# Emergent $\mathbb{Z}_{2}$ symmetry near a CDW multicritical point

Steven A. Kivelson  Department of Physics, Stanford University, Stanford, CA
94305, USA  Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Oxford OX1 3PU, United Kingdom  Stanford Institute for Materials and
Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road,
Menlo Park, California 94025, USA  Akshat Pandey  Department of Physics,
Stanford University, Stanford, CA 94305, USA  Rudolf Peierls Centre for
Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
Anisha G. Singh  Stanford Institute for Materials and Energy Sciences, SLAC
National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California
94025, USA  Department of Applied Physics, Stanford University, Stanford, CA
94305, USA  Aharon Kapitulnik  Department of Physics, Stanford University,
Stanford, CA 94305, USA  Stanford Institute for Materials and Energy Sciences,
SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park,
California 94025, USA  Department of Applied Physics, Stanford University,
Stanford, CA 94305, USA  Ian R. Fisher  Stanford Institute for Materials and
Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road,
Menlo Park, California 94025, USA  Department of Applied Physics, Stanford
University, Stanford, CA 94305, USA

###### Abstract

We consider the critical behavior associated with incommensurate
unidirectional charge-density-wave ordering in a weakly orthorhombic system
subject to uniaxial strain as an experimentally significant example of
$U(1)\times U(1)$ multicriticality. We show that, depending on microscopic
details, the phase diagram can have qualitatively different structures which
can involve a vestigial meta-nematic critical point, a pair of tricritical
points, a decoupled tetracritical point, or (at least at mean-field level) a
bicritical point. We analyze the emergent symmetries in the critical regime
and find that these can — at least in some cases — involve an emergent
$\mathbb{Z}_{2}$ order parameter symmetry.

From an electronic structure perspective, ErTe3 is very nearly tetragonal, but
from a structural perspective, due to the presence of a glide plane, it is
intrinsically orthorhombic [1]. Below a well-characterized transition
temperature, $T_{c}$, it exhibits unidirectional incommensurate charge-
density-wave (CDW) order with ordering vector along the orthorhombic $c$ axis.
In the presence of an in-plane unidirectional applied stress $s$ in excess of
a modest critical value, $s^{*}$, (or, more generally, anisotropic in-plane
strain greater than some critical value) the CDW ordering vector rotates by
$\pi/2$ to lie along the $a$ axis [2, 3, 4]. Moreover, for temperatures
slightly above $T_{c}$, the nematic susceptibility (as inferred from the
elasto-resistance) as a function of strain is strongly peaked at $s=s^{*}$
[3]. These observations motivate us to reconsider the possible multicritical
phase diagrams relevant to this general situation in which there are two
distinct $U(1)$ order parameters (associated with breaking of translational
symmetry in the two directions).

From a symmetry perspective, the two CDW states found for ErTe3 are
fundamentally distinct; this is an inescapable consequence of the orthorhombic
crystal structure. For instance, while the magnitudes of the ordering vectors
in the two directions are similar, they are not identical [3]. Similarly, the
magnitude of the slope of $T_{c}$ with respect to $s$ is different for the two
states [3]. Nonetheless, we identify circumstances when, in a sense that we
will define precisely, there is an emergent $\mathbb{Z}_{2}$ symmetry under
exchange of the two order parameters that characterizes the critical regime —
i.e. the system behaves as if it were truly tetragonal at a critical value of
the stress, $s=s^{\star}$, and for $T$ close to an appropriate critical point.

Specifically, we consider a Landau-Ginzburg-Wilson effective field theory with
two complex scalar order parameters, $\phi_{1}$ and $\phi_{2}$, corresponding
to the two components of the CDW order. We will consider the solution of this
problem in the multicritical regime first in the context of Landau mean-field
theory. Then, more qualitatively (and conjecturally) we will discuss the true
three-dimensional critical fluctuations.

For simplicity, we will start by treating the case in which the system is
actually tetragonal, in which case, for vanishing applied stress ($s=0$),
there is a set of discrete (point-group) symmetries which interchange
$\phi_{1}\ \leftrightarrow\ \phi_{2}$ and at the same time interchange
coordinates, $x\ \leftrightarrow\ y$ 111For simplicity, we will refer to this
as a $\mathbb{Z}_{2}$ symmetry reflecting its action on the order parameters,
but since the true symmetries at play are spatial symmetries, this
nomenclature should not be taken literally.. We will then analyze the effects
of including small terms consistent with an orthorhombic point group symmetry.
In Figs. 1 and 2 we show the possible structures of the phase diagram that
come from a saddle-point (Landau mean-field) solution of the model in various
ranges of couplings for the tetragonal system and the weakly orthorhombic
system respectively.

The effective field theories from which these results are derived are defined
in Sec. I. The mean-field treatment and the considerations concerning the true
asymptotic critical phenomena are discussed in Secs. II and III, respectively.
In Sec. IV, the results are discussed with emphasis on the existence (or not)
of additional emergent symmetries in the asymptotic near-critical regime.

##  I The Effective Field Theory

The effective (Landau-Ginzburg-Wilson) field theory in the neighborhood of a
multicritical point for two unidirectional incommensurate CDW orders is
described by a classical effective Hamiltonian density

${\cal H}[\phi_{1},\phi_{2}]=H+\tilde{H}=V+K+\tilde{V}+\tilde{K}$ (1)

where $H=V+K$ contains all the terms that are consistent with the point group
symmetry of a tetragonal system ($D_{4h}$) and an assumed emergent
translational symmetry in all three directions (${\mathbb{R}}^{3}$), while
$\tilde{H}=\tilde{V}+\tilde{K}$ contains the additional terms that are allowed
when (either due to applied strain or the intrinsic crystal structure) the
point-group symmetry is reduced to that of an orthorhombic system ($D_{2h}$)
222In unstrained Er$Te_{3}$, the relevant crystal symmetry group is actually
nonsymmorphic, but this does not change the symmetry analysis presented here..
In this treatment, because the CDW is assumed to be incommensurate,
translations also act on the order parameter fields, such that under
translation by $\bm{a}$, the order parameters transform as
$\phi_{1}(\bm{r})\to\phi_{1}(\bm{r}+\bm{a})e^{iQ_{x}a_{x}}$ and
$\phi_{2}(\bm{r})\to\phi_{2}(\bm{r}+\bm{a})e^{iQ_{y}a_{y}}$ (where $Q_{x,y}$
are the CDW ordering vectors in the $x$ and $y$ directions respectively),
while any point group element that exchanges $x$ and $y$ also exchanges
$\phi_{1}$ and $\phi_{2}$. In the field-theoretic description, the
translational symmetries in the $x$ and $y$ directions discussed above are
manifested as internal $U(1)$ symmetries associated with the order parameter
fields, $\phi_{1}$ and $\phi_{2}$, while the $\mathbb{Z}_{2}$ symmetry under
exchange of $\phi_{1}$ and $\phi_{2}$ requires the simultaneous exchange of
$x$ and $y$.

Here $V+\tilde{V}$ is the effective potential, which we will express as a
polynomial in powers of $\phi_{1}$ and $\phi_{2}$, keeping explicitly terms
only to the lowest necessary order, while $K+\tilde{K}$ are the lowest-order
gradient terms. Explicit expressions for these various terms in the effective
Hamiltonian follow:

$\displaystyle
V=\frac{\mu}{2}\left[|\phi_{1}|^{2}+|\phi_{2}|^{2}\right]+\frac{u}{4}\left[|\phi_{1}|^{2}+|\phi_{2}|^{2}\right]^{2}+\frac{\gamma}{2}|\phi_{1}|^{2}|\phi_{2}|^{2},$
$\displaystyle
K=\frac{\kappa_{L}}{2}\left[|\partial_{x}\phi_{1}|^{2}+|\partial_{y}\phi_{2}|^{2}\right]+\frac{\kappa_{T}}{2}\left[|\partial_{y}\phi_{1}|^{2}+|\partial_{x}\phi_{2}|^{2}\right]$
$\displaystyle\ \ \ \ \
+\frac{\kappa_{\perp}}{2}\left[|\partial_{z}\phi_{1}|^{2}+|\partial_{z}\phi_{2}|^{2}\right],$
(2)
$\displaystyle\tilde{V}=\frac{\tilde{\mu}}{2}\left[|\phi_{1}|^{2}-|\phi_{2}|^{2}\right]+\frac{\tilde{u}}{4}\left[|\phi_{1}|^{4}-|\phi_{2}|^{4}\right],$
$\displaystyle\tilde{K}=\frac{\tilde{\kappa}_{L}}{2}\left[|\partial_{x}\phi_{1}|^{2}-|\partial_{y}\phi_{2}|^{2}\right]+\frac{\tilde{\kappa}_{T}}{2}\left[|\partial_{y}\phi_{1}|^{2}-|\partial_{x}\phi_{2}|^{2}\right]$
$\displaystyle\ \ \ \ \
+\frac{\tilde{\kappa}_{\perp}}{2}\left[|\partial_{z}\phi_{1}|^{2}-|\partial_{z}\phi_{2}|^{2}\right].$
(3)

In most cases, we assume $u>\sqrt{\tilde{u}^{2}+\gamma^{2}}$,
$\kappa_{L}>|\tilde{\kappa}_{L}|$, $\kappa_{T}>|\tilde{\kappa}_{T}|$, and
$\kappa_{\perp}>|\tilde{\kappa}_{\perp}|$. However, when we consider the
tricritical phase diagrams in Figs. 1e and 2e, we will consider $u$ slightly
negative, in which case terms of order at least $\phi^{6}$ are required for
thermodynamic stability. Similarly, we shall see that for the special case
$\gamma=0$, higher order terms are necessary (at mean-field level) to fully
determine the phase diagram; where such higher-order terms enter our
discussion we will introduce them explicitly.

In general, all the coefficients appearing in these expressions are functions
of the temperature, $T$, and if we apply stress to the system, of the stress
tensor, $s_{ab}$. In an otherwise tetragonal system, all the coefficients with
a tilde vanish in the absence of shear strain, and are odd functions of the
shear stress, $s_{B_{1g}}=s_{xx}-s_{yy}$, while the coefficients that do not
carry tildes are even functions of all other components 333In Ref. [3], the
system is studied at fixed strain, $\epsilon_{ab}$. In many circumstances the
strain is proportional to the stress, so this distinction is of little
importance, but where first-order transitions arise, the phase diagrams at
fixed strain exhibits regions of macroscopic two-phase coexistence rather than
a single first-order phase boundary..

##  II Mean-field analysis

In a mean-field analysis, we look for field configurations that minimize
${\cal H}[\phi_{1},\phi_{2}]$. As $K+\tilde{K}$ is minimized by any uniform
field configuration, this means finding the minima of $V+\tilde{V}$.

###  II.1 The $\mathbb{Z}_{2}$ symmetric case

The mean-field phase diagram in the stress-temperature plane for the simple
case in which the underlying problem is tetragonal is shown in Fig. 1. In
making this figure, we have taken $\mu$ and $\tilde{\mu}$ to be linear in $T$
and $s$ respectively, $\mu=\mu_{0}[T-T_{0}]$ and
$\tilde{\mu}=\tilde{\mu}_{0}s$, and have ignored the $T$ and $s$ dependence of
all other parameters. In particular, this means we have set $\tilde{u}=0$.
Figs. 1a, b, and c follow directly from minimizing $V$ in Eq. (2), while Figs.
1d, e, and f involve consideration of additional terms.

Figure 1: Mean-field phase diagrams in the $\mathbb{Z}_{2}$-symmetric
(tetragonal) cases discussed in Sec. II.1: (a) bicritical point, (b)
tetracritical point, (c) decoupled tetracritical point, (d) $O(4)$-symmetric
tetracritical point, (e) tricritical points, and (f) vestigial nematic. Thin
and thick black lines denote, respectively, continuous and first-order
transitions. Thin and thick black lines denote, respectively, continuous and
first-order transitions.

The various cases are as follows:

  * •

Fig. 1a: For $u>0$ and $\gamma>0$ there is a bicritical point at
$\tilde{\mu}=\mu=0$. There is a first-order line between two unidirectional
phases that occurs along the half line $\tilde{\mu}=0$ with $\mu<0$. The phase
boundaries between the unidirectional ordered phases and the disordered high-
temperature phase correspond to $\tilde{\mu}=\pm\mu$ with $\mu\geq 0$.

  * •

Fig. 1b: For $u>0$ and $-2u<\gamma<0$ there is a tetracritical point at
$\mu=\tilde{\mu}=0$. For $\tilde{\mu}>0$, there is a transition from a
disordered state for $\mu>|\tilde{\mu}|$, to a state in which
$|\phi_{2}|>|\phi_{1}|=0$ at $\mu<|\tilde{\mu}|$. There is then a further
transition to a state with coexisting bidirectional order (but with $\phi_{2}$
dominant) at $\mu=-|\tilde{\mu}|(2u-|\gamma|)/|\gamma|$. For $\tilde{\mu}<0$,
the phase boundaries follow the corresponding lines, but with the roles of
$\phi_{1}$ and $\phi_{2}$ interchanged.

  * •

Fig. 1c: For $u>0$ and $\gamma=-u$, the system has the special feature of
exhibiting a “decoupled” tetracritical point in which the two phase boundaries
pass through each other without changing slope. At mean-field level, this
corresponds a third degree of fine-tuning, requiring tuning three parameters,
$\mu$, $\tilde{\mu}$, and $\gamma$ all to 0 simultaneously.

  * •

Fig. 1d: For $u>0$ and $\gamma=0$, $V$ exhibits an enhanced $O(4)$ symmetry.
Here there is no coexistence region. However, there is no latent heat
associated with crossing the phase boundary at $\mu<0$ between the two phases
(as $\tilde{\mu}$ changes sign) so the transition is not conventionally first
order despite the fact that the order parameter changes direction abruptly.
Indeed, as is discussed in Ref. [8], in this case, the nature of the phase
diagram below the multicritical point depends on higher order terms in powers
of $\phi_{j}$. Specifically, the tetracritical phase diagram shown in Fig. 1d
arises if we include next order terms

$\displaystyle V\to
V+\frac{u_{6}}{6}\left[|\phi_{1}|^{2}+|\phi_{2}|^{2}\right]^{3}$
$\displaystyle\ \ \ \ \
+\frac{\gamma_{6}}{3}\left[|\phi_{1}|^{2}+|\phi_{2}|^{2}\right]|\phi_{1}|^{2}|\phi_{2}|^{2}$
(4)

with $\gamma_{6}<0$ (and with $u_{6}>|\gamma_{6}|/2$ to preserve stability).
Again, at mean-field level, this form of multicriticality involves an
additional degree of fine tuning relative to a generic tetracritical point.

  * •

Fig. 1e: For $u<0$ the transition at zero stress is necessarily first order.
Generically the expansion in powers of $\phi$ is not justified at a first-
order transition unless the transition is only weakly first order, as happens
near a tricritical (or bicritical) point. Thus, in Fig. 1e, we have shown the
phase diagram that results from a case where $u$ changes sign in the vicinity
of a putative bicritical point. Specifically, we have included an implicit
(linear) $T$ dependence of $u$ taking the form $u=-u^{*}+u^{\prime}\mu$, so
that $u<0$ but of small magnitude as $\mu\to 0$, but $u>0$ for
$\mu>\mu^{*}\equiv u^{*}/u^{\prime}$. To ensure the stability of the free
energy, we are forced to include higher-order terms in the effective potential
as in Eq. 4. To be concrete, we have taken $\gamma>0$. Now, in the vicinity of
the point at which, for $u>0$, there would have been a bicritical point, we
instead find a pair of tricritical points at $\mu=\mu^{*}$ and
$\tilde{\mu}=\pm\mu^{\star}$.

  * •

In Fig. 1f we show a phase diagram that cannot be derived at mean-field level
from a Hamiltonian of the form shown, although a phase diagram of this sort —
with a “vestigial nematic” phase — can arise from fluctuation effects, as
discussed below and in Refs. [9] and [10]. To obtain this sort of phase
diagram in mean-field theory, we need to add in an additional nematic order
parameter, ${\cal N}$, which transforms as $B_{1g}$ ($x^{2}-y^{2}$) under the
point group symmetry of the tetragonal crystal. To the same order as we have
considered so far, this involves

$\displaystyle V\to V+$ $\displaystyle\tilde{\alpha}{\cal
N}+\frac{\nu}{2}{\cal N}^{2}+\frac{\tilde{w}}{3}{\cal N}^{3}+\frac{1}{4}{\cal
N}^{4}$ $\displaystyle+$ $\displaystyle\frac{\lambda}{2}{\cal
N}\left[|\phi_{1}|^{2}-|\phi_{2}|^{2}\right]$ (5)

where the nematic field has been normalized in such a way that the quartic-
coupling is equal to 1, and where $\tilde{\alpha}$ and $\tilde{w}$ both vanish
by symmetry in the absence of strain. To obtain the phase diagram in Fig. 1f,
we have assumed that $\nu$ changes sign at a temperature slightly above the
putative bicritical point, and in projecting the phase diagram onto the
$\mu-\tilde{\mu}$ plane we have taken $\nu=-\nu^{*}+\nu^{\prime}\mu$, with
$\nu^{*}$ small and positive. Coefficients of terms odd in $\cal N$ vary
linearly with $\tilde{\mu}$:
$\tilde{\alpha}=\tilde{\alpha}^{\prime}\tilde{\mu}$ and
$\tilde{w}=\tilde{w}^{\prime}\tilde{\mu}$. Now there is a nematic critical
point at $\tilde{\mu}=0$ and $\mu=\mu^{*}\equiv\nu^{*}/\nu^{\prime}$, while
the two critical endpoints (where the CDW ordering temperatures intersect a
first-order line) occur at a smaller value of $\mu$ which (for example) is at
$\mu=\mu^{*}\left[\lambda\nu/(1+\lambda\nu^{\prime})\right]$ in the limit
$\tilde{\mu}\to 0$.

###  II.2 Emergent $\mathbb{Z}_{2}$ symmetry: the orthorhombic case

Figure 2: Mean-field phase diagrams in the non-$\mathbb{Z}_{2}$-symmetric
(orthorhombic) cases discussed in Sec. II.2: (a) bicritical point, (b)
tetracritical point, (c) decoupled tetracritical point, (d) $O(4)$-symmetric
tetracritical point, (e) tricritical points, and (f) vestigial metanematic.
Thin and thick black lines denote, respectively, continuous and first-order
transitions. Dashed ovals indicate regimes of approximate emergent
$\mathbb{Z}_{2}$ symmetry. Grey lines are the corresponding phase boundaries
from Fig. 1.

We now consider modifications to the mean-field phase diagrams that result
from considering a system which is weakly orthorhombic, i.e. where
$\tilde{H}\neq 0$ even for $s=0$. Since at mean-field level, we can neglect
the gradient terms, to quartic order there are only two terms that break the
$\mathbb{Z}_{2}$ symmetry under exchange $\phi_{1}\leftrightarrow\phi_{2}$:
the quadratic term, $\tilde{\mu}$, and the quartic term, $\tilde{u}$.
Rescaling the order parameter fields such that $\phi_{1}\to Z\phi_{1}$ and
$\phi_{2}\to Z^{-1}\phi_{2}$ with
$Z=\left[(u+\tilde{u})/(u-\tilde{u})\right]^{-1/8}$ preserves the form of the
potential $V+\tilde{V}$, but with shifted parameters:

$\displaystyle\mu\to\left[(Z^{4}+1)\mu+(Z^{4}-1)\tilde{\mu}\right]/({2Z^{2}}),$
$\displaystyle\tilde{\mu}\to\left[(Z^{4}-1)\mu+(Z^{4}+1)\tilde{\mu}\right]/({2Z^{2}}),$
$\displaystyle u\to\sqrt{u^{2}-\tilde{u}^{2}},$ (6)
$\displaystyle\gamma\to\gamma+u-\sqrt{u^{2}-\tilde{u}^{2}},$
$\displaystyle\tilde{u}\to 0.$

We see therefore that (to the extent that we can neglect higher order terms in
powers of $\phi_{j}$) the theory possesses an emergent $\mathbb{Z}_{2}$
symmetry under the exchange of $\phi_{1}$ and $\phi_{2}$ and
$\tilde{\mu}\to-\tilde{\mu}$. In this sense, it behaves as if it were
tetragonal whether it is or not. Of course, the fact that this is not a true
symmetry will be reflected in the presence of higher order terms, such as
$\tilde{u}_{6}\left[|\phi_{1}|^{2}+|\phi_{2}|^{2}\right]\left[|\phi_{1}|^{4}-|\phi_{2}|^{4}\right]$,
which do not vanish under the rescaling transformation. Consequently, while an
appropriate coordinate transformation in the $T-s$ plane will make the phase
diagram of the orthorhombic system look like that of a tetragonal system close
enough to criticality that higher order terms in $V+\tilde{V}$ can be ignored,
further from criticality where the higher order terms begin to play a role,
the emergent $\mathbb{Z}_{2}$ symmetry will be increasingly violated, as
reflected in the schematic phase diagrams in Fig. 2.

The situation is somewhat subtle for the tricritical case in Fig. 2e since
higher order (at least sixth) terms play a role in the phase diagram near
criticality. Thus, the extent to which this system exhibits an approximate
$\mathbb{Z}_{2}$ symmetry can depend on other assumptions.

The case of the vestigial nematic, Fig. 2f, needs to be handled separately. In
the first place there is not, strictly speaking, a nematic phase defined by a
spontaneously broken symmetry, since the symmetry in question is explicitly
broken. This is reflected in the presence of odd terms, $\tilde{\alpha}$ and
$\tilde{w}$ in the in the effective potential in Eq. (5), where now no
symmetry requires that $\tilde{\mu}$, $\tilde{\alpha}$, and $\tilde{w}$ all
vanish simultaneously. Thus, the first-order line immediately below the
critical point corresponds to a line of meta-nematic transitions (where the
value of ${\cal N}$ jumps discontinuously — typically from a negative to a
positive value). The critical point at the end — what was formerly the nematic
critical point — is now analogous to the critical point that terminates the
liquid-gas coexistence line in the phase diagram of water or other common
liquids.

That the vicinity of the critical point possesses an emergent $\mathbb{Z}_{2}$
symmetry can be seen by making a shift of the order parameter, ${\cal
N}\to{\cal N}+\bar{{\cal N}}$, where $\bar{{\cal N}}=-\tilde{w}/3$ is chosen
to cancel the third order term. Then with proper rescaling of ${\cal N}$, this
replaces Eq. (5) with an expression of the same form, but with $\tilde{w}=0$.
The meta-nematic critical point at $\mu=\mu^{*}$ and
$\tilde{\mu}=\tilde{\mu}^{*}\equiv-\alpha^{*}/\alpha^{\prime}$ thus has the
same form as in the tetragonal case. However, as shown in the figure, the
critical end-points (where the CDW transitions end on the meta-nematic line)
are split by order $\lambda\bar{{\cal N}}$, thus spoiling the $\mathbb{Z}_{2}$
symmetry as one moves away from the critical point.

##  III Fluctuation Effects

Mean-field theory tends to work remarkably well in metallic systems —
presumably because the effective range of interactions is relatively long. The
coupling of the nematic component of the order parameter to the strain field
tends to further reduce fluctuations effects (see Ref. [11] and references
therein). It is, nonetheless, interesting in a more general context to analyze
what of the above structure survives (or may even be enhanced by) fluctuation
effects in the true asymptotic critical regime.

The most basic question to be addressed is the stability of the various mean-
field phase diagrams to fluctuations. The issue concerning the stability of
$O(N)\times O(N)$ (or more generally $O(N)\times O(M)$) multicritical points
has been addressed in several different ways, notably through the use of the
$\epsilon$ expansion 444For recent overviews of the $\epsilon$ expansion for
multicritical points, see Refs. [21, 22], the functional renormalization group
[13] and, more recently, using conformal bootstrap to address the problem
directly in three dimensions [14, 15].

In the context of the $\epsilon$ expansion, there are three fixed points (in
addition to the always unstable Gaussian fixed point) which correspond to the
decoupled tetracritical point, a higher symmetry $O(2N)$ symmetric
multicritical point, and an interacting “biconical” fixed point. However, for
given $N$, only one of these is ever stable with the other two being unstable
— i.e. the other two require at least one additional parameter to be fine
tuned (in addition to the usual two associated with “stable” multicritical
points).

A number of conclusions can been reached based on non-perturbative lines of
analysis [16]. For $N=1$, the $O(2)$ symmetric fixed point, familiar from
studies of the Ashkin-Teller model, is stable. 555Note that, throughout this
section, we discuss $O(2)$ order parameters interchangeably with those which
we have previously called $U(1)\equiv SO(2)$ ones. The distinction between the
two groups is of no consequence for our purposes.  This is one of the most
remarkable examples of an emergent symmetry. However, it is generally accepted
that the $O(2N)$ symmetric fixed point is (at least weakly) unstable for all
$N>1$ [18, 19].

For $N=2$ the situation is somewhat subtle. Certainly, the decoupled fixed
point is stable for $N\geq 2$. As viewed from the perspective of the
$\epsilon$ expansion, this suggests that this is the only stable multicritical
point, which among other things would imply that there is no stable
$O(N)\times O(N)$ bicritical point with $N\geq 2$. Recently, however, initial
evidence has emerged from conformal bootstrap of the existence of an
additional stable conformal field theory that might correspond to the long-
sought bicritical point. We will return to this below.

We now comment on the tricritical scenario. From the fact that there is no
stable bicritical fixed point in the context of the $\epsilon$ expansion, it
was suggested by Aharony et al. [20, 21, 22] that it was likely to be
fluctuation-driven first order. This further led to the suggestion that a
mean-field bicritical phase diagram should instead exhibit two tricritical
points, and thus be of the form shown in Figs. 1e and 2e.

The tricritical points in this case are readily characterized since $d=3$ is
the upper critical dimension, and so they should exhibit mean-field behavior
up to logarithmic corrections. Clearly, each tricritical point involves the
ordering of only one of the CDW components, so there can be no question of an
emergent $\mathbb{Z}_{2}$ symmetry.

Another possible form of a phase diagram with a fluctuation driven first-order
transition is that corresponding to a vestigial nematic phase, as in Figs. 1f
and 2f. It has been shown [10] that such a vestigial nematic phase arises from
fluctuations in a sufficiently layered (quasi-2D) system with $N>2$, and
strong arguments suggest that this is true as well for $N=2$. There thus
appear to be circumstances in which this form of a phase diagram arises
organically from CDW fluctuations, without need to introduce an explicit
nematic order parameter. (See also Refs. [23] and [9].)

In this case the critical point in question is in the 3D Ising universality
class. For the tetragonal case, this is obvious in that there is an Ising
symmetry — which corresponds to the $\mathbb{Z}_{2}$ symmetry under exchange
$\phi_{1}\leftrightarrow\phi_{2}$ — that is broken below the critical point.
But in the orthorhombic case, the analogy with the liquid-gas critical point
is more apt, in that there is no actual broken symmetry associated with the
first-order line below the critical point. However, the present discussion
puts a somewhat different perspective on this familiar problem — there is an
emergent $\mathbb{Z}_{2}$ symmetry at the critical point which is broken along
the first-order line below the critical point. Correspondingly, there is
presumably a Widom line, at which the emergent $\mathbb{Z}_{2}$ symmetry is
best defined, that extends above the critical point, and which should be
observable as a peak in the nematic susceptibility [3].

The conformal bootstrap offers a new approach to identifying possible critical
phenomena — especially in 3D [14, 15]. There is not yet any systematic
classification of all possible 3D conformal field theories, but (in many
cases) where such field theories have been identified and characterized from
this approach, these provide important complements to the $\epsilon$ expansion
and other more familiar approaches to critical phenomena. For instance, the
perturbative stability of the decoupled $O(2)\times O(2)$ tetracritical point
can be corroborated by knowing precisely the critical exponents of the $O(2)$
critical point [16].

In this context, it is interesting to note that there is preliminary
indication of the existence of at least one additional stable theory with
$O(2)\times O(2)$ symmetry, distinct from the decoupled theory [24, 25, 26].
The results here are not definitive, and the identification of the physical
meaning of this additional critical theory — if it indeed exists — is also not
established. It is tempting, however, to identify this as the long-sought
bicritical theory. In this context, it is interesting to note that the
theories bootstrapped in these works actually have a higher, $(O(2)\times
O(2))\rtimes\mathbb{Z}_{2}$ symmetry, so would correspond to a critical point
with emergent $\mathbb{Z}_{2}$ symmetry under exchange of the two order
parameters.

Finally, there is an issue concerning the presence or absence of an emergent
$SO(3)$ spatial rotational symmetry at criticality. This is a feature of an
$O(2)$-symmetric critical point, such as occurs along the narrow solid phase
boundaries in the figures. In particular, by appropriate rescaling of the
length scales in the $x$, $y$, and $z$ directions, the system at criticality —
even on an underlying orthorhombic lattice — can be described by an $SO(3)$
rotationally symmetric effective field theory. The same analysis can be
applied to the tricritical points or to the nematic or metanematic critical
points, such as those illustrated in Figs. 1f and 2f. It would presumably be
true at a bicritical point of the sort shown in Figs. 1e and 2e, if such a
critical point indeed arises.

In contrast, it is easy to see that rotational symmetry does not emerge at the
decoupled tetracritical points, because of the presence of an effective “spin-
orbit coupling” that reflects the fact that the broken symmetries involved are
spatial symmetries. Specifically, even in the tetragonal case, while the
effective field theory for $\phi_{1}$ can be made rotationally symmetric by
rescaling $x$, $y$, and $z$ appropriately, to achieve the same result for
$\phi_{2}$ requires interchanging the scale factors for $x$ and $y$. In
addition, if the system is orthorhombic, there is a different scale required
for the magnitude of fluctuations of $\phi_{1}$ and $\phi_{2}$ and for the $z$
coordinate that enters the two theories. Thus, even at criticality there is
only $C_{2}$ rotational symmetry for the orthorhombic system, while for the
tetragonal case the $C_{4}$ rotational symmetry is represented as a further
symmetry under $\phi_{1}\leftrightarrow\phi_{2}$ and $x\leftrightarrow y$.

##  IV Discussion

The analysis in the present paper was motivated by the observation of an
apparent bicritical phase diagram with an emergent tetragonal symmetry in
ErTe3, a weakly orthorhombic, quasi two-dimensional (layered) material with
unidirecitonal CDW order whose direction can be reoriented with the
application of uniaxial stress. Further studies of this material with the goal
of exploring the behavior closer to the putative bicritical point would
certainly be interesting. In particular, many of the theoretically most
interesting issues concern fluctuation effects not captured by mean-field
theory, while at present it is not clear whether the experiments approach
close enough to criticality to be sensitive to such effects.

It is worth noting that similar considerations apply to systems with
unidirectional spin density waves as well, with the modification that in that
case the order parameter symmetries associated with the two order parameter
fields is richer than in the present case. For instance, for a commensurate
unidirectional colinear antiferromagnet — i.e. the $(0,\pi)-(\pi,0)$ “stripe”
antiferromagnet seen in the Fe-based superconductors [27] — the two order
parameters fields (to the extent that spin-orbit coupling is negligible) are
three-component fields, so the relevant symmetry is $O(3)\times O(3)$ with,
for tetragonal systems, an additional $\mathbb{Z}_{2}$ under exchange. Many
other examples exist, with other symmetries or near symmetries, including
materials with still lower point-group symmetry, in various strongly
interacting electronic materials.

The analysis — with suitable modifications — is also of possible relevance to
unconventional superconductors under circumstances in which by tuning some
non-thermal parameter, e.g. pressure or alloy concentrations, the system can
be tuned from a regime in which there is one form of superconducting order to
a regime in which there is another. Here, again, one expects a multicritical
phase diagram at the point at which the two forms of superconductivity have
the same $T_{c}$. An example of this is an apparent change from an extended
s-wave to a d-wave superconductor that appears as a function of chemical
substitution in Ba1-xKxFe2As2 [28].

From a theory perspective, it remains to determine whether or not a stable
bicritical point not accessible from the $\epsilon$ expansion exists, what are
the best experimental tests of the existence of an emergent $\mathbb{Z}_{2}$
symmetry in the critical regime, and more generally how to nail down the
topology of the phase diagram in the asymptotic multicritical regime where
mean-field considerations are insufficient.

###### Acknowledgements.

We gratefully acknowledge useful discussions and correspondence with Vladimir
Calvera, Eduardo Fradkin, Andreas Stergiou and Gilles Tarjus. The work was
supported, in part, by the Department of Energy, Office of Basic Energy
Sciences, under contract DE-AC02-76SF00515. AP is supported by a Stanford
Graduate Fellowship. AGS was supported in part by an NSF GRFP and DOE SCGSR
award. SAK was further supported by a Leverhulme Trust International
Professorship grant number LIP-202-014 at Oxford.

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