{\chi}^{\pm}(k_{c}\xi^{\rm
FRG}_{\pm})$ (5.145)

where $f_{\chi}^{\pm}$ is the nonuniversal function defined in (5.129). This
constitutes an implicit equation that determines the parameter $k_{c}$ as a
nonuniversal function of $\chi^{\rm FRG}_{\pm}$ and $\xi^{\rm FRG}_{\pm}$ in
the entire range where (5.119) is valid. Thus $k_{c}$ can be exactly
eliminated in favor of $\chi^{\rm FRG}_{\pm}$ and $\xi^{\rm FRG}_{\pm}$, and
(5.119) can be expressed completely in terms of these two nonuniversal
temperature-dependent thermodynamic quantities, without any explicit cutoff
dependence even outside the standard asymptotic critical region. We consider
this nonuniversal structure to be due to the approximations made in hassel2007
; sinner2008  within the $\varphi^{4}$ model which are not expected to remain
universally valid for all systems in the Ising universality class.
Nevertheless, after the identification of the universal part of (5.119)
through (5.135) and (5.136), this result is a prediction that is of
substantial interest for a comparison with other systems of the $d=3$ Ising
universality class in the asymptotic critical region.

##  VI Multiparameter universality of critical bulk amplitude relations

The universal critical point bulk amplitude relations play an important role
in the traditional theory of critical phenomena priv  where, however, no
attention has been paid to amplitude relations in the subclass of weakly
anisotropic systems within a universality class. The failure of two-scale-
factor universality in this subclass was pointed out in cd2004 . Subsequently
several bulk amplitude relations have been shown to be valid within the weakly
anisotropic bulk $\varphi^{4}$ theory in $2<d<4$ dimensions dohm2006 ;
dohm2008 ; dohm2018 ; dohm2019  with the same universal constants as for the
isotropic case, such as
$Q_{c},Q_{1},R^{+}_{\xi},Q_{2},\widetilde{Q}_{3},P_{2},P_{3},W_{1},X_{-}(0)$
but where, in general, up to $d(d+1)/2+1$ independent nonuniversal parameters
are involved rather than only two nonuniversal parameters. These properties
were called multiparameter universality dohm2008  for bulk amplitude
relations. It was hypothesized dohm2018  that these properties are valid not
only for $\varphi^{4}$ models but also for all weakly anisotropic bulk systems
other than $\varphi^{4}$ models. So far no proof exists for this hypothesis.
Our generalized shear transformation introduced in Sec. IV enables us to
present such a proof. As a representative of a weakly anisotropic system other
than the $\varphi^{4}$ model we take the $n$-vector model.

###  VI.1 Proof of multiparameter universality

Application of the generalized shear transformation (4.89) to the volume $V$
of the anisotropic system yields the transformed volume of the isotropic
system

$\displaystyle\widehat{V}\equiv V^{\rm iso}$ $\displaystyle=$
$\displaystyle(\det{\mbox{\boldmath$\widehat{\lambda}$}})^{-1/2}\;V=\big{(}\xi^{\text{iso}}_{0\pm}/\bar{\xi}^{\rm
sp}_{0\pm}\big{)}^{d}\;V$ (6.1)

where we have used (4.94). This implies the invariance of the volume ratios

$\displaystyle\frac{V^{\rm iso}}{V^{\rm iso}_{\rm cor,+}}$ $\displaystyle=$
$\displaystyle\frac{V}{V_{\rm cor,+}},\;\frac{V^{\rm iso}}{V^{\rm iso}_{\rm
cor,-}}=\frac{V}{V_{\rm cor,-}}$ (6.2)

where $V^{\rm iso}_{\rm cor,\pm}=\big{(}\xi^{\text{iso}}_{0\pm}\big{)}^{d}$
and $V_{\rm cor\pm}=\prod^{d}_{\alpha=1}\xi_{0\pm}^{{\rm
sp}(\alpha)}=\big{(}\bar{\xi}^{\rm sp}_{0\pm}\big{)}^{d}$ are the isotropic
(spherical) and anisotropic (ellipsoidal) correlation volumes above and below
$T_{c}$, respectively. This is analogous to the invariance (4.46) in the
$\varphi^{4}$ theory.

We consider the singular part ${\cal F}^{\rm sp}_{s}$ of total free energy
(2.26) of the anisotropic $n$-vector model in the volume $V$. The generalized
shear transformation (4.89) generates the singular part ${\cal F}^{\rm
iso}_{s}$ of the transformed isotropic system. We have argued in Sec. II.B.
that since this transformation involves only a smooth change of the positions
${\bf x}\to\widehat{\bf x}$ of the lattice points it does not change its
singular part, thus we have the invariance

$\displaystyle{\cal F}^{\rm iso}_{s}={\cal F}^{\rm sp}_{s}.$ (6.3)

This is parallel to (4.23) of the special shear transformation.

From (6.3) and (6.1) we obtain the relation between the singular parts of the
bulk free-energy densities of the isotropic and anisotropic $n$-vector model

$\displaystyle f^{\rm sp}_{b,s,\pm}$ $\displaystyle=$
$\displaystyle\lim_{V\to\infty}\frac{{\cal F}^{\rm
sp}_{s}}{V}=\Big{(}\frac{\xi^{\text{iso}}_{0\pm}}{\bar{\xi}^{\rm
sp}_{0\pm}}\Big{)}^{d}\;\lim_{V^{\rm iso}\to\infty}\frac{{\cal F}^{\rm
iso}_{s}}{V^{\rm iso}}$ (6.4) $\displaystyle=$
$\displaystyle\Big{(}\frac{\xi^{\text{iso}}_{0\pm}}{\bar{\xi}^{\rm
sp}_{0\pm}}\Big{)}^{d}\;\;f^{\text{iso}}_{b,s,\pm}.$

Together with the isotropic relation (3.14) this yields the singular part of
the bulk free-energy density of the anisotropic system

$\displaystyle f^{\rm
sp}_{b,s,\pm}(t)=\left\\{\begin{array}[]{r@{\quad\quad}l}\;A_{\pm}|t|^{d\nu}&\mbox{for}\;2<d<4\;,\\\
\;\frac{1}{2}A_{\pm}|t|^{2}\ln|t|&\mbox{for}\;d=2,\end{array}\right.$ (6.7)

where the amplitudes $A_{\pm}$ of the anisotropic system are given by

$\displaystyle A_{\pm}$ $\displaystyle=$
$\displaystyle\Big{(}\frac{\xi^{\text{iso}}_{0\pm}}{\bar{\xi}^{\rm
sp}_{0\pm}}\Big{)}^{d}\;A^{\text{iso}}_{\pm},\;\;\;d\geq 2,$ (6.8)
$\displaystyle\frac{f^{\rm sp}_{b,s,+}(t)}{f^{\rm sp}_{b,s,-}(t)}$
$\displaystyle=$ $\displaystyle\frac{A_{+}}{A_{-}}=\frac{A^{\rm
iso}_{+}}{A^{\rm iso}_{-}}=\;\;\rm{universal},\;\;d\geq 2.\;\;.$ (6.9)

Substituting the isotropic relation (3.16) we arrive at the universal relation
for the anisotropic amplitude $A_{+}$ above $T_{c}$ of the $n$-vector model

$\displaystyle\left(\bar{\xi}^{\rm
sp}_{0+}\right)^{d}A_{+}=\left(\xi_{0+}^{\text{iso}}\right)^{d}A_{+}^{\text{iso}}=Q_{1},\;d\geq
2,$ (6.10)

where in the anisotropic case the same universal constant $Q_{1}$ appears as
in the isotropic case. Thus we obtain the singular part of the free-energy
density of the anisotropic $n$-vector model for $t>0$

$\displaystyle f^{\rm
sp}_{b,s,+}(t)=\left\\{\begin{array}[]{r@{\quad\quad}l}\;Q_{1}\left(\bar{\xi}^{\rm
sp}_{0+}\right)^{-d}t^{d\nu},&\;2<d<4\;,\\\
\;\frac{1}{2}Q_{1}\left(\bar{\xi}^{\rm sp}_{0+}\right)^{-2}\;t^{2}\ln
t,&\;d=2,\end{array}\right.$ (6.13)

and obtain $f^{\rm sp}_{b,s,-}(t)$ for $t<0$ from (6.9) as

$\displaystyle f^{\rm
sp}_{b,s,-}(t)=\left\\{\begin{array}[]{r@{\quad\quad}l}\;\frac{A_{-}}{A_{+}}Q_{1}\left(\bar{\xi}^{\rm
sp}_{0+}\right)^{-d}|t|^{d\nu},&\;2<d<4\;,\\\
\;\frac{1}{2}Q_{1}\left(\bar{\xi}^{\rm
sp}_{0+}\right)^{-2}\;t^{2}\ln|t|,&\;d=2,\end{array}\right.$ (6.16)

where we have used (3.20) for $d=2$. We see that for $d=2$ dimensions $f^{\rm
sp}_{b,s,\pm}$ depends only on $|t|$ rather than $t$.

Our final step is to reformulate our result for the free-energy density in a
scaling form for the singular bulk part of the free energy of the anisotropic
$n$-vector model

$\displaystyle{\cal F}^{\rm sp}_{b,s,\pm}(t,V)$ $\displaystyle=$
$\displaystyle Vf^{\rm sp}_{b,s,\pm}(t).$ (6.17)

This is achieved by means of the observation that the scaling variable
$\widetilde{x}$, (3.22), introduced for the isotropic system can be expressed
in two different ways as

$\displaystyle\widetilde{x}$ $\displaystyle=$ $\displaystyle t[V^{\rm
iso}/(\xi_{0+}^{\rm iso})^{d}]^{1/(d\nu)}$ (6.18) $\displaystyle=$
$\displaystyle t[V/(\bar{\xi}_{0+}^{\rm sp})^{d}]^{1/(d\nu)}$ (6.19)

where in (6.19) we have used the invariance of the volume ratios above $T_{c}$
in (6.2) under the shear transformation (4.89). A corresponding observation
was already made for the finite-size scaling variable $\hat{x}$ defined in Eq.
(6.12) of dohm2018  in the context of renormalized perturbation theory for the
anisotropic $\varphi^{4}$ model in a finite geometry in $2<d<4$ dimensions.
From (6.9)-(6.19) and (3.23)-(3.25) we derive the scaling form for the
singular bulk part of the free energy of the anisotropic $n$-vector model in
$2<d<4$ dimensions

$\displaystyle{\cal F}^{\rm sp}_{b,s,+}(t,V)$ $\displaystyle=$ $\displaystyle
Q_{1}\;\widetilde{x}^{d\nu}={\cal F}^{\rm iso}_{b,s,+}(t,V^{\rm
iso}),\,\,t>0,\;\;\;\;\;$ (6.20) $\displaystyle{\cal F}^{\rm sp}_{b,s,-}(t,V)$
$\displaystyle=$ $\displaystyle\frac{A^{\text{iso}}_{-}}{A^{\rm
iso}_{+}}Q_{1}\;|\widetilde{x}|^{d\nu}={\cal F}^{\rm iso}_{b,s,-}(t,V^{\rm
iso}),\,\,t<0,\;\;\;\;\;\;\;\;\;$ (6.21)

and the corresponding expression for the ($d=2,n=1$) universality class

$\displaystyle{\cal F}^{\rm sp}_{b,s,\pm}(t,V)$ $\displaystyle=$
$\displaystyle\frac{1}{2}Q_{1}\;|\widetilde{x}|^{2}\ln|t|={\cal F}^{\rm
iso}_{b,s,\pm}(t,V^{\rm iso}),\,\,\,\,\,\,\,\,$ (6.22)

with a nonscaling logarithmic factor $\ln|t|$. The nonuniversality due to weak
anisotropy enters only the scaling variable $\widetilde{x}$. For $d=2$, ${\cal
F}^{\rm sp}_{b,s,\pm}$ is a function of $|t|$.

Eqs. (6.20)-(6.22) explicitly reflect the invariance of the singular bulk
parts under the shear transformation of the $n$-vector model and constitute
the anisotropic extension of the singular bulk parts of the free energy of
isotropic systems presented in (3.23)-(3.25). Multiparameter universality
manifests itself by the fact that the corresponding result for the anisotropic
$\varphi^{4}$ model is obtained from (6.18)-(6.22) simply by the substitutions
$\bar{\xi}_{0+}^{\rm sp}\to\bar{\xi}_{0+},V^{\rm iso}\to
V^{\prime},\xi_{0+}^{\rm iso}\to\xi_{0+}^{\prime}$ in $\widetilde{x}$, (6.19)
which corresponds to the scaling variable (4.47) of the $\varphi^{4}$ theory,
with the same universal constants $Q_{1}$ and $A^{\rm iso}_{-}/A^{\rm
iso}_{+}$, where $\bar{\xi}_{0+}$ is defined in (4.34). This was shown already
previously in Eq. (3.33) of dohm2008  and Eq. (5.16) of dohm2018 . Since
$\bar{\xi}^{\rm sp}_{0+}$ and $\bar{\xi}_{0+}$ depend on $d$ independent
principal correlation lengths $\xi^{\rm sp(\alpha)}_{0+}$ and
$\xi_{0+}^{(\alpha)}$, respectively, our results for $f^{\rm sp}_{b,s,\pm}$
and ${\cal F}^{\rm sp}_{b,s,\pm}$ violate two-scale-factor universality.

In a similar way one can prove the validity of multiparameter universality for
the result of the singular bulk part of the free energy of the anisotropic
Gaussian model for $t>0$

$\displaystyle{\cal F}^{\rm
G}_{b,s,+}(t,V)=\left\\{\begin{array}[]{r@{\quad\quad}l}\;Q^{\rm
G}_{1}\;(\widetilde{x}^{\rm G})^{d/2}&\mbox{for}\;d>2,\\\ \;\frac{1}{2}Q^{\rm
G}_{1}\;\widetilde{x}^{\rm G}\ln t&\mbox{for}\;d=2,\end{array}\right.$ (6.25)

with the Gaussian scaling variable

$\displaystyle\widetilde{x}^{\rm G}$ $\displaystyle=$ $\displaystyle t[V^{\rm
iso}/(\xi_{0+}^{\rm G,iso})^{d}]^{2/d}=t[V/(\bar{\xi}_{0+}^{\rm
G})^{d}]^{2/d},$ (6.26)

with the Gaussian mean correlation length (5.117), and with the same universal
constant $Q^{\rm G}_{1}$, (3.30) and (3.31), as in the isotropic case. Eqs.
(6.25) and (6.26) are the anisotropic extension of (3.38) and (3.39) for the
isotropic Gaussian model.

There exist further universal bulk amplitude relations of isotropic systems
that involve the correlation length which are affected by anisotropy. As an
example we consider the relation (3.7). The amplitude of the order parameter
${\cal M}^{\rm sp}=B^{\rm sp}|t|^{\beta}$ of the anisotropic system is left
invariant under the generalized shear transformation (4.89),

$B^{\rm sp}=B^{\text{iso}},$ (6.27)

in contrast to (4.18) and (4.41) for the special shear transformation.
Together with the amplitude relation
$\widehat{\Gamma}_{+}\equiv\Gamma_{+}^{\rm iso}=\Big{(}\xi^{\rm
iso}_{0+}/\bar{\xi}^{\rm sp}_{0+}\Big{)}^{d}\;\Gamma_{+}^{\rm sp}$, (5.28),
this yields

$\displaystyle(B^{\text{iso}})^{2}(\Gamma_{+}^{\text{iso}})^{-1}(\xi^{\text{iso}}_{0+})^{d}=\big{(}B^{\rm
sp}\big{)}^{2}\big{(}\Gamma^{\rm sp}_{+}\big{)}^{-1}\big{(}\bar{\xi}^{\rm
sp}_{0+}\big{)}^{d}=Q_{c},$

with the same universal constant $Q_{c}$ for both isotropic and anisotropic
systems. On the basis of the special shear transformation (4.1)-(4.3) the same
relation with the same constant $Q_{c}$ was recently proven within the
anisotropic $\varphi^{4}$ model in $d=2$ dimensions dohm2019  and was verified
to be valid also in the exactly solvable anisotropic Ising model WuCoy ;
Vaidya1976  where $Q_{c}$ was identified as given in Eq. (76) of dohm2019 .
From (5.65), (5.68), and (5.87) we confirm that (VI.1) is also valid in the
large-$n$ limit, with the universal constant

$Q_{c}=A_{d}/(4-d),\;\;\;\;n=\infty,\;\;2<d<4,$ (6.29)

with $A_{d}$ given by (5.66).

In deriving the exact critical bulk amplitude relations (6.2), (6.13), and
(VI.1) for the $n$-vector model we have not made any assumptions other than
the validity of two-scale-factor universality for isotropic systems and the
existence of principal axes and correlation lengths for weakly anisotropic
systems. No specific properties of the $n$-vector model were needed in the
derivation. Thus these results are proven to be valid for arbitrary weakly
anisotropic systems. In a similar way, the validity of all the other bulk
amplitude relations considered in dohm2008 ; dohm2018  for general $n$ and $d$
within anisotropic $\varphi^{4}$ model including those at finite external
field (such as Eq. (3.35) of dohm2008 ) can be proven for all weakly
anisotropic systems. This feature is called multiparameter universality since
these relations contain the same universal constants that appear already in
the isotropic case but in the anisotropic case up to $d(d+1)/2+1$ independent
nonuniversal parameters (rather than only two independent parameters) are
involved whose definition depends on the nonuniversal orientation of the
principal axes and on the nonuniversal amplitudes of the principal correlation
lengths.

###  VI.2 Bulk specific heat of anisotropic systems

The universal bulk amplitude relations for the free energy derived above are
relevant for the analysis of the singular part

$\displaystyle C^{\rm sp}_{b,s\pm}(t)$ $\displaystyle=$
$\displaystyle-\partial^{2}f^{\rm sp}_{b,s\pm}(t)/\partial t^{2},$ (6.30)

of the bulk specific heat per unit volume (divided by $k_{B}$) of anisotropic
systems above and below $T_{c}$. Above $T_{c}$ it can be expressed in terms of
the mean correlation length $\bar{\xi}^{\rm sp}_{0+}$ defined in (4.92). We
obtain from (6.4)-(6.13)

$\displaystyle C^{\rm sp}_{b,s+}(t)$ $\displaystyle=$
$\displaystyle\frac{\big{(}R^{+}_{\xi}\big{)}^{d}}{\alpha\;(\bar{\xi}^{\rm
sp}_{0+})^{d}}\;t^{-\alpha},\;\;\;d>2,$ (6.31) $\displaystyle C^{\rm
sp}_{b,s+}(t)$ $\displaystyle=$
$\displaystyle-\;\frac{Q_{1}}{\big{(}\bar{\xi}^{\rm sp}_{0+}\big{)}^{2}}\;\ln
t,\;\;d=2,\;\;\;\;$ (6.32)

above $T_{c}$ and

$\displaystyle C^{\rm sp}_{b,s-}(t)$ $\displaystyle=$
$\displaystyle\frac{A_{-}}{A_{+}}\;C^{\rm sp}_{b,s+}(t),\;\;d\geq 2,$ (6.33)

below $T_{c}$. Eqs. (6.31) and (6.32) are the anisotropic extensions of (3.65)
and (3.66).

Eqs. (6.31)-(6.33) demonstrate that thermodynamic measurements of the
amplitude of specific heat of anisotropic systems can determine the mean
correlation length $\bar{\xi}^{\rm sp}_{0+}$ of real anisotropic systems.
Together with thermodynamic measurements of the amplitude $\Gamma_{+}$ of the
susceptibility of anisotropic systems one obtains the two amplitudes
determining the overall amplitude of the bulk correlation function (5.42).
Together with the universal ratio (4.93) this also determines $\bar{\xi}^{\rm
sp}_{\pm}(t)$ in the argument of $\Psi_{\pm}$ in (5.42).

These predictions are valid for any weakly anisotropic bulk system, e.g., for
superconductors and magnetic materials. Relations equivalent to (6.31) have
been employed previously schneider2004  in the analysis of experimental data
of superconductors whose critical behavior belongs to the $(d=3,n=2)$ $XY$
universality class. In the earlier work schneider2004  the anisotropic
fluctuations were treated within a Gaussian approximation for the case of a
diagonal anisotropy matrix (effective mass tensor). Here we have provided a
general and exact foundation for the bulk critical specific heat in weakly
anisotropic systems.

##  VII Correlation function of the two-dimensional Ising universality class

The proof of multiparameter universality presented in the preceding sections
has a significant impact on the anisotropic bulk correlation function of the
$(d=2,n=1)$ universality class dohm2019  with the exact critical exponents

$\nu=1,\;\;\eta=1/4.$ (7.1)

This includes both anisotropic Ising and anisotropic scalar $\varphi^{4}$
models.

###  VII.1 Isotropic scaling form and exact universal bulk scaling function
$\Psi_{\pm}(y_{\pm})$

There exists a large variety of interactions and lattice structures of
$(d=2,n=1)$ systems that have an isotropic bulk correlation function in the
scaling region near $T_{c}$. For example, for the two-dimensional
$\varphi^{4}$ lattice model (2.1) with short-range pair interactions the
condition of isotropy is

$\displaystyle{\bf A}^{\rm iso}=\left(\begin{array}[]{ccc}a&c\\\ c&b\\\
\end{array}\right)=c^{\rm iso}_{0}\left(\begin{array}[]{ccc}1&0\\\ 0&1\\\
\end{array}\right)$ (7.6)

with $c^{\rm iso}_{0}>0$ which, for given lattice structure, is a condition
for the couplings $K_{i,j}$ according to (2.3). The same condition holds for
the two-dimensional Gaussian lattice model (4.57). A corresponding general
condition of isotropy for the couplings $E_{i,j}$ of two-dimensional Ising
models (with the Hamiltonian $H^{\rm Is}$ given in (7.14) below) is unknown.
If the principal correlation lengths $\xi^{\rm Is(\alpha)}_{0+}$ of Ising
models are known as a function of the couplings $E_{i,j}$ the condition of
isotropy reads

$\xi^{\rm Is(1)}_{0+}=\xi^{\rm Is(2)}_{0+}.$ (7.7)

There exists an unlimited number of lattice structures and couplings
satisfying these conditions of isotropy. There exist even two-dimensional
systems with short-range multi-spin interactions within the Ising universality
class that have isotropic correlation functions. Owing to the principle of
two-scale-factor universality all of these isotropic systems have the same
structure of the bulk order-parameter correlation function above and below
$T_{c}$ dohm2019

$\displaystyle G^{\rm iso}({\bf x},t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma^{\rm iso}_{+}(\xi_{0+}^{\rm iso})^{-7/4}}{|{\bf
x}|^{1/4}}\;\Psi_{\pm}\Big{(}\frac{|{\bf x}|}{\xi^{\rm
iso}_{\pm}(t)}\Big{)},\;$ (7.8) $\displaystyle\xi^{\rm iso}_{\pm}(t)$
$\displaystyle=$ $\displaystyle\xi^{\rm iso}_{0\pm}\;|t|^{-1},$ (7.9)

with the universal scaling function $\Psi_{\pm}$. To determine this function
$\Psi_{\pm}$ it suffices to consider the simplest nontrivial example of this
universality class. This is not the isotropic two-dimensional $\varphi^{4}$
model but the exactly solvable isotropic Ising model with equal nearest-
neighbor (NN) couplings $E>0$ on a square lattice with the Hamiltonian

$H^{\rm
Is,iso}=-E\sum_{j,k}[\sigma_{j,k}\sigma_{j,k+1}+\sigma_{j,k}\sigma_{j+1,k}],\;\;\sigma_{j,k}=\pm
1,$ (7.10)

and with the lattice spacing $\tilde{a}=1$. The bulk correlation function of
this model is isotropic in the scaling limit (but not in the range $|{\bf
x}|/\xi^{\rm iso}_{\pm}\gg 1$ cd2000-1 ; cd2000-2 ; dan-2 ; dohm2008 ). It was
calculated exactly in WuCoy  where it was presented in a nonuniversal scaling
form without identifying the universal part of the scaling function. Recently
dohm2019  we have written this correlation function in the universal form
(7.8) and have identified the universal scaling function $\Psi_{\pm}(y_{\pm})$
as well as the arguments $y_{\pm}$, together with the two nonuniversal
amplitudes, specialized to the square lattice with equal NN couplings,

$\displaystyle\xi_{0+}^{\rm squ,iso}$ $\displaystyle=$ $\displaystyle
2\big{[}\ln(1+2^{1/2})\big{]}^{-1},$ (7.11) $\displaystyle\Gamma^{\rm
squ,iso}_{+}$ $\displaystyle=$ $\displaystyle 2^{19/8}\pi\big{(}\xi_{0+}^{\rm
squ,iso}\big{)}^{7/4}p_{+},$ (7.12)

where the constant $p_{+}$ is expressed in terms of a Painlevé function of the
third kind, as defined in Eqs. (30)-(36) of dohm2019  and in the text after
these equations. The exact universal amplitude ratio of the exponential
correlation lengths is WuCoy ; pelissetto

$\xi^{\rm squ,iso}_{0+}/\xi^{\rm squ,iso}_{0-}=X_{\xi}=2.$ (7.13)

The scaling function $\Psi_{\pm}(y_{\pm})$ applies to all systems in the
subclass of isotropic systems in the $(d=2,n=1)$ universality class, in
particular, to Ising models with isotropic bulk correlation functions on two-
dimensional lattices other than the square lattice. So far the exact results
for such Ising models on ”isotropic lattices” Perk2  (e.g. honeycomb lattice,
see Fig. 1 in pri ) have not been discussed in the light of the predictions of
two-scale-factor universality with only two (rather than four Perk2 )
nonuniversal parameters, e.g., two independent nonuniversal amplitudes
corresponding to (7.11) and (7.12) which are different for different lattice
structures.

###  VII.2  Exact anisotropic bulk scaling form

We apply our general analysis of Secs. IV and V to systems in the $(d=2,n=1)$
universality class. Examples for these systems are Ising models on two-
dimensional Bravais lattices with short-range pair interactions described by
the Hamiltonian

$\displaystyle H^{\rm
Is}=-\sum_{i,j}E_{i,j}\sigma_{i}\cdot\sigma_{j},\;\;\sigma_{i}=\pm 1.$ (7.14)

In the following all equations are given for these examples as indicated by
the superscript $"\rm Is"$ for the nonuniversal quantities but these equations
apply to all weakly anisotropic systems of the $(d=2,n=1)$ universality class.
We assume the existence of two principal axes and two principal correlation
lengths in a range of couplings $E_{i,j}$ where the model (7.14) displays
weakly anisotropic critical behavior. The orientation of the principal axes in
the direction of the angles $\Omega^{\rm Is}$ and $\Omega^{\rm Is}+\pi/2$ is
described by the principal unit vectors

${\bf e}^{(1){\rm Is}}={\bf e}(\Omega^{\rm
Is})=\left(\begin{array}[]{c}\cos\Omega^{\rm Is}\\\ \sin\Omega^{\rm Is}\\\
\end{array}\right),$ (7.15) ${\bf e}^{(2){\rm Is}}={\bf e}(\Omega^{\rm
Is}+\pi/2)=\left(\begin{array}[]{c}-\sin\Omega^{\rm Is}\\\ \cos\Omega^{\rm
Is}\\\ \end{array}\right).\;$ (7.16)

Correspondingly the rotation matrix of the generalized shear transformation
for a clockwise rotation is

$\displaystyle{\bf{U}}^{\rm Is}={\bf{U}}(\Omega^{\rm
Is})=\left(\begin{array}[]{ccc}\cos\;\Omega^{\rm Is}&\sin\;\Omega^{\rm Is}\\\
-\sin\;\Omega^{\rm Is}&\cos\;\Omega^{\rm Is}\\\ \end{array}\right).\;\;\;\;$
(7.19)

A counterclockwise rotation provided by the matrix

$\displaystyle{\bf{U}}^{\rm Is}_{cc}={\bf{U}}_{cc}(\Omega^{\rm
Is})=\left(\begin{array}[]{ccc}\sin\;\Omega^{\rm Is}&-\cos\;\Omega^{\rm Is}\\\
\cos\;\Omega^{\rm Is}&\sin\;\Omega^{\rm Is}\\\ \end{array}\right)\;\;\;\;$
(7.22)

would lead to equivalent results. The two principal correlation lengths

$\displaystyle\xi_{\pm}^{(\alpha){\rm Is}}(t)$ $\displaystyle=$
$\displaystyle\xi^{(\alpha){\rm Is}}_{0\pm}|t|^{-1},\alpha=1,2,$ (7.23)

have the nonuniversal ratio

$q^{\rm Is}=\xi^{(1){\rm Is}}_{0+}/\xi^{(2){\rm Is}}_{0+}=\xi^{(1){\rm
Is}}_{0-}/\xi^{(2){\rm Is}}_{0-}.$ (7.24)

The mean correlation length is

$\displaystyle\bar{\xi}^{\rm Is}_{\pm}(t)$ $\displaystyle=$
$\displaystyle\bar{\xi}^{\rm Is}_{0\pm}|t|^{-1},$ (7.25)
$\displaystyle\bar{\xi}^{\rm Is}_{0\pm}$ $\displaystyle=$
$\displaystyle\big{[}\xi_{0\pm}^{(1){\rm Is}}\xi_{0\pm}^{(2){\rm
Is}}\big{]}^{1/2},$ (7.26)

which can be used to express the principal correlation lengths as

$\xi^{(1){\rm Is}}_{0\pm}=\bar{\xi}^{\rm Is}_{0\pm}(q^{\rm
Is})^{1/2},\;\;\;\xi^{(2){\rm Is}}_{0\pm}=\bar{\xi}^{\rm Is}_{0\pm}(q^{\rm
Is})^{-1/2}.$ (7.27)

The reduced rescaling matrix of the generalized shear transformation reads

$\displaystyle{\mbox{\boldmath$\bar{\lambda}$}^{\rm
Is}}=\left(\begin{array}[]{ccc}\bar{\lambda}^{\rm Is}_{1}&0\\\
0&\;\bar{\lambda}^{\rm Is}_{2}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}q^{\rm Is}&0\\\ 0&\;(q^{\rm
Is})^{-1}\\\ \end{array}\right)\;\;\;\;\;\;\;$ (7.32)

with the diagonal elements

$\bar{\lambda}^{\rm Is}_{1}=[\xi^{(1){\rm Is}}_{0\pm}/\bar{\xi}^{\rm
Is}_{0\pm}]^{2}=q^{\rm Is},\;\;\;\;\bar{\lambda}^{\rm Is}_{2}=[\xi^{(2){\rm
Is}}_{0\pm}/\bar{\xi}^{\rm Is}_{0\pm}]^{2}=(q^{\rm Is})^{-1}\;.$ (7.33)

The exact scaling form of the anisotropic correlation function of the model
(7.14) is obtained from the general result (5.42) as

$\displaystyle G^{\rm Is}({\bf x},t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma^{\rm Is}_{+}\;(\bar{\xi}^{\rm
Is}_{0+})^{-7/4}}{\big{[}{\bf x}\cdot\big{(}{\bf\bar{A}}^{\rm
Is}\big{)}^{-1}{\bf x}\big{]}^{1/8}}\;\Psi_{\pm}\Big{(}\frac{[{\bf
x}\cdot\big{(}{\bf\bar{A}}^{\rm Is}\big{)}^{-1}{\bf x}]^{1/2}}{\bar{\xi}^{\rm
Is}_{\pm}(t)}\Big{)}\;\;\;\;\;\;\;\;\;$ (7.34)

with the same universal scaling function $\Psi_{\pm}$ as in (7.8). According
to (4.96)-(4.98) the reduced anisotropy matrix is given by

$\displaystyle{\bf\bar{A}}^{\rm Is}(q^{\rm Is},\Omega^{\rm Is})$
$\displaystyle=$ $\displaystyle({\bf U}^{\rm
Is})^{-1}{\bf\bar{\mbox{\boldmath$\lambda$}}}^{\rm Is}{\bf U}^{\rm Is}$ (7.35)
$\displaystyle=$ $\displaystyle{\bf\bar{A}}(q^{\rm Is},\Omega^{\rm Is})$
(7.36)

where ${\bf\bar{A}}(q,\Omega)$ has a universal structure given by

$\displaystyle{\bf\bar{A}}(q,\Omega)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}q\;c_{\Omega}^{2}+q^{-1}s_{\Omega}^{2}&\;\;\;(q-q^{-1})\;c_{\Omega}\;s_{\Omega}\\\
(q-q^{-1})\;c_{\Omega}\;s_{\Omega}&q\;s_{\Omega}^{2}+q^{-1}\;c_{\Omega}^{2}\end{array}\right)$
(7.39)

with the abbreviations
$c_{\Omega}\equiv\cos\Omega,s_{\Omega}\equiv\sin\Omega$. The result
(7.34)-(7.39) has the same form as derived recently dohm2019  by means of the
special shear transformation (4.1)-(4.3) for the two-dimensional anisotropic
scalar $\varphi^{4}$ model (2.1). The universality of $\Psi_{\pm}$ and of the
structure of ${\bf\bar{A}}(q,\Omega)$ confirms explicitly the validity of
multiparameter universality for the bulk correlation functions of all weakly
anisotropic systems in the $(d=2,n=1)$ universality class.

We add the following comments. Two-scale-factor universality is violated as
$G^{\rm Is}({\bf x},t)$, (7.34), depends on the four independent nonuniversal
parameters $\Gamma^{\rm Is}_{+},\bar{\xi}^{\rm Is}_{0+},q^{\rm Is},\Omega^{\rm
Is}$. The angle $\Omega^{\rm Is}$ of the principal axes of Ising models
depends in an unknown way on the microscopic interactions $E_{i,j}$ and needs
to be determined for each special Ising model under consideration. This is in
contrast to the known dependence dohm2019  of the corresponding angle $\Omega$
of the anisotropic $d=2$ $\varphi^{4}$ model (2.1) on the couplings $K_{i,j}$
through the matrix elements of the anisotropy matrix ${\bf A}$ (2.3). A
corresponding anisotropy matrix for the Ising model is unknown. Likewise a
general condition for weak anisotropy for the Ising model (7.14) analogous to
(2.4) for the $\varphi^{4}$ model is unknown. The significance of our general
result (7.34)-(7.39) is that the universal validity of the structure of the
correlation function (7.34) and of the reduced anisotropy matrix
${\bf\bar{A}}^{\rm Is}(q^{\rm Is},\Omega^{\rm Is})$ no longer rests upon exact
calculations within special models on special lattices (such as square,
triangular, or honeycomb lattices) or upon the hypothesis of multiparameter
universality but is a proven fact that applies to all two-dimensional weakly
anisotropic systems of the $(d=2,n=1)$ universality class. This implies that
the calculation of the correlation function of any two-dimensional weakly
anisotropic system of the Ising universality class no longer requires a new
calculation of a scaling function and of an anisotropy matrix but can be
restricted to the much simpler task of determining four nonuniversal
parameters, namely the thermodynamic amplitude $\Gamma^{\rm Is}_{+}$ of the
susceptibility above $T_{c}$, the two principal correlation lengths, and the
angle $\Omega^{\rm Is}$ of the principal axes. This conclusion constitutes a
fundamental simplification in the analytic theory of two-dimensional
anisotropic systems as well as in the analysis of numerical or experimental
data of anisotropic correlation functions.

We substantiate our findings by the correlation function of the anisotropic
triangular-lattice Ising model stephenson ; WuCoy ; Vaidya1976  for which the
exact universal and nonuniversal properties have been identified in most cases
dohm2019  from the explicit exact results that were given in WuCoy ;
Vaidya1976  as a function of the couplings $E_{i}$. The Hamiltonian of this
model reads

$H^{\rm
tr}=\sum_{j,k}[-E_{1}\sigma_{j,k}\sigma_{j,k+1}-E_{2}\sigma_{j,k}\sigma_{j+1,k}-E_{3}\sigma_{j,k}\sigma_{j+1,k+1}]$
(7.41)

with horizontal, vertical, and diagonal couplings $E_{1},E_{2},E_{3}$ on a
square lattice (see Fig. 2 of dohm2019 ). The condition for a ferromagnetic
critical point with weak anisotropy is Berker

$E_{1}+E_{2}>0,E_{1}+E_{3}>0,E_{2}+E_{3}>0.$ (7.42)

In this range of the couplings the applicability of our proof of
multiparameter universality is guaranteed because of the existence of the
principal axes and principal correlation lengths, as shown in dohm2019 .
Specifically, (i) the exact form of ${\bf\bar{A}}^{\rm tr}(q^{\rm
tr},\Omega^{\rm tr})$ was identified in dohm2019  in terms of
$\hat{S}_{i}=\sinh 2\beta_{c}^{\rm tr}E_{i}$ for general $E_{i}$, $i=1,2,3$,
(ii) the existence of the two principal axes was verified by the exact
determination of the angle $\Omega^{\rm Is}$ from the two extrema of the
angular-dependent correlation length, (iii) the exact ratio of $q^{\rm tr}$ of
the principal correlation lengths was determined in dohm2019  in terms of
$\hat{S}_{i}$ for general $E_{i}$, and (iv) the existence of the mean
correlation length $\bar{\xi}^{\rm tr}_{\pm}(t)$ follows from its
identification through Eqs. (52)-(54) of dohm2019  in terms of the exact
”scaled variable” $t^{\rm Vai}$ of Vaidya Vaidya1976  (denoted by $t$ in Eq.
(10) of Vaidya1976 ). The determination of $\bar{\xi}^{\rm tr}_{\pm}(t)$ of
the Ising model in the full range of (7.42) can be obtained by expanding the
scaled variable in Eq. (10) of Vaidya1976  around $T^{\rm tr}_{c}$ to leading
order in $|t|=|T-T^{\rm tr}_{c}|/T^{\rm tr}_{c}$, as noted in dohm2019 . More
explicitly, the scaled variable of Vaidya is identified according to Eq. (53)
of dohm2019  for small $0<t\ll 1$ as

$t^{\rm Vai}=\frac{R^{\rm tr}(t)}{\bar{\xi}^{\rm
tr}_{+}(t)}\longrightarrow\frac{R^{\rm tr}(0)}{\bar{\xi}^{\rm tr}_{0+}}t$
(7.43)

where the distance variable $R^{\rm tr}(t)$ in Eq. (11) of Vaidya1976  has a
finite limit $R^{\rm tr}(0)=\lim_{t\to 0}R^{\rm tr}(t)$. We have verified that
the mean correlation length $\bar{\xi}^{\rm tr}_{0+}$ is determined uniquely
as a function of $E_{i}$ from (7.43) and Eq. (10) of Vaidya1976  for general
$E_{i}$. This proves the existence of the two principal correlation lengths
$\xi^{(\alpha){\rm tr}}_{0+}$ according to (7.27) via the combination of
$(q^{\rm tr})^{1/2}$, $(q^{\rm tr})^{-1/2}$, and $\bar{\xi}^{\rm tr}_{0+}$ for
general $E_{i}$. The amplitudes below $T_{c}$ follow from $\bar{\xi}^{\rm
tr}_{0+}/\bar{\xi}^{\rm tr}_{0-}=2$. The expressions of $\bar{\xi}^{\rm
tr}_{0+}$ and of $\xi^{(\alpha){\rm tr}}_{0+}$ in terms of $\hat{S}_{i}$ are
derived in DKW2023 . In retrospect, the verification of multiparameter
universality of the structure of the correlation function of the special Ising
model (7.41) achieved in dohm2019  was to be expected in view of the general
proof of the present paper.

##  VIII Angular-dependent correlation vector

In dohm2019  the notion of an angular-dependent correlation length was
introduced. In the following we further develop this notion by introducing the
vector of the angular-dependent correlation length. Since such vectors exists
in all weakly $d$-dimensional anisotropic systems including $\varphi^{4}$,
Gaussian, and fixed-length spin models we suppress the superscripts $"{\rm
Is}"$, $"{\rm sp}"$, and ”G” in the following. Here we confine ourselves to
two dimensions. It is of interest to introduce a correlation length as a
measure of the spatial range of the critical correlations in a certain spatial
direction with an angle $\theta$ described by a unit vector

$\displaystyle{\bf e}(\theta)=\left(\begin{array}[]{c}\cos\theta\\\
\sin\theta\\\ \end{array}\right).\;$ (8.3)

This can be done by introducing polar coordinates ${\bf x}=r\;{\bf e}(\theta)$
for the argument ${\bf x}$ of the anisotropic bulk correlation function
$G({\bf x},t)$. As expected on physical grounds such a correlation length
should not depend on the absolute value of $\theta$ but on the angle
$\theta-\Omega$ relative to $\Omega$ describing the direction of the principal
correlation lengths. This is indeed the case. By rewriting the argument of the
scaling function $\Psi_{\pm}$ in (7.34) as

$\displaystyle\frac{[{\bf x}\cdot{\bf\bar{A}}(q,\Omega)^{-1}{\bf
x}]^{1/2}}{\bar{\xi}_{\pm}(t)}=\frac{r}{\xi_{\pm}(t,\theta-\Omega,q)}$ (8.4)

we obtain from (7.39) the angular-dependent correlation length

$\displaystyle\xi_{\pm}(t,\theta-\Omega,q)=\xi_{0\pm}(\theta-\Omega,q)|t|^{-1},$
(8.5)
$\displaystyle\xi_{0\pm}(\theta-\Omega,q)=\frac{\bar{\xi}_{0\pm}}{f(\theta-\Omega,q)},$
(8.6) $\displaystyle
f(\theta-\Omega,q)=[q\sin^{2}(\theta-\Omega)+q^{-1}\cos^{2}(\theta-\Omega)]^{1/2}.\;\;\;\;\;\;\;\;\;\;\;\;$
(8.7)

Similarly the prefactor in (7.34) can be rewritten in this form. This yields
the alternative representation of the correlation function in polar
coordinates

$\displaystyle G({\bf x},t)=$
$\displaystyle\frac{\Gamma_{+}\big{(}\bar{\xi}_{0+}\big{)}^{-7/4}}{[rf(\theta-\Omega,q)]^{1/4}}\;\Psi_{\pm}\Big{(}\frac{rf(\theta-\Omega,q)}{\bar{\xi}_{\pm}(t)}\Big{)}=$
$\displaystyle\frac{\Gamma_{+}}{\big{(}\bar{\xi}_{0+}\big{)}^{2}}\Bigg{(}\frac{\xi_{0\pm}(\theta-\Omega,q)}{r}\Bigg{)}^{1/4}\;\Psi_{\pm}\Big{(}\frac{r}{\xi_{\pm}(t,\theta-\Omega,q)}\Big{)}.\;\;\;\;\;\;\;\;\;\;$
(8.8)

The exact angular dependence described by the function $f(\theta-\Omega,q)$
was first found within the $\varphi^{4}$ model for general $K_{i,j}$ and the
triangular-lattice Ising model (7.41) in terms of $\hat{S}_{i}$ for general
$E_{i}$ dohm2019 . Because of the universality of the structure of the reduced
anisotropy matrix ${\bf\bar{A}}$, the function $f(\theta-\Omega,q)$ describes
a universal structure of the $(\theta-\Omega)$-dependence for all weakly
anisotropic two-dimensional systems including $\varphi^{4}$, Gaussian, and
Ising models. The usefulness of this correlation length is its property of
having two extrema with respect to $\theta$ at the angles

$\theta^{(1)}=\Omega,\;\theta^{(2)}=\Omega+\pi/2$ (8.9)

which determine the two principal directions, i. e.,

$\displaystyle\xi_{0\pm}(0,q)$ $\displaystyle=$
$\displaystyle\bar{\xi}_{0\pm}q^{1/2}=\xi_{0\pm}^{(1)},$ (8.10)
$\displaystyle\xi_{0\pm}(\pi/2,q)$ $\displaystyle=$
$\displaystyle\bar{\xi}_{0\pm}q^{-1/2}=\xi_{0\pm}^{(2)}.$ (8.11)

Here we extend the definition of the amplitude $\xi_{0\pm}(\theta-\Omega,q)$
to a definition of an angular-dependent correlation vector for two-dimensional
weakly anisotropic systems

$\displaystyle{\mbox{\boldmath$\xi$}}_{\pm}(t,\theta,\Omega,q)$
$\displaystyle=$
$\displaystyle{\mbox{\boldmath$\xi$}}_{0\pm}(\theta,\Omega,q)|t|^{-\nu},$
(8.12) $\displaystyle{\mbox{\boldmath$\xi$}}_{0\pm}(\theta,\Omega,q)$
$\displaystyle=$ $\displaystyle\xi_{0\pm}(\theta-\Omega,q)\;{\bf e}(\theta)$
(8.13)

that is oriented in the direction of the angle $\theta$. This is a
generalization of the principal correlation vectors (4.60) and (4.81) which
are obtained from (8.6)-(8.13) for $\theta=\theta^{(1)}$ and
$\theta=\theta^{(2)}$. It can be verified that the generalized shear
transformation of Sec. IV. C with ${\bf\widehat{U}}(\Omega)={\bf U}(\Omega)$
defined in (7.19) indeed transforms the angular-dependent correlation vector
(8.13) to a vector with a rescaled isotropic length $\xi^{\rm iso}_{0\pm}$,

$\displaystyle{\mbox{\boldmath$\widehat{\lambda}$}}^{-1/2}$
$\displaystyle{\bf\widehat{U}}(\Omega)\;{\mbox{\boldmath$\xi$}}_{0\pm}(\theta,\Omega,q)=\xi^{\rm
iso}_{0\pm}\;{\bf e}(\theta-\Omega,q)$ (8.14)

with a unit vector ${\bf e}(\theta-\Omega,q)$ whose orientation depends on the
relative angle $\theta-\Omega$,

$\displaystyle{\bf
e}(\theta-\Omega,q)=\frac{1}{\big{[}1+q^{2}\tan^{2}(\theta-\Omega)\big{]}^{1/2}}\left(\begin{array}[]{c}1\\\
q\tan(\theta-\Omega)\end{array}\right),$ (8.17) $|{\bf
e}(\theta-\Omega,q)|=1.$ (8.19)

This agrees with the transformation of the principal correlation vectors
discussed in Sec. IV. An analogous result is obtained within the $\varphi^{4}$
and Gaussian models if the special shear transformation (4.1) is applied to
${\mbox{\boldmath$\xi$}}_{0\pm}(\theta,\Omega,q)$ where $\xi^{\rm iso}_{0\pm}$
is replaced by $\xi^{\prime}_{0\pm}$ (or $\xi_{0+}^{\prime G}$, respectively),
i.e.,

$\displaystyle{\mbox{\boldmath$\lambda$}}^{-1/2}$ $\displaystyle{\bf
U}(\Omega)\;{\mbox{\boldmath$\xi$}}_{0\pm}(\theta,\Omega,q)=\xi^{\prime}_{0\pm}\;{\bf
e}(\theta-\Omega,q)$ (8.20)

with the same unit vector ${\bf e}(\theta-\Omega,q)$. The same angular-
dependent representation can be employed in the shear transformation applied
to any lattice point ${\bf x}$ of the anisotropic system. For the generalized
shear transformation (4.89) this yields the transformed vector

$\displaystyle{\bf\widehat{x}}=$
$\displaystyle{\mbox{\boldmath$\widehat{\lambda}$}}^{-1/2}$
$\displaystyle{\bf\widehat{U}}(\Omega)\;{\bf x}=\frac{\xi^{\rm
iso}_{0\pm}}{\xi_{0\pm}(\theta-\Omega,q)}\;|{\bf x}|\;{\bf
e}(\theta-\Omega,q)\;\;\;\;\;\;\;\;\;\;\;\;$ (8.21) $\displaystyle=$
$\displaystyle\frac{\xi^{\rm iso}_{0\pm}}{\bar{\xi}_{0\pm}}\;|{\bf
x}|\;f(\theta-\Omega,q)\;{\bf e}(\theta,q,\Omega)$ (8.22)

where the factor $\xi^{\rm iso}_{0\pm}/\xi_{0\pm}(\theta-\Omega,q)$ describes
the amount of rescaling in the direction of $\theta$. For the special shear
transformation (4.1) of the $\varphi^{4}$ and Gaussian models the
corresponding representation of the vector ${\bf x^{\prime}}$ reads

$\displaystyle{\bf x^{\prime}}=$
$\displaystyle{\mbox{\boldmath$\lambda$}}^{-1/2}$ $\displaystyle{\bf
U}(\Omega)\;{\bf
x}=\frac{\xi^{\prime}_{0\pm}}{\xi_{0\pm}(\theta-\Omega,q)}\;|{\bf x}|\;{\bf
e}(\theta-\Omega,q)\;\;\;\;\;\;\;\;\;\;\;\;$ (8.23) $\displaystyle=$
$\displaystyle\frac{\xi^{\prime}_{0\pm}}{\bar{\xi}_{0\pm}}\;|{\bf
x}|\;f(\theta-\Omega,q)\;{\bf e}(\theta-\Omega,q)$ (8.24)

where the rescaling factor is
$\xi^{\prime}_{0\pm}/\xi_{0\pm}(\theta-\Omega,q)$ (or $\xi^{\prime
G}_{0\pm}/\xi^{G}_{0\pm}(\theta-\Omega,q)$ for the Gaussian model).

Although the function $f(\theta-\Omega,q)$ has a universal form describing the
angular dependence of the correlation length of all weakly anisotropic two-
dimensional systems one should keep in mind that it contains a substantial
source of an intrinsic diversity through the dependence on the nonuniversal
angle $\Omega$ and the ratio $q$. Both $\Omega$ and $q$ depend on all
microscopic details. Within the $\varphi^{4}$ theory $q$ and $\Omega$ are well
defined via the eigenvalues and the orientation of the eigenvectors of the
anisotropy matrix ${\bf A}$ dohm2019  but they are unknown for, e. g., an
unlimited number of weakly anisotropic Ising models on various lattices with
short range interactions.

We are not aware of an analytic approach to determining the principal axes of
such systems. This serious lack of knowledge is a nontrivial source of
nonuniversality in the physics of weakly anisotropic systems. It causes a
directional nonuniversality of two- and three-dimensional correlation
functions dohm2019  that has not been anticipated in the traditional theory
where it was believed that weak anisotropy enters only the amplitudes of power
laws zia . As long as the principal axes and correlation lengths are unknown
for an anisotropic system no appropriate rotation matrix can be defined, and
not only the required amount of rescaling is unknown but also the directions
are unknown along which the rescaling should be performed. Thus, in general,
the effects of weak anisotropy cannot be simply ”transformed away” by a
”trivial rescaling”. This problem has not been adequately addressed in the
earlier literature on weakly anisotropic bulk and confined systems cardy1987 ;
cardybuch ; zia ; diehl-chamati ; diehl2010 . In particular in confined
systems these anisotropy effects may become unexpectedly complex DW2021 .

The notion of an angular-dependent representation of correlation vectors and
lattice points is applicable to both bulk and confined systems with arbitrary
boundary conditions. This is of relevance to the application of the shear
transformation to the boundaries of finite systems as will be further
discussed in Sec. IX. A.

##  IX Critical Casimir forces in anisotropic systems

###  IX.1 Proof of multiparameter universality of the critical Casimir
amplitude in two dimensions

Recently exact results have been derived for the critical free energy and the
ensuing critical Casimir amplitude of the anisotropic $\varphi^{4}$ model at
$T_{c}$ on a finite rectangle with periodic boundary conditions DW2021 .
Surprisingly complex finite-size effects were found near the instability where
weak anisotropy breaks down. These exact results were based on conformal field
theory franc1997  and the principle of two-scale-factor universality priv ;
pri  for finite isotropic systems combined with the special shear
transformation (4.1)-(4.3) of the anisotropic $\varphi^{4}$ model on a
rectangle to an isotropic $\varphi^{4}$ model on a parallelogram.
Corresponding predictions were presented for the finite anisotropic
triangular-lattice Ising model Vaidya1976 ; dohm2019  on the basis of the
assumption that multiparameter universality dohm2018  is valid for this model
but no proof was given. A quantitative test was performed DWKS2021  by high-
precision Monte Carlo simulations for a special Ising model on a square with a
diagonal anisotropic coupling and remarkable agreement was found with the
predicted critical amplitude ${\cal F}_{c}^{\rm Is}$ of the free energy at
$T_{c}$.

It was noted DW2021  that the assumption of multiparameter universality for
the anisotropic triangular-lattice Ising model is equivalent to an ”effective
shear transformation” between the isotropic Ising model on a parallelogram and
the anisotropic Ising model on a rectangle but this effective shear
transformation was not specified. Here we shall identify this effective shear
transformation together with a proof for the validity of multiparameter
universality for critical free energy ${\cal F}_{c}^{\rm Is}$ of the
anisotropic Ising model. Our proof is based on (a) the generalized shear
transformations (4.89) and (8.21) applied to the boundaries of the Ising
model, (b) the invariance of the critical free energy under this shear
transformation as discussed in Sec. II. B, (c) the inversion of this shear
transformation. The strategy is to perform the generalized shear
transformation from the anisotropic rectangle to an isotropic parallelogram,
to take the exact critical free energy on the parallelogram from isotropic CFT
franc1997 , and to determine the critical free energy on the anisotropic
rectangle by means of inverting the shear transformation. This can be done
without recourse to the $\varphi^{4}$ model and no assumptions are needed
other than the existence of weakly anisotropic behavior, i.e., the existence
of principal axes with the angle $\Omega^{\rm Is}$ and the ratio $q^{\rm Is}$
of the principal correlation lengths. We recall that both $\Omega^{\rm Is}$
and $q^{\rm Is}$ are known exactly for the triangular-lattice Ising model
(7.41) dohm2019 .

We assume a finite $L_{\parallel}\times L_{\perp}$ rectangle spanned by the
vectors ${\bf L}_{\parallel}=L_{\parallel}\;(1,0)$ and ${\bf
L}_{\perp}=L_{\perp}\;(0,1)$ in the horizontal and vertical directions with
the aspect ratio

$\rho=L_{\perp}/L_{\parallel}.$ (9.1)

We apply our generalized shear transformation (4.89) directly to the
anisotropic Ising model on the rectangle in order to obtain an isotropic Ising
model on a parallelogram. The transformation applied to the boundaries of the
rectangle reads

$\displaystyle{\bf L}_{p\parallel}$ $\displaystyle=$
$\displaystyle({\mbox{\boldmath$\lambda$}}^{\rm Is})^{-1/2}{\bf U}(\Omega^{\rm
Is}){\bf L_{\parallel}},$ (9.2) $\displaystyle{\bf L}_{p\perp}$
$\displaystyle=$ $\displaystyle({\mbox{\boldmath$\lambda$}}^{\rm
Is})^{-1/2}{\bf U}(\Omega^{\rm Is}){\bf L}_{\perp},$ (9.3)

where ${\bf U}(\Omega^{\rm Is})$ is given by (7.19) and

$\displaystyle{\mbox{\boldmath$\lambda$}^{\rm
Is}}=\left(\begin{array}[]{ccc}\lambda^{\rm Is}_{1}&0\\\ 0&\;\lambda^{\rm
Is}_{2}\\\ \end{array}\right)$ (9.6)

with the diagonal elements

$\lambda^{\rm Is}_{1}=[\xi^{{\rm Is}(1)}_{0\pm}/\xi^{\rm
iso}_{0\pm}]^{2},\;\;\;\;\lambda^{\rm Is}_{2}=[\xi^{{\rm
Is}(2)}_{0\pm}/\xi^{\rm iso}_{0\pm}]^{2}\;,$ (9.7)

(compare (4.88)). This generates an isotropic Ising model with the bulk
correlation-length amplitude $\xi^{\rm iso}_{0\pm}$ on a parallelogram spanned
by the vectors ${\bf L}_{p\parallel}$ and ${\bf L}_{p\perp}$. From the general
transformation formulae (8.21), (8.6), (8.7), and (8.17) we obtain for
$\theta=0$ and $\theta=\pi/2$

$\displaystyle{\bf L}_{p\parallel}$ $\displaystyle=$ $\displaystyle
L_{\parallel}\;\frac{\xi^{\rm iso}_{0\pm}/\xi^{\rm Is}_{0\pm}(-\Omega^{\rm
Is},q^{\rm Is})}{\big{[}1+(q^{\rm Is})^{2}\tan^{2}\Omega^{\rm
Is}\big{]}^{1/2}}\left(\begin{array}[]{c}1\\\ -q^{\rm Is}\tan\Omega^{\rm
Is}\end{array}\right),\;\;\;\;\;\;\;\;$ (9.10) $\displaystyle{\bf L}_{p\perp}$
$\displaystyle=$ $\displaystyle L_{\perp}\;\frac{\xi^{\rm iso}_{0\pm}/\xi^{\rm
Is}_{0\pm}(\pi/2-\Omega^{\rm Is},q^{\rm Is})}{\big{[}1+(q^{\rm
Is})^{2}\cot^{2}\Omega^{\rm Is}\big{]}^{1/2}}\left(\begin{array}[]{c}1\\\
q^{\rm Is}\cot\Omega^{\rm Is}\end{array}\right),\;\;\;\;\;\;$ (9.13)

with

$\displaystyle|{\bf L}_{p\parallel}|$ $\displaystyle=$ $\displaystyle
L_{\parallel}\;\xi^{\rm iso}_{0\pm}/\xi^{\rm Is}_{0\pm}(-\Omega^{\rm
Is},q^{\rm Is}),$ (9.14) $\displaystyle|{\bf L}_{p\perp}|$ $\displaystyle=$
$\displaystyle L_{\perp}\;\xi^{\rm iso}_{0\pm}/\xi^{\rm
Is}_{0\pm}(\pi/2-\Omega^{\rm Is},q^{\rm Is}),$ (9.15)

where we have used $\tan(\pi/2-\Omega)=\cot\Omega$. We see that the ratio
$\xi^{\rm iso}_{0\pm}/\xi^{\rm Is}_{0\pm}(\theta-\Omega^{\rm Is},q^{\rm Is}))$
determines the rescaling of the lengths in the directions $\theta=0$ and
$\theta=\pi/2$. The parallelogram is characterized by the transformed aspect
ratio

$\rho_{\rm p}=|{\bf L}_{p\perp}|/|{\bf L}_{p\parallel}|$ (9.16)

and by the angle $\alpha$ between the vectors ${\bf L}_{p\perp}$ and ${\bf
L}_{p\parallel}$ (Fig.2 of DW2021 ). This angle is related to ${\bf
L}_{p\perp}$ and ${\bf L}_{p\parallel}$ by

$\cos\alpha=\frac{{\bf L}_{p\perp}\cdot{\bf L}_{p\parallel}}{|{\bf
L}_{p\perp}||{\bf L}_{p\parallel}|}.$ (9.17)

Substituting (9.10) and (9.13) into (9.16) and (9.17) and using (8.6) and
(8.7) we obtain the transformed aspect ratio

$\displaystyle\rho_{\rm{p}}(\rho,q^{\rm{Is}},\Omega^{\rm{Is}})$
$\displaystyle=$
$\displaystyle\frac{L_{\perp}/\xi^{\rm{Is}}_{0+}(\pi/2-\Omega^{\rm{Is}},q^{\rm{Is}})}{L_{\parallel}/\xi^{\rm{Is}}_{0+}(-\Omega^{\rm{Is}},q^{\rm{Is}})}$
(9.18) $\displaystyle=$ $\displaystyle\rho\;\Bigg{[}\frac{\tan^{2}\Omega^{\rm
Is}+(q^{\rm Is})^{2}}{1+(q^{\rm Is})^{2}\tan^{2}\Omega^{\rm
Is}}\Bigg{]}^{1/2}\;\;\;$ (9.19) $\displaystyle=$
$\displaystyle\rho\;\Bigg{[}\frac{{\bf\bar{A}}(q^{\rm Is},\Omega^{\rm
Is})_{11}}{{\bf\bar{A}}(q^{\rm Is},\Omega^{\rm Is})_{22}}\Bigg{]}^{1/2}$
(9.20)

and the angle $\alpha$ determined by

$\displaystyle\cot\alpha(q^{\rm Is},\Omega^{\rm Is})$ $\displaystyle=$
$\displaystyle\frac{(q^{\rm Is})^{-1}-q^{\rm Is}\tan\Omega^{\rm
Is}\;\tan(\pi/2-\Omega^{\rm Is})}{\tan\Omega^{\rm Is}+\tan(\pi/2-\Omega^{\rm
Is})}\;\;\;\;\;\;\;\;\;\;$ (9.21) $\displaystyle=$ $\displaystyle[(q^{\rm
Is})^{-1}-q^{\rm Is}]\cos\Omega^{\rm Is}\sin\Omega^{\rm Is}$ (9.22)
$\displaystyle=$ $\displaystyle-{\bf\bar{A}}(q^{\rm Is},\Omega^{\rm Is})_{12}$
(9.23)

where the matrix elements ${\bf\bar{A}}(q^{\rm Is},\Omega^{\rm
Is})_{\alpha\beta}$ are defined in (7.35)-(7.39). We note that the free
parameter $\xi^{\rm iso}_{0\pm}$ in (9.10) and (9.13) is canceled in all
subsequent equations, as expected.

No specific properties of the weakly anisotropic Ising model were needed in
the derivation of (9.18)-(9.23), thus these relations have a universal
structure. In Eqs. (7) and (8) of DW2021  analogous formulae corresponding to
(9.19), (9.20) and (9.22), (9.23), but with $q^{\rm Is},\Omega^{\rm Is}$
replaced by $q,\Omega$, were first presented for the $\varphi^{4}$ model.
These formulae were obtained on the basis of geometric considerations as an
extended version of the derivation in the context of Fig. 2 in dohm2006  for
the $\varphi^{4}$ model (see also Eqs. (16) and (17) of dohm2006  for the
special shear transformation from an anisotropic square to an isotropic
rhombus for $\Omega=\pi/4$). These formulae for the $\varphi^{4}$ model were
then adopted in DW2021  for the Ising model by the substitution $q\to q^{\rm
Is},\Omega\to\Omega^{\rm Is}$ on the basis of the assumption that
multiparameter universality is valid for the Ising model. Here we have
provided an exact analytic derivation for this substitution directly within
the Ising model, without recourse to the $\varphi^{4}$ model, through the
general transformation formula (8.21) derived from our generalized shear
transformation (4.89). Thus this transformation between anisotropic and
isotropic Ising models specifies what was called ”effective shear
transformation” in Fig. 1 of DW2021 .

The critical free energy of the transformed isotropic Ising model on the
parallelogram is denoted by ${\cal F}^{\rm Is,iso}_{c}$. As noted in Sec. III.
D, two-scale-factor universality implies that ${\cal F}^{\rm Is,iso}_{c}$ is
universal, i.e., it has a universal dependence on the geometric parameters
$\alpha$ and $\rho_{\rm p}$ of the parallelogram. Since the critical free
energy ${\cal F}^{\rm Is}_{c}$ on the anisotropic rectangle remains invariant
under this pure coordinate transformation (see Sec. II. B) we obtain

$\displaystyle{\cal F}^{\rm Is}_{c}={\cal F}^{\rm Is,iso}_{c}(\alpha,\rho_{\rm
p}).$ (9.24)

As pointed out in DW2021  the exact result for ${\cal F}^{\rm
Is,iso}_{c}(\alpha,\rho_{\rm p})$ for periodic boundary conditions can be
taken directly from the critical free energy ${\cal F}_{c}^{\rm{CFT}}(\tau)$
of conformal field theory for the isotropic Ising model on a torus franc1997

$\displaystyle{\cal F}^{\rm Is,iso}_{c}(\alpha,\rho_{\rm p})$ $\displaystyle=$
$\displaystyle{\cal F}_{c}^{\rm{CFT}}(\tau)=-\ln Z^{\rm{CFT}}(\tau),$ (9.25)
$\displaystyle\tau(\alpha,\rho_{\rm{p}})$ $\displaystyle=$
$\displaystyle\mbox{Re}\;\tau+i\;\mbox{Im}\;\tau=\rho_{\rm{p}}\exp(i\;\alpha)\;\;\;\;\;$
(9.26)

which is characterized by the complex torus modular parameter
$\tau(\alpha,\rho_{\rm{p}})$. The $\tau$-dependence of ${\cal
F}_{c}^{\rm{CFT}}(\tau)$ is universal. The partition function
$Z^{\rm{CFT}}(\tau)$ is expressed in terms of Jacobi theta functions
$\theta_{i}(0|\tau)\equiv\theta_{i}(\tau)$ as franc1997

$Z^{\rm{CFT}}(\tau)=\big{(}{|\theta_{2}(\tau)|+|\theta_{3}(\tau)|+|\theta_{4}(\tau)|}\big{)}/\big{(}{2|\eta(\tau)|}\big{)},$
(9.27)

with
$\eta(\tau)=\Big{[}\frac{1}{2}\theta_{2}(\tau)\theta_{3}(\tau)\theta_{4}(\tau)\Big{]}^{1/3}$.
The crucial step is to transfer this exact information directly from the
isotropic Ising model to the anisotropic Ising model by means of inverting the
generalized shear transformation, without recourse to the $\varphi^{4}$ model.
This is achieved by defining the $(q^{\rm Is},\Omega^{\rm Is})$-dependent
quantity

$\tau(\rho,q^{\rm Is},\Omega^{\rm Is})=\tau\big{(}\alpha(q^{\rm
Is},\Omega^{\rm Is}),\rho_{\rm{p}}(\rho,q^{\rm Is},\Omega^{\rm Is})\big{)}$
(9.28)

with $\alpha(q^{\rm Is},\Omega^{\rm Is})$ and $\rho_{\rm{p}}(\rho,q^{\rm
Is},\Omega^{\rm Is})$ given by (9.22) and (9.19) and by substituting (9.28)
into the isotropic formula (9.25). Using (9.24) we then obtain the exact
result for the critical free energy ${\cal F}^{\rm Is}_{c}$ and the Casimir
amplitude $X^{\rm Is}_{c}$ of the anisotropic Ising model on the rectangle as

$\displaystyle{\cal F}^{\rm Is}_{c}(\rho,q^{\rm Is},\Omega^{\rm Is})$
$\displaystyle=$ $\displaystyle-\ln Z^{\rm{CFT}}\big{(}\tau(\rho,q^{\rm
Is},\Omega^{\rm Is})\big{)},$ (9.29) $\displaystyle X^{\rm Is}_{c}(\rho,q^{\rm
Is},\Omega^{\rm Is})$ $\displaystyle=$ $\displaystyle-\rho^{2}\;\partial{\cal
F}^{\rm Is}_{c}(\rho,q^{\rm Is},\Omega^{\rm Is})/\partial\rho.$ (9.30)

with two nonuniversal parameters $q^{\rm Is},\Omega^{\rm Is}$. Going from
(9.25) to (9.29) is equivalent to performing a nonuniversal inverse shear
transformation from the isotropic to the anisotropic Ising model. The
difference between (9.25) and (9.29) is that the former relation contains a
universal function of the geometric variables $\alpha$ and $\rho_{\rm p}$ in
agreement with two-scale factor universality whereas (9.29) contains a
universal function reflecting multiparameter universality with the two
nonuniversal anisotropy parameters $q^{\rm Is}$ and $\Omega^{\rm Is}$. They
are introduced via the nonuniversal anisotropy parameters contained in the
shear transformation formulae (9.18)-(9.23). The same structure was derived
for the $\varphi^{4}$ model in DW2021 . This proves the validity of the
predictions of DW2021  not only for the triangular lattice Ising model but
more generally for all other weakly anisotropic systems that belong to the
$d=2,n=1$ universality class. In particular this establishes the validity of
the self-similar structures of the critical free energy and the Casimir
amplitude discovered in DW2021  for all weakly anisotropic systems with
periodic boundary conditions in this universality class.

As noted in the context of the anisotropic bulk correlation function, there
exists a corresponding intrinsic diversity in confined anisotropic systems as
compared to confined isotropic systems: It arises from the nonuniversal
parameters $q,\Omega$ and $q^{\rm Is},\Omega^{\rm Is}$ that do not exist in
isotropic systems. Furthermore there is a basic difference between
$q(\\{K_{i,j}\\})$ and $\Omega(\\{K_{i,j}\\})$ of the $\varphi^{4}$ model
(which are known exactly as functions of $K_{i,j}$ dohm2019 ) and $q^{\rm
Is}(\\{E_{i,j}\\})$ and $\Omega^{\rm Is}(\\{E_{i,j}\\})$ of the Ising model
which are generically unknown for the general Ising model (7.14). We conclude
that weak anisotropy destroys the universality of the critical Casimir
amplitude $X_{c}$ of the subclass of isotropic systems and makes $X_{c}$ to
become an unknown quantity for general anisotropic systems whose principal
angles and correlation lengths are unknown. The triangular-lattice Ising model
Vaidya1976 ; dohm2019  is a very rare example for a system other than the
$\varphi^{4}$ model for which these parameters are known exactly as a function
of the microscopic couplings. It would be worthwhile to extend this knowledge
to other anisotropic Ising models Perk1 ; Perk2 ; Perk3 ; Perk4  whose
relevant anisotropy parameters are as yet unknown.

###  IX.2 Critical Casimir forces in anisotropic superconductors

An experimental verification of critical Casimir forces has been achieved so
far only in isotropic systems, most prominently in superfluid 4He films garcia
. It has been pointed out by Williams wil-1  that a measurable critical
Casimir force should occur also in superconducting films. Superconductors
belong to the same universality class as superfluid 4He and have the same
(Dirichlet) boundary conditions but are anisotropic. Williams’ scenario is the
following: a superconducting film is connected to a bulk sample of the same
material. He argues that below the bulk critical temperature the film-bulk-
system can lower its free energy by a transfer of electrons (Cooper pairs)
from the film to the bulk system which is analogous to helium atoms moving
from the film to the superfluid bulk reservoir. While in the helium system
this leads to a thinning of the film the corresponding effect in the
superconducting film-bulk-system is a transfer of negative electrical charge
from the film to the bulk system. Williams wil-1  argues that this gives rise
to an electrical potential difference which can be related to the free-energy
difference (per unit area) between the film and the bulk and from which a
Casimir force can be derived. He estimates the voltage difference to have a
measurable magnitude.

The critical Casimir force is an observable only if the ordering degrees of
freedom can enter and leave the system. Therefore it has been claimed in the
literature toldin ; diehl2009 ; diehl2010 ; DDreview  that this force can be
active only in isotropic fluids and that the issue of spatial anisotropy is
not relevant in the context of the critical Casimir force. We argue that,
unlike the localized degrees of freedom (magnetic moments) of the order
parameter of an anisotropic magnetic material, the ordering degrees of freedom
(Cooper pairs) of a superconductor are not localized at lattice points but
play the role of an electrical superfluid in an anisotropic environment that
can leave and enter the film connected to the bulk of the same material, as
anticipated by Williams wil-1 . So far no specific objection has been raised
in the literature against this specific argumentation for superconductors, and
in a comment comment  on wil-1  the measurability of the critical Casimir
force in superconductors has not been questioned. Furthermore, we point to the
largely unexplored area of thermodynamic Casimir forces in liquid crystals
singh  which exhibit a wide variety of spatial anisotropy and whose ordering
degrees of freedom can leave and enter the system.

In closing we note that experimental studies, Monte Carlo simulations, and
further theoretical research are called for in view of the fact that at
present no experimental or Monte Carlo data and no analytic predictions are
available for the critical Casimir force in anisotropic systems with realistic
boundary conditions. Also theoretical efforts based on the functional
renormalization group metzner2021  applied to anisotropic confined systems
could yield important contributions to this matter. Analytic results for the
critical Casimir force in anisotropic films of finite thickness have been
presented previously dohm2018  for the case of periodic boundary conditions
which refute earlier results wil-1  where no anisotropy effect in anisotropic
superconductors near $T_{c}$ was found. An analytic renormalization-group
study in three dimensions with realistic Dirichlet boundary conditions below
$T_{c}$ without adjustable parameters was performed dohm2014  that explains
the depth and position of the deep minimum of the Casimir force scaling
function observed in isotropic 4He films garcia  in the temperature regime
$T_{\rm c,film}<T<T_{c}$ on a semiquantitative level. It is conceivable that
this isotropic study can be extended to anisotropic film systems with
Dirichlet boundary conditions which could lead to quantitative predictions of
the critical Casimir force in real superconductors.

##  X Summary

We have presented a general theory of bulk critical phenomena in weakly
$d$-dimensional $O(n)$-symmetric anisotropic systems in $2\leq d<4$ dimensions
where our only assumptions are the existence of $d$ principal axes and
correlation lengths and the validity of two-scale-factor universality for
isotropic systems. Our general conclusions with regard to the validity of
multiparameter universality confirms and specifies the early ”belief in some
form of universality not only for the two-dimensional Ising model but also for
a large class of two-dimensional models with short-range interactions” WuCoy .
Our findings are supported by exact results for the $d=2,n=1$ universality
class and for the spherical and Gaussian universality classes in $d\geq 2$
dimensions. On the other hand our theory reveals a high degree of intrinsic
diversity in weakly anisotropic systems even in the asymptotic critical
region. This limitation of universality was not anticipated in the traditional
theory of critical phenomena and in the more recent development of the
functional renormalization group. Furthermore we have applied our theory to
finite-size effects at $T_{c}$ of the critical free energy and Casimir
amplitude of anisotropic systems on a rectangle with periodic boundary
conditions of the $d=2,n=1$ universality class studied recently DW2021 . Our
main results are summarized in more detail as follows.

(i) After defining the anisotropic $O(n)$-symmetric $\varphi^{4}$ and
$n$-vector models in Sec. II and summarizing several aspects of two-scale-
factor universality in Sec. III we have introduced in Sec. IV a generalized
shear transformation that provides exact relations between weakly anisotropic
systems and isotropic systems in the same universality class. We have
identified a temperature-independent universal structure of a reduced
anisotropy matrix ${\bf\bar{A}}$ that depends on the ratios of principal
correlation lengths and on the principal unit vectors describing the principal
axes. The latter quantities depend on microscopic details such as coupling
constants and the lattice structure, thus ${\bf\bar{A}}$ is a nonuniversal
quantity that does not exist in isotropic systems.

(ii) In Sec. V a proof has been presented for the validity of multiparameter
universality of the bulk order-parameter correlation function in weakly
anisotropic systems. It implies that the traditional notion of a universality
class of critical phenomena must be revised in that it must be divided into
subclasses of isotropic and weakly anisotropic systems. The latter have up to
$d(d+1)/2+1$ independent nonuniversal parameters. Only two of these parameters
can be expressed in terms of thermodynamic amplitudes whereas the remaining
$d(d+1)/2-1$ parameters enter the matrix ${\bf\bar{A}}$. We have also
determined the exact structure of the anisotropic bulk order-parameter
correlation function for $n>1$ in the Goldstone regime below $T_{c}$. Exact
anisotropic results are presented in the large-$n$ limit and for the Gaussian
model. From the nonasymptotic nonuniversal result of the functional
renormalization group hassel2007 ; sinner2008  we have identified the
universal part of the isotropic correlation function of the $n=1$ Ising
universality class in three dimensions. We have refuted the claim that an
extended universality hassel2007 ; sinner2008  is valid in the nonasymptotic
region. Our theory provides the opportunity of a quantitative comparison with
the universal parts of numerical or experimental data in two-dimensional and
three-dimensional anisotropic systems after the nonuniversal parameters have
been determined.

(iii) In Sec. VI a bulk scaling variable $\widetilde{x}$, (6.18), has been
introduced for general $n$ in $2\leq d<4$ dimensions which is invariant under
the shear transformation. It permits us to represent the singular bulk part of
the free energy of isotropic and anisotropic systems in a compact form. We
have also shown the validity of multiparameter universality of several
critical bulk amplitude relations. In particular the amplitude of the bulk
specific heat of anisotropic systems is shown to be universally related to the
mean correlation length.

(iv) In Sec. VII our theory has been applied to two dimensions. The
significance of our general results is that the universal validity of the
structure of the correlation function no longer rests upon exact calculations
within special models on special lattices or upon the hypothesis of
multiparameter universality but is a proven fact that applies to all two-
dimensional weakly anisotropic systems of the $(d=2,n=1)$ universality class.
This constitutes a fundamental simplification in the analytic theory of two-
dimensional anisotropic systems as well as in the analysis of numerical or
experimental data.

(v) In Sec. VIII the anisotropic correlation function is written in terms of
polar coordinates. An angular-dependent correlation vector is introduced for
all two-dimensional weakly anisotropic systems. Angular-dependent formulae of
the shear transformation are derived that are applicable to any lattice vector
in bulk and confined systems.

(vi) Application of these formulae to the boundaries of a finite anisotropic
rectangle in Sec. IX provides an analytic derivation of the aspect ratio and
the angle of the transformed isotropic parallelogram. Combining this result
with the exact partition function of conformal field theory of the isotropic
two-dimensional Ising model on a torus at $T_{c}$ proves the validity of the
predictions of DW2021  with regard to multiparameter universality and self-
similar structures of the critical free energy and the Casimir amplitude not
only for the triangular lattice Ising model but more generally for all weakly
anisotropic Ising models and other systems with periodic boundary conditions
that belong to the $d=2,n=1$ universality class. In particular this identifies
the previous ”effective shear transformation” DW2021  between anisotropic and
isotropic two-dimensional Ising models.

(vii) We have not made progress in developing a systematic approach to
determining the principal axes of weakly anisotropic systems other than
$\varphi^{4}$ models. These axes are of fundamental physical importance in
real systems. So far the angles describing the principal directions, e.g., of
the $n$-vector model, depend in an unknown way on the microscopic anisotropic
interactions which must be determined for each special anisotropic system
under consideration. This lack of knowledge is not of a harmless kind and
constitutes a major challenge to future research.

Significant issues of weak anisotropy were not yet addressed in this paper and
call for further research in several directions, e.g., extensions to

(a) bulk and finite-size properties of other lattice systems Perk1 ; Perk2 ;
Perk3 ; Perk4 ; CoyWu  and other models Baxter1982  with both isotropic and
weakly anisotropic interactions, to be analyzed in the framework of two-scale-
factor universality and multiparameter universality,

(b) angular-dependent representations of correlation lengths and correlation
functions in three dimensions for general $n$,

(c) other geometries beyond rectangles,

(d) effects of an ordering field,

(e) anisotropic effects near the Kosterlitz-Thouless transition of systems in
the $d=2,n=2$ universality class,

(f) finite-size effects in weakly anisotropic systems away from $T_{c}$,

(g) crossover from weak to strong anisotropy,

(h) finite-size effects in anisotropic systems in the presence of non-periodic
boundary conditions.

We consider item (h) to be most important as it is relevant for applications
to real systems such as magnetic materials and superconductors which require a
description with free or Dirichlet boundary conditions.

##  Appendix A: Relation between $Q_{3}$ and $\widetilde{Q}_{3}$

In the following the relation (3.56) between the universal constants $Q_{3}$
and $\widetilde{Q}_{3}$ is derived Kastening . The isotropic bulk correlation
functions (3.49) and (3.55) at $T_{c}$

$\displaystyle G_{c}(|{\bf x}|)$ $\displaystyle=$
$\displaystyle\frac{D_{c}}{|{\bf x}|^{d-2+\eta}}\;,\;$ (A.1)
$\displaystyle\hat{G}_{c}(|{\bf k}|)$ $\displaystyle=$
$\displaystyle\frac{\hat{D}_{c}}{|{\bf k}|^{2-\eta}}\;,$ (A.2)
$\displaystyle\frac{D_{c}}{\hat{D}_{c}}$ $\displaystyle=$
$\displaystyle\frac{\widetilde{Q}_{3}}{Q_{3}}\;,$ (A.3)

are related by the Fourier transformation

$\displaystyle G_{c}(|{\bf x}|)=\int_{\bf k}\;e^{i{\bf k}\cdot{\bf
x}}\hat{G}_{c}(|{\bf k}|)\;,$ (A.4)

where $\int_{\bf k}$ stands for $(2\pi)^{-d}\int d^{d}k$ with an infinite
cutoff. We decompose ${\bf k}={\bf k_{0}}+{\bf q}$ where ${\bf k_{0}}$ and
${\bf q}$ are parallel and perpendicular to ${\bf x}$, respectively. Then the
ratio $D_{c}/\hat{D}_{c}$ is determined by

$\displaystyle\int_{\bf k}\frac{e^{i{\bf k}\cdot{\bf x}}}{|{\bf
k}|^{2-\eta}}=\int^{\infty}_{-\infty}\frac{dk_{0}}{2\pi}e^{ik_{0}|{\bf
x}|}\int\frac{d^{d-1}q}{(2\pi)^{d-1}}\frac{1}{(k_{0}^{2}+q^{2})^{(2-\eta)/2}}$
$\displaystyle=\frac{1}{|{\bf
x}|^{d-2+\eta}}\int^{\infty}_{-\infty}\frac{dy}{2\pi}\frac{e^{iy}}{|y|^{3-d-\eta}}\int\frac{d^{d-1}q}{(2\pi)^{d-1}}\frac{1}{(1+q^{2})^{(2-\eta)/2}}$
$\displaystyle=\frac{1}{|{\bf
x}|^{d-2+\eta}}\;\frac{-2\Gamma(d-2+\eta)\;\cos[(d+\eta)\pi/2]}{2\pi}$
$\displaystyle\times\frac{2}{(4\pi)^{(d-1)/2}\Gamma[(d-1)/2]}\;\frac{\Gamma[(d-1)/2]\;\Gamma[(3-d-\eta)/2]}{2\Gamma(1-\eta/2)}$
$\displaystyle=\frac{\sin[(3-d-\eta)\pi/2]\;\Gamma[(3-d-\eta)/2]\;\Gamma(d-2+\eta)}{|{\bf
x}|^{d-2+\eta}\;\pi(4\pi)^{(d-1)/2}\;\Gamma(1-\eta/2)}$
$\displaystyle=\frac{1}{|{\bf
x}|^{d-2+\eta}}\;\frac{\Gamma(d-2+\eta)}{(4\pi)^{(d-1)/2}\;\Gamma(1-\eta/2)\;\Gamma[(d-1+\eta)/2]}$
$\displaystyle=\frac{1}{|{\bf
x}|^{d-2+\eta}}\;\frac{2^{d-2+\eta}\;\Gamma[(d-2+\eta)/2]}{(4\pi)^{d/2}\;\Gamma[(2-\eta)/2]}$
(A.5) $\displaystyle=\frac{D_{c}}{\hat{D}_{c}}\;\frac{1}{|{\bf
x}|^{d-2+\eta}}\;,$ (A.6)

which yields (3.56).

## ACKNOWLEDGMENT

I thank F. Kischel and S. Wessel for useful discussions and J. H. H. Perk for
calling attention to Refs. Perk1 ; Perk2 ; Perk3 ; Perk4 .

## References

  * (1) X.S. Chen and V. Dohm, Phys. Rev. E 70, 056136 (2004).
  * (2) V. Dohm, in Encyclopedia of Physics, edited by R.G.Lerner and G.L. Trigg (Wiley-VCH Verlag, Weinheim, 2005), Vol. 2, p. 1901.
  * (3) W. Selke and L.N. Shchur, J. Phys. A 38, L 739 (2005).
  * (4) V. Dohm, J. Phys. A 39, L 259 (2006).
  * (5) X.S. Chen and H.Y. Zhang, Int. J. Mod. Phys. B 21, 4212 (2007).
  * (6) V. Dohm, Phys. Rev. E 77, 061128 (2008).
  * (7) V. Dohm, Physik Journal 8(11), 37 (2009).
  * (8) W. Selke and L.N. Shchur, Phys. Rev. E 80, 042104 (2009).
  * (9) D. Dantchev and D. Grüneberg, Phys. Rev. E 79, 041103 (2009).
  * (10) B. Kastening and V. Dohm, Phys. Rev. E 81, 061106 (2010).
  * (11) V. Dohm, Phys. Rev. E 84, 021108 (2011).
  * (12) V. Dohm, arXiv:1708.06272.
  * (13) V. Dohm, Phys. Rev. E 97, 062128 (2018).
  * (14) V. Dohm, Phys. Rev. E 100, 050101(R) (2019).
  * (15) V. Dohm and S. Wessel, Phys. Rev. Lett. 126, 060601 (2021).
  * (16) V. Dohm, S. Wessel, B. Kalthoff, and W. Selke, J. Phys. A 54, 23LT01 (2021).
  * (17) V. Privman, A. Aharony, and P.C. Hohenberg, in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, New York, 1991), Vol. 14, p. 1.
  * (18) A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002).
  * (19) A.D. Bruce, J. Phys. C 7, 2089 (1974).
  * (20) F. Wegner, in Phase Transitions and Critical Phenomena, edited by C. Domb and M.S. Green (Academic, New York, 1976), Vol. 6, p. 7.
  * (21) A. Aharony, in Phase Transitions and Critical Phenomena, edited by C. Domb and M.S.Green (Academic, New York, 1976), Vol. 6, p. 357.
  * (22) V. Privman and M.E. Fisher, Phys. Rev. B 30, 322 (1984).
  * (23) B. Schmittmann and R. K. P. Zia, in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, New York, 1995), Vol. 17, p. 1.
  * (24) D. Stauffer, M. Ferer, and M. Wortis, Phys. Rev. Lett. 29, 345 (1972).
  * (25) A. Aharony, Phys. Rev. B 9, 2107 (1975).
  * (26) P. R. Gerber, J. Phys. A: Math. Gen. 8, 67 (1975).
  * (27) P.C. Hohenberg, A. Aharony, B.I. Halperin, and E.D. Siggia, Phys. Rev. B 13, 2986 (1976).
  * (28) V. Privman, in Finite Size Scaling and Numerical Simulation of Statistical Systems, edited by V. Privman (World Scientific, Singapore, 1990), p.1.
  * (29) V. Privman, Phys. Rev. B 38, 9261 (1988).
  * (30) M. Henkel, in Finite Size Scaling and Numerical Simulation of Statistical Systems, edited by V. Privman (World Scientific, Singapore, 1990), p. 353.
  * (31) J. L. Cardy, J. Phys. A 16, 3617 (1983).
  * (32) M.N. Barber, I. Peschel, ans P.A. Pearce, J. Stat. Phys. 37, 497 (1984).
  * (33) J. O. Indekeu, M.P. Nightingale, and W. V. Wang, Phys. Rev. B 34, 330 (1986).
  * (34) J. L. Cardy, in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, New York, 1987), Vol. 11, p. 55.
  * (35) K. Binder and J.S. Wang, J. Stat. Phys. 55, 87 (1989).
  * (36) M. P. Nightingale, in Finite Size Scaling and Numerical Simulation of Statistical Systems, edited by V. Privman (World Scientific, Singapore (1990), p. 287.
  * (37) J. L. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press,1996).
  * (38) T.T. Wu, B.M. McCoy, C.A. Tracy, and E. Barouch, Phys. Rev. B 13, 316 (1976).
  * (39) H.G. Vaidya, Phys. Lett. A 57, 1 (1976).
  * (40) A. Aharony and M.E. Fisher, Phys. Rev. Lett. 45, 679 (1980).
  * (41) H. Au-Yang and J.H.H. Perk, Physica A 144, 44 (1987).
  * (42) H. Au-Yang and J.H.H. Perk, in: MathPhys Odyssey 2001: Integrable Models and Beyond, edited by M. Kashiwara and T. Miwa, Birkäuser, Boston, 2002), p. 23.
  * (43) H. Au-Yang and J.H.H. Perk, Physica A 321, 81 (2003).
  * (44) J. H. H. Perk and H. Au-Yang, J. Phys.: Conf. Ser Vol. 42, 231 (2006).
  * (45) B.M. McCoy and T.T. Wu, The Two-dimensional Ising Model (second edition, Dover, 2014) and references therein.
  * (46) P. Nightingale and H. Blöte, J. Phys. A 16, L 657 (1983).
  * (47) D. Kim and P.A. Pearce, J. Phys. A 20, L 451 (1987).
  * (48) J.L. Cardy, in Finite-size scaling, edited by J.L. Cardy (North Holland, Amsterdam, 1988), p. 325.
  * (49) M.A. Yurishchev, Phys. Rev. B 50, 13533 (1994).
  * (50) C. Nash and D. O’Connor, Ann. Phys. 273, 72 (1999).
  * (51) N. Sh. Izmailian, J. Phys. A 45, 494009 (2012).
  * (52) H. Hobrecht and A. Hucht, SciPost Phys. 7, 026 (2019).
  * (53) J. Berges, N. Tetradis, and C. Wetterich, Phys. Rep. 363, 323 (2002).
  * (54) N. Hasselmann, A. Sinner, and P. Kopietz, Phys. Rev. E 76, 040101(R) (2007).
  * (55) A. Sinner, N. Hasselmann, and P. Kopietz, J. Phys.: Conden. Matter 20, 075208 (2008).
  * (56) A. Rançon, L.-P. Henry, F. Rose, D.L. Cardozo, N. Dupuis, P.C.W. Holdsworth, and T. Roscilde, Phys. Rev. B 94, 140506 (R) (2016).
  * (57) N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J.M. Pawlowski, M. Tissier, N. Wschebor, Phys. Rep. 910, 1 (2021).
  * (58) I. Baloq, A. Rançon, and B. Delamotte, Phys. Rev. Lett. 129, 210602 (2022).
  * (59) T. Machado and N. Dupuis, Phys. Rev. E 82, 041128 (2010); N. Dupuis and K. Sengupta, Eur. Phys. J. B 66, 271 (2008).
  * (60) H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986).
  * (61) M. Krech, The Casimir Effect in Critical Systems World Scientific, Singapore (1994).
  * (62) F. P. Toldin, M. Tröndle, and S. Dietrich, Phys. Rev. E 88, 052110 (2013).
  * (63) H.W. Diehl, D. Grüneberg, M. Hasenbusch, A. Hucht, S.B. Rutkevich, and F. Schmidt, Phys Rev. E 89, 062123 (2014).
  * (64) D.M. Dantchev and S. Dietrich, Phys. Rep. 1005, 1 (2023), and references therein.
  * (65) V. Dohm, Z. Phys. B 60, 61 (1985); B 61, 193 (1985).
  * (66) R. Schloms and V. Dohm, Nucl. Phys. B 328, 639 (1989); R. Schloms and V. Dohm, Phys. Rev. B 42, 6142 (1990).
  * (67) P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer-Verlag New York, 1997); P. Di Francesco, H. Saleur, and J. B. Zuber, Nucl. Phys. B 290, 527 (1987).
  * (68) T. H. Berlin and M. Kac, Phys. Rev 86, 821 (1952).
  * (69) H. E. Stanley, Phys. Rev. Lett. 20, 589 (1968); Introduction to phase transitions and critical phenomena (Oxford University Press, Oxford 1971).
  * (70) G. S. Joyce, in Phase Transitions and Critical Phenomena, edited by C. Domb and M.S.Green (Academic, London, 1976), Vol. 2, p. 375.
  * (71) R. J. Baxter, Exactly solved models in Statistical Mechanics (Academic, London, 1982).
  * (72) J.G. Brankov, D.M. Dantchev, and N.S. Tonchev, Theory of Critical Phenomena in Finite-Size Systems: Scaling and Quantum Effects (World Scientific, Singapore, 2000).
  * (73) J. Stephenson, J. Math. Phys. 5, 1009 (1964).
  * (74) B.I. Halperin and P.C. Hohenberg, Phys. Rev. 177, 952 (1968).
  * (75) P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
  * (76) M.E. Fisher and A. Aharony, Phys. Rev. B 10, 2818 (1974).
  * (77) H.B. Tarko and M.E. Fisher, Phys. Rev. B 11, 1217 (1975).
  * (78) X.S. Chen and V. Dohm, Eur. Phys. J. B 10, 687 (2000).
  * (79) X.S. Chen and V. Dohm, Eur. Phys. J. B 15, 283 (2000).
  * (80) D. Dantchev, Eur. Phys. J. B 23, 211 (2001).
  * (81) E. Brézin, J.C. Le Guillou, and J. Zinn-Justin, Phys. Lett. 47A, 285 (1974).
  * (82) M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974).
  * (83) E. Brézin, J.C. Le Guillou, J. Zinn-Justin, in Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, (Academic, New York, 1976), Vol. 6, p. 125.
  * (84) J. Zinn-Justin, Phase Transitions and Renormalization Group (Oxford University Press, 2007).
  * (85) H.W. Diehl and H. Chamati, Phys. Rev. B 79, 104301 (2009).
  * (86) M. Burgsmüller, H.W. Diehl, and M.A. Shpot, J. Stat. Mech. P11020 (2010).
  * (87) X.S. Chen and V. Dohm, Eur. Phys. J. B 5, 529 (1998).
  * (88) V. Dohm, F. Kischel, and S. Wessel, manuscript in preparation.
  * (89) T. Schneider, in The Physics of Superconductors, edited by K.H. Bennemann and J.B. Ketterson (Springer-Verlag, Berlin, 2004), Vol. II, p. 111; T. Schneider and J.M. Singer, Phase Transition Approach to High Temperature Superconductivity (Imperial College Press, London, 2000); T. Schneider, Phys. Rev. B 75, 174517 (2007); S. Weyeneth, T. Schneider, and E Giannini, Phys. Rev. B 79, 214504 (2009).
  * (90) R. M. F. Houtappel, Physica 16, 425 (1950); A. N. Berker and K. Hui, Phys. Rev. B 48, 12393 (1993).
  * (91) R. Garcia and M.H.W. Chan, Phys. Rev. Lett. 83, 1187 (1999).
  * (92) G.A. Williams, Phys. Rev. Lett. 92, 197003 (2004).
  * (93) H.W. Diehl and D. Grüneberg, Nucl. Phys. B 822, 517 (2009).
  * (94) F. P. Toldin and S. Dietrich, J. Stat. Mech. P11003 (2010).
  * (95) D.M. Dantchev and S. Dietrich, arXiv: 2203.15050v1.
  * (96) D. Dantchev, M. Krech, and S. Dietrich, Phys. Rev. Lett. 95, 259701 (2005).
  * (97) S. Singh, Phys. Rep. 324, 107 (2000).
  * (98) V. Dohm, Phys. Rev. E 90, 030101(R) (2014).
  * (99) B. Kastening, unpublished.