#  Multiparameter universality and intrinsic diversity of critical phenomena
in weakly anisotropic systems

Volker Dohm  Institute for Theoretical Physics, RWTH Aachen University, 52056
Aachen, Germany

(July 4, 2023)

###### Abstract

Recently a unified hypothesis of multiparameter universality for the critical
behavior of bulk and confined anisotropic systems has been formulated [V.
Dohm, Phys. Rev. E 97, 062128 (2018)]. We prove the validity of this
hypothesis in $d\geq 2$ dimensions on the basis of the principle of two-scale-
factor universality for isotropic systems. We introduce an angular-dependent
correlation vector and a generalized shear transformation that transforms
weakly anisotropic systems to isotropic systems. As examples we consider the
$O(n)$-symmetric $\varphi^{4}$ model, Gaussian model, and $n$-vector model. By
means of the inverse of the shear transformation we determine the general
structure of the bulk order-parameter correlation function, of the singular
bulk part of the critical free energy, and of critical bulk amplitude
relations of anisotropic systems at and away from $T_{c}$. It is shown that
weakly anisotropic systems exhibit a high degree of intrinsic diversity due to
$d(d+1)/2-1$ independent parameters that cannot be determined by thermodynamic
measurements. Exact results are derived for the $d=2$ Ising universality class
and for the spherical and Gaussian universality classes in $d\geq 2$
dimensions. For the $d=3$ Ising universality class we identify the universal
scaling function of the isotropic bulk correlation function from the
nonuniversal result of the functional renormalization group. A proof is
presented for the validity of multiparameter universality of the exact
critical free energy and critical Casimir amplitude in a finite rectangular
geometry of weakly anisotropic systems with periodic boundary conditions in
the Ising universality class. This confirms the validity of recent predictions
of self-similar structures of finite-size effects in the $(d=2,n=1)$
universality class at $T=T_{c}$ derived from conformal field theory [V. Dohm
and S. Wessel, Phys. Rev. Lett. 126, 060601 (2021)]. This also substantiates
the previous notion of an effective shear transformation for anisotropic two-
dimensional Ising models. Our theory paves the way for a quantitative theory
of nonuniversal critical Casimir forces in anisotropic superconductors for
which experiments have been proposed by G.A. Williams, Phys. Rev. Lett.
92,197003 (2004).

##  I Introduction

Spatial anisotropy is a fundamental property that is omnipresent in condensed
matter physics where it is the origin of a wide variety of nonuniversal
effects. Substantial evidence has emerged over the last two decades cd2004 ;
dohm2005 ; dohm-arXiv ; selke2005 ; selke2009 ; dohm2006 ; chen-zhang ;
dohm2008 ; dohmphysik2009 ; DG ; dohm2011 ; kastening-dohm ; dohm2018 ;
dohm2019 ; DW2021 ; DWKS2021  that part of this nonuniversal diversity
persists near ordinary critical points and in the near-critical Goldstone
regime of so-called weakly anisotropic systems. This includes magnetic
materials, superconductors, alloys, solids with structural phase transitions,
liquid crystals, compressible solids, and ultracold gases in anisotropic
optical lattices. Recently an unexpected complex form of self-similarity of
anisotropy effects in finite weakly anisotropic systems with periodic boundary
conditions has been found DW2021  near the instability where weak anisotropy
breaks down.

Ordinary bulk critical phenomena can be divided into universality classes
characterized by the dimension $d$ and the symmetry properties of the ordered
state priv ; pelissetto . As an example we consider $O(n)$-symmetric systems
with short-range interactions and with an $n$-component order parameter.
Within each $(d,n)$ universality class, all systems have the same universal
quantities (critical exponents, amplitude ratios, and scaling functions) which
includes isotropic and weakly anisotropic systems since spatial anisotropy is
only a marginal perturbation in the renormalization-group sense pri ; priv ;
Aharony1976 ; weg-1 ; bruce ; zia . The principle of two-scale-factor
universality (or hyperuniversality) stau ; priv ; aharony1975 ; hohenberg1976
; pri ; weg-1 ; gerber ; privmanbuch  predicts that, once the universal
quantities of a universality class are known, the asymptotic critical behavior
of any particular system of this universality class is known completely
provided that only two nonuniversal amplitudes are given. This principle was
stated to be valid for all systems in a universality class priv ;
henkelprivmanbuch ; privman1988 . Furthermore it was asserted cardy1987 ;
cardy1983 ; cardybuch ; Indekeu ; binder-wang ; nightingaleprivmanbuch ;
barber1984 ; zia  that asymptotic isotropy can be restored in weakly
anisotropic systems by a suitable anisotropic scale transformation and that
universality can be restored Indekeu , reintroduced nightingaleprivmanbuch ,
or repaired priv  in some cases. However the question was left unanswered how
the various nonuniversal effects due to non-cubic anisotropy in bulk CoyWu ;
bruce ; cardybuch ; Aharony1976 ; aharony1980 ; Vaidya1976 ; WuCoy ; Perk1 ;
Perk2 ; Perk3 ; Perk4  and confined Izmailian ; HH2019 ; cardyfinite ;
cardy1987 ; kim1987 ; Yurishchev ; Indekeu ; night1983 ;
nightingaleprivmanbuch ; NO1999 ; barber1984 ; binder-wang  systems could be
reconciled with the principle of two-scale-factor universality. So far the
issue of weak anisotropy has not been addressed in the approach based on the
nonperturbative functional renormalization-group (FRG) wetterich2002 ;
hassel2007 ; sinner2008 ; AdamR ; metzner2021 ; baloq ; dupuis2008  where an
extended universality was claimed to be valid for the isotropic bulk order-
parameter correlation function in the nonasymptotic critical region with only
two nonuniversal parameters hassel2007 ; sinner2008 .

A systematic study of the effect of non-cubic spatial anisotropy on the
critical behavior in bulk and confined systems was begun cd2004  by
introducing a nondiagonal anisotropy matrix ${\bf A}$ into the
$O(n)$-symmetric $d$-dimensional $\varphi^{4}$ field theory. An anisotropy-
induced nonuniversality was discovered for the bulk correlation function as
well as for the critical Casimir amplitude DDPhysRep ; bloete ; krech ;
toldin2013 ; diehl2014  and the critical Binder cumulant ratio priv  which
were the hallmarks of finite-size universality. The analytic results cd2004
were based on the exact large-$n$ limit and on the $\varphi^{4}$ theory in the
minimal subtraction scheme in three dimensions dohm1985 ; schl . These results
demonstrated that restoring isotropy relates the critical behavior of
anisotropic systems to that of isotropic systems but does not restore two-
scale-factor universality. The violation of two-scale-factor universality in
weakly anisotropic bulk and confined systems was subsequently confirmed and
substantiated selke2005 ; dohm2006 ; DG ; dohm2008 ; dohm2018 ; dohm2019 ;
chen-zhang ; selke2009 ; kastening-dohm ; dohm-arXiv , most recently by exact
analytic results for the critical free energy and Casimir amplitude in two
dimensions DW2021  based on conformal field theory franc1997 , as well as by
Monte Carlo simulations DWKS2021 .

The main analytical and conceptional progress was achieved within the
anisotropic $\varphi^{4}$ theory in three dohm2018  and two dohm2019 ; DW2021
dimensions. A central role is played by a shear transformation between
anisotropic and isotropic systems which is formulated in terms of eigenvectors
and eigenvalues of the anisotropy matrix ${\bf A}$. It was shown that weakly
anisotropic $\varphi^{4}$ models do not have a unique single bulk correlation
length but rather an angular-dependent correlation length dohm2019  with $d$
independent nonuniversal amplitudes in the $d$ principal directions which are
determined by $d(d-1)/2$ nonuniversal angles. As a consequence, two-scale-
factor universality is absent in the anisotropic $\varphi^{4}$ theory and is
replaced by multiparameter universality which allows for up to $d(d+1)/2+1$
independent nonuniversal parameters. This implies a revised notion of a
universality class: it must be divided into subclasses dohm2008 ;
dohmphysik2009  of isotropic and weakly anisotropic systems with different
scaling forms for the bulk correlation function and for the free energy of
confined systems within the same universality class. These anisotropic scaling
forms are governed by a reduced anisotropy matrix ${\bf\bar{A}}$ which has a
universal structure in terms of principal correlation lengths and principal
directions.

It was hypothesized dohm2018  that this revised picture of a universality
class holds not only for the subclass of anisotropic $\varphi^{4}$ models but
is valid quite generally for all other weakly anisotropic systems beyond the
$\varphi^{4}$ theory. So far no general proof has been given for this
hypothesis.

It is the aim of this paper to present a general proof for the validity of
multiparameter universality of the bulk critical behavior of weakly
anisotropic $d$-dimensional systems at and away from $T_{c}$ for $d\geq 2$.
Our strategy is to perform a generalized shear transformation from the
anisotropic to an isotropic system, then to invoke two-scale-factor
universality for the isotropic system and to derive the anisotropic structures
by inverting the shear transformation. It is shown that among the $d(d+1)/2+1$
nonuniversal parameters of weakly anisotropic systems there are only two
parameters that can be determined by thermodynamic measurements whereas the
$d(d+1)/2-1$ independent parameters contained in ${\bf\bar{A}}$ cause a high
degree of intrinsic diversity arising from the nonuniversal angular dependence
of the critical correlations. In most cases their principal directions depend
in a generically unknown way on the anisotropic interactions dohm2019 ;
night1983 . This implies that in most cases, e.g., in anisotropic $n$-vector
models and for real systems, it is unknown a priori how to perform the
anisotropic scale transformation invoked in the literature cardy1987 ;
cardy1983 ; cardybuch ; Indekeu ; binder-wang ; nightingaleprivmanbuch ;
barber1984 ; zia , with the exception of anisotropic $\varphi^{4}$ models with
known eigenvectors of the anisotropy matrix ${\bf A}$.

Special attention is payed to the bulk order-parameter correlation function
whose universal scaling function $\Psi_{\pm}$ in the asymptotic critical
region is the same for both isotropic and weakly anisotropic systems in the
same universality class. This function is known exactly for the $d=2$ Ising
universality class WuCoy ; dohm2019 . For the $d=3$ Ising universality class
we determine the universal Fourier transform $\hat{\Psi}_{\pm}$ from an
approximate nonasymptotic scaling form that has been derived within the FRG
approach hassel2007 ; sinner2008 . We refute the claim hassel2007 ; sinner2008
that an extended universality is valid in the nonasymptotic region.

We also reanalyze the finite-size effects on the critical free energy and
Casimir amplitude at $T_{c}$ of anisotropic two-dimensional systems on a
rectangle with periodic boundary conditions DW2021 ; DWKS2021 . We prove the
validity of multiparameter universality for these quantities of the $d=2$
Ising universality class including the feature of self-similarity on the basis
of conformal field theory franc1997  without recourse to the anisotropic
$\varphi^{4}$ theory.

The outline of this paper is as follows. In Sec. II we give basic definitions
within the anisotropic $\varphi^{4}$ and $n$-vector models. In Sec. III we
recall important facts with regard to two-scale-factor universality of
isotropic systems. Secs. IV and V are the central sections of this paper. In
Sec. IV we introduce the notion of principal correlation vectors and a
generalized shear transformation that depends on a free parameter and that is
applicable to all weakly anisotropic systems, in contrast to the special shear
transformation for the $\varphi^{4}$ model introduced earlier cd2004 . In
particular we demonstrate the universality of the structure of the reduced
$d\times d$ anisotropy matrix ${\bf\bar{A}}$ as a function of the nonuniversal
angles of the principal axes and of the nonuniversal ratios of the principal
correlation lengths. In Sec. V we prove that multiparameter universality is
generally valid for the bulk order-parameter correlation function in the
scaling limit of weakly anisotropic systems in $d\geq 2$ dimensions. As
applications we consider in Sec. V the anisotropic correlation functions of
the Gaussian universality class for $d\geq 2$ and of the spherical
universality class for $d>2$. The latter universality class includes the
exactly solvable large-$n$ limit of the anisotropic $\varphi^{4}$ and
$n$-vector models as well as the exactly solvable spherical and the mean
spherical models berlin-kac ; joyce ; brankov ; Baxter1982 ; stanley1968 . We
also reanalyze the nonuniversal scaling form of the isotropic bulk correlation
function derived within the $d=3$ FRG approach hassel2007 ; sinner2008  and
invoke an exact sum rule that leads to an identification of the universal part
of this scaling form in the asymptotic region. Our proof of multiparameter
universality is extended in Sec. VI to the bulk part of the free energy and to
critical bulk amplitude relations of anisotropic systems for $d\geq 2$. As a
major application we discuss in Sec. VII the anisotropic bulk correlation
function of the $(d=2,n=1)$ universality class which includes both anisotropic
Ising and scalar $\varphi^{4}$ models. The exactly solvable triangular-lattice
Ising model dohm2019 ; Vaidya1976 ; stephenson  is discussed as an example. In
Sec. VIII we present the shear transformation of the two-dimensional angular-
dependent correlation vector and derive a general transformation formula that
is applicable to both bulk and confined systems. An application is given in
Sec. IX where we take up the analysis of finite-size effects on the critical
free energy and Casimir amplitude of the anisotropic two-dimensional Ising
model in DW2021 . We show that our generalized shear transformation
substantiates and specifies the notion of the ”effective shear transformation”
mentioned in DW2021 . We also discuss the critical Casimir force in
anisotropic superconducting films. Sec. X contains a summary.

##  II Basic definitions

We consider isotropic and weakly anisotropic systems near their critical
points. In order to discuss the issue of multiparameter universality it is
appropriate to analyze and compare different types of anisotropic models that
belong to the same universality classes. As examples we consider (i) the
$O(n)$-symmetric anisotropic ”soft-spin” $\varphi^{4}$ lattice model and (ii)
the $O(n)$-symmetric anisotropic $n$-vector model which is a fixed-length spin
model. We assume short-range interactions in a range of couplings where the
systems have an isotropic or weakly anisotropic critical point. Although the
models (i) and (ii) have the same critical exponents and therefore belong to
the same $(d,n)$ universality classes the analytic description of their
anisotropy properties require different approaches. The reason is that the
principal directions of the anisotropic $\varphi^{4}$ model are well defined
at the outset through the anisotropy matrix A whereas these axes are
generically unknown for the $n$-vector model. We also consider the
$O(n)$-symmetric Gaussian model which is defined only for $T\geq T_{c}$.

###  II.1 ${\bf\varphi^{4}}$ model

The $\varphi^{4}$ lattice Hamiltonian divided by $k_{B}T$ reads dohm2006 ;
dohm2008

$\displaystyle H$ $\displaystyle=$ $\displaystyle
v\Bigg{[}\sum_{i=1}^{N}\left(\frac{r_{0}}{2}\varphi_{i}^{2}+u_{0}(\varphi_{i}^{2})^{2}-{\bf
h}\cdot\varphi_{i}\right)$ (2.1) $\displaystyle+$
$\displaystyle\sum_{i,j=1}^{N}\frac{K_{i,j}}{2}(\varphi_{i}-\varphi_{j})^{2}\Bigg{]},\;$

with $u_{0}>0$ and an ordering field ${\bf h}=h\;{\bf e}_{h}$ with a unit
vector ${\bf e}_{h}$. The variables $\varphi_{i}\equiv\varphi({\bf x}_{i})$
are $n$-component vectors on $N$ lattice points ${\bf
x}_{i}\equiv(x_{i1},x_{i2},\ldots,x_{id})$ of a $d$-dimensional Bravais
lattice of volume $V=Nv$ where $v$ is the volume of the primitive cell. We
assume periodic boundary conditions. The components
$\varphi_{i}^{(\mu)}\;,\mu=1,2,\ldots,n$ of $\varphi_{i}$ vary in the
continuous range $-\infty\leq\varphi_{i}^{(\mu)}\leq\infty$. For an
appropriate class of pair interactions $K_{i,j}$ the model undergoes a phase
transition in the bulk limit at a finite bulk critical temperature $T_{c}$ for
$n=1,d>1$ and $n\geq 2,d>2$. For $n=2,d=2$ this is a Kosterlitz-Thouless
transition. No finite $T_{c}$ exists for $n>2$ in $d\leq 2$ dimensions. For
$n=1,2,3,\infty$ the $\varphi^{4}$ model belongs to the Ising, $XY$,
Heisenberg and spherical universality classes priv ; pelissetto ,
respectively. The distance from bulk criticality is described by the variable

$r_{0}(T)-r_{0c}=a_{0}t,\;\;\;t=(T-T_{c})/T_{c}$ (2.2)

with $a_{0}>0$ where $T_{c}$ is the bulk critical temperature. The critical
value $r_{0}(T_{c})=r_{0c}=r_{0c}(v,u_{0},K_{i,j},d,n)$ is determined by the
divergence of the bulk susceptibility at $h=0$ dohm2008  and depends on the
lattice structure, on $v$, on $u_{0}$, and on all couplings $K_{i,j}$. If
$u_{0}=0$ the model (2.1) with $r_{0c}=0$ and $r_{0}=a_{0}t\geq 0$ belongs to
the Gaussian universality class with a critical point at $r_{0}=0,h=0$ in
$d>0$ dimensions. The large-distance anisotropy is described by a
dimensionless symmetric anisotropy matrix ${\bf A}(\\{K_{i,j}\\})$ cd2004
whose matrix elements

$A_{\alpha\beta}(\\{K_{i,j}\\})=N^{-1}\sum^{N}_{i,j=1}(x_{i\alpha}-x_{j\alpha})(x_{i\beta}-x_{j\beta})\;K_{i,j}$
(2.3)

are determined by the second moments of the microscopic pair interactions
$K_{i,j}$ dohm2006 . A characteristic feature of weakly anisotropy systems is
that they have the same critical exponents as the isotropic system cd2004 .
This requires

$\det{\bf A}(\\{K_{i,j}\\})>0$ (2.4)

as a necessary (but not yet sufficient dohm2008 ; dohm2018 ) condition. The
condition for isotropy in the large-distance regime is

${\bf A}(\\{K_{i,j}\\})=c^{\rm iso}_{0}{\bf 1}$ (2.5)

with $c^{\rm iso}_{0}>0$ where ${\bf 1}$ is the unity matrix. Eq. (2.3) and
the criteria (2.4) and (2.5) are valid for general $d$ and $n$ dohm2008 ;
dohm2018  including the large-$n$ limit and the Gaussian model.

The dimensionless partition function $Z$, the total free energy ${\cal F}_{\rm
tot}$ (divided by $k_{B}T$), and the total free-energy density $f$ of the
$\varphi^{4}$ model are defined by

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\Big{[}\prod_{i=1}^{N}\frac{\int
d^{n}{\varphi}_{i}}{v^{n(2-d)/(2d)}}\Big{]}\exp\left(-H\right),$ (2.6)
$\displaystyle{\cal F}_{\rm tot}$ $\displaystyle=$ $\displaystyle-\ln Z,$
(2.7) $\displaystyle f$ $\displaystyle=$ $\displaystyle{\cal F}_{\rm{tot}}/V,$
(2.8)

and the bulk parts above $(+)$ and below $(-)$ $T_{c}$ by

$\displaystyle f_{b,\pm}$ $\displaystyle=$
$\displaystyle\lim_{V\to\infty}{\cal F}_{\rm tot}/V,$ (2.9)
$\displaystyle{\cal F}_{b,\pm}$ $\displaystyle=$ $\displaystyle Vf_{b,\pm}.$
(2.10)

Near $T_{c}$ we use the decompositions

$\displaystyle{\cal F}_{\rm tot}$ $\displaystyle=$ $\displaystyle{\cal
F}_{s}+{\cal F}_{ns},$ (2.11) $\displaystyle f$ $\displaystyle=$
$\displaystyle f_{s}+f_{ns},$ (2.12) $\displaystyle f_{b,\pm}$
$\displaystyle=$ $\displaystyle f_{b,s,\pm}+f_{b,ns,\pm}$ (2.13)

into singular and nonsingular parts. The singular bulk part $f_{b,s,\pm}(t,h)$
has the scaling form of Eq. (1.1) of dohm2008  with the two nonuniversal
constants $A_{1}$ and $A_{2}$, and it vanishes at $t=0,h=0$. We shall discuss
the leading singular bulk part at $h=0$

$\displaystyle{\cal F}_{b,s,\pm}$ $\displaystyle=$ $\displaystyle
Vf_{b,s,\pm}$ (2.14)

of ${\cal F}_{\rm tot}$ for $d\geq 2$ and, for $d=2$, the finite critical free
energy

${\cal F}_{c}=\lim_{h\to 0,T\to T_{c}}{\cal F}_{s}$ (2.15)

which is identical with the singular finite-size part of ${\cal F}_{\rm tot}$
at $T_{c}$, $h=0$ studied in DW2021 .

The bulk order-parameter correlation function is

$G_{\pm}({\bf x_{i}}-{\bf
x_{j}},t,h)=\lim_{V\to\infty}\big{[}<{\varphi}_{i}\cdot{\varphi}_{j}>-{\cal
M}^{2}\big{]}.$ (2.16)

where ${\cal M}^{2}=\lim_{|{\bf
x_{i}-x_{j}}|\to\infty}<\varphi_{i}\cdot\varphi_{j}>$ is the square of the
bulk order parameter. The static version of the dissipation-fluctuation
theorem halperin ; hohenberg  yields the exact sum rule dohm2008

$\displaystyle v\sum_{\bf x}\;G_{\pm}({\bf
x},t,h)=\chi_{\pm}(t,h)=\partial^{2}f_{b,\pm}(t,h)/\partial h^{2}\;\;\;$
(2.17)

where $\chi_{\pm}(t,h)$ is the bulk susceptibility. Here we have assumed
general $n\geq 1$ for $T\geq T_{c}$ and $n=1$ for $T<T_{c}$. For $n>1,T<T_{c}$
it is necessary to distinguish between longitudinal and transverse correlation
functions priv .

For the description of the asymptotic critical behavior on a long-distance
scale it suffices to study the continuum version of this model in terms of the
vector field $\varphi({\bf x})$. The continuum Hamiltonian reads cd2004 ;
dohm2008

$\displaystyle H^{\rm field}$ $\displaystyle=$
$\displaystyle\int_{V}d^{d}x\Bigg{[}\frac{r_{0}}{2}\varphi^{2}+\sum_{\alpha,\beta=1}^{d}\frac{A_{\alpha\beta}}{2}\frac{\partial\varphi}{\partial
x_{\alpha}}\frac{\partial\varphi}{\partial x_{\beta}}$ (2.18) $\displaystyle+$
$\displaystyle u_{0}(\varphi^{2})^{2}-h\varphi\Bigg{]}$

with a finite anisotropic cutoff in momentum space where now the sum rule has
the form

$\displaystyle\int d^{d}{\bf x}\;\;G_{\pm}(|{\bf
x}|,t,h)=\chi_{\pm}(t,h)=\partial^{2}f^{\rm field}_{b,\pm}(t,h)/\partial
h^{2}.\;\;\;\;\;$ (2.19)

The sum rules (2.17) and (2.19) have nothing to do with the existence of a
critical point and therefore are exactly valid in the entire range of $t$ and
$h$, not only in the asymptotic critical region. In particular the sum rule
(2.19) remains exactly valid for any choice of a finite cutoff and for both
isotropic and anisotropic $\varphi^{4}$ models. The susceptibility plays an
important role for the structure of the Fourier transform $\hat{G}_{\pm}({\bf
k},t,h)$ of the correlation function. It can be uniquely divided into a
thermodynamic part $\chi_{\pm}(t)$ and a correlation part $\hat{D}({\bf
k},t,h)$ in the Fisher-Aharony scaling form Fisher-Aharony-1974 ; tarko ;
hohenberg1976 ,

$\displaystyle\hat{G}_{\pm}({\bf k},t,h)$ $\displaystyle=$
$\displaystyle\chi_{\pm}(t,h)\;\hat{D}_{\pm}({\bf k},t,h)$ (2.20)

with the normalization

$\displaystyle\hat{D}_{\pm}({\bf 0},t,h)=1.$ (2.21)

Here $\chi_{\pm}(t,h)$ is determined entirely by the bulk free-energy density
defined at ${\bf k}={\bf 0}$ whereas $\hat{D}({\bf k},t,h)$ requires ${\bf
k}$-dependent investigations. Experimentally this means that $\chi_{\pm}(t,h)$
is determined by thermodynamic measurements whereas $\hat{D}({\bf k},t,h)$
requires spatially resolved scattering experiments. This physical distinction
is of fundamental importance in the asymptotic critical region where
$\hat{D}_{\pm}({\bf k},t,h)$ becomes a universal function whereas
$\chi_{\pm}(t,h)$ still contains a nonuniversal thermodynamic amplitude that
is independent of the correlation-length amplitude. Thus near criticality the
relevance of (2.20) is that it provides a unique decomposition into a
nonuniversal and a universal part hohenberg1976 ; tarko ; priv ; dohm2019 .
Right at criticality with $t=0,h=0$, $\chi_{\pm}(0,0)$ does not exist and two
alternative decompositions have been introduced pri ; dohm2019  that remain
applicable to the asymptotic critical region including the critical point
$t=0,h=0$, as discussed in Sec. III.C. These general considerations are valid
for both isotropic and weakly anisotropic systems. They are of relevance for a
discussion in Sec. V. E of the claims with regard to the universality
properties of the correlation function of the isotropic $\varphi^{4}$ model
derived in the framework of the FRG hassel2007 ; sinner2008  where
$\hat{G}_{\pm}({\bf k},t,0)$ was divided into two nonuniversal parts, as we
shall see.

In the anisotropic case, the principal correlation lengths and the principal
axes of $G_{\pm}$ are determined by the eigenvalues and the eigenvectors of
${\bf A}$ cd2004 ; dohm2006 , thus they are well-defined functions of the
couplings and the lattice structure, as shown explicitly for $d=2$ in dohm2019
. This differs fundamentally from the $n$-vector model.

###  II.2 Fixed-length spin model

We consider an anisotropic $O(n)$-symmetric fixed-length spin model defined on
the same lattice with the same boundary conditions as the $\varphi^{4}$ model.
It is called $n$-vector model and has the Hamiltonian stanley1968 ; priv

$\displaystyle H^{\rm sp}=-\sum_{i,j}E_{i,j}{\bf S}_{i}\cdot{\bf
S}_{j}-\sum_{i}{\bf h}^{\rm sp}\cdot{\bf S}_{i}$ (2.22)

with the statistical weight $\exp(-H^{\rm sp}/(k_{B}T))$. The classical
dimensionless spin variables ${\bf S}_{i}$ are $n$-component vectors with a
fixed length ${\bf S}_{i}^{2}=1$. They have components $S_{i}^{(\mu)}$,
$\mu=1,2,...,n$ which are continuous variables for $n>1$. For $n=2,3$ the
model is called $XY$ and Heisenberg model, respectively. For $n\to\infty$ the
model belongs to the spherical universality class. For $n=1$ (2.22) is the
anisotropic Ising model with the discrete variables $S_{i}\equiv\sigma_{i}=\pm
1$.

The dimensionless partition function $Z^{\rm sp}$, the total free energy
${\cal F}_{\rm tot}^{sp}$ (divided by $k_{B}T$), and the bulk correlation
function are defined by

$\displaystyle Z^{\rm sp}=\Big{[}\prod_{i=1}^{N}\int d^{n}{\bf
S}_{i}\Big{]}\exp\left(-\beta H^{\rm sp}\right),$ (2.23) $\displaystyle{\cal
F}^{\rm sp}_{\text{tot}}=-\ln Z^{\rm sp},$ (2.24) $\displaystyle G^{\rm
sp}_{\pm}({\bf x_{i}}-{\bf x_{j}},t,{\bf h}^{\rm
sp})=\lim_{V\to\infty}\big{[}<{\bf S}_{i}\cdot{\bf S}_{j}>-({\cal M}^{\rm
sp})^{2}\big{]}\;\;\;\;\;\;\;\;\;\;$ (2.25)

with $\beta=1/(k_{B}T)$ and the constraint ${\bf S}_{i}^{2}=1$. The
definitions of $f^{\rm sp}$, ${\cal F}^{\rm sp}_{b,\pm}$, ${\cal F}^{\rm
sp}_{c}$, $\chi^{\rm sp}_{\pm}$, etc. are analogous to (2.8)-(2.17). The
counterpart of the sum rule (2.17) in the $n$-vector model is formultated for
the example of the Ising model in Eq. (2.43) of WuCoy . If the variables
$S_{i}^{(\mu)}$ have an unrestricted range, $-\infty\leq
S_{i}^{(\mu)}\leq\infty$, the model (2.22) with $T\geq T_{c}$ belongs to the
Gaussian universality class. The general considerations with regard to the
structure of the correlation functions $G_{\pm}$ and $\hat{G}_{\pm}$ remain
valid also for $G^{\rm sp}_{\pm}$ and $\hat{G}^{\rm sp}_{\pm}$.

In view of the generalized shear transformation which is a pure coordinate
transformation to be introduced in Sec. IV we make the following general
remark on the fixed-length spin model (2.22). The definition of the
Hamiltonian $H^{\rm sp}$ and the partition function $Z^{\rm sp}$ in terms of
the dimensionless variables ${\bf S}_{i}$ implies that $Z^{\rm sp}$ and the
total free energy (2.24) do not depend on the details of the position of the
lattice points but only on the value and the topology of the pair interactions
$E_{i,j}$. This means that, for given $n$, given boundary conditions, and
given number of lattice points, two $n$-vector models $H^{\rm sp}$ with the
same coupling constants $E_{i,j}$ on two different lattices with the same
topology of the pair interactions have the same partition function and free
energy. Thus, if a transformation is performed at fixed boundary conditions
that involves only a smooth change of the coordinates of the lattice points
without changing the value and topology of the pair interactions $E_{i,j}$ and
without changing the amplitudes of the spin variables ${\bf S}_{i}$, the
partition function and total free energy are not affected and remain invariant
under such a transformation. This implies that also the decomposition into
singular and nonsingular parts

${\cal F}^{\rm sp}_{\rm tot}={\cal F}^{\rm sp}_{s}+{\cal F}^{\rm sp}_{ns}$
(2.26)

remains unchanged and that ${\cal F}^{\rm sp}_{s}$ remains invariant. In
particular the critical free energy

${\cal F}^{\rm sp}_{c}=\lim_{{\bf h}^{\rm sp}\to{\bf 0},T\to T_{c}}{\cal
F}^{\rm sp}_{s}$ (2.27)

of the finite system remains invariant. Furthermore such a pure coordinate
transformation changes only the spatial argument of the correlation function
$G_{\pm}^{\rm sp}$ but leaves its amplitude invariant.

No general approach to an analytic construction of the principal axes and
correlation lengths has been developed so far for the model (2.22).
Furthermore, no analytic condition for criticality is known for general
couplings $E_{i,j}$, and general criteria for weak anisotropy and isotropy
analogous to the general conditions (2.4) and (2.5) for $\varphi^{4}$ models
are as yet unknown for the $n$-vector model. Apart from anisotropic $d=2$
Ising models, very little is known with regard to the anisotropy properties of
$n$-vector models for $d>2$ as compared to those of $\varphi^{4}$ models where
quantitative renormalized perturbation calculations can be performed within
the minimal subtraction scheme at fixed dimension $d$ schl ; dohm1985 ;
dohm2008 ; dohm2018 . Nevertheless it is possible to derive general structural
properties of the anisotropic critical behavior of the model (2.22) for
general $d$ and $n$ in the scaling limit, as will be shown below.

In the remainder of this paper we consider only the case of vanishing external
field.

##  III Isotropic case: Two-scale-factor universality

For any given Bravais lattice of the models (2.1) and (2.22) the couplings
$K_{i,j}$ and $E_{i,j}$ can be chosen such that the bulk system has isotropic
correlations in the large-distance scaling regime near $T_{c}$. Since the bulk
critical behavior of isotropic systems plays a fundamental role for the
development of the theory of weakly anisotropic systems we recall and
reformulate important definitions and facts in an appropriate way such that
they can be extended to weakly anisotropic systems. A detailed exposition of
the universal isotropic reformulation is also necessary for a comparison with
the nonuniversal formulation for the correlation function of the FRG
hassel2007 ; sinner2008  discussed in Sec. V.E.

###  III.1 Isotropic bulk correlation function

In the isotropic case, the correlation function (2.16) has the established
Privman-Fisher scaling form in the asymptotic critical scaling region above
$(+)$ and below $(-)$ $T_{c}$ below $d=4$ dimensions pri

$\displaystyle G^{\rm iso}_{\pm}(|{\bf x}|,t)$ $\displaystyle=$
$\displaystyle\frac{D^{\rm iso}_{1}}{|{\bf
x}|^{d-2+\eta}}\;\Phi_{\pm}\Big{(}\frac{|{\bf x}|}{\xi^{\rm
iso}_{\pm}(t)}\Big{)}\;,\;\;\;\;\;\;$ (3.1) $\displaystyle\xi^{\rm
iso}_{\pm}(t)$ $\displaystyle=$ $\displaystyle\xi^{\rm
iso}_{0\pm}\;|t|^{-\nu}$ (3.2)

with the universal critical exponents $\eta$ and $\nu$ and the universal
scaling function $\Phi_{\pm}$. For an alternative form see (3.45). The
constant $D^{\rm iso}_{1}$ is the same above, at, and below $T_{c}$. The
scaling region is defined by $|{\bf x}|\gg v^{1/d}$ and $\xi^{\rm
iso}_{\pm}\gg v^{1/d}$ at fixed finite ratio $|{\bf x}|/\xi^{\rm iso}_{\pm}$.
The function $G^{\rm iso}_{\pm}$ is finite and continuous at $T_{c}$ with
$\Phi_{+}(0)=\Phi_{-}(0)$. The correlation lengths $\xi^{\rm iso}_{\pm}$ may
be chosen to be either the ”true” (exponential) pri ; pelissetto ; dohm2019
or the second-moment dohm2008  correlation lengths. The ratios $Q^{+}_{\xi}$
and $Q^{-}_{\xi}$ of these correlation lengths above and below $T_{c}$ defined
in Table 2 of pelissetto  are universal. In this paper we employ the
exponential correlation length defined by the exponential decay
$\sim\exp(-|{\bf x}|/\xi^{\rm iso}_{\pm})$ for large but finite $|{\bf
x}|/\xi^{\rm iso}_{\pm}$ within the scaling region. We recall, however, that
the universal scaling form (3.1) is not valid for $|{\bf x}|/\xi^{\rm
iso}_{\pm}\gg 1$ where two-scale-factor universality and scaling in the sense
of (3.1) is violated even arbitrarily close to $T_{c}$ cd2000-1 ; cd2000-2 ;
dan-2 ; dohm2008 ; dohm2011 , see the shaded region in Fig. 2 of dohm2008 .

Among the three nonuniversal amplitudes $D^{\rm iso}_{1}$, $\xi^{\rm
iso}_{0+}$, and $\xi^{\rm iso}_{0-}$ there are only two independent amplitudes
since the amplitudes $\xi^{\rm iso}_{0\pm}$ are universally related by

$\xi^{\rm iso}_{0+}/\xi^{\rm iso}_{0-}=X_{\xi}={\rm universal}.$ (3.3)

In pelissetto  $X_{\xi}$ is denoted by $U_{\xi_{\rm gap}}$ for the exponential
correlation length. It corresponds to $1/X_{-}(0)$ in dohm2008 ; dohm2018
(denoted by $U_{\xi}$ in pelissetto ) where the second-moment correlation
length is employed. We shall also consider the asymptotic susceptibility above
and below $T_{c}$

$\displaystyle\chi^{\rm iso}_{\pm}(t)=\Gamma^{\rm iso}_{\pm}|t|^{-\gamma},$
(3.4)

with $\gamma=(2-\eta)\nu$ and the bulk order parameter

${\cal M}^{\rm iso}(t)=B^{\rm iso}|t|^{\beta}$ (3.5)

below $T_{c}$ with the nonuniversal amplitudes $\Gamma^{\rm iso}_{\pm}$ and
$B^{\rm iso}$ with the universal ratio pelissetto

$\displaystyle\Gamma^{\rm iso}_{+}/\Gamma^{\rm iso}_{-}=U_{2}\;=\;{\rm
universal}.$ (3.6)

The amplitude $B^{\rm iso}$ can be expressed in terms of $\Gamma^{\rm
iso}_{+}$ and $\xi^{\rm iso}_{0+}$ through the universal relation priv ;
pelissetto ; dohm2019

$\displaystyle(B^{\rm iso})^{2}(\Gamma_{+}^{\rm iso})^{-1}(\xi^{\rm
iso}_{0+})^{d}=Q_{c}=\text{universal},$ (3.7)

for general $n$ with the universal constant $Q_{c}$.

For $n>1,T<T_{c}$ the transverse bulk correlation function of isotropic
systems has the algebraic large-distance behavior for $d>2$ priv

$\displaystyle G^{\rm iso}_{\rm T}(|{\bf x}|,t)$ $\displaystyle=$
$\displaystyle{\cal C}_{\rm T}[{\cal M}^{\rm iso}(t)]^{2}\big{[}\xi^{\rm
iso}_{\rm T}(t)/|{\bf x}|\big{]}^{d-2},$ (3.8) $\displaystyle{\cal C}_{\rm T}$
$\displaystyle=$ $\displaystyle\Gamma(d/2)/[2\pi^{d/2}(d-2)].$ (3.9)

The bulk transverse correlation length near $T_{c}$ is

$\xi^{\rm iso}_{\rm T}(t)=\xi^{\rm iso}_{0\rm T}|t|^{-\nu}$ (3.10)

with the nonuniversal amplitude $\xi^{\rm iso}_{0\rm T}$ and the universal
amplitude ratio

$\xi^{\rm iso}_{0+}/\xi^{\rm iso}_{0\rm T}=X_{{\rm T}\xi}={\rm universal}.$
(3.11)

###  III.2 Singular part of the bulk free-energy density

The following relations apply to isotropic systems of both the $(d,n)$
universality class (including $n$-vector and $\varphi^{4}$ models) and the
Gaussian universality class for $t>0$ (with $\nu=1/2$ for general $d$ and
$n$). The singular part of the bulk free-energy density has the asymptotic
form priv ; pelissetto

$\displaystyle
f^{\text{iso}}_{b,s,\pm}(t)=\left\\{\begin{array}[]{r@{\quad\quad}l}\;A^{\text{iso}}_{\pm}|t|^{d\nu}&\mbox{for}\;2<d<4\;,\\\
\;\frac{1}{2}A^{\rm
iso}_{\pm}|t|^{2\nu}\ln|t|&\mbox{for}\;d=2.\end{array}\right.$ (3.14)

The nonuniversal amplitudes $A^{\rm iso}_{\pm}$ have the universal ratio

$f^{\rm iso}_{b,s+}(t)/f^{\rm iso}_{b,s-}(t)=A^{\rm iso}_{+}/A^{\rm
iso}_{-}=\;\;\rm{universal}.$ (3.15)

Due to two-scale-factor universality pri ; priv ; weg-1 ; stau  these
amplitudes are universally related to the correlation-length amplitude
$\xi_{0+}^{\rm iso}$ through

$\displaystyle\left(\xi_{0+}^{\rm iso}\right)^{d}A_{+}^{\rm
iso}=\;Q_{1}=\;\;\rm{universal},\;d\geq 2$ (3.16)

with the universal constant $Q_{1}$, with $Q_{1}=1/(2\pi)$ for the $d=2$ Ising
universality class (called $(R^{+}_{\xi})^{2}$ in Eq. (6.31) of priv ) for the
case that the true correlation length is used. The validity of (3.16) has been
established by the renormalization-group theory weg-1 ; hohenberg1976 ;
aharony1975 ; gerber . Thus we obtain in $2<d<4$ dimensions

$\displaystyle f^{\text{iso}}_{b,s,+}(t)$ $\displaystyle=$ $\displaystyle
Q_{1}\left(\xi_{0+}^{\rm iso}\right)^{-d}t^{d\nu},$ (3.17) $\displaystyle
f^{\text{iso}}_{b,s,-}(t)$ $\displaystyle=$ $\displaystyle\frac{A_{-}^{\rm
iso}}{A_{+}^{\rm iso}}Q_{1}\left(\xi_{0+}^{\rm iso}\right)^{-d}|t|^{d\nu},$
(3.18)

whereas for the ($d=2,n=1$) universality class $f^{\text{iso}}_{b,s,\pm}$
depends on $|t|$ rather than $t$

$\displaystyle f^{\rm iso}_{b,s,\pm}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}Q_{1}\left(\xi_{0+}^{\rm
iso}\right)^{-2}\;t^{2}\ln|t|\;,\;d=2,\;\;\;\;\;\;\;\;\;\;$ (3.19)

because of

$A^{\rm iso}_{+}/A^{\rm iso}_{-}=1,\;\;\nu=1,\;\;d=2$ (3.20)

for this universality class priv ; pelissetto . We note that the amplitude
$\xi_{0+}^{\rm iso}$ defined above $T_{c}$ appears in the above relations for
both $t>0$ and $t<0$.

For the purpose of a structural analysis of critical bulk amplitude relations
and a later extension to weakly anisotropic systems it is advantageous to
express the relations (3.17)-(3.19) for the bulk free-energy density
$f^{\text{iso}}_{b,s,\pm}$ in terms of the singular bulk part ${\cal F}^{\rm
iso}_{b,s,\pm}$ of the free energy in a arbitrary finite volume $V^{\rm iso}$
according to the definition (2.14),

$\displaystyle{\cal F}^{\rm iso}_{b,s,\pm}(t,V^{\rm iso})$ $\displaystyle=$
$\displaystyle V^{\rm iso}f^{\rm iso}_{b,s,\pm}(t).$ (3.21)

This leads in a natural way to the scaling variable

$\displaystyle\widetilde{x}$ $\displaystyle=$ $\displaystyle t[V^{\rm
iso}/(\xi_{0+}^{\rm iso})^{d}]^{1/(d\nu)}$ (3.22)

appearing in the scaling form obtained from (3.17)- (3.21) in $2<d<4$
dimensions

$\displaystyle{\cal F}^{\rm iso}_{b,s,+}(t,V^{\rm iso})$ $\displaystyle=$
$\displaystyle Q_{1}\;\widetilde{x}^{d\nu},\,\,\;\;\;\;\;$ (3.23)
$\displaystyle{\cal F}^{\rm iso}_{b,s,-}(t,V^{\rm iso})$ $\displaystyle=$
$\displaystyle\frac{A^{\text{iso}}_{-}}{A^{\rm
iso}_{+}}Q_{1}\;|\widetilde{x}|^{d\nu},\,\,\;\;\;\;\;\;\;\;\;$ (3.24)

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

$\displaystyle{\cal F}^{\rm iso}_{b,s,\pm}(t,V^{\rm iso})$ $\displaystyle=$
$\displaystyle\frac{1}{2}Q_{1}\;|\widetilde{x}|^{2}\ln|t|,\,\,\;\;\;$ (3.25)

with a nonscaling logarithmic factor. Unlike $f^{\text{iso}}_{b,s,\pm}$, both
${\cal F}^{\rm iso}_{b,s,\pm}$ and $\widetilde{x}$ satisfy important
invariance properties with respect to the shear transformations between
isotropic and anisotropic systems as we shall see in Eqs. (6.20)-(6.22).

The singular part of the Gaussian bulk free-energy density per component
divided by $k_{B}T$ has the asymptotic behavior above $T_{c}$ kastening-dohm

$\displaystyle f^{\rm
G,iso}_{b,s,+}(t)=\left\\{\begin{array}[]{r@{\quad\quad}l}\;A^{\rm
G,iso}_{+}t^{d/2}&\mbox{for}\;2<d<4\;,\\\ \;\frac{1}{2}A^{\rm G,iso}_{+}t\ln
t&\mbox{for}\;d=2.\end{array}\right.$ (3.28)

Due to two-scale-factor universality the amplitude $A^{\rm G,iso}_{+}$ is
universally related to the correlation-length amplitude $\xi_{0+}^{\rm G,iso}$
through

$\displaystyle\left(\xi_{0+}^{\rm G,iso}\right)^{d}A_{+}^{\rm G,iso}$
$\displaystyle=$ $\displaystyle Q^{\rm G}_{1}\;\;={\rm universal}$ (3.29)

where the universal constant is kastening-dohm

$Q^{\rm G}_{1}=-\frac{\Gamma(-d/2)}{2(4\pi)^{d/2}}\;\;,2<d<4,$ (3.30)

with $Q^{\rm G}_{1}=-1/(12\pi)$ for $d=3$, and kastening-dohm

$Q^{\rm G}_{1}=-1/(4\pi),\;\;d=2,$ (3.31)

thus we obtain for $t>0$

$\displaystyle f^{\rm
G,iso}_{b,s,+}(t)=\left\\{\begin{array}[]{r@{\quad\quad}l}\;Q^{\rm
G}_{1}\left(\xi_{0+}^{\rm G,iso}\right)^{-d}t^{d/2}&\mbox{for}\;d>2,\\\
\;\frac{1}{2}Q^{\rm G}_{1}\left(\xi_{0+}^{\rm G,iso}\right)^{-2}\;t\ln
t&\mbox{for}\;d=2.\end{array}\right.$ (3.34)

For the singular bulk part

$\displaystyle{\cal F}^{\rm G,iso}_{b,s,+}(t,V^{\rm iso})$ $\displaystyle=$
$\displaystyle V^{\rm iso}f^{\rm G,iso}_{b,s,+}(t)$ (3.35)

of the free energy of the isotropic Gaussian model in a finite volume $V^{\rm
iso}$ this leads to the scaling form

$\displaystyle{\cal F}^{\rm G,iso}_{b,s,+}(t,V^{\rm
iso})=\left\\{\begin{array}[]{r@{\quad\quad}l}\;Q^{\rm
G}_{1}\;(\widetilde{x}^{\rm G})^{d/2}&\mbox{for}\;d>2,\\\ \;\frac{1}{2}Q^{\rm
G}_{1}\;\widetilde{x}^{\rm G}\ln t&\mbox{for}\;d=2,\end{array}\right.$ (3.38)

with a nonscaling logarithmic factor for $d=2$ and with the Gaussian scaling
variable

$\displaystyle\widetilde{x}^{\rm G}$ $\displaystyle=$ $\displaystyle t[V^{\rm
iso}/(\xi_{0+}^{\rm G,iso})^{d}]^{2/d}\;\;\;{\rm for}\;\;d\geq 2.$ (3.39)

###  III.3 Sum rule and thermodynamic amplitudes of isotropic systems

Two-scale-factor universality implies that the scaling form of the isotropic
order-parameter correlation function is fully determined once two independent
nonuniversal parameters are given. In the correlation function $G^{\rm
iso}_{\pm}({\bf x},t)$, (3.1), these parameters were chosen to be the
correlation-length amplitude $\xi^{\rm iso}_{0+}$ (or $\xi^{\rm iso}_{0-}$)
and the amplitude $D^{\rm iso}_{1}$. However, in view of the sum rule (2.19)

$\displaystyle\chi^{\rm iso}_{\pm}(t)=\int d^{d}{\bf x}\;\;G^{\rm
iso}_{\pm}(|{\bf x}|,t)$ (3.40)

it is advantageous to express $D^{\rm iso}_{1}$ in terms of the amplitude
$\Gamma_{\pm}^{\rm iso}$ of the directly obervable isotropic susceptibility
(3.4). From (3.1),(3.4), and (3.40) we obtain two relations

$\displaystyle D_{1}^{\rm iso}\;$ $\displaystyle=$ $\displaystyle\Gamma^{\rm
iso}_{+}(\xi^{\rm iso}_{0+})^{-2+\eta}\;\;\widetilde{\Phi}_{+}^{-1}\;\;\;{\rm
for\;general}\;n,$ (3.41) $\displaystyle D_{1}^{\rm iso}\;$ $\displaystyle=$
$\displaystyle\Gamma^{\rm iso}_{-}(\xi^{\rm
iso}_{0-})^{-2+\eta}\;\;\widetilde{\Phi}_{-}^{-1}\;\;\;{\rm
for}\;n=1,\;\;\;\;\;$ (3.42)

with the two universal constants

$\displaystyle\widetilde{\Phi}_{\pm}$ $\displaystyle=$ $\displaystyle
2\pi^{d/2}\Gamma(d/2)^{-1}\int_{0}^{\infty}dss^{1-\eta}\Phi_{\pm}(s).$ (3.43)

Thus the susceptibility can be expressed as

$\displaystyle\chi^{\rm iso}_{\pm}(t)=D_{1}^{\rm iso}[\xi^{\rm
iso}_{\pm}(t)]^{2-\eta}\widetilde{\Phi}_{\pm}.$ (3.44)

This shows that $\chi^{\rm iso}_{\pm}(t)$ and $\xi^{\rm iso}_{\pm}(t)$ are not
universally related. Equations (3.1) and (3.41) yield the alternative
representation of the correlation function above and below $T_{c}$ for general
$n$ in terms of the observable amplitudes $\Gamma^{\rm iso}_{+}$ and
$\xi_{0+}^{\rm iso}$,

$G^{\rm iso}_{\pm}(|{\bf x}|,t)=\frac{\Gamma^{\rm iso}_{+}(\xi_{0+}^{\rm
iso})^{-2+\eta}}{|{\bf x}|^{d-2+\eta}}\;\Psi_{\pm}\Big{(}\frac{|{\bf
x}|}{\xi^{\rm iso}_{\pm}(t)}\Big{)}\;,$ (3.45)

with the universal scaling function $\Psi_{\pm}$ above and below $T_{c}$,

$\displaystyle\Psi_{+}(y_{+})=\frac{\Phi_{+}(y_{+})}{\widetilde{\Phi}_{+}},\;\;\;\Psi_{-}(y_{-})=\frac{\Phi_{-}(y_{-})}{\widetilde{\Phi}_{+}}.\;\;\;\;\;$
(3.46)

The value of $\Psi_{\pm}$ at $T_{c}$ is a universal constant dohm2008

$\displaystyle\Psi_{+}(0)=\Psi_{-}(0)=\widetilde{Q}_{3}={\rm universal},$
(3.47)

compare Eqs. (3.15) and (A16) of dohm2008  and Eq. (5.32) of dohm2018 . The
sum rule (3.40) implies the normalization

$2\pi^{d/2}\big{[}\Gamma(d/2)\big{]}^{-1}\int_{0}^{\infty}dyy^{1-\eta}\Psi_{\pm}(y)=1$
(3.48)

which provides a unique separation of the universal and nonuniversal parts of
the correlation function. The scaling function $\Psi_{\pm}$ depends on the
universality class. Its exact analytic form is known for the $d=2$ Ising
universality class WuCoy ; Vaidya1976 ; dohm2019  and is presented in Secs.
V.C and V.D. for the spherical universality class corresponding to the
large-$n$ limit and for the Gaussian universality class. It is remarkable
that, although no finite bulk correlation length and bulk susceptibility exist
at $T_{c}$, the isotropic bulk correlation function at $T_{c}$

$\displaystyle G^{\rm iso}_{\pm}(|{\bf x}|,0)$ $\displaystyle=$
$\displaystyle\widetilde{Q}_{3}\;\frac{\Gamma^{\rm iso}_{+}(\xi_{0+}^{\rm
iso})^{-2+\eta}}{|{\bf x}|^{d-2+\eta}}$ (3.49)

is expressed in terms of the two independent bulk amplitude $\xi_{0+}^{\rm
iso}$ and $\Gamma^{\rm iso}_{+}$ defined above $T_{c}$ for general $n$.
(According to (3.42), $G^{\rm iso}({\bf x},0)$ can also be expressed in terms
of the bulk amplitudes $\xi_{0-}^{\rm iso}$ and $\Gamma^{\rm iso}_{-}$ below
$T_{c}$ but this is applicable only for $n=1$.) The representation (3.49)
demonstrates that, together with $\Gamma^{\rm iso}_{+}$, the amplitude
$\xi_{0+}^{\rm iso}$ constitutes an important reference length for the spatial
decay of the correlation function at $T_{c}$ which is not apparent in the form
(3.1) with the single amplitude $D^{\rm iso}_{1}$. The physical significance
of $\xi_{0\pm}^{\rm iso}$ right at $T_{c}$ is relevant for the applicability
of our generalized shear transformation at $T_{c}$.

We shall also need the Fourier transform

$\displaystyle\hat{G}^{\rm iso}_{\pm}(|{\bf k}|,t)$ $\displaystyle=$
$\displaystyle\int d^{d}{\bf x}\;e^{-i{\bf k}\cdot{\bf x}}G^{\rm
iso}_{\pm}(|{\bf x}|,t)$ (3.50) $\displaystyle=$
$\displaystyle\frac{\Gamma^{\rm iso}_{+}}{\big{(}|{\bf k}|\;\xi^{\rm
iso}_{0+}\big{)}^{2-\eta}}\;\hat{\Psi}_{\pm}\Big{(}|{\bf k}|\xi^{\rm
iso}_{\pm}(t)\Big{)}\;$ (3.51)

with the universal scaling function

$\displaystyle\hat{\Psi}_{\pm}(y_{\pm})=\frac{2\pi^{(d-1)/2}}{\Gamma\big{(}(d-1)/2\big{)}}\int\limits_{0}^{\infty}ds\;s^{1-\eta}$
$\displaystyle\times\int\limits_{-1}^{1}d(\cos\vartheta)(\sin\vartheta)^{d-3}\;e^{-is\cos\vartheta}\Psi_{\pm}(s/y_{\pm}).\;\;\;$
(3.52)

The existence of the scaling function $\hat{\Psi}_{+}\Big{(}|{\bf k}|\xi^{\rm
iso}_{+}(t)\Big{)}$ at infinite cutoff has been shown in $4-\varepsilon$
dimensions up to $O(\varepsilon^{2})$ for general $n$ Fisher-Aharony-1974 .
Because of the sum rule (3.40) we have for $t\neq 0$

$\displaystyle\hat{G}_{\pm}^{\rm iso}(0,t)=\chi^{\rm iso}_{\pm}(t)=\Gamma^{\rm
iso}_{\pm}|t|^{-\gamma}.$ (3.53)

The scaling function $\hat{\Psi}_{\pm}$ has a universal value at $T_{c}$
corresponding to $y_{\pm}=\infty$ tarko ; priv ,

$\hat{\Psi}_{+}(\infty)=\hat{\Psi}_{-}(\infty)=Q_{3}={\rm universal}.$ (3.54)

This implies the asymptotic universal scaling form at $T_{c}$

$\displaystyle\hat{G}_{\pm}^{\rm iso}(|{\bf k}|,0)=Q_{3}\frac{\Gamma^{\rm
iso}_{+}}{\big{(}|{\bf k}|\;\xi^{\rm iso}_{0+}\big{)}^{2-\eta}}\;\;$ (3.55)

where again two independent bulk amplitudes $\Gamma^{\rm iso}_{+}$ and
$\xi^{\rm iso}_{0+}$ appear. The universal constants $Q_{3}$ and
$\widetilde{Q}_{3}$ are related by a Fourier transformation which yields
dohm2008 ; dohm2019  (see App. A)

$\displaystyle\widetilde{Q}_{3}=\frac{2^{d-2+\eta}\Gamma[(d-2+\eta)/2]}{(4\pi)^{d/2})\Gamma[(2-\eta)/2]}\;Q_{3}.\;\;\;\;$
(3.56)

Both $Q_{3}$ and $\widetilde{Q}_{3}$ are known exactly for the $d=2$ Ising
universality class dohm2019 ; tarko  and approximately for $d=3,n=1$ priv ;
tarko ; brezin1974 . In Sec. V. E we shall show how to derive
$\hat{\Psi}_{\pm}$ and $Q_{3}$ from an approximate result of the FRG
hassel2007 ; sinner2008 .

Owing to the sum rule the nonuniversal overall amplitudes of
$\hat{G}_{\pm}^{\rm iso}$ can be expressed completely in terms of $\chi^{\rm
iso}_{\pm}(t)$ if $t\neq 0$. Using (3.51), (3.53), (3.3), (3.4)and (3.6)) we
obtain the representation for $t\neq 0$

$\displaystyle\hat{G}_{\pm}^{\rm iso}(|{\bf k}|,t)$ $\displaystyle=$
$\displaystyle\Gamma^{\rm iso}_{\pm}|t|^{-\gamma}\;\hat{D}_{\pm}\Big{(}|{\bf
k}|\xi^{\rm iso}_{\pm}(t)\Big{)},$ (3.57)

where the universal single-parameter scaling functions $\hat{D}_{\pm}$ are
universally related to the universal functions $\hat{\Psi}_{\pm}$ as

$\displaystyle\hat{D}_{+}(y_{+})$ $\displaystyle=$ $\displaystyle
y_{+}^{-2+\eta}\hat{\Psi}_{+}(y_{+}),$ (3.58)
$\displaystyle\hat{D}_{-}(y_{-})$ $\displaystyle=$ $\displaystyle
U_{2}\;(X_{\xi})^{-2+\eta}\;y_{-}^{-2+\eta}\hat{\Psi}_{-}(y_{-}),$ (3.59)

with the universal constants $U_{2}$ and $X_{\xi}$. These scaling functions
satisfy the normalization condition

$\displaystyle\hat{D}_{\pm}(0)$ $\displaystyle=$ $\displaystyle 1$ (3.60)

in agreement with the scaling form (2.20 and (2.21). This condition is
equivalent to our normalization condition (3.48). The same scaling form was
derived within the framework of two-scale-factor universality by Hohenberg et
al. hohenberg1976  where $\hat{D}_{\pm}$ was denoted by $\widetilde{Z}$. Thus,
in the asymptotic region where $\hat{D}_{\pm}$ is calculated at infinite
cutoff, $\hat{G}_{\pm}^{\rm iso}(|{\bf k}|,t)$ depends on two independent
nonuniversal amplitudes $\Gamma_{+}^{\rm iso}$ and $\xi^{\rm iso}_{0+}$. Since
(3.57) does not exist at $t=0$ our representations (3.45) and (3.51) have the
advantage that they are applicable to the asymptotic region including the
critical point $t=0$. In particular our scaling functions $\Psi_{\pm}$ and
$\hat{\Psi}_{\pm}$ immediately capture the universal constants
$\widetilde{Q}_{3}=\Psi_{\pm}(0)$ and $Q_{3}=\hat{\Psi}_{\pm}(\infty)$, in
contrast to the scaling functions $\hat{D}_{\pm}$ and its inverse Fourier
transformed counterpart $D_{\pm}$ in real space.

The advantage of the representations (3.45) and (3.48) over (3.1) is that,
unlike the amplitude $D_{1}^{\rm iso}$, both the amplitude $\Gamma^{\rm
iso}_{+}$ of the bulk susceptibility $\chi^{\rm iso}_{+}$ above $T_{c}$ and
$\xi_{0+}^{\rm iso}$ can be determined directly from independent thermodynamic
measurements as follows. The bulk susceptibility

$\displaystyle\chi^{\rm iso}_{+}(t)$ $\displaystyle=$
$\displaystyle-\lim_{h\to 0}\partial^{2}f^{\rm iso}_{b,+}(t,h)/\partial h^{2}$
(3.61) $\displaystyle=$ $\displaystyle\lim_{h\to 0}\partial m^{\rm
iso}(t,h)/\partial h$ (3.62)

can be determined via a measurement of the order parameter $m^{\rm iso}(t,h)$
for small $h\to 0$. Furthermore, owing to two-scale-factor universality, the
bulk correlation length

$\displaystyle\xi_{0+}^{\rm iso}$ $\displaystyle=$
$\displaystyle(Q_{1}/A_{+}^{\rm iso})^{1/d}$ (3.63)

is determined by the amplitude $A_{+}^{\rm iso}$ of the free energy density
according to (3.16). The latter amplitude can be measured via the singular
part of the bulk specific heat per unit volume (divided by $k_{B}$) above
$T_{c}$

$\displaystyle C^{\rm iso}_{b,s,+}(t)$ $\displaystyle=$
$\displaystyle-\partial^{2}f^{\rm iso}_{b,s,+}(t)/\partial t^{2}$ (3.64)

of isotropic systems where $f^{\rm iso}_{b,s,+}(t)$ is given by (3.14). This
yields the dependence on $\xi^{\rm iso}_{0+}$

$\displaystyle C^{\rm iso}_{b,s+}(t)$ $\displaystyle=$
$\displaystyle\frac{\big{(}R^{+}_{\xi}\big{)}^{d}}{\alpha\;(\xi^{\rm
iso}_{0+})^{d}}\;t^{-\alpha},\;\;\;d>2,$ (3.65) $\displaystyle C^{\rm
iso}_{b,s+}(t)$ $\displaystyle=$ $\displaystyle-\;\frac{Q_{1}}{\big{(}\xi^{\rm
iso}_{0+}\big{)}^{2}}\;\ln t,\;\;d=2,\;\;\;\;$ (3.66)

with the universal constant

$\displaystyle\frac{(R^{+}_{\xi})^{d}}{\alpha(1-\alpha)(2-\alpha)}=-\;Q_{1},\;\;\;d>2.\;\;$
(3.67)

Thus the important consequence of two-scale-factor universality for isotropic
bulk systems is that all nonuniversal parameters of the isotropic bulk
correlation function can be completely determined by the thermodynamic
amplitudes of the bulk susceptibility and the bulk specific heat without
involvement of measurements of the spatial dependence of the correlation
function via scattering experiments. We shall show in Sec. V that weak
anisotropy destroys this important feature due to the intrinsic
nonuniversality of the correlation function.

###  III.4 Critical free energy in finite isotropic systems

In Sec. IX we shall also consider the singular part of the free energy at
$T_{c}$ of a finite anisotropic system on the basis of its structure in the
isotropic case. For finite isotropic systems, e.g., in a $d$-dimensional
parallelepiped of volume $V^{\rm iso}$ with given aspect ratios and given
angles between the confining surfaces, the hypothesis of two-scale-factor
universality for finite systems priv ; pri  predicts that, for given boundary
conditions, the singular part ${\cal F}^{\rm iso}_{s}$ of the free energy
${\cal F}_{\rm tot}^{\rm iso}$ has a universal value at $T=T_{c}$ with the
finite critical amplitude

$\displaystyle{\cal F}^{\rm iso}_{c}=\lim_{T\to T_{c}}{\cal F}_{s}^{\rm
iso}=\text{universal}$ (3.68)

which depends on $d,n,$ and is a universal function of the aspect ratios and
angles.

The universal structure of (3.1), (3.7), (3.45), (3.51), and (3.65)-(3.68) as
well as the scaling relations (3.22)-(3.25) and (3.38) are a consequence of
the principle of two-scale-factor universality for isotropic systems. In the
subsequent sections we address the question how weak anisotropy affects the
structure of these results. We shall show that multiparameter universality
with up to $d(d+1)/2+1$ independent nonuniversal parameters replaces two-
scale-factor universality not only within the anisotropic $\varphi^{4}$ theory
and the anisotropic Gaussian model dohm2008 ; dohm2018 ; dohm2019  but quite
generally in all weakly anisotropic systems.

##  IV Shear transformations of anisotropic systems

Although the analysis of this section is formulated primarily for
$O(n)$-symmetric systems with general $n\geq 1,T\geq T_{c}$ and $n=1,T<T_{c}$
all definitions and results can be extended to systems with $n>1,T<T_{c}$
where transverse correlation lengths come into play dohm2018 .

Our strategy is to perform a shear transformation from the anisotropic to an
isotropic system, then to invoke two-scale-factor universality for the
isotropic system with the known structures of the correlation function, of the
free energy, and of universal amplitude relations, and finally to derive the
anisotropic structures by inverting the shear transformation. However, the
choice of the shear transformation is not unique.

###  IV.1 Special shear transformation of the $\varphi^{4}$ theory

For the subsequent development of the anisotropic theory it is indispensable
to complement and reformulate the special shear transformation that is most
suitable for the weakly anisotropic $\varphi^{4}$ model cd2004 ; dohm2006 ;
dohm2008 ; dohm2018 . In the absence of an external field this transformation
is defined by

$\displaystyle{\bf x^{\prime}}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath$\lambda$}}^{-1/2}{\bf U}{\bf x},$ (4.1)
$\displaystyle\varphi^{\prime}({\bf x^{\prime}})$ $\displaystyle=$
$\displaystyle(\det{\bf A})^{1/4}\varphi({\bf x}),$ (4.2) $\displaystyle
u^{\prime}_{0}$ $\displaystyle=$ $\displaystyle(\det{\bf A})^{-1/2}u_{0},$
(4.3)

where ${\bf A}={\bf A}(\\{K_{i,j}\\})$ is given by (2.3). This $T$-independent
transformation is defined at fixed couplings $K_{i,j}$ for arbitrary
temperatures above, at, and below $T_{c}$ and leaves the distance
$r_{0}-r_{0c}$ from criticality invariant dohm2008 . It consists of a rotation
of the lattice points ${\bf x}$ provided by the orthogonal matrix ${\bf U}$
and a subsequent spatial rescaling by the diagonal rescaling matrix $\lambda$
such that the $O(\bf k^{2})$ part of the Fourier transform $\delta\hat{K}({\bf
k})$ of the anisotropic interaction of the Hamiltonian (2.1) dohm2006 ;
dohm2008  is brought into the isotropic form

$\displaystyle\delta\hat{K}({\bf k})$ $\displaystyle=$
$\displaystyle\delta\hat{K}({\bf U}^{-1}{\mbox{\boldmath$\lambda$}}^{-1/2}{\bf
k^{\prime}})\equiv\delta\hat{K}^{\prime}({\bf k^{\prime}})={\bf
k^{\prime}}\cdot{\bf k^{\prime}},\;\;\;\;$ (4.4) $\displaystyle{\bf
k^{\prime}}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath$\lambda$}}^{1/2}{\bf U}{\bf k}.$ (4.5)

Note, however, that $O(\bf k^{4})$ parts remain anisotropic in general and
give rise to anisotropic corrections outside the isotropic scaling regime of
the transformed system (see Fig. 2 of dohm2008 ). The matrix elements of

${\bf U}={\bf U}\big{(}\\{{\bf e}^{(\alpha)}\\}\big{)}$ (4.6)

are $U_{\alpha\beta}=e_{\beta}^{(\alpha)}$ which are determined by the $d$
eigenvectors ${\bf e}^{(\alpha)}$ of the matrix ${\bf A}$ whose Cartesian
components are denoted by $e_{\beta}^{(\alpha)}$. We call these vectors
principal unit vectors. They satisfy the eigenvalue equation

$\displaystyle{\bf A}(\\{K_{i,j}\\})\;{\bf e}^{(\alpha)}$ $\displaystyle=$
$\displaystyle{\mbox{$\lambda$}_{\alpha}}\;{\bf
e}^{(\alpha)},\;\alpha=1,2,...,d,\;\;$ (4.7)

with ${\bf e}^{(\alpha)}\cdot{\bf e}^{(\beta)}=\delta_{\alpha\beta}$. The
matrix ${\bf U}$ diagonalizes the matrix ${\bf A}$ according to

${\mbox{\boldmath$\lambda$}}(\\{K_{i,j}\\})={\bf U}{\bf A}(\\{K_{i,j}\\}){\bf
U}^{-1}$ (4.8)

whose diagonal elements are the eigenvalues $\lambda_{\alpha}>0$ of ${\bf A}$.
Thus we have

$\det{\bf
A}=\det{\mbox{\boldmath$\lambda$}}=\prod^{d}_{\alpha=1}{\mbox{$\lambda$}_{\alpha}}.$
(4.9)

The eigenvalue equation (4.7) determines the eigenvalues $\lambda_{\alpha}$
and the principal unit vectors ${\bf e}^{(\alpha)}$ as a function of the
couplings $K_{i,j}$. The latter determine the directions of the principal axes
of the large-distance correlations of the anisotropic system above, at, and
below $T_{c}$ cd2004 ; dohm2008 ; dohm2018 . For the formulation of bulk and
finite-size properties of anisotropic systems it will be necessary to define
the reduced anisotropy matrix

$\displaystyle{\bf\bar{A}}(\\{K_{i,j}\\})$ $\displaystyle=$ $\displaystyle{\bf
A}/(\det{\bf A})^{1/d}.$ (4.10)

An important alternative definition is

$\displaystyle{\bf\bar{A}}$ $\displaystyle=$ $\displaystyle{\bf
U}^{-1}{\bar{\mbox{\boldmath$\lambda$}}}{\bf U}$ (4.11)

where ${\bar{\mbox{\boldmath$\lambda$}}}$ is the reduced rescaling matrix

$\displaystyle{\bar{\mbox{\boldmath$\lambda$}}}={{\mbox{\boldmath$\lambda$}}}/(\det{\mbox{\boldmath$\lambda$}})^{1/d}$
(4.12)

as follows from (4.8). According to (4.11) and (4.12), ${\bf\bar{A}}$ is known
if the ratios of the eigenvalues $\lambda_{\alpha}$ of ${\bf A}$ and the
principal unit vectors ${\bf e}^{(\alpha)}$ are given. This makes possible to
determine ${\bf\bar{A}}$ even if the matrix ${\bf A}$ is not known. It is the
definition (4.11) rather than (4.10) that is used in deriving the shear
transformation of the bulk correlation functions in Sec. V [compare (5.19) and
(5.37)].

The anisotropic continuum Hamiltonian (2.18) is invariant under the special
shear transformation (4.1)-(4.3) and is transformed to the Hamiltonian
$H_{\text{field}}^{\prime}$ which has the standard isotropic form

$\displaystyle H_{\text{field}}$ $\displaystyle=$ $\displaystyle
H_{\text{field}}^{\prime}$ (4.13) $\displaystyle=$
$\displaystyle\int_{V^{\prime}}d^{d}x^{\prime}\Big{[}\frac{r_{0}}{2}\varphi^{\prime}({\bf
x}^{\prime})^{2}+\frac{1}{2}(\nabla^{\prime}\varphi^{\prime})^{2}+u^{\prime}_{0}(\varphi^{\prime
2})^{2}\Big{]},\;\;\;\;\;\;\;\;\;\;\;$

with the transformed volume

$\displaystyle V^{\prime}$ $\displaystyle=$
$\displaystyle\int_{V^{\prime}}d^{d}x^{\prime}=\det{\mbox{\boldmath$\lambda$}}^{-1/2}\;\int_{V}d^{d}x$
(4.14) $\displaystyle=$
$\displaystyle(\det{\mbox{\boldmath$\lambda$}})^{-1/2}\;V=(\det{\bf
A})^{-1/2}\;V$ (4.15)

and with a transformed cutoff which in general is still anisotropic. The
anisotropic bulk correlation function (2.16), its Fourier transform, and the
bulk order parameter ${\cal M}=<\varphi>$ are transformed as

$\displaystyle G_{\pm}^{\prime}({\bf x^{\prime}},t)$ $\displaystyle=$
$\displaystyle(\det{\bf A})^{1/2}G_{\pm}({\bf x},t),\;\;\;\;\;$ (4.16)
$\displaystyle\hat{G}_{\pm}^{\prime}({\bf k^{\prime}},t)$ $\displaystyle=$
$\displaystyle\hat{G}_{\pm}({\bf k},t),$ (4.17) $\displaystyle{\cal
M^{\prime}}(t)$ $\displaystyle=$ $\displaystyle(\det{\bf A})^{1/4}{\cal
M}(t).$ (4.18)

The bulk susceptibility remains invariant as follows from the sum rules for
the bulk correlation functions

$\displaystyle\chi_{\pm}^{\prime}(t)=\int d^{d}{\bf
x^{\prime}}\;\;G_{\pm}^{\prime}({\bf x^{\prime}},t)$ (4.19)
$\displaystyle=\int d^{d}{\bf x}\;\;G_{\pm}({\bf
x},t)=\chi_{\pm}(t).\;\;\;\;\;\;$ (4.20)

The partition function and the total free energy are not invariant but are
transformed as dohm2008

$\displaystyle Z^{\prime}$ $\displaystyle=$ $\displaystyle(\det{\bf
A})^{nN/(2d)}\;Z,$ (4.21) $\displaystyle{\cal F}^{\prime}_{\rm tot}$
$\displaystyle=$ $\displaystyle{\cal F}_{\rm tot}\;-\;[nN/(2d)]\ln(\det{\bf
A})\;$ (4.22)

where the last term is a nonsingular bulk contribution (which is absent in our
generalized shear transformation of the $n$-vector model to be defined below.)
Thus the singular part ${\cal F}_{s}$ of the total free energy is invariant
under the shear transformation,

${\cal F}^{\prime}_{s}={\cal F}_{s},$ (4.23)

and the singular part $f_{b,s}(t)$ of the bulk-free energy density is
transformed as

$\displaystyle f^{\prime}_{b,s,\pm}(t)$ $\displaystyle=$
$\displaystyle\lim_{V^{\prime}\to\infty}{\cal
F}^{\prime}_{s}/V^{\prime}=(\det{\bf A})^{1/2}f_{b,s,\pm}(t).\;\;\;$ (4.24)

This implies that the singular bulk part ${\cal F}_{b,s,\pm}$ of the free
energy

${\cal F}^{\prime}_{b,s,\pm}=V^{\prime}f^{\prime}_{b,s}=Vf_{b,s,\pm}={\cal
F}_{b,s,\pm}$ (4.25)

is invariant under the shear transformation.

All of the above relations are exact and a consequence of the special shear
transformation (4.1)-(4.3). They are valid for arbitrary $t$ above, at, and
below $T_{c}$. So far the quantities ${\bf e}^{(\alpha)}$,
${\bar{\mbox{\boldmath$\lambda$}}}$, ${\bar{\mbox{\boldmath$\lambda$}}}$,
${\bf A}$, and ${\bf\bar{A}}$ governing the transformations are nonuniversal
quantities which are defined as a function of the couplings $K_{i,j}$
according to (2.3). This was called parametrization (i) in dohm2018 . They are
independent of $n$ and $u_{0}$ and remain valid in the large-$n$ limit and for
the Gaussian model.

For the purpose of a later extension of the anisotropic theory to systems
other than the $\varphi^{4}$ model we confine ourselves in the following to
the asymptotic critical scaling region where this parametrization in terms of
$K_{i,j}$ can be reformulated in favor of a parametrization in terms of
correlation lengths. This was called parametrization (ii) in Sec. V. of
dohm2018 . This critical scaling region is defined by large $|{\bf
x^{\prime}}|$, large $\xi^{\prime}_{\pm}(t)$, but finite $|{\bf
x^{\prime}}|/\xi^{\prime}_{\pm}(t)\geq 0$, where

$\xi^{\prime}_{\pm}(t)=\xi^{\prime}_{0\pm}|t|^{-\nu}.$ (4.26)

is the asymptotic isotropic correlation length defined through the exponential
decay of the correlation function $G_{\pm}^{\prime}({\bf x^{\prime}},t)$ of
the Hamiltonian (4.13). In this scaling region $G_{\pm}^{\prime}({\bf
x^{\prime}},t)$ depends only on $|{\bf x^{\prime}}|$ rather than ${\bf
x^{\prime}}$ thus we shall make the replacement

$G_{\pm}^{\prime}({\bf x^{\prime}},t)\longrightarrow G_{\pm}^{\prime}(|{\bf
x^{\prime}}|,t)$ (4.27)

in the remainder of this paper, and correspondingly

$\hat{G}_{\pm}^{\prime}({\bf
k^{\prime}},t)\longrightarrow\hat{G}_{\pm}^{\prime}(|{\bf k^{\prime}}|,t).$
(4.28)

The amplitudes $\xi^{\prime}_{0+}$ and $\xi^{\prime}_{0-}$ are universally
related by

$\xi^{\prime}_{0+}/\xi^{\prime}_{0-}=X_{\xi}$ (4.29)

where $X_{\xi}$ is the same universal constant as in (3.3). The principal
correlation lengths above and below $T_{c}$ have been determined as cd2004

$\displaystyle\xi^{(\alpha)}_{\pm}(t)$ $\displaystyle=$
$\displaystyle\xi^{(\alpha)}_{0\pm}|t|^{-\nu}={\mbox{$\lambda$}_{\alpha}}^{1/2}\xi^{\prime}_{\pm}(t).$
(4.30)

As a consequence their amplitude ratios are independent of the direction
$\alpha$ and are universal quantities given by

$\displaystyle\xi^{(\alpha)}_{0+}/\xi^{(\alpha)}_{0-}=\xi^{\prime}_{0+}/\xi^{\prime}_{0-}=X_{\xi}\;\;\text{for
each}\;\;\alpha$ (4.31)

where $X_{\xi}$ is independent of $\alpha$ and the same universal constant as
in (3.3). This holds for arbitrary short-range interactions and lattice
structures of weakly anisotropic $\varphi^{4}$ models. Together with the fact
that the critical exponents above, at, and below $T_{c}$ of weakly anisotropic
systems are the same as those of isotropic systems in the same universality
class we consider the result (4.31) for the amplitude ratio
$\xi^{(\alpha)}_{0+}/\xi^{(\alpha)}_{0-}$ to be an additional characteristic
feature of weakly anisotropic systems.

According to (4.30) the eigenvalues can be expressed in terms of ratios of
correlation lengths as

$\displaystyle{\mbox{$\lambda$}_{\alpha}}^{1/2}$ $\displaystyle=$
$\displaystyle\xi^{(\alpha)}_{0+}/\xi^{\prime}_{0+}=\xi^{(\alpha)}_{0-}/\xi^{\prime}_{0-}$
(4.32)

which are the same above and below $T_{c}$. This implies

$\displaystyle(\det{\mbox{\boldmath$\lambda$}})^{1/2}$ $\displaystyle=$
$\displaystyle\prod^{d}_{\alpha=1}\big{(}\xi^{(\alpha)}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}=\big{(}\bar{\xi}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}^{d}$
(4.33)

with the mean correlation length

$\displaystyle\bar{\xi}_{\pm}(t)$ $\displaystyle=$
$\displaystyle\big{[}\prod^{d}_{\alpha=1}\xi_{\pm}^{(\alpha)}(t)\big{]}^{1/d}=\bar{\xi}_{0\pm}|t|^{-\nu},$
(4.34) $\displaystyle\bar{\xi}_{0\pm}$ $\displaystyle=$
$\displaystyle\big{[}\prod^{d}_{\alpha=1}\xi_{0\pm}^{(\alpha)}\big{]}^{1/d},$
(4.35) $\displaystyle\bar{\xi}_{0+}/\bar{\xi}_{0-}$ $\displaystyle=$
$\displaystyle X_{\xi}.$ (4.36)

Equations (4.33) and (4.9) yield the desired representation of $\det{\bf A}$
in terms of correlation lengths as

$\displaystyle(\det{\bf A})^{1/2}$ $\displaystyle=$
$\displaystyle\prod^{d}_{\alpha=1}\big{(}\xi^{(\alpha)}_{0+}/\xi^{\prime}_{0+}\big{)}=\big{(}\bar{\xi}_{0+}/\xi^{\prime}_{0+}\big{)}^{d}$
(4.37) $\displaystyle=$
$\displaystyle\prod^{d}_{\alpha=1}\big{(}\xi^{(\alpha)}_{0-}/\xi^{\prime}_{0-}\big{)}=\big{(}\bar{\xi}_{0-}/\xi^{\prime}_{0-}\big{)}^{d}.$
(4.38)

This enables us to reformulate the special shear transformations (4.14)-(4.18)
and (4.24) in terms of correlation lengths as

$\displaystyle V^{\prime}$ $\displaystyle=$
$\displaystyle\big{(}\bar{\xi}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}^{-d}\;V,$
(4.39) $\displaystyle G^{\prime}_{\pm}(|{\bf x^{\prime}}|,t)$ $\displaystyle=$
$\displaystyle\big{(}\bar{\xi}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}^{d}\;G_{\pm}({\bf
x},t),\;\;\;\;\;$ (4.40) $\displaystyle{\cal M^{\prime}}(t)$ $\displaystyle=$
$\displaystyle\big{(}\bar{\xi}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}^{d/2}\;{\cal
M}(t),$ (4.41) $\displaystyle f^{\prime}_{b,s}(t)$ $\displaystyle=$
$\displaystyle\big{(}\bar{\xi}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}^{d}\;f_{b,s}(t),$
(4.42)

with the $T$-independent transformation factor
$\big{(}\bar{\xi}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}^{d}$. This reformulation
is needed for the derivation of the anisotropic physical properties within the
$\varphi^{4}$ theory in the subsequent sections.

The factor $\big{(}\bar{\xi}_{0\pm}/\xi^{\prime}_{0\pm}\big{)}^{d}$ can be
interpreted as the ratio of the ellipsoidal correlation volume

$V_{\rm
cor,\pm}=\prod^{d}_{\alpha=1}\xi_{0\pm}^{(\alpha)}=\big{(}\bar{\xi}_{0\pm}\big{)}^{d}$
(4.43)

of the anisotropic system and the spherical correlation volume

$V^{\prime}_{\rm cor,\pm}=\big{(}\xi^{\prime}_{0\pm}\big{)}^{d}$ (4.44)

of the transformed isotropic system, with the ratio

$V_{\rm cor,+}/V_{\rm cor,-}=V^{\prime}_{\rm cor,+}/V^{\prime}_{\rm
cor,-}=(X_{\xi})^{d}.$ (4.45)

Eq. (4.39) can be rewritten as

$\displaystyle\frac{V^{\prime}}{V^{\prime}_{\rm cor,+}}=\frac{V}{V_{\rm
cor,+}},\;\;\;\frac{V^{\prime}}{V^{\prime}_{\rm cor,-}}$ $\displaystyle=$
$\displaystyle\frac{V}{V_{\rm cor,-}}.$ (4.46)

This means that the ratios of the geometric volumes and correlation volumes
remain invariant under the special shear transformation (4.1) above and below
$T_{c}$. This implies that the quantity

$\displaystyle
t[V^{\prime}/(\xi^{\prime}_{0+})^{d}]^{1/(d\nu)}=t[V/(\bar{\xi}_{0+})^{d}]^{1/(d\nu)},$
(4.47)

appearing as the scaling variable in the bulk and finite-size theory of the
$\varphi^{4}$ theory, remains invariant under the special shear
transformation. Eqs. (4.37) and (4.38) can be rewritten as

$\displaystyle\big{(}\xi^{\prime}_{0\pm}\big{)}^{d}$ $\displaystyle=$
$\displaystyle(\det{\bf A})^{-1/2}\big{(}\bar{\xi}_{0\pm}\big{)}^{d},$ (4.48)

which describes the shear transformation of the anisotropic correlation volume
to the isotropic correlation volume.

The invariance of the susceptibility (4.19) yields in the asymptotic critical
region

$\displaystyle\chi_{\pm}(t)$ $\displaystyle=$
$\displaystyle\Gamma_{\pm}|t|^{-\gamma}=\chi_{\pm}^{\prime}(t)=\Gamma^{\prime}_{\pm}|t|^{-\gamma},$
(4.49) $\displaystyle\Gamma^{\prime}_{\pm}$ $\displaystyle=$
$\displaystyle\Gamma_{\pm}.$ (4.50)

This implies the invariance of the combination

$\displaystyle\big{(}\xi^{\prime}_{0\pm}\big{)}^{d}G_{\pm}^{\prime}(|{\bf
x^{\prime}}|,t)/\Gamma^{\prime}$ $\displaystyle=$
$\displaystyle\big{(}\bar{\xi}_{0\pm}\big{)}^{d}G_{\pm}({\bf x},t)/\Gamma$
(4.51)

under the special shear transformation. We shall show that the invariance of
both the combination (4.51) and the volume ratio (4.46) are universal features
of weakly anisotropic systems.

The two independent nonuniversal parameters $\xi^{\prime}_{0\pm}$ and
$\Gamma^{\prime}_{\pm}$ will not be specified further. They can be determined
from the correlation function and the susceptibility as functions of the
parameter $a_{0}$ in (2.2) and the coupling $u^{\prime}_{0}$ of the isotropic
Hamiltonian (4.13) where, according to (4.3), $u^{\prime}_{0}$ depends on the
four-point coupling $u_{0}$ and on $K_{i,j}$ through the anisotropy matrix
${\bf A}$ of the original anisotropic $\varphi^{4}$ Hamiltonian. We note that
$\xi^{\prime}_{0\pm}$ plays the role as a reference length in the framework of
renormalized perturbation theory dohm2008 ; dohm2018  based on the transformed
$\varphi^{4}$ Hamiltonian (4.13).

It is important to note that the ratios of the eigenvalues can be expressed in
terms of the ratio of the principal correlation lengths

$\displaystyle{\mbox{$\lambda$}_{\alpha}}/{\mbox{$\lambda$}_{\beta}}=\Big{(}\xi^{(\alpha)}_{0\pm}/\xi^{(\beta)}_{0\pm}\Big{)}^{2}$
(4.52)

as follows from (4.30). This makes possible to reexpress the diagonal elements
of the reduced rescaling matrix (4.12) in terms of ratios of principal
correlation lengths as

$\displaystyle\bar{\lambda}_{\alpha}\big{(}\\{\xi_{0\pm}^{(\alpha)}\\}\big{)}$
$\displaystyle=$
$\displaystyle\prod^{d}_{\beta=1,\;\beta\neq\alpha}\Big{(}\frac{\xi_{0+}^{(\alpha)}}{\xi_{0+}^{(\beta)}}\Big{)}^{2/d}=\Big{(}\frac{\xi_{0+}^{(\alpha)}}{\bar{\xi}_{0+}}\Big{)}^{2}$
(4.53) $\displaystyle=$
$\displaystyle\prod^{d}_{\beta=1,\;\beta\neq\alpha}\Big{(}\frac{\xi_{0-}^{(\alpha)}}{\xi_{0-}^{(\beta)}}\Big{)}^{2/d}=\Big{(}\frac{\xi_{0-}^{(\alpha)}}{\bar{\xi}_{0-}}\Big{)}^{2}\;.\;\;\;\;$
(4.54)

Here the isotropic correlation length $\xi^{\prime}_{0\pm}$ has been canceled.
The rescaling and reduced rescaling matrices are related by

$\displaystyle{\mbox{\boldmath$\lambda$}}^{1/2}\xi^{\prime}_{0\pm}$
$\displaystyle=$
$\displaystyle{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}\bar{\xi}_{0\pm}.$ (4.55)

From the alternative definition (4.11) and from (4.53) and (4.54) we then
obtain the reduced anisotropy matrix in terms of ratios of principal
correlation lengths as

${\bf\bar{A}}={\bf\bar{A}}\big{(}\\{\xi_{0\pm}^{(\alpha)},{\bf
e}^{(\alpha)}\\}\big{)}={\bf U(\\{{\bf
e}^{(\alpha)}\\})}^{-1}{\bf\bar{\mbox{\boldmath$\lambda$}}}\big{(}\\{\xi_{0\pm}^{(\alpha)}\\}\big{)}{\bf
U(\\{{\bf e}^{(\alpha)}\\})}$ (4.56)

which is the parametrization (ii) of ${\bf\bar{A}}$ as given in Eqs. (5.12)
and (5.30) of dohm2018 . Both ${\bf\bar{\mbox{\boldmath$\lambda$}}}$ and
${\bf\bar{A}}$ are nonuniversal quantities since they depend on nonuniversal
parameters $\xi_{0\pm}^{(\alpha)}$ and ${\bf e}^{(\alpha)}$. We shall show in
Sec. IV.C, however, that in this parametrization (ii) the dependence of the
matrices
${\bf\bar{\mbox{\boldmath$\lambda$}}}\big{(}\\{\xi_{0\pm}^{(\alpha)}\\}\big{)}$
and ${\bf\bar{A}}\big{(}\\{\xi_{0\pm}^{(\alpha)},{\bf e}^{(\alpha)}\\}\big{)}$
on these parameters attains a universal structure, unlike the parametrization
(i) in terms of the couplings.

All of the relations presented above for the $\varphi^{4}$ theory at finite
$n$ remain applicable also in the large $n$-limit. Furthermore they remain
applicable to the Gaussian lattice model for $T\geq T_{c}$,

$\displaystyle H^{G}$ $\displaystyle=$ $\displaystyle
v\Bigg{[}\sum_{i=1}^{N}\frac{r_{0}}{2}\varphi_{i}^{2}+\sum_{i,j=1}^{N}\frac{K_{i,j}}{2}(\varphi_{i}-\varphi_{j})^{2}\Bigg{]},\;\;\;\;\;\;\;\;$
(4.57)

$r_{0}(T)=a_{0}t\geq 0$, which is obtained from (2.1) by setting $u_{0}=0$ and
$r_{0c}=0$. We shall consider the continuum version of this model in the form

$\displaystyle
H^{G}_{\text{field}}=\int_{V}d^{d}x\Big{[}\frac{r_{0}}{2}\varphi^{2}+\sum_{\alpha,\beta=1}^{d}\frac{A_{\alpha\beta}}{2}\frac{\partial\varphi}{\partial
x_{\alpha}}\frac{\partial\varphi}{\partial x_{\beta}}\Big{]}\;\;\;\;\;\;$
(4.58)

with the same anisotropy matrix ${\bf A}$ (2.3) as for the $\varphi^{4}$
model. Correspondingly the isotropic Gaussian Hamiltonian obtained after the
special shear transformation (4.1) and (4.2) reads

$\displaystyle H^{\prime G}_{\text{field}}$ $\displaystyle=$
$\displaystyle\int_{V^{\prime}}d^{d}x^{\prime}\Big{[}\frac{r_{0}}{2}\varphi^{\prime}({\bf
x}^{\prime})^{2}+\frac{1}{2}(\nabla^{\prime}\varphi^{\prime})^{2}\Big{]}.\;\;\;\;\;\;\;\;$
(4.59)

The advantage of the special shear transformation (4.1)-(4.3) is that the
transformed Hamiltonian (4.13) has the form of the standard isotropic field-
theoretic Landau-Ginzburg-Wilson Hamiltonian. This implies that the same
renormalization factors ($Z$ factors) can be employed for the transformed
system as for the established isotropic $\varphi^{4}$ field theory. This has
permitted us to perform renormalized perturbation theory dohm2008 ; dohm2018
for the transformed isotropic system in $2<d<4$ dimensions.

The disadvantage of this shear transformation is, however, that it is not
directly applicable to other weakly anisotropic systems such as the $n$-vector
model where it is unknown how to construct an appropriate continuum
Hamiltonian with a large-distance anisotropy matrix as a function of the
couplings $E_{i,j}$ that plays the same role as ${\bf A}$ in the $\varphi^{4}$
theory. In the following we introduce a generalized shear transformation that
does not need an anisotropy matrix and that is applicable to all weakly
anisotropic systems.

###  IV.2 Generalized shear transformation of the $\varphi^{4}$ theory

We aim at introducing a shear transformation of the $\varphi^{4}$ theory in
the scaling region that differs from the special shear transformation in that
the explicit use of the anisotropy matrix ${\bf A}$ is avoided. The principal
correlation lengths of the $\varphi^{4}$ model above, at, and below $T_{c}$
are directed along the $T$-independent principal unit vectors ${\bf
e}^{(\alpha)}$ . Accordingly we introduce the orthogonal set of $d$ principal
correlation vectors

$\displaystyle{\mbox{\boldmath$\xi$}}_{\pm}^{(\alpha)}(t)={\mbox{\boldmath$\xi$}}_{0\pm}^{(\alpha)}|t|^{-\nu}=\xi^{(\alpha)}_{0\pm}|t|^{-\nu}{\bf
e}^{(\alpha)},\;\;\alpha=1,...,d\;\;\;$ (4.60)

where $\xi^{(\alpha)}_{0\pm}$ are the amplitudes of the principal correlation
lengths. We use a Cartesian coordinate system with orthogonal unit vectors
${\mbox{\boldmath$\epsilon$}}^{(\alpha)}$, $\alpha=1,...,d$ along the $d$
Cartesian axes. The shear transformation is achieved by a rotation of the
vectors ${\mbox{\boldmath$\xi$}}_{0\pm}^{(\alpha)}$ by means of the orthogonal
matrix ${\bf U}\big{(}\\{{\bf e}^{(\alpha)}\\}\big{)}$ with matrix elements
$U_{\alpha\beta}=e_{\beta}^{(\alpha)}$ (which are the same as in the special
shear transformation) such that these vectors point along the direction of the
Cartesian axes,

$\displaystyle{\bf U}{\bf
e}^{(\alpha)}={\mbox{\boldmath$\epsilon$}}^{(\alpha)},\;\;{\bf
U}{\mbox{\boldmath$\xi$}}_{0\pm}^{(\alpha)}=\xi^{(\alpha)}_{0\pm}{\mbox{\boldmath$\epsilon$}}^{(\alpha)},$
(4.61)

and by a subsequent rescaling of their lengths by means of a diagonal matrix
${\mbox{\boldmath$\widetilde{\lambda}$}}^{-1/2}$. This matrix can be chosen to
be independent of the temperature, i.e., it can be chosen to be the same
above, at, and below $T_{c}$ as will be specified below. This yields the $d$
transformed correlation vectors

${\mbox{\boldmath$\widetilde{\xi}$}}_{0\pm}^{(\alpha)}={\mbox{\boldmath$\widetilde{\lambda}$}}^{-1/2}{\bf
U}{\mbox{\boldmath$\xi$}}_{0\pm}^{(\alpha)}=\widetilde{\lambda}_{\alpha}^{-1/2}\xi^{(\alpha)}_{0\pm}{\mbox{\boldmath$\epsilon$}}^{(\alpha)}$
(4.62)

where $\widetilde{\lambda}_{\alpha}$ are the diagonal elements of
$\widetilde{\lambda}$. To obtain a system that is isotropic in the scaling
region it is necessary and sufficient that
$\widetilde{\lambda}_{\alpha}^{1/2}$ is proportional to
$\xi^{(\alpha)}_{0\pm}$,

$\widetilde{\lambda}_{\alpha}^{1/2}=\widetilde{c}_{\pm}\;\xi^{(\alpha)}_{0\pm},$
(4.63)

where $\widetilde{c}_{+}>0$ and $\widetilde{c}_{-}>0$ are independent of the
direction $\alpha$. This rescaling guarantees that the lengths
$|{\mbox{\boldmath$\widetilde{\xi}$}}_{0\pm}^{(\alpha)}|$ of the transformed
correlation vectors (4.62)

$|{\mbox{\boldmath$\widetilde{\xi}$}}_{0\pm}^{(\alpha)}|=|{\mbox{\boldmath$\widetilde{\lambda}$}}^{-1/2}{\bf
U}{\mbox{\boldmath$\xi$}}_{0\pm}^{(\alpha)}|=\widetilde{\lambda}_{\alpha}^{-1/2}\xi^{(\alpha)}_{0\pm}=\widetilde{c}_{\pm}^{-1}$
(4.64)

become independent of the direction $\alpha$. This implies that isotropic
correlations are obtained in the large-distance scaling regime where the
isotropic lengths $\widetilde{c}_{\pm}^{-1}$ are as yet unspecified. The
transformation (4.1)-(4.3) corresponds to the special choice
$\widetilde{c}_{\pm}=(\xi^{\prime}_{0\pm})^{-1}$ where $\xi^{\prime}_{0\pm}$
is the correlation-length amplitude of the isotropic Hamiltonian
$H_{\text{field}}^{\prime}$, (4.13) and thus depends on the parameters $a_{0}$
and $(\det{\bf A})^{1/2}u_{0}$ of the original anisotropic $\varphi^{4}$ model
through (4.3). For the explicit expression of $\xi^{\prime}_{0\pm}$ see Eq.
(2.82) of dohm-arXiv .

Here we make a more general and conceptually different choice of
$\widetilde{c}_{\pm}$ that is independent of any parameter of the original
anisotropic system. We choose

$\displaystyle\widetilde{c}_{+}^{-1}$ $\displaystyle=$
$\displaystyle\xi^{\text{ iso}}_{0+},\;\;\;\widetilde{c}_{-}^{-1}=\xi^{\text{
iso}}_{0-},$ (4.65)

thus

$\displaystyle{\mbox{\boldmath$\widetilde{\xi}$}}_{0\pm}^{(\alpha)}$
$\displaystyle=$ $\displaystyle\xi^{\text{
iso}}_{0\pm}{\mbox{\boldmath$\epsilon$}}^{(\alpha)},\;\;\;|{\mbox{\boldmath$\widetilde{\xi}$}}_{0\pm}^{(\alpha)}|=\xi^{\text{
iso}}_{0\pm},\;\;\alpha=1,...,d\;\;\;$ (4.66)

where $\xi^{\text{ iso}}_{0+}$ and $\xi^{\text{ iso}}_{0-}$ are free
parameters, together with the requirement that

$\xi^{\text{ iso}}_{0+}/\xi^{\text{ iso}}_{0-}=X_{\xi}={\rm universal}.$
(4.67)

This requirement is necessary in order to comply with the fact implied by two-
scale-factor universality that the ratio of the correlation lengths above and
below $T_{c}$ of any isotropic system near $T_{c}$ must satisfy the universal
relation (3.3). According to (4.31) this implies
$\xi^{(\alpha)}_{0+}/\xi^{(\alpha)}_{0-}=\xi^{\text{ iso}}_{0+}/\xi^{\text{
iso}}_{0-}$ or

$\xi^{(\alpha)}_{0-}/\xi^{\text{ iso}}_{0-}=\xi^{(\alpha)}_{0+}/\xi^{\text{
iso}}_{0+},$ (4.68)

thus we obtain from (4.63)-(4.67) $d$ different constants

$\displaystyle\widetilde{\lambda}_{\alpha}^{1/2}$ $\displaystyle=$
$\displaystyle\xi^{(\alpha)}_{0-}/\xi^{\text{
iso}}_{0-}=\xi^{(\alpha)}_{0+}/\xi^{\text{ iso}}_{0+}$ (4.69)

which are the same above and below $T_{c}$. By continuity, this
$T$-independent identification of $\widetilde{\lambda}_{\alpha}$ is applicable
also at $T=T_{c}$. This property is analogous to the relation (4.32) where the
$T$-independent eigenvalues $\lambda_{\alpha}(K_{i,j})$ of the anisotropy
matrix ${\bf A}(K_{i,j})$ can be expressed in terms of
$\xi^{(\alpha)}_{0\pm}/\xi^{\prime}_{0\pm}$, thus both diagonal matrices
$\widetilde{\lambda}$ and $\lambda$ are $T$-independent and are related by

$\widetilde{\lambda}$ $\displaystyle=$
$\displaystyle\Bigg{(}\frac{\xi^{\prime}_{0-}}{\xi^{\text{
iso}}_{0-}}\Bigg{)}^{2}{\mbox{\boldmath$\lambda$}}=\Bigg{(}\frac{\xi^{\prime}_{0+}}{\xi^{\text{
iso}}_{0+}}\Bigg{)}^{2}{\mbox{\boldmath$\lambda$}}.$ (4.70)

We note that the mean correlation length can be used to express
$\det{\mbox{\boldmath$\widetilde{\lambda}$}}$ as

$\displaystyle\det{\mbox{\boldmath$\widetilde{\lambda}$}}=(\bar{\xi}_{0\pm}/\xi^{\rm
iso}_{0\pm})^{2d}.$ (4.71)

We apply this generalized shear transformation to the lattice vectors ${\bf
x}$ of the anisotropic system

$\displaystyle{\bf\widetilde{x}}={\mbox{\boldmath$\widetilde{\lambda}$}}^{-1/2}{\bf
U}{\bf x},$ (4.72)

with $\widetilde{\lambda}$ given by (4.69). This generates lattice vectors
${\bf\widetilde{x}}$ of an isotropic system with the correlation-length
amplitudes $\xi^{\text{ iso}}_{0\pm}$ in the scaling region. This
transformation is applicable not only in the presence of finite correlations
lengths $\xi^{\text{ iso}}_{\pm}(t)$ for $T\neq T_{c}$ but also right at
$T=T_{c}$. We may also consider the inverse shear transformation

${\bf x}={\bf
U}^{-1}{\mbox{\boldmath$\widetilde{\lambda}$}}^{1/2}{\bf\widetilde{x}}$ (4.73)

of the isotropic system to the anisotropic system with given principal axes
and given principal correlation lengths. Eq. (4.73) describes a rescaling of
the coordinates of the isotropic system in the direction of the Cartesian axes
followed by a rotation of these axes into the direction of the principal axes
of the anisotropic system. Our generalized shear transformation differs
significantly from the special transformation (4.1)-(4.3) in that our
transformation is a pure coordinate transformation without transforming the
field $\varphi$ and the coupling $u_{0}$. As a consequence, this
transformation leaves the amplitude of the order-parameter correlation
function $G^{\rm sp}$ invariant (as discussed in Sec. II. B) but not of the
susceptibility.

Within this generalized transformation, we are primarily interested in
determining the reduced anisotropy matrix ${\bf\bar{A}}^{\rm gen}={\bf
U}^{-1}{\bar{\mbox{\boldmath$\lambda$}}^{\rm gen}}{\bf U}$ via the alternative
definition given in (4.11) in terms of a reduced rescaling matrix
${\mbox{\boldmath$\bar{\lambda}$}^{\rm gen}}$. In this transformation the
reduced rescaling matrix is defined by

$\displaystyle{\mbox{\boldmath$\bar{\lambda}$}^{\rm gen}}$ $\displaystyle=$
$\displaystyle\frac{{\mbox{\boldmath$\widetilde{\lambda}$}}}{(\det{\mbox{\boldmath$\widetilde{\lambda}$}})^{1/d}}\;\;,$
(4.74) $\displaystyle\det{\mbox{\boldmath$\widetilde{\lambda}$}}$
$\displaystyle=$
$\displaystyle\prod^{d}_{\alpha=1}\widetilde{\lambda}_{\alpha},$ (4.75)

where the diagonal elements $\widetilde{\lambda}_{\alpha}$ of
$\widetilde{\lambda}$ are given by (4.69). The rescaling matrix
$\widetilde{\lambda}$ and the reduced rescaling matrix
${\mbox{\boldmath$\bar{\lambda}$}^{\rm gen}}$ are related by

$\displaystyle{\mbox{\boldmath$\widetilde{\lambda}$}}^{1/2}\xi^{\rm
iso}_{0\pm}$ $\displaystyle=$
$\displaystyle\big{(}{\mbox{\boldmath$\bar{\lambda}$}^{\rm
gen}}\big{)}^{1/2}\bar{\xi}_{0\pm}.$ (4.76)

We see that the dependence on $\xi^{\rm iso}_{0\pm}$ is canceled in the
diagonal matrix (4.74). This implies that the matrix (4.74) is the same as
that defined in (4.12),

${\mbox{\boldmath$\bar{\lambda}$}^{\rm
gen}}={\mbox{\boldmath$\bar{\lambda}$}},$ (4.77)

with the same diagonal elements as given in (4.53) and (4.54). As a
consequence, the reduced anisotropy matrix in the generalized shear
transformation

${\bf\bar{A}}^{\rm gen}={\bf\bar{A}}\big{(}\\{\xi_{0\pm}^{(\alpha)},{\bf
e}^{(\alpha)}\\}\big{)}$ (4.78)

is identical with that given in (4.56) within the special shear transformation
(4.1)-(4.3). This result demonstrates that the generalized shear
transformation is capable deriving the structure of the reduced anisotropy
matrix ${\bf\bar{A}}$ of the anisotropic $\varphi^{4}$ model without explicit
knowledge of the anisotropy matrix ${\bf A}$. Furthermore the generalized
shear transformation is simpler than the special shear transformation in that
it is a pure coordinate transformation. For this reason this transformation is
applicable also to other weakly anisotropic systems.

The generalized shear transformation introduced above for the $\varphi^{4}$
theory at finite $n$ remains applicable also in the large $n$-limit of the
$\varphi^{4}$ theory and to the Gaussian model for $T\geq T_{c}$. We note,
however, that there is no advantage within the $\varphi^{4}$ theory to work
with the generalized shear transformation (4.72) since the corresponding
transformed Hamiltonian $\widetilde{H}_{\text{field}}$ would not have the form
of a standard isotropic $\varphi^{4}$ theory. For this reason it is more
convenient within the $\varphi^{4}$ theory to employ the special shear
transformation (4.1)-(4.3) which allows one to use ordinary renormalized
isotropic perturbation theory within the transformed system on the basis of
the standard isotropic Landau-Ginzburg-Wilson Hamiltonian (4.13).

###  IV.3 Generalized shear transformation of the ${\bf n}$-vector model

As a representative of weakly anisotropic systems other than the $\varphi^{4}$
model we take the fixed-length $n$-vector Hamiltonian $H^{\rm sp}$, (2.22),
for general $d$ and $n$. We consider only the scaling region. Our only
assumptions for this model are (i) that there exist $d$ principal unit vectors
${\bf e}^{{\rm sp}(\alpha)}$ in the directions of $d$ principal axes together
with $d$ principal correlation lengths above and below $T_{c}$

$\xi^{{\rm sp}(\alpha)}_{\pm}(t)=\xi^{{\rm sp}(\alpha)}_{0\pm}|t|^{-\nu}$
(4.79)

along these directions where $\nu$ is the critical exponent of the isotropic
$n$-vector model, and (ii) that the amplitude ratios satisfy

$\xi^{{\rm sp}(\alpha)}_{0+}/\xi^{{\rm sp}(\alpha)}_{0-}=X_{\xi}\;\;\text{for
each}\;\;\alpha$ (4.80)

where $X_{\xi}$ is the same universal constant as in (3.3) for isotropic
systems of the $(d,n)$ universality class. These assumptions are supported by
our exact results (4.31) for the $\varphi^{4}$ model for general $(d,n)$ and
for the anisotropic two-dimensional Ising model dohm2019 , as discussed in
Sec. VIII. B. We shall show that for deriving the universal structures of the
correlation function and of bulk amplitude relations it is not necessary to
know the dependence of $\xi^{{\rm sp}(\alpha)}_{0\pm}$ and of the principal
unit vectors ${\bf e}^{{\rm sp}(\alpha)}$ on the microscopic couplings
$E_{i,j}$. For most of our results and conclusions the assumption (4.80) is
not needed.

Our introduction of the generalized shear transformation of the $n$-vector
model is parallel to (4.60) - (4.73). Accordingly we introduce the orthogonal
set of $d$ principal correlation vectors

${\mbox{\boldmath$\xi$}}_{0\pm}^{{\rm sp}(\alpha)}=\xi^{{\rm
sp}(\alpha)}_{0\pm}{\bf e}^{{\rm sp}(\alpha)},\;\;\;\alpha=1,...,d$ (4.81)

and perform a rotation by means of

${\bf\widehat{U}}={\bf U}\big{(}\\{{\bf e}^{{\rm sp}(\alpha)}\\}\big{)}$
(4.82)

with matrix elements $\widehat{U}_{\alpha\beta}=e_{\beta}^{{\rm sp}(\alpha)}$
such that these vectors point along the direction of the Cartesian axes,

$\displaystyle{\bf\widehat{U}}{\bf e}^{{\rm
sp}(\alpha)}={\mbox{\boldmath$\epsilon$}}^{(\alpha)},\;\;{\bf\widehat{U}}{\mbox{\boldmath$\xi$}}_{0\pm}^{{\rm
sp}(\alpha)}$ $\displaystyle=$ $\displaystyle\xi^{{\rm
sp}(\alpha)}_{0\pm}{\mbox{\boldmath$\epsilon$}}^{(\alpha)}.$ (4.83)

As for the $\varphi^{4}$ model, the orthogonal matrix ${\bf\widehat{U}}$
contains $d(d-1)/2$ independent matrix elements which are needed to specify
the directions ${\bf e}^{{\rm sp}(\alpha)}$ of the principal axes. These $d$
mutually orthogonal axes can be described by $d(d-1)/2$ independent angles
$\Omega_{i}$, $i=1,2,...,d(d-1)/2$, i.e., one angle $\Omega$ in two
dimensions, three angles in three dimensions, six angles in four dimensions,
etc.,

$\displaystyle{\bf U}\big{(}\\{{\bf e}^{{\rm sp}(\alpha)}\\}\big{)}={\bf
U}\big{(}\Omega_{1},\Omega_{2},\Omega_{3},...\big{)}.$ (4.84)

A subsequent rescaling of the lengths $\xi^{{\rm sp}(\alpha)}_{0\pm}$ by means
of a diagonal matrix ${\mbox{\boldmath$\widehat{\lambda}$}}^{-1/2}$ yields the
$d$ transformed vectors

$\displaystyle{\mbox{\boldmath$\widehat{\xi}$}}_{0\pm}^{(\alpha)}$
$\displaystyle=$
$\displaystyle{\mbox{\boldmath$\widehat{\lambda}$}}^{-1/2}{\bf\widehat{U}}{\mbox{\boldmath$\xi$}}_{0\pm}^{{\rm
sp}(\alpha)}=\widehat{\lambda}_{\alpha}^{-1/2}\xi^{{\rm
sp}(\alpha)}_{0\pm}{\mbox{\boldmath$\epsilon$}}^{(\alpha)}.$ (4.85)

Here we make the choice, similar to (4.63),

$\widehat{\lambda}_{\alpha}^{1/2}=\widetilde{c}_{\pm}\;\xi^{{\rm
sp}(\alpha)}_{0\pm},$ (4.86)

together with the same choice for $\widetilde{c}_{\pm}$ as specified in (4.65)
and (4.67), with free parameters $\xi^{\text{ iso}}_{0\pm}$. Because of (4.67)
and (4.80) this implies $\xi^{{\rm sp}(\alpha)}_{0+}/\xi^{{\rm
sp}(\alpha)}_{0-}=\xi^{\text{ iso}}_{0+}/\xi^{\text{ iso}}_{0-}$ or

$\xi^{{\rm sp}(\alpha)}_{0-}/\xi^{\text{ iso}}_{0-}=\xi^{{\rm
sp}(\alpha)}_{0+}/\xi^{\text{ iso}}_{0+},$ (4.87)

thus the diagonal elements of $\widehat{\lambda}$ are defined to be
$T$-independent parameters given by

$\displaystyle\widehat{\lambda}_{\alpha}$ $\displaystyle=$
$\displaystyle\Big{(}\xi^{{\rm sp}(\alpha)}_{0-}/\xi^{\text{
iso}}_{0-}\Big{)}^{2}=\Big{(}\xi^{{\rm sp}(\alpha)}_{0+}/\xi^{\text{
iso}}_{0+}\Big{)}^{2},$ (4.88)

similar to (4.32) and (4.69). This transforms the $d$ different principal
correlation vectors (4.81) to the $d$ vectors
${\mbox{\boldmath$\widehat{\xi}$}}_{0\pm}^{(\alpha)}=\xi^{\text{
iso}}_{0\pm}{\mbox{\boldmath$\epsilon$}}^{(\alpha)}$ with the angular-
independent lengths $\xi^{\text{ iso}}_{0+}$ and $\xi^{\text{ iso}}_{0-}$
above and below $T_{c}$, respectively, representing an isotropic system.

Correspondingly we define the $T$-independent shear transformation of the
lattice points ${\bf x}\to{\bf\widehat{x}}$ at fixed couplings $E_{i,j}$ of
the anisotropic $n$-vector model by

$\displaystyle{\bf\widehat{x}}={\mbox{\boldmath$\widehat{\lambda}$}}^{-1/2}{\bf\widehat{U}}{\bf
x},$ (4.89)

with $\widehat{\lambda}$ given by (4.88) and ${\bf\widehat{U}}$ defined by
(4.82). This generates an $n$-vector model on a transformed lattice with
lattice points ${\bf\widehat{x}}$ and with isotropic correlations
characterized by the correlation length $\xi^{\text{ iso}}_{0\pm}$ for which
two-scale-factor universality can be invoked. In constructing this
transformation no assumption has been made other than the existence of the
principal correlation lengths (4.79) and, for the application to $T<T_{c}$,
the universality of the amplitude relation (4.80).

Now we have arrived at the position to derive the structure of the reduced
anisotropy matrix ${\bf\bar{A}}^{\rm sp}$ that will enter the anisotropic
correlation function $G^{\rm sp}({\bf x},t)$ of the $n$-vector model to be
derived in Sec. V. B. We define the reduced rescaling matrix

$\displaystyle{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{(}\\{\xi_{0\pm}^{{\rm sp}(\alpha)}\\}\big{)}$ $\displaystyle=$
$\displaystyle{\mbox{\boldmath$\widehat{\lambda}$}}/\big{(}\det{\mbox{\boldmath$\widehat{\lambda}$}}\big{)}^{1/d}\;\;$
(4.90)

which has the diagonal elements

$\displaystyle\bar{\lambda}^{\rm
sp}_{\alpha}=\prod^{d}_{\beta=1,\;\beta\neq\alpha}\Big{(}\frac{\xi_{0\pm}^{{\rm
sp}(\alpha)}}{\xi_{0\pm}^{{\rm
sp}(\beta)}}\Big{)}^{2/d}=\Big{(}\frac{\xi_{0\pm}^{{\rm
sp}(\alpha)}}{\bar{\xi}^{\rm sp}_{0\pm}}\Big{)}^{2}\;\;\;$ (4.91)

with the amplitude $\bar{\xi}^{\rm sp}_{0\pm}$ of the mean correlation length

$\displaystyle\bar{\xi}^{\rm sp}_{\pm}(t)=\bar{\xi}^{\rm
sp}_{0\pm}|t|^{-\nu},\;\;\bar{\xi}^{\rm
sp}_{0\pm}=\big{[}\prod^{d}_{\alpha=1}\xi_{0\pm}^{{\rm
sp}(\alpha)}\big{]}^{1/d},$ (4.92) $\displaystyle\bar{\xi}^{\rm
sp}_{0+}/\bar{\xi}^{\rm sp}_{0-}=X_{\xi}.$ (4.93)

In the ratios (4.91) the dependence on the free parameters $\xi^{\rm
iso}_{0\pm}$ is canceled. Similar to (4.71) and (4.76), we have the relations

$\displaystyle\det{\mbox{\boldmath$\widehat{\lambda}$}}=(\bar{\xi}^{\rm
sp}_{0\pm}/\xi^{\rm iso}_{0\pm})^{2d}$ (4.94)

and

$\displaystyle{\mbox{\boldmath$\widehat{\lambda}$}}^{1/2}\xi^{\rm iso}_{0\pm}$
$\displaystyle=$ $\displaystyle\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}\bar{\xi}^{\rm sp}_{0\pm}.$ (4.95)

Finally we obtain the reduced anisotropy matrix via the definition

$\displaystyle{\bf\bar{A}}^{\rm sp}$ $\displaystyle=$
$\displaystyle\widehat{\bf U}^{-1}{\bf\bar{\mbox{\boldmath$\lambda$}}}^{\rm
sp}\widehat{\bf U}$ (4.96) $\displaystyle=$
$\displaystyle{\bf\bar{A}}\big{(}\\{\xi_{0\pm}^{{\rm sp}(\alpha)},{\bf
e}^{{\rm sp}(\alpha)}\\}\big{)}$ (4.97) $\displaystyle=$ $\displaystyle{\bf
U(\\{{\bf e}^{{\rm
sp}(\alpha)}\\})}^{-1}{\bf\bar{\mbox{\boldmath$\lambda$}}}^{\rm
sp}\big{(}\\{\xi_{0\pm}^{{\rm sp}(\alpha)}\\}\big{)}{\bf U(\\{{\bf e}^{{\rm
sp}(\alpha)}\\})}\;\;\;\;\;$ (4.98)

where ${\bf\bar{A}}^{\rm sp}$ has the same structure as the matrix
${\bf\bar{A}}$ in (4.56) for the $\varphi^{4}$ model but with the arguments
$\xi_{0\pm}^{(\alpha)}$ and ${\bf e}^{(\alpha)}$ replaced by $\xi_{0\pm}^{{\rm
sp}(\alpha)}$ and ${\bf e}^{{\rm sp}(\alpha)}$. This can be extended to any
weakly anisotropic system beyond the $\varphi^{4}$ theory without explicit
knowledge of an anisotropy matrix ${\bf A}$ as a function of the couplings.

This means that we have identified a temperature-independent universal
structure of the reduced anisotropy matrix
${\bf\bar{A}}(\\{\xi_{0\pm}^{(\alpha)},{\bf e}^{(\alpha)}\\}\big{)}$ whose
dependence either on the nonuniversal ratios
$\xi_{0\pm}^{(\alpha)}/\xi_{0\pm}^{(\beta)}$ and nonuniversal principal unit
vectors ${\bf e}^{(\alpha)}$ or on $\xi_{0\pm}^{{\rm
sp}(\alpha)}/\xi_{0\pm}^{{\rm sp}(\beta)}$ and ${\bf e}^{{\rm sp}(\alpha)}$
has a universal functional form that is the same for all weakly anisotropic
systems including the $\varphi^{4}$ model and the $n$-vector model. We have
shown that for the construction of this reduced anisotropy matrix the
existence and knowledge of an anisotropy matrix ${\bf A}$ is not required,
thus our construction of the structure of ${\bf\bar{A}}$ is applicable to any
weakly anisotropic system. This is an important ingredient for the general
validity of multiparameter universality of weakly anisotropic systems.

##  V Multiparameter universality of the anisotropic correlation function

Besides the bulk free energy density the most fundamental physical quantity
describing the critical fluctuations is the bulk order-parameter correlation
function. However, in the traditional theory of critical phenomena fish-1 ;
bre-1 ; pelissetto ; priv ; pri ; hohenberg1976 ; cardybuch ; zinn2007
including the nonperturbative functional renormalization group metzner2021 ;
hassel2007 ; sinner2008  the issue of the general universality properties of
weakly anisotropic correlation functions was not addressed. Only special
anisotropic models were studied whose bulk correlation functions bruce ;
Vaidya1976 ; WuCoy ; CoyWu ; Perk1 ; Perk2 ; Perk3 ; Perk4  were presented in
a nonuniversal form. Recently a general scaling form of the anisotropic bulk
correlation function of the $O(n)$-symmetric $d$-dimensional $\varphi^{4}$
model has been presented dohm2018 ; dohm2019  in terms of a reduced anisotropy
matrix ${\bf\bar{A}}$ which was found to violate two-scale-factor universality
in both two and three dimensions. Instead, within the $\varphi^{4}$ theory,
these results exhibit the feature of multiparameter universality dohm2018  and
it was hypothesized that this structure is valid quite generally for weakly
anisotropic systems beyond $\varphi^{4}$ models. In this section we prove the
validity of this hypothesis for the bulk correlation function and point to the
intrinsic diversity of this scaling structure.

###  V.1 Anisotropic correlation function of the $\varphi^{4}$ theory

In dohm2008 ; dohm2018  the representation (3.1) of the isotropic correlation
function was used in order to derive the anisotropic correlation function
$G_{\pm}({\bf x},t)$ of the $\varphi^{4}$ theory in the form presented in Eq.
(1.6) of dohm2018 . In the following we shall work with the alternative
representation (3.45) and its Fourier transform (3.51) and present a more
transparent derivation. Application of (3.45) to the isotropic correlation
function $G^{\prime}_{\pm}(|{\bf x^{\prime}}|,t)$ of the transformed isotropic
Hamiltonian (4.13) yields

$G^{\prime}_{\pm}(|{\bf
x^{\prime}}|,t)=\frac{\Gamma^{\prime}_{+}(\xi^{\prime}_{0+})^{-2+\eta}}{|{\bf
x^{\prime}}|^{d-2+\eta}}\;\Psi_{\pm}\Big{(}\frac{|{\bf
x^{\prime}}|}{\xi^{\prime}_{\pm}(t)}\Big{)}\;$ (5.1)

which can be rewritten as

$G^{\prime}_{\pm}(|{\bf
x^{\prime}}|,t)=\frac{\Gamma^{\prime}_{+}}{(\xi^{\prime}_{0+})^{d}}\Big{(}\frac{\xi^{\prime}_{0+}}{|{\bf
x^{\prime}}|}\Big{)}^{d-2+\eta}\;\Psi_{\pm}\Big{(}\frac{|{\bf
x^{\prime}}|}{\xi^{\prime}_{\pm}(t)}\Big{)}\;.$ (5.2)

Now we derive the anisotropic correlation function from the isotropic
correlation function (5.2) by inverting (4.40),

$\displaystyle G_{\pm}({\bf
x},t)=\big{(}\xi^{\prime}_{0\pm}/\bar{\xi}_{0\pm}\big{)}^{d}\;G_{\pm}^{\prime}(|{\bf
x^{\prime}}|,t),$ (5.3)

and expressing $|{\bf x^{\prime}}|$ in terms of ${\bf x}$. Using the inverse
of (4.11)

${\mbox{\boldmath$\bar{\lambda}$}}^{-1}={\bf U}{\bf\bar{A}}^{-1}{\bf U}^{-1}$
(5.4)

we first rewrite $|{\bf x^{\prime}}|$ as

$\displaystyle|{\bf x^{\prime}}|=\big{[}{\bf
x^{\prime}}\cdot{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\mbox{\boldmath$\bar{\lambda}$}}^{-1}{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf
x^{\prime}}\big{]}^{1/2}$ (5.5) $\displaystyle=\big{[}{\bf
x^{\prime}}\cdot{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf
U}{\bf\bar{A}}^{-1}{\bf U}^{-1}{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf
x^{\prime}}\big{]}^{1/2}$ (5.6)
$\displaystyle=\big{[}({\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf U})^{T}{\bf
x^{\prime}}\cdot{\bf\bar{A}}^{-1}{\bf
U}^{-1}{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf
x^{\prime}}\big{]}^{1/2}\;\;\;\;$ (5.7) $\displaystyle=\big{[}{\bf
U}^{-1}{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf
x^{\prime}}\cdot{\bf\bar{A}}^{-1}{\bf
U}^{-1}{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf
x^{\prime}}\big{]}^{1/2}\;\;\;\;$ (5.8)

where we have used

$({\mbox{\boldmath$\bar{\lambda}$}}^{1/2}{\bf U})^{T}={\bf
U}^{T}({\mbox{\boldmath$\bar{\lambda}$}}^{1/2})^{T}={\bf
U}^{-1}{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}.$ (5.9)

Then we express ${\mbox{\boldmath$\bar{\lambda}$}}^{1/2}$ in terms of
${\mbox{\boldmath$\lambda$}}^{1/2}$ according to (4.55),

$\displaystyle{\mbox{\boldmath$\bar{\lambda}$}}^{1/2}=\frac{\xi^{\prime}_{0\pm}}{\bar{\xi}_{0\pm}}\;{\mbox{\boldmath$\lambda$}}^{1/2}$
(5.10)

which yields the identity

$\displaystyle|{\bf x^{\prime}}|$ $\displaystyle=$
$\displaystyle\frac{\xi^{\prime}_{0\pm}}{\bar{\xi}_{0\pm}}\;[{\bf
U}^{-1}{\mbox{\boldmath$\lambda$}}^{1/2}{\bf
x^{\prime}}\cdot{\bf\bar{A}}^{-1}{\bf
U}^{-1}{\mbox{\boldmath$\lambda$}}^{1/2}{\bf
x^{\prime}}]^{1/2}.\;\;\;\;\;\;\;\;\;\;$ (5.11)

From the original shear transformation (4.1) we obtain the inverse shear
transformation

${\bf U}^{-1}{\mbox{\boldmath$\lambda$}}^{1/2}{\bf x^{\prime}}={\bf x}.$
(5.12)

This leads to the relation

$\displaystyle\frac{{\bf
x^{\prime}}^{2}}{\big{(}\xi^{\prime}_{0\pm}\big{)}^{2}}$ $\displaystyle=$
$\displaystyle\frac{{\bf x}\cdot{\bf\bar{A}}^{-1}{\bf
x}}{\big{(}\bar{\xi}_{0\pm}\big{)}^{2}}.$ (5.13)

Here the representation (4.56) of the reduced anisotropy matrix ${\bf\bar{A}}$
is to be inserted as a function of the principal unit vectors and principal
correlation lengths. Eq. (5.13) describes how the isotropic structure is
transferred to the anisotropic structure by means of the inverse shear
transformation. Using (5.3) together with the invariance (4.50) and
substituting (5.13) into (5.2) we arrive at the anisotropic bulk order-
parameter correlation function above, at, and below $T_{c}$ of the
$\varphi^{4}$ model for $2\leq d<4$

$\displaystyle G_{\pm}({\bf
x},t)=\frac{\Gamma_{+}(\bar{\xi}_{0+})^{-2+\eta}}{({\bf
x}\cdot{\bf\bar{A}}^{-1}{\bf x})^{(d-2+\eta)/2}}\Psi_{\pm}\Big{(}\frac{[{\bf
x}\cdot{\bf\bar{A}}^{-1}{\bf
x}]^{1/2}}{\bar{\xi}_{\pm}(t)}\Big{)}\;\;\;\;\;\;\;\;\;$ (5.14)

with the universal scaling function $\Psi_{\pm}$, (3.46), of the isotropic
system. At $T_{c}$ we obtain

$\displaystyle G_{\pm}({\bf
x},0)=\widetilde{Q}_{3}\frac{\Gamma_{+}\big{(}\bar{\xi}_{0+}\big{)}^{-2+\eta}}{({\bf
x}\cdot{\bf\bar{A}}^{-1}{\bf x})^{(d-2+\eta)/2}}\;\;\;\;\;\;\;\;\;$ (5.15)

with the universal constant $\widetilde{Q}_{3}$ defined in (3.56). These
results are equivalent to the results of the $\varphi^{4}$ theory in Eqs.
(1.6) and (5.32) of dohm2018 .

An analogous anisotropic representation can be derived in Fourier space based
on the shear transformation (4.5). Application of (3.51) to the Fourier
transformed isotropic correlation function $\hat{G}^{\prime}_{\pm}(|{\bf
k^{\prime}}|,t)$ of the transformed isotropic Hamiltonian (4.13) yields

$\displaystyle\hat{G}^{\prime}_{\pm}(|{\bf k^{\prime}}|,t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma^{\prime}_{+}}{\big{(}|{\bf
k^{\prime}}|\;\xi^{\prime}_{0+}\big{)}^{2-\eta}}\;\hat{\Psi}_{\pm}\Big{(}|{\bf
k^{\prime}}|\xi^{\prime}_{\pm}(t)\Big{)},$ (5.16)
$\displaystyle\hat{G}^{\prime}(|{\bf k}|,0)$ $\displaystyle=$ $\displaystyle
Q_{3}\frac{\Gamma^{\prime}_{+}}{\big{(}|{\bf
k}|\;\xi^{\prime}_{0+}\big{)}^{2-\eta}}\;.\;$ (5.17)

Because of the sum rule (3.53) we have

$\displaystyle\lim_{|{\bf k^{\prime}}|\to 0}\hat{G}^{\prime}_{\pm}(|{\bf
k^{\prime}}|,t)=\chi^{\prime}_{\pm}(t)=\Gamma^{\prime}_{\pm}|t|^{-\gamma}.$
(5.18)

Using (4.11) in the form

${\mbox{\boldmath$\bar{\lambda}$}}={\bf U}{\bf\bar{A}}{\bf U}^{-1}$ (5.19)

we rewrite $|{\bf k^{\prime}}|$ in a way analogous to $|{\bf x^{\prime}}|$ and
obtain the relation

$\displaystyle|{\bf k}^{\prime}|^{2}(\xi_{0\pm}^{\prime})^{2}$
$\displaystyle=$ $\displaystyle({\bf k}\cdot\;{\bf\bar{A}}{\bf
k})(\bar{\xi}_{0\pm})^{2}.$ (5.20)

Using the invariance (4.17) together with the invariance (4.50) and
substituting (5.20) into (5.16) we obtain the anisotropic correlation function
of the $\varphi^{4}$ model in ${\bf k}$ space

$\displaystyle\hat{G}_{\pm}({\bf k},t)$ $\displaystyle=$
$\displaystyle\Gamma_{+}\;\frac{\hat{\Psi}_{\pm}\Big{(}[{\bf
k}\cdot\;{\bf\bar{A}}{\bf k}]^{1/2}\bar{\xi}_{\pm}(t)\Big{)}}{\big{(}[{\bf
k}\cdot\;{\bf\bar{A}}{\bf k}]^{1/2}\;\bar{\xi}_{0+}\big{)}^{2-\eta}}\;$ (5.21)

with

$\displaystyle\hat{G}_{\pm}({\bf k},0)=Q_{3}\frac{\Gamma_{+}}{\big{(}[{\bf
k}\cdot\;{\bf\bar{A}}{\bf k}]^{1/2}\bar{\xi}_{0+}\big{)}^{2-\eta}}\;\;$ (5.22)

at $T_{c}$, with the universal constant $Q_{3}$, (3.54). A discussion of these
results for the $\varphi^{4}$ theory is given in Sec. V.B. in the context of
analogous results for the $n$-vector model.

###  V.2 Proof of multiparameter universality of the anisotropic correlation
function

Our proof is formulated for the anistropic $n$-vector model as an example for
a system other than the anisotropic $\varphi^{4}$ model. It is based on the
following properties: (a) The universal structure of the scaling form (3.1) or
(3.45) of the isotropic correlation function, (b) the general applicability of
the generalized shear transformations (4.72) or (4.89) to any weakly
anisotropic system including the $n$-vector model, (c) the universality of the
structure of the reduced anisotropy matrix ${\bf\bar{A}}^{\rm sp}$, (4.98),
(d) the sum rule for the anisotropic correlation function near $T_{c}$

$\displaystyle\chi_{\pm}^{\rm sp}(t)=\int d^{d}{\bf x}\;\;G_{\pm}^{\rm
sp}({\bf x},t)=\Gamma^{\rm sp}_{\pm}|t|^{-\gamma}$ (5.23)

where $\Gamma^{\rm sp}_{\pm}$ is the nonuniversal critical amplitude of the
susceptibility above and below $T_{c}$, respectively, and (e) the invariance
of the bulk correlation function $G_{\pm}^{\rm sp}({\bf x},t)$ under our shear
transformation.

The latter issue is due to the fact that, unlike the special shear
transformation of the $\varphi^{4}$ theory cd2004 ; dohm2008  where the
transformation (4.2) of the variable $\varphi$ is involved, our generalized
shear transformation (4.89) transforms only the spatial coordinates ${\bf
x}\to{\bf\widehat{x}}$ of the lattice points without changing the topology of
the interactions. As discussed in Sec. II. B, this leaves the amplitude of the
order-parameter correlation function $G_{\pm}^{\rm sp}({\bf x},t)$ invariant.
Thus our generalized shear transformation implies that the special
transformations (4.16), (4.40), and (5.3) are replaced by the invariance

$\displaystyle G_{\pm}^{\rm sp}({\bf x},t)=G_{\pm}^{\rm
sp,iso}(|{\bf\widehat{x}}|,t)=G_{\pm}^{\rm
sp,iso}(|{\mbox{\boldmath$\widehat{\lambda}$}}^{-1/2}{\bf\widehat{U}}{\bf
x}|,t)$ (5.24)

for the $n$-vector model where the transformed isotropic correlation function
$G_{\pm}^{\rm sp,iso}$ has the isotropic structure given in (3.45). The
susceptibility, however, is not invariant under the generalized shear
transformation. Eq. (5.23) can be combined with the sum rule of the isotropic
system and with (4.89), (4.94), and (5.24) to obtain the susceptibility of the
isotropic system as

$\displaystyle\chi_{\pm}^{\rm sp,iso}(t)$ $\displaystyle=$ $\displaystyle\int
d^{d}{\bf\widehat{x}}\;\;G_{\pm}^{\rm
sp,iso}(|{\bf\widehat{x}}|,t)=\Gamma^{\rm
sp,iso}_{\pm}\;|t|^{-\gamma}\;\;\;\;\;\;$ (5.25) $\displaystyle=$
$\displaystyle(\det{\mbox{\boldmath$\widehat{\lambda}$}})^{-1/2}\int d^{d}{\bf
x}\;\;G_{\pm}^{\rm sp}({\bf x},t)$ (5.26) $\displaystyle=$
$\displaystyle(\xi^{\rm iso}_{0\pm}/\bar{\xi}^{\rm
sp}_{0\pm})^{d}\chi_{\pm}^{\rm sp}(t)$ (5.27)

where $\bar{\xi}^{\rm sp}_{0\pm}$ is the amplitude of the mean correlation
length. This implies the relation between the amplitudes of the
susceptibilities of the isotropic and anisotropic systems

$\Gamma^{\rm sp,iso}_{\pm}=\Big{(}\xi^{\rm iso}_{0\pm}/\bar{\xi}^{\rm
sp}_{0\pm}\Big{)}^{d}\;\Gamma_{\pm}^{\rm sp}$ (5.28)

which can be interpreted as the invariance of the ratio of the correlation
volume and the susceptibility amplitude

$\big{(}\xi^{\rm iso}_{0\pm}\big{)}^{d}/\Gamma^{\rm
sp,iso}_{\pm}=\;\big{(}\bar{\xi}^{\rm sp}_{0\pm}\big{)}^{d}/\Gamma_{\pm}^{\rm
sp}$ (5.29)

under the generalized shear transformation. We note that also the combination

$\displaystyle\big{(}\xi^{\rm iso}_{0\pm}\big{)}^{d}G_{\pm}^{\rm
sp,iso}(|{\bf\widehat{x}}|,t)/\Gamma^{\rm
sp,iso}_{\pm}=\;\big{(}\bar{\xi}^{\rm sp}_{0\pm}\big{)}^{d}G_{\pm}^{\rm
sp}({\bf x},t)/\Gamma_{\pm}^{\rm sp}\;\;\;\;\;\;\;\;\;\;$ (5.30)

remains invariant which shows the universality of the same feature (4.51) of
the special shear transformation of the $\varphi^{4}$ theory. From (3.45) we
have the representation of the isotropic bulk correlation function

$G_{\pm}^{\rm sp,iso}(|{\bf\widehat{x}}|,t)=\frac{\Gamma_{+}^{\rm
sp,iso}\big{(}\xi^{\rm
iso}_{0+}\big{)}^{-2+\eta}}{|{\bf\widehat{x}}|^{(d-2+\eta)/2}}\;\Psi_{\pm}\Big{(}\frac{|{\bf\widehat{x}}|}{\xi^{\rm
iso}_{\pm}(t)}\Big{)}.\;$ (5.31)

or

$G_{\pm}^{\rm sp,iso}({\bf\widehat{x}},t)=\frac{\Gamma_{+}^{\rm
sp,iso}}{\big{(}\xi^{\rm iso}_{0+}\big{)}^{d}}\Bigg{(}\frac{\xi_{0+}^{\rm
iso}}{|{\bf\widehat{x}}|}\Bigg{)}^{d-2+\eta}\;\Psi_{\pm}\Big{(}\frac{|{\bf\widehat{x}}|}{\xi^{\rm
iso}_{\pm}(t)}\Big{)}.\;$ (5.32)

The derivation of the anisotropic bulk correlation function $G_{\pm}^{\rm
sp}({\bf x},t)$, (5.24) from (5.32) is analogous to that in (5.2)-(5.14) for
the $\varphi^{4}$ model. Thus we rewrite

$\displaystyle|{\bf\widehat{x}}|=\big{[}{\bf\widehat{x}}\cdot\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{-1}\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}{\bf\widehat{x}}\big{]}^{1/2}$ (5.33)
$\displaystyle=\big{[}{\bf\widehat{x}}\cdot\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}{\bf\widehat{U}}\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf\widehat{U}}^{-1}\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}{\bf\widehat{x}}\big{]}^{1/2}$ (5.34)
$\displaystyle=\big{[}{\bf\widehat{U}}^{-1}\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}{\bf\widehat{x}}\cdot\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf\widehat{U}}^{-1}\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}{\bf\widehat{x}}\big{]}^{1/2}\;\;\;\;$ (5.35)
$\displaystyle=\frac{\xi^{\rm iso}_{0\pm}}{\bar{\xi}^{\rm
sp}_{0\pm}}\;[{\bf\widehat{U}}^{-1}{\mbox{\boldmath$\widehat{\lambda}$}}^{1/2}{\bf\widehat{x}}\cdot\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf\widehat{U}}^{-1}{\mbox{\boldmath$\widehat{\lambda}$}}^{1/2}{\bf\widehat{x}}]^{1/2}$
(5.36)

where we have used

$\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{-1}={\bf\widehat{U}}\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf\widehat{U}}^{-1}$ (5.37)

which is the inverse of (4.96), and

$\displaystyle\big{(}{\mbox{\boldmath$\bar{\lambda}$}}^{\rm
sp}\big{)}^{1/2}=\frac{\xi^{\rm iso}_{0\pm}}{\bar{\xi}^{\rm
sp}_{0\pm}}\;{\mbox{\boldmath$\widehat{\lambda}$}}^{1/2}$ (5.38)

according to (4.95). Finally we employ the inverse of the transformation
(4.89),

${\bf\widehat{U}}^{-1}{\mbox{\boldmath$\widehat{\lambda}$}}^{1/2}{\bf\widehat{x}}={\bf
x}$ (5.39)

and obtain from (5.36) the relation

$\displaystyle\frac{|{\bf\widehat{x}}|}{\xi^{\text{iso}}_{0\pm}}$
$\displaystyle=$ $\displaystyle\frac{[{\bf x}\cdot\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf x}]^{1/2}}{\bar{\xi}^{\rm sp}_{0\pm}}\;,$ (5.40)

compare (5.13). A similar transformation holds for

$\frac{\xi^{\rm iso}_{\rm T}(t)}{|{\bf\widehat{x}}|}=\frac{\bar{\xi}^{\rm
sp}_{\rm T}(t)}{\big{[}{\bf x}\cdot\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf x}\big{]}^{1/2}}$ (5.41)

in (3.8). Together with (5.28) this leads to the scaling form of the
anisotropic bulk order-parameter correlation function (2.25) above, at, and
below $T_{c}$ of the $n$-vector model (2.22) for $2\leq d<4$

$\displaystyle G_{\pm}^{\rm sp}({\bf x},t)=$
$\displaystyle\frac{\Gamma_{+}^{\rm sp}\big{(}\bar{\xi}^{\rm
sp}_{0+}\big{)}^{-2+\eta}}{\big{[}{\bf x}\cdot\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf
x}\big{]}^{(d-2+\eta)/2}}\Psi_{\pm}\Bigg{(}\frac{\big{[}{\bf
x}\cdot\big{(}{\bf\bar{A}}^{\rm sp}\big{)}^{-1}{\bf
x}\big{]}^{1/2}}{\bar{\xi}^{\rm sp}_{\pm}(t)}\Bigg{)}\;\;\;\;\;\;\;\;\;$
(5.42)

with the universal scaling function $\Psi_{\pm}$, (3.46), of the isotropic
system. Here ${\bf\bar{A}}^{\rm sp}$ is to be inserted in the form of (4.98)
as a function of the principal unit vectors and principal correlation lengths.
Right at $T_{c}$ we have the purely algebraic behavior

$\displaystyle G_{\pm}^{\rm sp}({\bf
x},0)=\widetilde{Q}_{3}\frac{\Gamma_{+}^{\rm sp}\big{(}\bar{\xi}^{\rm
sp}_{0+}\big{)}^{-2+\eta}}{\big{[}{\bf x}\cdot\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf x}\big{]}^{(d-2+\eta)/2}}.\;\;\;\;\;\;\;\;\;$ (5.43)

Eqs. (5.42) and (5.43) are the anisotropy-dependent generalizations of Eqs.
(3.45) and (3.49). Similarly we obtain the anisotropic counterpart of the
transverse correlation function (3.8) below $T_{c}$ for $n>1,d>2$

$\displaystyle G^{\rm sp}_{\rm T}({\bf x},t)$ $\displaystyle=$
$\displaystyle{\cal C}_{\rm T}[{\cal M}^{\rm
sp}(t)]^{2}\Bigg{(}\frac{\bar{\xi}^{\rm sp}_{\rm T}(t)}{\big{[}{\bf
x}\cdot\big{(}{\bf\bar{A}}^{\rm sp}\big{)}^{-1}{\bf
x}\big{]}^{1/2}}\Bigg{)}^{d-2}\;\;\;\;$ (5.44)

where ${\cal M}^{\rm sp}(t)={\cal M}^{\rm iso}(t)$ is invariant under the
generalized shear transformation and where ${\bf\bar{A}}^{\rm sp}$ is
represented as a function of $\xi^{{\rm sp}(\alpha)}_{0\rm T}/\xi^{{\rm
sp}(\beta)}_{0\rm T}$ and ${\bf e}^{{\rm sp}(\alpha)}$. In a similar way we
obtain from (3.1), with $D^{\rm iso}_{1}$ replaced by

$\widehat{D}_{1}=\Big{(}\xi^{\rm iso}_{0+}/\bar{\xi}^{\rm
sp}_{0+}\Big{)}^{d-2+\eta}\;D^{\rm sp}_{1},$ (5.45)

the representation above, at, and below $T_{c}$

$\displaystyle G_{\pm}^{\rm sp}({\bf x},t)=$ (5.46)
$\displaystyle\frac{D_{1}^{\rm sp}}{[{\bf x}\cdot\big{(}{\bf\bar{A}}^{\rm
sp}\big{)}^{-1}{\bf x}]^{(d-2+\eta)/2}}\;\Phi_{\pm}\Bigg{(}\frac{\big{[}{\bf
x}\cdot\big{(}{\bf\bar{A}}^{\rm sp}\big{)}^{-1}{\bf
x}\big{]}^{1/2}}{\bar{\xi}^{\rm sp}_{\pm}(t)}\Bigg{)}\;\;\;\;\;\;\;\;\;$

where the universal scaling function $\Phi_{\pm}$ is the same as in (3.1) for
the isotropic system.

The analysis in Fourier space is parallel to that in Sec. V. A. Defining the
generalized shear transformation in ${\bf k}$ space [compare (4.5)]

$\displaystyle{\bf\widehat{k}}={\mbox{\boldmath$\widehat{\lambda}$}}^{1/2}{\bf\widehat{U}}{\bf
k},$ (5.47)

and the Fourier transforms

$\displaystyle\hat{G}^{\rm sp,iso}_{\pm}(|{\bf\widehat{k}}|,t)$
$\displaystyle=$ $\displaystyle\int
d^{d}{\bf\widehat{x}}\;e^{-i{\bf\widehat{k}}\cdot{\bf\widehat{x}}}G^{\rm
sp,iso}_{\pm}(|{\bf\widehat{x}}|,t),$ (5.48) $\displaystyle\hat{G}^{\rm
sp}_{\pm}({\bf k},t)$ $\displaystyle=$ $\displaystyle\int d^{d}{\bf
x}\;e^{-i{\bf k}\cdot{\bf x}}G^{\rm sp}_{\pm}({\bf x},t),$ (5.49)

with ${\bf\widehat{k}}\cdot{\bf\widehat{x}}={\bf k}\cdot{\bf x}$ and

$\displaystyle\hat{G}^{\rm sp}_{\pm}({\bf k},t)$ $\displaystyle=$
$\displaystyle(\det{\mbox{\boldmath$\widehat{\lambda}$}})^{1/2}\hat{G}^{\rm
sp,iso}_{\pm}(|{\bf\widehat{k}}|,t),$ (5.50)
$\displaystyle|{\bf\widehat{k}}|^{2}(\xi_{0\pm}^{\rm iso})^{2}$
$\displaystyle=$ $\displaystyle({\bf k}\cdot\;{\bf\bar{A}^{\rm sp}}{\bf
k})(\bar{\xi}_{0\pm}^{\rm sp})^{2},$ (5.51)

we obtain the isotropic and anisotropic correlation function in ${\bf k}$
space for the $n$-vector model

$\displaystyle\hat{G}_{\pm}^{\rm sp,iso}(|{\bf\widehat{k}}|,t)$
$\displaystyle=$ $\displaystyle\Gamma^{\rm
sp,iso}_{+}\;\frac{\hat{\Psi}_{\pm}\Big{(}|{\bf\widehat{k}}|\xi^{\rm
sp,iso}_{\pm}(t)\Big{)}}{\big{(}|{\bf\widehat{k}}|\xi^{\rm
sp,iso}_{0,+}\big{)}^{2-\eta}}\;,$ (5.52) $\displaystyle\hat{G}_{\pm}^{\rm
sp}({\bf k},t)$ $\displaystyle=$ $\displaystyle\Gamma^{\rm
sp}_{+}\;\frac{\hat{\Psi}_{\pm}\Big{(}[{\bf k}\cdot\;{\bf\bar{A}}^{\rm sp}{\bf
k}]^{1/2}\bar{\xi}^{\rm sp}_{\pm}(t)\Big{)}}{\big{(}\big{[}{\bf
k}\cdot\;{\bf\bar{A}}^{\rm sp}{\bf k}\big{]}^{1/2}\;\bar{\xi}^{\rm
sp}_{0+}\big{)}^{2-\eta}},\;$ (5.53)

where $\hat{\Psi}_{\pm}$ is the same function as in (3.51) and (5.16), with

$\displaystyle\hat{G}^{\rm sp}_{\pm}({\bf k},0)=Q_{3}\frac{\Gamma^{\rm
sp}_{+}}{\big{(}\big{[}{\bf k}\cdot\;{\bf\bar{A}}^{\rm sp}{\bf
k}\big{]}^{1/2}\bar{\xi}^{\rm sp}_{0+}\big{)}^{2-\eta}}\;\;$ (5.54)

at $T_{c}$, with the same universal constant $Q_{3}$ as in (5.22).

The effect of the inverse shear transformations on the arguments of the
correlation functions is condensed into the exact relations (5.13), (5.20),
(5.40), and (5.51) which describe how the isotropic structure is transferred
to the anisotropic structure. No specific properties of the $n$-vector model
have been used, thus the derivation given above can be extended to any weakly
anisotropic system beyond the $\varphi^{4}$ theory without explicit knowledge
of an anisotropy matrix ${\bf A}$ as a function of the couplings. Our results
(5.42), (5.44), (5.46), and (5.53) for the anisotropic $n$-vector model have
the same form as derived for the anisotropic $\varphi^{4}$ model given in
(5.14) and (5.21) and in Eqs. (1.6) and (5.42) of dohm2018 , with the same
critical exponents $\nu$ and $\eta$, the same universal structure of the
reduced anisotropy matrix ${\bf\bar{A}}$ or ${\bf\bar{A}}^{\rm sp}$, the same
universal scaling functions $\Psi_{\pm}$ (or $\Phi_{\pm}$) and
$\hat{\Psi}_{\pm}$, and with the same universal constants $Q_{3}$ and
$\widetilde{Q}_{3}$ for a given $(d,n)$ universality class. This proves the
validity of multiparameter universality for the anisotropic bulk correlation
function for general $d$ and $n$ with up to $d(d+1)/2+1$ independent
nonuniversal parameters. This is the central result of this section. Here we
have not made any assumptions other than the validity of two-scale-factor
universality for isotropic systems and the existence of principal correlation
lengths and principal axes for weakly anisotropic systems together with
(4.80). Our derivation cannot, of course, make any prediction about the
dependence of the nonuniversal parameters on the couplings $E_{i,j}$ of the
$n$-vector model, unlike the dependence on the couplings $K_{i,j}$ within the
$\varphi^{4}$ model dohm2019 ; dohm2018 .

Among the $d(d+1)/2+1$ nonuniversal parameters of weakly anisotropic systems
there are only two independent parameters that can be determined by
thermodynamic measurements, namely the amplitude $\Gamma^{\rm sp}_{+}$ of the
susceptibility and the amplitude $\bar{\xi}_{0+}$ of the mean correlation
length which enters the amplitude of the specific heat above $T_{c}$, as will
be shown in Sec. VI. B. While two-scale-factor universality implies that
isotropic correlation functions can be expressed in terms of purely
thermodynamic amplitudes this property is destroyed by weak anisotropy. The
remaining $d(d+1)/2-1$ independent nonuniversal parameters $\xi_{0\pm}^{{\rm
sp}(\alpha)}/\xi_{0\pm}^{{\rm sp}(\beta)}$ or $\xi_{0{\rm T}}^{{\rm
sp}(\alpha)}/\xi_{0{\rm T}}^{{\rm sp}(\beta)}$ and ${\bf e}^{{\rm
sp}(\alpha)}$ are contained in the reduced anisotropy matrix
${\bf\bar{A}}^{\rm sp}$. Although this matrix has a universal structure in
terms of ratios of principal correlation lengths and in terms of principal
angles the latter are nonuniversal quantities describing the nonuniversal
angular dependence of the critical correlations. Thus there exists a high
degree of intrinsic diversity in the asymptotic critical region of weakly
anisotropic systems with up to five intrinsic parameters in three dimensions.
On the experimental side, the determination of these parameters requires
spatially resolved scattering measurements. On the theoretical side, it has
not been widely recognized in the literature cardy1987 ; cardy1983 ; cardybuch
; Indekeu ; barber1984 ; zia ; diehl-chamati ; diehl2010  that in most cases
the directions of the principal axes depend in a generically unknown way on
the anisotropic couplings dohm2019 ; night1983 . Thus, in practice, it is by
no means simple to identify the appropriate principal axes of anisotropic two-
and three-dimensional $n$-vector models or of real systems before the invoked
anisotropic scale transformations can be performed. For example, even for the
two-dimensional anisotropic Ising model the principal axes and correlation
lengths are known only in a few special cases. We shall substantiate our
findings in the next sections by exact results in the spherical and Gaussian
universality classes and by an approximate result derived from the FRG for
$d=3,n=1$ hassel2007 ; sinner2008  as well as in Sec. VII by exact results for
the $d=2$ Ising universality class.

###  V.3 Exact anisotropic correlation function in the large-${\bf n}$ limit

Exact analytic results can be derived for systems belonging to the spherical
universality class which includes the $O(n)$-symmetric $\varphi^{4}$ model in
the large-$n$ limit, the $n$-vector model in the large-$n$ limit, the
spherical model, and the mean spherical model berlin-kac ; joyce ; brankov ;
Baxter1982 ; stanley1968 . So far the exact scaling form of the anisotropic
bulk correlation function of these models has not been given in the
literature. As a representative of this universality class we take the
large-$n$ limit of the continuum version of the $\varphi^{4}$ model (2.18). We
consider the anisotropic bulk correlation function per component $G_{\infty}$
for $T\geq T_{c}$ in the large-$n$ limit at fixed $u_{0}n$ cd2000-2 ; cd1998

$\displaystyle G_{\infty}({\bf x},t)=\lim_{n\to\infty}\frac{1}{n}<\varphi({\bf
x})\varphi^{\prime}({\bf 0})>$ (5.55) $\displaystyle=\int_{\bf
k}\;\hat{G}_{\infty}({\bf k},t)e^{i{\bf k}\cdot{\bf x}}.$ (5.56)

After the shear transformation (4.1)-(4.3) the isotropic $\varphi^{4}$
Hamiltonian (4.13) and the correlation functions

$\displaystyle G^{\prime}_{\infty}({\bf x^{\prime}},t)$ $\displaystyle=$
$\displaystyle(\det{\bf A})^{1/2}G_{\infty}({\bf x},t),$ (5.57)
$\displaystyle\hat{G}^{\prime}_{\infty}(|{\bf k}^{\prime}|,t)$
$\displaystyle=$
$\displaystyle\hat{G}^{\prime}_{\infty}(|{\mbox{\boldmath$\lambda$}}^{1/2}{\bf
U}{\bf k}|,t)=\hat{G}_{\infty}({\bf k},t),$ (5.58)

are obtained. They are given by cd2000-2 ; cd1998

$\displaystyle\hat{G}^{\prime}_{\infty}(|{\bf k}^{\prime}|,t)$
$\displaystyle=$ $\displaystyle\\{[\chi^{\prime}_{\infty}(t)]^{-1}+({\bf
k^{\prime}})^{2}\\}^{-1}\;\;\;,$ (5.59) $\displaystyle
G^{\prime}_{\infty}({\bf x}^{\prime},t)$ $\displaystyle=$
$\displaystyle\int_{\bf k^{\prime}}\;\frac{e^{i{\bf k^{\prime}}\cdot{\bf
x}^{\prime}}}{[\chi^{\prime}_{\infty}(t)]^{-1}+({\bf k^{\prime}})^{2}}\;\;,$
(5.60)

where the inverse of the bulk susceptibility per component is determined
implicitly by

$[\chi^{\prime}_{\infty}(t)]^{-1}=r_{0}+4u^{\prime}_{0}n\int_{\bf
k^{\prime}}\\{[\chi^{\prime}_{\infty}(t)]^{-1}+({\bf
k^{\prime}})^{2}\\}^{-1}.$ (5.61)

The same equation determines the square of the bulk correlation length above
$T_{c}$ because cd1998

$\displaystyle\chi^{\prime}_{\infty}(t)$ $\displaystyle=$
$\displaystyle[\xi^{\prime}_{\infty}(t)]^{2}.$ (5.62)

Here $\int_{\bf k^{\prime}}$ stands for $(2\pi)^{-d}\int d^{d}k^{\prime}$ with
a transformed cutoff. This cutoff dependence becomes negligible in the
asymptotic region near $T_{c}$ where $G^{\prime}_{\infty}({\bf
x^{\prime}},t)\to G^{\prime}_{\infty}(|{\bf x^{\prime}}|,t)$ becomes
isotropic. For $t>0$ and $2<d<4$ the asymptotic behavior is cd1998

$\displaystyle\chi^{\prime}_{\infty}(t)$ $\displaystyle=$
$\displaystyle\Gamma^{\prime}_{\infty}\;t^{-\gamma_{\infty}},$ (5.63)
$\displaystyle\xi^{\prime}_{\infty}(t)$ $\displaystyle=$
$\displaystyle\xi^{\prime}_{\infty,0}t^{-\nu_{\infty}},$ (5.64)
$\displaystyle\xi^{\prime}_{\infty,0}$ $\displaystyle=$
$\displaystyle[4u^{\prime}_{0}nA_{d}/(\varepsilon a_{0})]^{1/(d-2)},$ (5.65)
$\displaystyle A_{d}\;$ $\displaystyle=$
$\displaystyle\;\frac{\Gamma(3-d/2)}{2^{d-2}\pi^{d/2}(d-2)}$ (5.66)

with $\varepsilon=4-d$, with the critical exponents

$\displaystyle\gamma^{\prime}_{\infty}=2\nu_{\infty},\;\nu_{\infty}=1/(d-2),\;\eta_{\infty}=0.\;\;$
(5.67)

In particular the amplitude of the susceptibility

$\Gamma^{\prime}_{\infty}=\big{(}\xi^{\prime}_{\infty,0}\big{)}^{2}$ (5.68)

is independent of the cutoff of the $\varphi^{4}$ Hamiltonian. We note that
this simple relation is not valid for all members of the spherical
universality class, e.g., it is not valid for the spherical model dan-2 .

The universal isotropic scaling form for $t\geq 0$ in ${\bf k^{\prime}}$ space
is simply

$\displaystyle\hat{G}^{\prime}_{\infty}(|{\bf k}^{\prime}|,t)$
$\displaystyle=$ $\displaystyle({\bf
k^{\prime}})^{-2}\hat{\Psi}_{\infty}(|{\bf
k^{\prime}}|\xi^{\prime}_{\infty}(t)),$ (5.69)
$\displaystyle\hat{\Psi}_{\infty}(y)$ $\displaystyle=$
$\displaystyle(1+y^{-2})^{-1},$ (5.70) $\displaystyle Q_{\infty,3}$
$\displaystyle=$ $\displaystyle\hat{\Psi}_{\infty}(\infty)=1.$ (5.71)

To derive $G^{\prime}_{\infty}(|{\bf x}^{\prime}|,t)$ in real space we use the
integral representation $w^{-1}=\int_{0}^{\infty}dse^{-ws}$ of the quantity
$w=(\xi^{\prime}_{\infty})^{-2}+({\bf k^{\prime}})^{2}$. Then we obtain from
(5.60)

$G^{\prime}_{\infty}(|{\bf
x}^{\prime}|,t)=\int_{0}^{\infty}dse^{-s/[\xi^{\prime}_{\infty}(t)]^{2}}\int_{\bf
k^{\prime}}\;e^{-s({\bf k^{\prime}})^{2}+i{\bf k^{\prime}}\cdot{\bf
x}^{\prime}}.$ (5.72)

Performing the Gaussian ${\bf k^{\prime}}$-integration at infinite cutoff
cd1998 ; dohm2008  we obtain the scaling form

$\displaystyle G^{\prime}_{\infty}(|{\bf x}^{\prime}|,t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{ds}{(4\pi
s)^{d/2}}e^{-s/(\xi^{\prime}_{\infty})^{2}-|{\bf
x}^{\prime}|^{2}/(4s)}\;\;\;\;\;\;\;\;\;$ (5.73) $\displaystyle=$
$\displaystyle\frac{1}{|{\bf x}^{\prime}|^{d-2}}\Psi_{\infty}(|{\bf
x}^{\prime}|/\xi^{\prime}_{\infty})\;\;\;\;\;\;\;\;\;$ (5.74)

where the exact universal scaling function is given by

$\displaystyle\Psi_{\infty}(y)$ $\displaystyle=$
$\displaystyle\frac{y^{d-2}}{(4\pi)^{d/2}}\int_{0}^{\infty}dss^{-d/2}e^{-s-y^{2}/(4s)},\;\;\;$
(5.75) $\displaystyle\widetilde{Q}_{\infty,3}$ $\displaystyle=$
$\displaystyle\Psi_{\infty}(0)=\frac{\Gamma[(d-2)/2]}{4\pi^{d/2}}\;.\;\;\;\;$
(5.76)

This function depends only on the scalar scaling variable $y=|{\bf
x}^{\prime}|/\xi^{\prime}_{\infty}$ because of isotropy. $G^{\prime}_{\infty}$
has indeed the form of (3.45) with $\eta=0$ since, according to (5.68), we may
replace 1 in (5.74) by
$1=\Gamma^{\prime}_{\infty}(\xi^{\prime}_{\infty,0})^{-2}$. Only owing to the
simple relation (5.68) the isotropic correlation function in the large-$n$
limit has a single-parameter scaling form which is not universally valid for
finite $n$ and $\eta\neq 0$.

On the basis of the exact isotropic results (5.69)-(5.75) it is straight
forward to determine the anisotropic correlation functions in real space and
${\bf k}$ space from (5.57) and (5.58) by substituting the general results of
Secs. IV and V. Using (4.9), (4.33), (4.50) and the general relations (5.13)
and (5.20) we obtain the exact anisotropic correlation function of the
$\varphi^{4}$ model in the large-$n$ limit for $d>2$ and $t\geq 0$

$\displaystyle G_{\infty}({\bf x},t)$ (5.77)
$\displaystyle\;=\frac{\Gamma_{\infty}\;(\bar{\xi}_{\infty,0})^{-2}}{[{\bf
x}\cdot({\bf\bar{A}}^{-1}{\bf x})]^{(d-2)/2}}\;\Psi_{\infty}\Big{(}\frac{[{\bf
x}\cdot({\bf\bar{A}}^{-1}{\bf
x})]^{1/2}}{\bar{\xi}_{\infty}(t)}\Big{)},\;\;\;\;\;\;\;$
$\displaystyle\hat{G}_{\infty}({\bf
k},t)=\frac{\Gamma_{\infty}\;\;(\bar{\xi}_{\infty,0})^{-2}}{{\bf
k}\cdot\;{\bf\bar{A}}{\bf k}}\hat{\Psi}_{\infty}\Big{(}[{\bf
k}\cdot\;{\bf\bar{A}}{\bf
k}]^{1/2}\bar{\xi}_{\infty}(t)\Big{)},\;\;\;\;\;\;\;\;\;$ (5.79)
$\displaystyle\Gamma_{\infty}\;\;(\bar{\xi}_{\infty,0})^{-2}=\Big{(}\prod^{d}_{\alpha=1}{\mbox{$\lambda$}_{\alpha}}\Big{)}^{-1/d}=(\det{\bf
A})^{-1/d},$

where $\Psi_{\infty}$ and $\hat{\Psi}_{\infty}$ are the same scaling functions
as in the isotropic case. Here
$\Gamma_{\infty}=\Gamma_{\infty}^{\prime}=(\bar{\xi}_{\infty,0})^{2}$ and
$\bar{\xi}_{\infty,0}$ are the amplitudes of the susceptibility per component
and mean correlation length of the anisotropic system above $T_{c}$,

$\displaystyle\chi_{\infty}(t)$ $\displaystyle=$
$\displaystyle\Gamma_{\infty}\;t^{-\gamma_{\infty}},$ (5.80)
$\displaystyle\bar{\xi}_{\infty}(t)$ $\displaystyle=$
$\displaystyle\bar{\xi}_{\infty,0}\;t^{-\nu_{\infty}},$ (5.81)
$\displaystyle\bar{\xi}_{\infty,0}$ $\displaystyle=$
$\displaystyle\big{[}\prod^{d}_{\alpha=1}\xi_{\infty,0}^{(\alpha)}\big{]}^{1/d}$
(5.82)

where the amplitudes $\xi_{\infty,0}^{(\alpha)}$ of the principal correlation
lengths

$\displaystyle\xi^{(\alpha)}_{\infty
0}={\mbox{$\lambda$}_{\alpha}}^{1/2}\xi^{\prime}_{\infty,0}$ (5.83)

are determined by the eigenvalues ${\mbox{$\lambda$}_{\alpha}}$ of the
anisotropy matrix ${\bf A}$ and by the amplitude $\xi^{\prime}_{\infty,0}$ of
the isotropic correlation length above $T_{c}$ of the Hamiltonian (4.13).

Below $T_{c}$ the algebraic large-distance behavior of the transverse
correlation function of the isotropic $\varphi^{4}$ Hamiltonian (4.13) in the
large-$n$ limit reads for $d>2$,

$\displaystyle G^{\prime}_{\infty,\rm T}(|{\bf x}^{\prime}|,t)$
$\displaystyle=$ $\displaystyle{\cal C}_{\rm T}\big{[}{\cal
M}^{\prime}_{\infty}(t)\big{]}^{2}\big{[}\xi^{\prime}_{\infty,\rm T}(t)/|{\bf
x}^{\prime}|\big{]}^{d-2},\;\;\;\;$ (5.84) $\displaystyle\big{[}{\cal
M}^{\prime}_{\infty}(t)\big{]}^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{\prime}_{\infty,0c}-r_{0}}{4u^{\prime}_{0}n}=(B^{\prime}_{\infty})^{2}|t|^{2\beta_{\infty}},$
(5.85) $\displaystyle{\beta_{\infty}}$ $\displaystyle=$ $\displaystyle
1/2,\;\;\;\;\;$ (5.86) $\displaystyle(B^{\prime}_{\infty})^{2}$
$\displaystyle=$ $\displaystyle a_{0}/(4u^{\prime}_{0}n),$ (5.87)
$\displaystyle r^{\prime}_{\infty,0c}$ $\displaystyle=$
$\displaystyle-4u^{\prime}_{0}n\int_{\bf k^{\prime}}{\bf k^{\prime}}^{-2},$
(5.88)

according to (3.8) and to Eq. (28) of cd1998 . The transverse correlation
length $\xi^{\prime}_{\infty,\rm T}(t)=\xi^{\prime}_{\infty,\rm
T,0}|t|^{-\nu_{\infty}}$ below $T_{c}$ is universally related to the
correlation length $\xi^{\prime}_{\infty}(t)$ above $T_{c}$

$\xi^{\prime}_{\infty,\rm T,0}/\xi^{\prime}_{\infty,0}=X_{\infty\rm
T}=\text{universal},$ (5.89)