# Different critical behaviors in cubic to trigonal and tetragonal perovskites

Amnon Aharony<EMAIL_ADDRESS>School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 6997801, Israel  Ora Entin-Wohlman<EMAIL_ADDRESS>School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
Andrey Kudlis<EMAIL_ADDRESS>ITMO University, Kronverkskiy prospekt
49, Saint Petersburg 197101, Russia

###### Abstract

Perovskites like LaAlO3 (or SrTiO3) undergo displacive structural phase
transitions from a cubic crystal to a trigonal (or tetragonal) structure. For
many years, the critical exponents in both these types of transitions have
been fitted to those of the isotropic three-components Heisenberg model.
However, field theoretical calculations showed that the isotropic fixed point
of the renormalization group is unstable, and renormalization group iterations
flow either to a cubic fixed point or to a fluctuation-driven first-order
transition. Here we show that these two scenarios correspond to the cubic to
trigonal and to the cubic to tetragonal transitions, respectively. In both
cases, the critical behvior is described by slowly varying effective critical
exponents, which exhibit universal features. For the trigonal case, we predict
a crossover of the effective exponents from their Ising values to their cubic
values (which are close to the isotropic ones). For the tetragonal case, the
effective exponents can have the isotropic values over a wide temperature
range, before exhibiting large changes en route to the first-order transition.
New renormalization group calculations near the isotropic fixed point in three
dimensions are presented and used to estimate the effective exponents, and
dedicated experiments to test these predictions are proposed. Similar
predictions apply to cubic magnetic and ferroelectric systems.

##  I Introduction

Perovskite materials exhibit intriguing physical properties, and have been
extensively explored for both practical applications and theoretical modeling
[1]. In particular, perovskites like SrTiO3 and LaAlO3 play important roles in
modern solid state applications [2]. At high temperatures, perovskites usually
have a cubic structure (left panel, Fig. 1). As the temperature $T$ decreases,
some perovskites undergo an antiferrodistortive structural transition from the
cubic to a lower-symmetry structure, via a rotation of the oxygen (or
fluorine) octahedra: SrTiO3, KMnF3, RbCaF3 and others undergo a cubic to
tetragonal transition, see Fig. 1. The octahedra rotate around a cubic axis
and the order-parameter vector ${\bf Q}$ (aka the rotation vector) is along
that axis (with a length proportional to the rotation angle). Similar
rotations occur in double perovskites, e.g., the tetragonal to orthorhombic
transition in La2CuO4 [3]. In contrast, other perovskites, e.g., LaAlO3,
PrAlO3, and NdAlO3, undergo a cubic to trigonal transition, with ${\bf Q}$
along one of the cubic diagonals.

Figure 1: (color online) The cubic (left) and tetragonal (right) unit cells in
SrTiO3 (the latter shows only half the cell: neighboring cells rotate in
opposite directions). Large, intermediate and small spheres correspond to Sr,
O and Ti ions, respectively. The dashed lines represent the octahedra, which
rotate around the vector ${\bf Q}$, lying along the vertical axis (the O ions
in the central horizontal plane move as indicated by the arrow). When ${\bf
Q}$ is along a diagonal of the cube, the octahedra rotate around that
diagonal, and the unit cell is stretched along ${\bf Q}$, causing a cubic to
trigonal transition (as in LaAlO3).

The behavior of a system at the vicinity of its transition temperature $T_{c}$
can be expressed by critical exponents. When the transition at $T_{c}$ is
continuous, the correlation length diverges as $\xi\propto|t|^{-\nu}$ and the
order-parameter approaches zero (for temperatures $T<T_{c}$) as $|\langle{\bf
Q}\rangle|\propto|t|^{\beta}$, where $t=T/T_{c}-1$. The critical exponents
$\nu$ and $\beta$ are expected to be universal, i.e., having the same values
for many physical systems. The exponents describing other physical properties,
e.g., $\alpha$ and $\gamma$ for the specific heat and for the order parameter
susceptibility, are obtained via scaling relations,
$d\nu=2-\alpha=2\beta+\gamma$, where $d$ is the dimensionality.

The $T-$dependence of the order-parameters of SrTiO3 and LaAlO3 had been
measured by Müller and Berlinger [4], who obtained a collapse of the scaled
order-parameters in both materials onto a single line, showing a crossover
from the mean-field exponent $\beta_{MF}=1/2$, valid relatively far away from
$T_{c}$ (i.e., at relatively large $|t|$) to an apparently universal critical
exponent $\beta_{c}=0.33\pm 0.02$ close to $T_{c}$. Consequently, it was
concluded erroneously that both materials share the same exponents, and so
belong to the same universality class and have the same critical behavior. In
fact it turned out that both samples were single domains, due to a uniaxial
anisotropy caused by polishing [5]. Therefore both systems ordered along a
single axis, and the data were consistent with Ising ($n=1$, a single-
component order-parameter) exponents [6, 7, 8]. Later experiments gave a wide
range of results, e.g. $\beta\simeq 0.40\pm 0.03,~{}1/3,~{}0.27,~{}0.17\pm
0.02$ for SrTiO3 [9], KMnF3 [10], RbCaF3 and NaNbO3 [11], respectively.
Furthermore, some experiments hint that SrTiO3 may be close to a tricritical
point [12, 13, 14], while both RbCaF3 and KMnF3 have first-order transitions
[15, 16]. More experiments are reviewed e.g. in Refs. 17, 18, 19.

Here we show theoretically that the cubic to trigonal and the cubic to
tetragonal phase transitions do not share the same critical behavior. In fact
these two transitions are characterized by different behaviors of the
renormalization-group (RG) iterations. We find that systems undergoing a cubic
to trigonal transitions have second-order transitions, with the universal
critical exponents of the cubic fixed point (FP). However, since the cubic and
the isotropic fixed points turn out to be very close to each other (see
below), it is indeed difficult to distinguish between their asymptotic
critical exponents even without any uniaxial symmetry breaking. As we show,
these asymptotic exponents are expected only very close to $T_{c}$. For a
large range of $t$ we predict effective exponents, which vary with $t$. In
contrast, the cubic to tetragonal transitions become fluctuation-driven first-
order at small $|t|$, with different effective exponents over an intermediate
range of $t$. These effective exponents are characterized by the isotropic
fixed point, which dominates the crossover to the first-order transition. This
conclusion extends to quite a number of other systems as well.

Our predictions for the effective exponents come from a novel way to obtain
analytic solutions to the RG recursion relations in the vicinity of the
isotropic and cubic fixed points. After a short review of the theory of the RG
on cubic systems in Sec. II, Sec. III presents our new calculation. Our
predictions are compared with the above experimental information in Sec. IV,
where we also propose dedicated measurements to test our results and list
other cubic systems, which are expected to exhibit similar behaviors. Section
V contains our conclusions. The Appendices describe the resummation technique
and the analytic solution of the recursion relations.

##  II  The renormalization group on cubic systems

Fifty years ago, Wilson [20] showed that at $T$ very close to $T_{c}$ the
short-length details can be eliminated on scales below $1/e^{\ell}$ ($\ell$
counts the number of iterations in the elimination process), and that
rescaling the unit length by the factor $e^{\ell}$ yields a renormalized
effective (dimensionless) Hamiltonian (or free-energy density) $\overline{\cal
H}(\ell)$, which ‘flows’ in the space spanned by all such Hamiltonians. These
flows represent the RG. Critical points are associated with fixed points of
these flows, which are invariant under the RG iterations. Near a fixed point,
the singular part of the corresponding free-energy density obeys the
homogeneous scaling form

$\displaystyle{\cal F}(\\{\mu_{i}\\})$ $\displaystyle=e^{-d\ell_{f}}{\cal
F}(\\{\mu_{i}(0)e^{\lambda_{i}\ell_{f}}\\})$
$\displaystyle\equiv|t|^{d\nu}{\cal
W}(h|t|^{-\nu\lambda_{2}},~{}\mu_{3}(0)|t|^{-\nu\lambda_{3}},~{}\dots),$ (1)

where the $\mu_{i}(0)$’s are parameters that measure deviations from the fixed
point, and the $\lambda_{i}$ are exponents describing their variation as
function of $\ell$. The first two parameters are the temperature $\mu_{1}=t$
and the ordering field $\mu_{2}=h$. Continuing the RG flow until
$\xi(\ell_{f})=\xi(0)e^{-\ell_{f}}\sim 1$, with $\xi(0)\sim|t(0)|^{-\nu}$,
yields $\nu=1/\lambda_{1}$. Derivatives of ${\cal F}$ w.r.t. $t$ and $h$ yield
the critical exponents for the measurable quantities, e.g. the scaling
relations $\beta=\nu(d-\lambda_{2})$, $\gamma=\nu(d-2\lambda_{2})$ and
$\alpha=2-d\nu$. The exponents and the scaling function ${\cal W}$ for systems
near a specific FP are fully determined by the FP itself, and not by the
initial effective Hamiltonian, which encompasses the short-scales behavior.
Therefore they are universal [21, 22, 23, 24]. All the physical systems which
flow to that FP then belong to its universality class, and exhibit the same
critical exponents. A stable fixed point has only two relevant variables, $t$
and $h$, with $\lambda_{1},~{}\lambda_{2}>0$. All the other parameters are
irrelevant, with $\lambda_{i}<0$. In contrast, when a third parameter,
$\mu_{3}$, is also relevant, $\lambda_{3}>0$, the fixed point is unstable, and
the RG trajectories flow to another, stable, fixed point, via a crossover
region, or flow to a region where the renormalized Hamiltonian has a first
order transition.

The RG analysis of cubic systems has been mainly based on the Landau theory
[25, 26], which expands the free-energy density in powers of the (small)
order-parameter components $Q_{i}$ ($i=1,2,\ldots,n$). The terms in this
expansion are determined by the symmetries of the system above the transition.
For the isotropic $n-$component order-parameter vector ${\bf Q}$, this free-
energy is $U_{0}({\bf Q})=r|{\bf Q}|^{2}/2+u|{\bf Q}|^{4}+{\cal O}[|{\bf
Q}|^{6}]$, where $u$ is a system-dependent parameter, while $r=T/T^{\rm
MF}_{c}-1$, with the mean-field transition temperature $T^{\rm MF}_{c}$ (the
parameter $t$, mentioned above, contains the downwards shift from $T^{\rm
MF}_{c}$ to $T_{c}$ by the fluctuations). The cubic symmetry is characterized
by adding the term $U_{v}({\bf Q})=v\Sigma^{n}_{i=1}Q^{4}_{i}$ to the free
energy, with $v$ a system-dependent coefficient [28, 27]. Long wave-length
fluctuations in ${\bf Q}({\bf r})$ are introduced via a gradient term [29],
$|{\boldmath{\nabla}}{\bf Q}({\bf r})|^{2}$, whose coefficient is normalized
to $1$. The effective Hamiltonian is then written as $\int
d^{d}r\overline{\cal H}({\bf r})$, where

$\displaystyle\overline{\cal H}({\bf r})\equiv|{\boldmath{\nabla}}{\bf Q}({\bf
r})|^{2}/2+U_{0}[{\bf Q}({\bf r})]+U_{v}[{\bf Q}({\bf r})].$ (2)

In the absence of the ordering field $h$, the parameters which flow under the
RG iterations are $r~{},u$ and $v$.

The Wilson-Fisher RG at $d=4-\epsilon$ is performed in Fourier space,
eliminating large momentum (small length scale) components of the order-
parameter [30, 21]. As we discuss below, it is not trivial to extrapolate the
results to $d=3$, i.e., $\epsilon=1$. Generally, the recursion relations in
the $u-v$ plane have the form

$\displaystyle\frac{\partial u}{\partial\ell}$
$\displaystyle=\bar{\beta}_{u}[\epsilon,u(\ell),v(\ell)],\ \ \ \frac{\partial
v}{\partial\ell}=\bar{\beta}_{v}[\epsilon,u(\ell),v(\ell)],$ (3)

and the $\bar{\beta}$ functions (not to be confused with the critical exponent
$\beta$) are expanded in powers of their arguments. The fixed points
$u^{\ast},v^{\ast}$ are found as the zeroes of these functions, with values
which are series in $\epsilon$. For $v=0$, this procedure gives two FP’s, one
at $u^{\ast}_{G}=0$, termed Gaussian, and the other, termed the isotropic FP,
with $u^{\ast}_{I}(n)={\cal O}[\epsilon]>0$. For $d<4$ the Gaussian FP is
unstable, i.e., $\lambda^{G}_{u}>0$, so that systems for which $u>0$ flow
towards the stable isotropic FP [31] (where $\lambda^{I}_{u}<0$) and those
with $u<0$ flow to a region in which the mean-field analysis of the
renormalized free-energy yields a first-order transition (stabilized by the
positive sixth-order terms) [32, 33]. The Gaussian FP is thus identified as a
tricritical point, separating between a first- and a second-order transitions.

The RG analysis of the cubic model, Eq. (2), for general $n$ and in dimension
$d=4-\epsilon$ [27, 34, 35] yielded four FP’s of order $\epsilon$: the
Gaussian ($G$, $u^{\ast}_{G}=v^{\ast}_{G}=0$), isotropic ($I$,
$v^{\ast}_{I}=0,~{}u^{\ast}_{I}>0$), decoupled Ising ($D$,
$u^{\ast}_{D}=0,~{}v^{\ast}_{D}>0$, for which the different $Q_{i}$’s decouple
from each other and exhibit the Ising model behavior), and ‘cubic’ ($C$) FP’s.
The location of the cubic FP, $(u^{\ast}_{C},~{}v^{\ast}_{C})$, depends on the
number of the order-parameter components, $n$: for small (large) $n$, it is in
the lower (upper) half plane, as shown on the left (right) panel of Fig. 2.
This figure [24] has been reproduced by other authors, e.g., Refs. 36, 37, 38,
and included in textbooks [39]. In these references, the FP values were
calculated using various approximations, but (except at lowest order in
$\epsilon$) the flow lines were drawn schematically. The figure shows the
critical surface of the effective Hamiltonian in the $u-v$ plane, on which
$|t|=h=0$ and $\xi=\infty$ [40]. At a finite (but very small) $|t|$ the RG
flow starts very close to this surface, and as the RG is iterated the flow
stays close to the arrows in the figure. When the flow reaches the vicinity of
a FP the system exhibits the critical exponents of that FP.

Figure 2: (color online) Schematic flow diagram and fixed points for the cubic
model, Eq. (2), adapted from Ref. 24. $G=$Gaussian, $I=$isotropic,
$D=$Decoupled (Ising) and $C=$Cubic FP’s. S=initial point for SrTiO3.
L=initial point for LaAlO3. The dashed lines represent the stability edges,
$u+v=0$ (for $v<0$) and $u+v/n=0$ (for $v>0$), below which the free energy in
Eq. (2) is stabilized by the terms of order $|{\bf Q}|^{6}$, and the
transitions are first-order. The shaded areas are the regions of attraction of
the stable FP’s ($I$ on left and $C$ on right).

As seen in Fig. 2, the Gaussian FP is doubly unstable; both $u$ and $v$ are
relevant in its vicinity. The decoupled FP is singly unstable, with $u$ being
relevant [24, 41]. The stability of the isotropic and cubic FP’s depends on
the borderline value $n_{c}(3)$. It was clear that $2<n_{c}(3)<4$, but
different approximations yielded conflicting answers to the question whether
$n=d=3$ is above or below $n_{c}(3)$ (for the history, see Refs. 8, 42, 36,
37, 43, 38 and references therein). For instance, a third-order
$\epsilon-$expansion gave [24] $n_{c}(3)\approx 3.128$ at $\epsilon=1$. The
result $3<n_{c}(d=3)$ was also obtained by the scaling-field method [44]. If
this were true then all the second order transitions cubic systems would be
decsribed by the universality class of the isotropic FP, and the cubic
deviatoions from rotational symmetry would decay at criticality.

However, this scenario is now known to be wrong. Four accurate methods (Monte
Carlo simulations of lattice $O(n)$ models [42], six-loop recursion relations
[36] at $d=3$, the good old $\epsilon$-expansion, recently expanded to order
$\epsilon^{6}$ [37] and the very recent bootstrap method, which calculates
exponents at any dimension [43]) find $2.85<n_{c}(3)<3$. Therefore, the RG
flows are as in the right panel in Fig. 2: the isotropic FP is unstable [with
a small but positive exponent for the flow of $v$,
$0<\lambda^{I}_{v}\backsimeq 0.02$, see Eq. (1)], while the cubic FP has a
small but positive FP value $v^{\ast}_{C}>0$, and is fully stable (deviations
of both $u$ and $v$ from it decay under the RG iterations).

Without even looking at the specific numerical values of the locations of the
FP’s, the right panel of Fig. 2 yields qualitative crucial consequences: since
the cubic to trigonal and cubic to tetragonal transitions correspond to
opposite signs of $v$, they have different flow trajectories. For the former,
$v<0$. If the initial $|v|$ is small, and if the initial $u$ is positive, so
that $u+v>0$, then the respective effective Hamiltonian (shown by the blue
trajectory leaving S in Fig. 2) first flows closer to the isotropic FP, and
may then exhibit $t-$dependent effective exponents associated with that FP,
but eventually it must turn downwards, and cross the stability line $u+v=0$ \-
turning the transition fluctuation-driven first-order [45]. As
$\lambda^{I}_{v}$ is small at $n=d=3$ [42, 36, 37, 43], this flow is slow, so
that the first-order transition will occur only close to $T_{c}$, with a small
discontinuity. For larger initial $|v|$’s the transition becomes first-order
at larger $|t|$. In contrast, the cubic to trigonal transition has $v>0$, and
therefore its Hamiltonian must flow to the stable cubic FP, resulting in a
second-order transition with cubic exponents.

As mentioned above, the cubic and the isotropic FP’s are close to each other
at $n=d=3$ [42, 36, 37, 43]. Indeed, the calculated asymptotic critical
exponents (expected only at very small $|t|$) are $\nu^{I}\backsimeq
0.706,~{}\nu^{C}\backsimeq 0.700$,
$\beta^{I}\backsimeq.366,~{}\beta^{C}\backsimeq 0.368$. This closeness also
implies that the stability exponent of the stable cubic FP,
$\lambda^{C}_{3}\lesssim 0$ and that of the unstable isotropic FP,
$\lambda^{I}_{v}\gtrsim 0$, are small, indicating slow flows towards and away
from these FP’s. The experimental exponents should therefore be compared with
effective exponents, e.g., $\beta_{\rm eff}\equiv\partial\log|{\bf
Q}|/\partial\log|t|$, which depend on $|t|$. For $v>0$, the cubic FP is
stable, with two negative stability exponents,
$\lambda^{C}_{4}<\lambda^{C}_{3}<0$ (for flows in the $u-v$ plane). Therefore
$\beta_{\rm eff}$ approaches the asymptotic $\beta^{C}$ slowly, with
corrections of order $|t|^{|\varphi^{C}_{3}|}$, where
$\varphi^{C}_{3}=\nu^{C}\lambda^{C}_{3}$ is small. For $v<0$, $\beta_{\rm
eff}$ first approaches the isotropic FP value $\beta^{I}$, but then (for
smaller $|t|$, i.e., larger number of iterations $\ell$), it moves away from
that value, and $|{\bf Q}|$ has a discontinuity.

##  III RG flow near the isotropic and cubic fixed points

To quantify the above qualitative statements, we used existing
$\epsilon^{6}-$order expansions [37] to derive RG flow equations in the
vicinity of the isotropic and cubic fixed points at $d=n=3$. As discussed in
Ref. 37 (and also in other references, e.g., Refs. 46, 47, 48, 49), these
series are divergent, and numerical estimates of the quantities of interest at
$\epsilon=1$ were obtained employing resummation techniques, see Appendix A.

Critical exponents are found from the RG recursion relations for the various
scaling fields. To linear order in $\mu_{i}(\ell)$ these have the form

$\displaystyle\frac{\partial\mu_{i}}{\partial\ell}=\lambda_{i}(\epsilon,u,v)\mu_{i}(\ell),$
(4)

where the effective exponent $\lambda_{i}$ is expanded in powers of its
arguments and then resummed. At a fixed point, $u^{\ast}$ and $v^{\ast}$ are
replaced by their $\epsilon-$expansions, and the series for $\lambda_{i}$ are
resummed to give asymptotic values at $d=n=3$ [37].

As mentioned, $v$ varies slowly near the isotropic and cubic FP’s. Therefore,
we need to solve the recursion equations (3) as functions of the number of
iterations $\ell$, and then estimate the effective critical exponents
$\lambda_{i}(\ell)$ at finite values of $\ell$, up to the total number of
iterations $\ell_{f}$. Since $t(\ell)=t(0)e^{\lambda_{1}\ell}$, a larger
$\ell_{f}$ implies a smaller $t(0)$, i.e., an initial state closer to the
critical point. The divergence of the $u-$ and $v-$dependent series for
$\beta_{u}$, $\beta_{v}$ and $\lambda_{i}$ prevents their use in solving the
differential equations (3). Instead, we derived approximate expressions of
$\beta_{u}$ and $\beta_{v}$, by expanding them to second order near the
isotropic FP, and resumming the $\epsilon-$expansions of the resulting
coefficients. Since the cubic FP is expected to be close to the isotropic one,
we expect these expansions to give reasonable results near both FP’s. Hence,

$\displaystyle\frac{\partial\delta u}{\partial\ell}=\lambda_{u}\delta
u+a_{01}v+a_{11}v\delta u+\frac{1}{2}[a_{20}\delta u^{2}+a_{02}v^{2}],$ (5)
$\displaystyle\frac{\partial v}{\partial\ell}=\lambda_{v}v+b_{11}v\delta
u+\frac{1}{2}b_{02}v^{2},$ (6)

where $\delta u=u-u^{\ast}_{I}$ and the coefficients $a_{ij}$ and $b_{ij}$,
which are given in Table 1, are found from

$\displaystyle\left.a_{ij}=\widehat{\textup{Rsm}}\Bigg{[}\dfrac{\partial^{(i+j)}\beta_{u}(u,v)}{\partial
u^{i}\partial v^{j}}\Bigg{]}\right\rvert_{u=u^{\ast}_{I},v=0},$
$\displaystyle\left.b_{ij}=\widehat{\textup{Rsm}}\Bigg{[}\dfrac{\partial^{(i+j)}\beta_{v}(u,v)}{\partial
u^{i}\partial v^{j}}\Bigg{]}\right\rvert_{u=u^{\ast}_{I},v=0}.$ (7)

The operation $\widehat{\textup{Rsm}}$ denotes resummation of the $\epsilon$
expansions at $\epsilon=1$ (and $n=3$) of the derivatives, using the procedure
described in Appendix A. Specifically,
$\lambda^{I}_{u}\equiv\lambda_{u}=a_{10}$ and
$\lambda^{I}_{v}\equiv\lambda_{v}=b_{01}$ are the stability exponents of the
isotropic FP, known accurately from resummations of the $\epsilon^{6}$ series
[37]. Similarly, the isotropic exponents are identical to those calculated
before, e.g. in Ref. 37. The values in Table 1 are taken from Refs. 48, 49,
which use the same resummation as described in Appendix A.

Table 1: Numerical estimates of the coefficients entering Eqs. (5), (6) and (15). The numbers are found by means of the resummation procedure described in Appendix A.  Quantity | Value | Quantity | Value
---|---|---|---
$u^{\ast}_{I}$ | $0.39273(63)$ | $v^{\ast}_{\textup{I}}$ | $0$
$a_{01}$ | $-0.4791(19)$ | $a_{10}$ | $-0.7967(57)$
$a_{11}$ | $-0.938(36)$ | $a_{02}$ | $-0.037(21)$
$a_{20}$ | $-3.423(58)$ | $b_{01}$ | $0.0083(15)$
$b_{11}$ | $-1.854(15)$ | $b_{02}$ | $-2.971(18)$
$\beta^{I}$ |  $0.3663(12)$[48] | $\gamma^{I}$ |  $1.385(4)$[48]
$\varphi^{I}$ |  $1.263(13)$[49] |  |
$c_{10}$ | $0.431(17)$ | $c_{01}$ | $0.258(10)$
$c_{11}$ | $0.49(11)$ | $c_{20}$ | $0.82(20)$
$c_{02}$ | $0.29(11)$ | $d_{10}$ | $1.177(62)$
$d_{01}$ | $0.706(38)$ | $d_{11}$ | $0.820(54)$
$d_{20}$ | $1.367(90)$ | $d_{02}$ | $0.26(12)$
Table 2: Numerical estimates of the cubic fixed point and the exponents obtained by our approximation, compared to those found in Ref. 37 and those calculated by means of the resummation strategy suggested in Ref. 48 and used in Appendix A. The error bars on the approximate values are based on Eq. (15). Quantity | Effective | Ref. 37 | Using
---|---|---|---
|  |  | Ref. 48
$u^{*}_{C}$ | $0.3791(27)$ | - | $0.376(19)$
$v^{*}_{C}$ | $0.0226(43)$ | - | $0.028(11)$
$\gamma^{C}$ | $1.3849(61)$ | $1.368(12)$ | $1.387(9)$
$\beta^{C}$ | $0.3663(21)$ | $0.3684(13)$ | $0.3669(12)$

To solve the recursion relations (5) and (6), it is convenient to define the
non-linear scaling field [50, 51],

$\displaystyle g_{u}=\delta u+z_{01}v+z_{20}\delta u^{2}+z_{11}v\delta
u+z_{02}v^{2}.$ (8)

Here and below we keep only quadratic terms in $\delta u$ and $v$, since the
recursion relations used that approximation. Using the coefficients

$\displaystyle z_{01}=a_{01}/(\lambda_{u}-\lambda_{v})\approx 0.595,$
$\displaystyle
z_{11}=-\Big{(}a_{11}-\frac{a_{01}a_{20}}{\lambda_{u}}+\frac{a_{01}b_{11}}{\lambda_{u}-\lambda_{v}}\Big{)}\Big{/}\lambda_{v}\approx-2.051$
$\displaystyle z_{20}=-a_{20}/(2\lambda_{u})\approx-2.148,$ $\displaystyle
z_{02}=\frac{a_{02}/2+a_{01}z_{11}+b_{02}z_{01}/2}{\lambda_{u}-2\lambda_{v}}\approx-0.0983,$
(9)

this scaling field obeys the linear equation

$\displaystyle\frac{\partial g_{u}}{\partial\ell}=\lambda_{u}g_{u},$ (10)

with the solution

$\displaystyle g_{u}(\ell)=g_{u}(0)e^{\lambda_{u}\ell}.$ (11)

Since $\lambda_{u}=a_{10}=-0.7967(57)$, the exponential factor decreases fast
with $\ell$. Indeed, we show below that after a relatively small number of
iterations $\ell_{1}=\ln[\delta/g_{u}(0)]/\lambda_{u}$, with, say
$g_{u}(\ell_{1})=\delta=10^{-3}$, during which the RG trajectories undergo a
transient non-universal flow, the trajectories approach a universal asymptotic
line, on which they either flow to the cubic fixed point [$v(0)>0$] or to the
first order region [$v(0)<0$]. Examples of solutions are shown in Fig. 3,
where the asymptotic line is shown in red. This figure should replace the
schematic Fig. 2 near the isotopic and cubic fixed points.

Figure 3: (color online) Flow trajectories in the $u-v$ plane (blue) for
several initial points. The dots indicate integer values of $\ell$. The red
line is the universal asymptotic line, Eq. (12). The green line is the
asymptotic line in the linear approximation, $v=-\delta u/z_{01}$. The small
circles denote the isotropic ($v^{\ast}_{I}=0$) and cubic fixed points.

The explicit solution of the differential equations is presented in Appendix
B. We first express $\delta u(\ell)$ in terms of $g_{u}(\ell)$ and $v(\ell)$,
from Eq. (8). Since $g_{u}(\ell)$ quickly decays to zero, after a relatively
small number of iterations $\ell_{1}$, the flow quickly approaches an
asymptotic line,

$\displaystyle\delta u(\ell)$ $\displaystyle\rightarrow-
z_{01}v(\ell)-[z_{02}+z^{2}_{01}z_{20}-z_{01}z_{11}]v(\ell)^{2}$
$\displaystyle\approx-.595v-.3613v^{2}.$ (12)

This line is universal, since its coefficients are fully determined by the
isotropic fixed point. Keeping terms up to quadratic order in $g_{u}(\ell)$
and $v(\ell)$ yields a single differential equation, with the solution

$\displaystyle\frac{1}{v(\ell)}=e^{-Ae^{\lambda_{u}\ell}}e^{-\lambda_{v}\ell}\Big{[}\frac{e^{A}}{v(0)}+\frac{(-A)^{-\lambda_{v}/\lambda_{u}}B}{\lambda_{u}}\Big{(}\Gamma[\lambda_{v}/\lambda_{u},-A]-\Gamma[\lambda_{v}/\lambda_{u},-Ae^{\lambda_{u}\ell}]\Big{)}\Big{]},$
(13)

where $A=b_{11}g_{u}(0)/\lambda_{u}$, $B=-(b_{11}/a+b_{02}/2)$ and
$\Gamma[s,z]$ is the incomplete gamma function.

Figure 3 shows the resulting RG trajectories, beginning at several initial
points. We have checked that these analytic solutions coincide with a direct
numerical solution of Eqs. (5)-(6) for the range shown in the figure. As seen,
each RG trajectory has two (or three) main parts. In the first $\ell_{1}$
iterations, $g_{u}(\ell)$ decays quickly to zero, implying a fast non-
universal transient flow towards the asymptotic line (12). In this part, the
points at integer values of $\ell$ are rather far from each other, indicating
the fast flow. In the second part, the trajectory practically coincides with
the asymptotic line. On this line, the points at integer values of $\ell$
become dense, indicating a slow variation with $\ell$. For $v>0$, this implies
a slow approach to the cubic fixed point. For $v<0$, this slow flow is
followed by a third part, in which the flow gradually speeds up as the
trajectory moves towards more negative values of $v$.

As explained in Appendix B, the flow at large $\ell$ can be described by

$\displaystyle
v(\ell)=\frac{v(\ell_{1})e^{\lambda_{v}(\ell-\ell_{1})}}{1+Bv(\ell_{1})\big{[}e^{\lambda_{v}(\ell-\ell_{1})}-1\big{]}/\lambda_{v}}.$
(14)

Since $\lambda_{v}$ is very small, the variation of the second term in the
denominator with $\ell$ is slow, explaining the second part above.. Since
$B=0.3706>0$, this implies a slow variation in $v$ for positive $v$,
approaching the cubic fixed point $v^{\ast}_{C}=\lambda_{v}/B$. However, for
$v(0)<0$ $v(\ell)$ becomes more negative. At first it decreases slowly, but
when $e^{\lambda_{v}\ell}$ becomes of order unity this decrease becomes faster
(the points at integer $\ell$ become less dense), and $v(\ell)$ diverges at
$\ell=\ell_{2}$, when
$\big{[}e^{\lambda_{v}(\ell_{2}-\ell_{1})}-1\big{]}/\lambda_{v}\approx\ell_{2}-\ell_{1})=-1/[Bv(\ell_{1})]$.
This value is larger for smaller $|v(\ell_{1})|$ [and therefore also for
smaller $|v(0)|$]. Within our quadratic approximation, we are not allowed to
follow this solution beyond some finite value, say $v<-.2$. However, it is
reasonable that a full solution will also continue downwards, on the
asymptotic trajectory, and reach the first order transition at
$v(\ell)=-u(\ell)$.

The effective critical exponents $\beta$ and $\gamma$ are given by

$\displaystyle\beta(u,v)=\beta^{I}+c_{10}\delta u+c_{01}v+c_{11}v\delta
u+\frac{c_{20}}{2}\delta u^{2}+\frac{c_{02}}{2}v^{2},$
$\displaystyle\gamma(u,v)=\gamma^{I}+d_{10}\delta u+d_{01}v+d_{11}v\delta
u+\frac{d_{20}}{2}\delta u^{2}+\frac{d_{02}}{2}v^{2},$ (15)

with the coefficients listed in Table 1. Figure 4 shows these effective
exponents, calculated with $v(\ell)$ and $\delta u(\ell)$ from Eqs. (13) and
(20). As expected, the exponents with $v(0)>0$ (dashed lines) approach the
asymptotic values of the cubic fixed point (which are practically the same as
those of the isotropic one). The rate of these approaches depends on the
initial value $g_{u}(0)$, and the corresponding effective exponents are
smaller (larger) than the asymptotic ones if $g_{u}(0)<0~{}(>0)$. Both
trajectories I and IV in Fig. 3 have $g_{u}(0)>0$), and therefore both start
at similar large effective exponents, and reach the vicinity of the isotropic
(=cubic) fixed point values after a few iterations ($\ell_{1}\sim 5$).
Generally, each value of $\ell$ is related to the initial value of $t$ via
$t(\ell)\sim t(0)e^{\ell/\nu}$. Assuming that the RG iterations end after
$\ell_{f}$ iterations, when $t(\ell_{f})\sim 1$, the asymptotic exponents can
be observed only if $|t(0)|<e^{-\ell_{1}/\nu}\sim e^{-5/\nu}\sim 10^{-3}$.
Trajectories II and III start at similar negative values of $g_{u}(0)$, and
therefore they both approach the asymptotic lines from below, resulting with
effective exponents smaller than the asymptotic isotropic ones. After reaching
the asymptotic line, the flow on that line is slow. For trigonal systems, with
$v(0)>0$, the effective exponents vary slowly on that line until they approach
the asymptotic cubic values.

In the tetragonal case, $v(0)<0$, the beginning of the trajectories is similar
to that described above, depending only on the sign of the initial $g_{u}(0)$,
which is not universal. Once the asymptotic line is reached, the initially
slow growth of $|v|$, causes the effective exponents to approach slowly
(mostly below) the isotropic asymptotic values, but then they turn downwards,
diverging at a value $\ell=\ell_{2}$ that depends on the value of
$v(\ell_{1})$. Since the decrease in the exponents in part 3 of the flow is
fully determined by the flow on the universal asymptotic line, and depends
only on $v(\ell_{1})$, one can collapse the two full curves in Fig. 4 onto
each other just by shifting them along the $\ell$ axis. In particular, the
decreasing part of line of curve I accurately overlaps that of line II when we
shift $\ell\rightarrow\ell-15$. This reflects a universality of these
effective exponents. We are not aware of earlier discussions of such a
universality.

We stopped our calculation when the trajectory left the region $|v|<0.2$,
where our quadratic approximation may fail. However, we expect that the
asymptotic universal line will eventually cross the line $v=-u$, and the
transition will become first order. Given Eq. (14), this will happen at larger
$\ell$, i.e., smaller $t$, when $|v(0)|$ is smaller. Since SrTiO3 is supposed
to have a very small negative $v$ (see below), this may explain why its first
order transition happens at a very small $t$, which has not yet been reached
experimentally. In contrast, KMnF3 and RbCaF3 do show first order transitions
at some finite $t$, implying that they start at larger values of $|v(0)|$.

Figure 4: (color online) Effective exponents $\beta$ and $\gamma$ for the
trajectories shown in Fig. 3, as functions of $\ell$. The horizontal axes are
at the asymptotic values of the isotropic fixed point (which are also very
close to those at the cubic fixed point). The exponents corresponding to
trajectories with $v(0)>0$ (III and IV, dashed lines) approach this asymptotic
value. In contrast, those with $v(0)<0$ (I and II, full lines) initially come
close to these values, but then turn downwards to smaller values.

##  IV Experiments

Based on the mean-field region of the experiments, Müller et al. [52]
estimated the ‘initial’ Landau parameters to be $\\{u,v\\}_{\rm
S}\cong\\{1.91,-0.068\\}$ for SrTiO3 and $\\{u,v\\}_{\rm L}\cong 0.06\pm
0.06,~{}0.68\pm 0.06\\}$ for LaAlO3, all in cgs units divided by $10^{43}$.
These rough values (which should be improved!) fit beautifully with our
expectations: SrTiO3 (S in Fig. 2) has a small initial negative $v$, and we
predict that it will flow quickly parallel to the horizontal axis towards the
universal asymptotic line and the isotropic FP before turning downwards, as
for trajectory I in Fig. 3.

On the other hand, LaAlO3 (L in Fig. 2) starts close to the positive $v-$axis,
in a region which is probably not covered by our quadratic approximation. In
principle, one could perform an expansion of the RG recursion relations near
the decoupled Ising fixed point. Since the instability exponent at that fixed
point, $\lambda^{D}_{u}=\alpha^{D}/\nu^{D}$ (with the Ising values of $\alpha$
and $\nu$ [24]) is small, we again expect two parts for the flows which begin
close to the $v$-axis. At first, the trajectories will flow quickly to the
vicinity of the decoupled fixed point, dominated by the large negative
exponent $\lambda^{D}_{v}$(=the stability exponent of the Ising FP),
approaching a universal asymptotic line on which they will flow slowly away
from the decoupled FP. Eventually, this asymptotic line will coincide with the
one we calculated near the isotropic FP, and the flows will approach the cubic
fixed point along that line, as in trajectories III and IV in Fig. 4.
Interestingly, the line (12) reached the $v-$axis at $v\approx 0.5$, quite
close to the decoupled FP value $v^{\ast}_{D}=0.482(9)$ which we calculated
for $n=1$ using the resummation technique of Appendix A. Therefore, Eq. (12)
may give a good approximation for the whole universal line connecting the D, C
and I fixed points.

To the best of our knowledge, there are no experiments showing first-order
transitions into the trigonal phase [53], but – as listed in Sec. I – several
experiments found such transitions into the tetragonal phase. Interestingly,
the experiments listed in Sec. I exhibit intermediate regions (before the
first-order transitions) with effective critical exponents, which are smaller
than their isotropic values. This is consistent with our Fig. 4. The smaller
values of $\beta$ in NaNbO3 have also been attributed to the inter-plane weak
correlations along the rotation axis [54].

To test our predictions in detail, we propose a repeated analysis of existing
experiments, and new dedicated experiments, in which the critical exponents
will be measured over several separate ranges of $t$, to check how they vary
with $t$ (which is equivalent to their dependence on the number of iterations
$\ell$).

As mentioned in Sec. I, polishing the surfaces of the crystals SrTiO3 and
LaAlO3 caused a crossover from the $n=3$ critical behavior to that of the
Ising ($n=1$) critical behavior. This crossover is due to an axially
anisotropic Hamiltonian, ${\cal
H}^{\alpha\beta}_{g}=g\big{(}Q_{\alpha}Q_{\beta}-\delta_{\alpha\beta}{\bf
Q}^{2}/3\big{)}$, where $g$ represents the uniaxial stress. Such uniaxial
stress has also been applied directly, yielding information on the
corresponding exponent $\lambda_{g}$. Existing experiments gave a range of
values for $\lambda_{g}$. A detailed application of our theory to these
exponents will be presented separately.

One way to vary $v$ experimentally is to use mixed crystals, e.g. Sr1-xCaxTiO3
[55] or a mixture of SrTiO3 with LaAlO3, which is expected to be easy to grow
due to their matching lattice constants [2]. Since both the isotropic and
cubic FP’s have $d\nu>2$, randomness is irrelevant [56, 57] and one expects
the same competition predicted above. It is interesting to note that
KMn1-xCaxF3 seems to approach a second-order transition as $x$ increases [58].
If the transition is still into the tetragonal structure, this may represent a
smaller value of the initial $|v|$ in the dilute case. This may be caused by
the larger dimension of the parameter space in the dilute case, which involves
many transient iterations until the flow reaches the $u-v$ plane [57].

Three-component order-parameter vectors ${\bf Q}$ with cubic symmetry are
abundant. Examples include ferroelectric transitions from cubic to tetragonal
or trigonal, ferromagnets and antiferromagnets with the magnetization ordering
along an axis or a diagonal. Close to their transitions, these cubic crystals
may be described by Eq. (2), and therefore their critical behavior is expected
to split into the two types described above. In particular, ordering of the
$n=3$ cubic system along a cube axis must be fluctuation-driven first-order,
with intermediate effective isotropic exponents! This important prediction can
be tested experimentally [59].

##  V Conclusions

In conclusion, we have shown that for systems having cubic symmetry,
universality is not determined merely by the symmetry above the transition.
Rather, the critical behavior of such systems depends on their symmetry below
the transition. Cubic systems split into two groups - those that become
trigonal, which belong to the cubic FP universality class, and those that
become tetragonal, which undergo a fluctuation driven first-order transition.
In both cases, there is a wide temperature range in which slow-varying
effective exponents, which exhibit universal features, are expected. For
systems like SrTiO3, the cubic to tetragonal transition exhibits effective
exponents close to those of the isotropic FP, but then cross over to a
fluctuation-driven first-order transition. This crossover is accompanied by
large changes in the effective exponents, which await experimental detection.
These results resolve a long standing confusion about the universality of the
displacive phase transitions in the perovskites, and leaves its complete
confirmation to future dedicated experiments, concentrating on the $t-$
dependence of the effective exponents.

###### Acknowledgements.

This research was initiated by a very stimulating discussion with Slava
Rychkov, who drew our attention to the new accurate values of $n_{c}$ (see
also Ref. 8). A.K. gratefully acknowledges Mikhail Kompaniets for the helpful
discussion and Andrey Pikelner for his help with RG expansions from Ref. 60,
and the support of Foundation for the Advancement of Theoretical Physics
”BASIS” through Grant 18-1-2-43-1.

##  Appendix A Details of the resummation

Since the $\epsilon-$expansions, e.g.
$f(\epsilon)=\sum_{k=0}^{\infty}f_{k}\epsilon^{k}$, are divergent, numerical
estimates of the quantities of interest at $d=3$ are obtained employing proper
resummation techniques [46, 47]. Such resummations were performed for the
isotropic and cubic FP’s in Ref. 37, which used the basic resummation
procedure of the Padé-Borel-Leroy method. Here we use the
$\epsilon-$expansions based on those in Ref. 37, but apply the more advanced
resummation strategy proposed in Ref. 48. The main idea of this strategy can
be formulated simply: the asymptotic behavior of the series coefficients for
large order is written as

$\displaystyle f_{k}\xrightarrow[k\rightarrow\infty]{}c\,k!k^{b_{0}}(-a)^{k},$
(16)

where $1/a$ is the radius of convergence and $b_{0}$ is fixed by the high-
order asymptotic behavior of the series. However, in practice we only have a
limited number of terms in the series, and consequently this asymptotic
behavior is not known. Therefore, the authors of Ref. 48 proposed to treat the
Leroy parameter $b=b_{0}+3/2$ as a free parameter, to be determined
variationally (see below).

Once a Borel transformation, based on this modified asymptotic form, is
performed, the variable $\epsilon$ is conformally mapped onto

$\displaystyle
w(\epsilon)=\dfrac{\sqrt{1+a\epsilon}-1}{\sqrt{1+a\epsilon}+1},\quad\epsilon(w)=\dfrac{4w}{a(1-w)^{2}},$
(17)

and it is assumed that the function has the strong asymptotic behavior
$f(\epsilon)\sim\epsilon^{\lambda},\ \ \epsilon\rightarrow\infty$. The results
are then improved by a preliminary homogeneous homographic transformation,

$\displaystyle\epsilon(\epsilon^{\prime})\rightarrow\dfrac{\epsilon^{\prime}}{1+q\epsilon^{\prime}},\quad\epsilon^{\prime}(\epsilon)\rightarrow\dfrac{\epsilon}{1-q\epsilon},$
(18)

and the final approximate estimates are found by applying the steps mentioned
above to the new $\epsilon^{\prime}$ expansion. Finally, the optimal values of
the parameters $b,~{}\lambda$ and $q$ (for each series) are determined by the
least sensitivity to their variation (for details see Ref. 48). This technique
was applied to find all the coefficients listed in Table 1.

##  Appendix B Solution of the differential equations

Solving Eq. (8) for $\delta u(\ell)$ yields

$\displaystyle
du(\ell)=\Big{[}-1-z_{11}v(\ell)+\sqrt{[1+z_{11}v(\ell)]^{2}-4z_{20}[z_{01}v(\ell)+z_{02}v(\ell)^{2}-g_{u}(\ell)]}\Big{]}\Big{/}(2z_{20}).$
(19)

Expanding to quadratic order in $v(\ell)$ and $g_{u}(\ell)$ gives

$\displaystyle\delta u(\ell)\approx
g_{u}(\ell)-z_{20}g_{u}(\ell)^{2}-\big{[}z_{01}+\big{(}z_{11}-2z_{01}z_{20}\big{)}g_{u}(\ell)\big{]}v(\ell)-\big{[}z_{02}+z^{2}_{01}z_{20}-z_{01}z_{11}\big{]}v(\ell)^{2}.$
(20)

After a few iterations $g_{u}(\ell)$ decays to zero, and this solution reaches
the asymptotic line (12).

We now return to Eq. (6). Substituting Eq. (20), and stopping at quadratic
order, this equation becomes

$\displaystyle\frac{\partial v}{\partial\ell}$
$\displaystyle\approx\lambda_{v}v(\ell)+b_{11}\big{[}g_{u}(\ell)-z_{01}v(\ell)\big{]}v(\ell)+b_{02}v(\ell)^{2}/2$
$\displaystyle=\lambda_{v}v(\ell)+b_{11}g_{u}(\ell)v(\ell)-Bv(\ell)^{2},$ (21)

where $B\equiv b_{11}z_{01}-b_{02}/2$. This can be written as

$\displaystyle\frac{\partial}{\partial\ell}\Big{[}\frac{1}{v}\Big{]}=-\big{[}\lambda_{v}+b_{11}g_{u}(0)e^{\lambda_{u}\ell}\big{]}\Big{[}\frac{1}{v}\Big{]}+B.$
(22)

To solve this equation, we write $x=e^{\lambda_{u}\ell}$ and

$\displaystyle\frac{1}{v(x)}=e^{-Ax}x^{-\lambda_{v}/\lambda_{u}}W(x),$ (23)

where $A=b_{11}g_{u}(0)/\lambda_{u}$. This yields

$\displaystyle\frac{dW}{dx}=\frac{B}{\lambda_{u}}e^{Ax}x^{\lambda_{v}/\lambda_{u}-1},$
(24)

with the solution (13), where

$\displaystyle\Gamma[s,z]=\int_{z}^{\infty}t^{s-1}e^{-t}dt.$ (25)

is the incomplete gamma function.

For large $\ell$, $x=e^{\lambda_{u}\ell}$ is small, and
$\Gamma[s,z]=-z^{s}/s+{\cal O}[1]$, so that
$\Gamma[\lambda_{v}/\lambda_{u},-Ax]\propto e^{\lambda_{v}\ell}$. This result
can be obtained directly: For $\ell>\ell_{1}$ we can neglect $g_{u}(\ell)$ in
Eq. (22). The solution to this equation is then given in Eq. (14).

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