# Critical Matter

Leo Radzihovsky  Department of Physics, University of Colorado, Boulder, CO
80309, USA

###### Abstract

I review a class of novel ordered states of “critical matter”, that exhibit
strongly fluctuating universal power-law orders, controlled by an infra-red
attractive, non-Gaussian fixed point. I will illustrate how RG methods
pioneered by Wilson and Fisher can be used to deduce critical phenomenology of
such critical phases, resembling that of a critical point of second order
phase transitions, but requiring no fine tuning.

##  I Introduction

Michael Fisher is a towering figure in theoretical physics, most notably
recognized for his seminal development (in collaboration with Ken Wilson and
building on the works of Kadanoff, Migdal, Widom, Larkin, Pokrovsky, and
others) of the renormalization group (RG)epsilonWilsonFisher ; MEFrmp74 ;
WilsonKogutPR  and its numerous early applications to critical phenomena of
continuous phase transitionsChaikinLubensky . In these works Michael
transformed Wilson’s deep ideasMEFrmp74 ; WilsonKogutPR  into a practical and
powerful theoretical tool for controlled calculationsepsilonWilsonFisher
(complementing more formal developments in quantum field theoryZinnJustin )
and demonstrated its power through numerous seminal applications to critical
phenomena. With this, he elevated RG into a central calculation tool in modern
theoretical physics.

Figure 1: Illustration of unimportance of fluctuations inside phases of
conventional systems, where qualitative effects of thermal fluctuations are
confined to a vicinity of a critical point.

Beginning with the earliest seminal analysis of the $\phi^{4}$ field theory to
describe criticality of the Ising paramagnet-ferromagnet transition, most
applications of RG are used to treat enhanced fluctuations near a critical
point of a continuous phase transition, where direct perturbation theory
breaks down. In such applications a critical behavior emerges after tuning a
set of parameters - e.g., temperature, pressure, magnetic field, etc, to a
(multi-) critical point in a phase diagram, as illustrated in Fig.1. It is
only then that fluctuations and nonlinearities (interactions) become important
(below the upper-critical dimension) and universal asymptotic critical
behavior emerges, e.g., power-law, scale-free correlation functions and
thermodynamic responses with universal exponentscriticalTcomment .

In contrast to critical points, as illustrated in Fig.1 phases of matter, by
their very definition are stable to weak generic perturbations that do not
explicitly break their symmetry. They are thus characterized by infrared
attractive fixed points, and thereby require no fine-tuning, in contrast to a
critical point that separates them. Fluctuations in ordered Landau phases of
matterLandau  (that is our focus here) are characterized by their Goldstone
modes associated with the spontaneous breaking of a continuous symmetry –
Landau phases that break discrete symmetries have no interesting fluctuations.
Despite being gapless, generically within ordered phases Goldstone-mode
fluctuations are small, with finite root-mean-squared fluctuations, latter
typically taken as the defining property of the ordered phase. In the RG
parlance, this corresponds to typical phases that are controlled by a Gaussian
attractive fixed point, as illustrated in Fig.1. Thus, conventional Landau
phases’ description is effectively trivial, with fluctuations (above the
lower-critical dimension, where the ordered phase is stable) only leading to
small corrections to their mean-field description.

In this chapter I instead focus on an exotic class of ordered phases of matter
– “critical phases”, e.g., smectics, cholesterics, columnar phases, membranes,
elastomers,…, illustrated in Fig.2 – where in stark contrast to conventional
phases, Goldstone-mode fluctuations are divergingly strong, interacting, and
thus are controlled by a non-Gaussian infrared attractive fixed point, as
illustrated in Fig.3.

Figure 2: Illustration of critical phases: (a) smectics realized as
conventional liquid crystalsdeGennes ; ChaikinLubensky ; GP ; RTaerogelPRL ;
RTaerogelPRB ; BRTCaerogelScience , 2d colloidsGlaser , quantum-Hall
systemsEisensteinSm ; Du99 ; Fogler ; Moessner ; MacDonaldFisherSm ;
FradkinKivelsonPRB ; LRDorseyQHN  and as striped “pair-density wave” FFLOFF ;
LO  superconducting phases in degenerate atomic gasesffloLR ; SRreview ;
PDWaop , p-wave resonantLR_ChoiPRL , spin-orbit coupledHuiZhai  and frustrated
superfluidshelicalBosonsHMR22 ,(b) cholesteric liquid crystalsdeGennes ;
ChaikinLubensky ; cholestericLubensky ; cholestericLR  and helical state of
frustrated magnetscholestericLR ; MnSi ; BalentsNature ; AParamakanti , (c)
columnar liquid crystals and spontaneous vortex latticesPhysicsToday ;
experiments ; Varma ; SRTcolumnar ; magneticSCprl ; magneticSCprb , (d)
nematic elastomersbookWarner ; XingRadzAOP , and (e) polymerized
membranesJWSmembranes ; NelsonPeliti ; AronovitzLubensky ; DG ;
LeDoussalRadzihovskyPRL ; LRphdThesis .

As such the resulting ordered critical phase, while breaking a continuous
symmetry is highly nontrivial and strongly interacting, with little
resemblance of its mean-field cartoon. It is characterized by universal
critical exponents, but in striking constrast to a critical point requires no
fine-tuning, protected by underlying spontaneously broken symmetries, with
importance of strong fluctuations and nonlinearities extending throughout the
ordered phase. A generic ingredient of such critical phases (often associated
with spontaneous breaking of continuous spatial symmetries) is that underlying
symmetry enforces an exact vanishing of a subset of elastic moduli, resulting
in particularly “soft” harmonic elasticity of the associated Goldstone modes,
described by $m$-Lifshitz modelsmLifshitz ; BakSOC .

Figure 3: Illustration of the importance of fluctuations inside ordered phases
of “critical matter”, in which even deep in the ordered phase the quartic
potential is missing the stabilizing quadratic contribution, resulting in
divergent fluctuations akin to a critical point. The bottom of the figure is a
schematic of a renormalization group flow, e.g., for a smectic state in $d<3$
dimensions, illustrating that at low $T$ it is a “critical phase” displaying
universal power-law phenomenology, controlled by a nontrivial infrared-stable
fixed point.

In subsequent sections of this chapter, I will illustrate above phenomenology
through a set of examples of critical phases, some illustrated in Fig.2. I
will begin with a detailed discussion of a quintessential critical phases –
uniaxially periodically modulated states that spontaneously break
translational and rotational symmetry, and are realized in numerous physical
contexts, including smectic liquid crystalsChaikinLubensky ; deGennes ; GP ,
cholestericscholestericLR , helical magnets and bosons on frustrated
latticescholestericLR ; MnSi ; BalentsNature ; helicalBosonsHMR22 , nonzero
momentum superfluids such as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)
superconductorsffloLR ; SRreview , p-wave resonantly paired bosonsLR_ChoiPRL ,
spin-orbit coupled bosonsHuiZhai , and quantum-Hall striped statesEisensteinSm
; Du99 ; Fogler ; Moessner ; MacDonaldFisherSm ; FradkinKivelsonPRB . In Sec.
3, I will describe a columnar states, that appear in discotic liquid
crystalsChaikinLubensky ; deGennes ; SRTcolumnar  and in its line-crystal
analogs, such as putative spontaneous vortex lattice in magnetic
superconductorsmagneticSCprl ; magneticSCprb . I will analyze another rich
example of a critical phase – a thermally fluctuating tensionless polymerized
membrane in Sec. 4.JWSmembranes ; NelsonPeliti ; AronovitzLubensky ; DG ;
LeDoussalRadzihovskyPRL  In Sec. 5, I will discuss a nematic elastomer – a
liquid crystal rubber, that, mathematically is a three-dimensional amalgam of
a smectic and columnar phases, displaying critical phenomenology throughout
its phasebookWarner ; XingRadzAOP .

##  II Uniaxially periodic critical states

There is a number of states of matter that spontaneously develop a periodic
modulation along an arbitrarily chosen single axis. Strictly speaking this can
only take place in a system with underlying rotational invariance, though can
be exhibited over an extended intermediate regime when explicitly broken
rotational symmetry couples only weakly, as e.g., in low density electronic
systems.EisensteinSm ; Du99

Beyond conventional liquid crystals, some of the novel systems include helical
states of frustrated and spin-orbit coupled magnets and bosonsMnSi ;
BalentsNature ; AParamakanti ; HuiZhai ; helicalBosonsHMR22 , and strongly
correlated electronic and bosonic systems (quite surprising for isotropic and
point constituents), such as FFLOffloLR  “striped” (“pair density wave”)
superconductorsPDWaop ; FradkinKivelsonPRB , finite momentum
superfluidsLR_ChoiPRL ; HuiZhai ; helicalBosonsHMR22 , and a two-dimensional
electron gas in the quantum Hall regime of half-filled Landau level
EisensteinSm ; Du99 ; Fogler ; Moessner ; MacDonaldFisherSm ;
FradkinKivelsonPRB ; LRDorseyQHN . The underlying unifying feature of these
diverse set of critical systems is the spontaneously broken rotational (in
contrast to e.g., density waves in crystalline solids) and translational
symmetries. Based on these symmetries that they break, we collectively refer
to this class of states as smectics akin to their conventional soft matter
liquid crystal realizations, that we turn to next.

###  II.1 Smectic liquid crystals

Figure 4: Illustration of the most ubiquitous nematic (orientationally ordered
uniaxial fluid), smectic-A and smectic-C (one-dimensional density wave with,
respectively isotropic and polar in-plane fluid orders) liquid crystal phases
and their associated textures in cross-polarized microscopy (N.A. Clark
laboratory).

Illustrated in Fig.4, the most ubiquitous liquid crystal phases are the
uniaxial nematic, that spontaneously breaks rotational symmetry of the parent
isotropic fluid and the smectic state, a uniaxial one-dimensional density wave
that further breaks translational symmetry along a single axis. The uniaxial
nematic is characterized by a quadrupolar order parameter
$Q_{ij}=S({\hat{n}}_{i}{\hat{n}}_{j}-\frac{1}{3}\delta_{ij})$, with $S$ the
strength of the orientational order along the principle uniaxial axis
$\hat{\bf n}$. A three-dimensional smectic-A is a periodic array of two-
dimensional fluids, characterized by a uniaxial periodic density modulation
with Fourier components that are integer multiples of the smectic ordering
wavevector ${\bf q_{0}}=\hat{{\bf n}}2\pi/a$ (with $a$ the layer spacing)
parallel to the nematic director $\hat{\bf n}$; other type of smectics, e.g.,
smectic-C, where $\hat{\bf n}$ makes a nonzero angle with ${\bf q}_{0}$ are
common.ChaikinLubensky ; deGennes  The dominant lowest Fourier component
$\psi({\bf x})\equiv\rho_{{\bf q}_{0}}$ can be taken as the local (complex
scalar) order parameter which distinguishes the smectic-A from the nematic
phasedeGennes . It is related to the molecular density $\rho({\bf x})$ by

$\rho({{\bf x}})=\mbox{Re}[\rho_{0}+\psi({{\bf x}})e^{i{\bf q_{0}}\cdot{{\bf
x}}}]\;,$ (1)

where $\rho_{0}$ is the mean density of the smectic, $\psi=\rho_{{\bf q}_{0}}$
is the Fourier amplitude of the density at wavevector ${\bf q}_{0}$ and Re is
a real part.

As first introduced by de GennesdeGennes , the effective Hamiltonian
functional $H_{dG}[\psi,\hat{\bf n}]$, that describes the nematic-to-smectic-A
(NA) transition at long length scales, is given by,

$\displaystyle H_{dG}[\psi,\hat{\bf n}]$ $\displaystyle=$
$\displaystyle\int\\!d^{d}x\bigg{[}c|({\bm{\nabla}}-iq_{0}{\bf\delta{\hat{n}}})\psi|^{2}+t_{0}|\psi|^{2}+{1\over
2}g_{0}|\psi|^{4}\bigg{]}+H_{F}[\bf{n}]\;,$ (2)

where $t_{0}\propto T-T_{NA}$ is the reduced temperature for transition at
$T_{NA}$,

$\delta\hat{\bf n}({\bf x})\equiv\hat{\bf n}({\bf x})-\hat{\bf
n}_{0}=\delta\hat{\bf n}_{\perp}+\hat{\bf n}_{0}(\sqrt{1-\delta\hat{\bf
n}_{\perp}^{2}}-1)$ (3)

is the fluctuation of the local nematic director $\hat{\bf n}({\bf x})$ away
from its average value $\hat{\bf n}_{0}$, which I take to be $\hat{\bf z}$,
and $H_{F}[\bf n]$ is the Frank effective Hamiltonian that describes the
elasticity of the nematic director:

$\displaystyle H_{F}[\hat{\bf n}]=\int\\!d^{d}x\;{1\over
2}\bigg{[}K_{s}({\bm{\nabla}}\cdot\hat{\bf n})^{2}+K_{t}(\hat{\bf
n}\cdot{\bm{\nabla}}\times\hat{\bf n})^{2}+K_{b}(\hat{\bf
n}\times{\bm{\nabla}}\times\hat{\bf n})^{2}\bigg{]}\;,$ (4)

where $K_{s}$, $K_{t}$, and $K_{b}$ are the bare elastic moduli for splay,
twist and bend of the nematic director field, respectively.deGennes ;
ChaikinLubensky

The “minimal” gauge-like coupling between $\bf n$ and $\psi$ is enforced by
the requirement of global rotational invariancedeGennes . It is important to
emphasize, however, that although de Gennes Hamiltonian $H_{dG}$ closely
resembles the Ginzburg-Landau model of a superconductor and the Abelian-Higgs
model, there are essential differences, that are discussed in the chapter by
Tom Lubensky. The physical reality of the nematic director ${\bf{\hat{n}}}$
and the smectic order parameter $\psi$ (in contrast to the gauge ambiguity in
the definition of the vector potential and the superconducting order
parameter), selects the liquid crystal gauge
${\delta\bf{\hat{n}}}_{\perp}\cdot{\bf{\hat{n}}}_{0}=0$ as the physical gauge
in which $\bf{\hat{n}}$ and $\psi$ are measured. The strict gauge invariance
is also explicitly broken in $H_{dG}$ by the splay term
$K_{s}({\bm{\nabla}}\cdot\hat{\bf n})^{2}$ of the Frank Hamiltonian
(contrasting with the Maxwell action that involves purely gauge invariant
derivatives, e.g., $({\bm{\nabla}}\times{\bf A})^{2}$), that, as we will see
below determines the smectic Goldstone-mode elasticity.

###  II.2 Smectic elasticity

####  II.2.1 Smectic from a nematic liquid crystal fluid

Within the ordered smectic phase, the fluctuations are conveniently described
in terms of the magnitude and phase of the smectic order parameter $\psi$. It
is easy to show that the fluctuations of the magnitude of $\psi$ around the
average value $|\psi_{0}|=\sqrt{t_{0}/g_{0}}=\mbox{const.}$ are “massive”, and
can therefore be safely integrated out of the partition function, leading to
only finite, unimportant shifts in the effective elastic moduli. In contrast,
the phase of $\psi$ is a U(1) massless Goldstone mode, corresponding to
spontaneously broken translational symmetry along ${\bf q}_{0}$. It is the
low-energy phonon degree of freedom of the smectic state, describing local
displacement of the smectic layers from its perfect periodic order. In accord
with this discussion, deep within the smectic phase, we can represent the
smectic order parameter as

$\displaystyle\psi({\bf x})=|\psi_{0}|e^{-iq_{0}u({\bf x})}\;,$ (5)

safely neglecting fluctuations in the magnitude $|\psi_{0}|$ at temperature
below the $T_{NA}$.

Using this low-temperature smectic description (5) inside the de Gennes
Hamiltonian (2) and dropping constants, one finds

$\displaystyle H[u,\delta{\bf{\hat{n}}}_{\perp}]$ $\displaystyle=$
$\displaystyle\int\\!d^{d}x\bigg{[}{B\over
2}({\bm{\nabla}}_{\perp}u+\delta{\bf{\hat{n}}}_{\perp})^{2}+{B\over
2}(\partial_{z}u-{\frac{1}{2}}\delta{\hat{n}}_{\perp}^{2})^{2}\;$ (6)
$\displaystyle+{K_{s}\over
2}({\bm{\nabla}}\cdot\delta{\bf{\hat{n}}})^{2}+{K_{t}\over 2}(\hat{\bf
z}\cdot{\bm{\nabla}}\times\delta{\bf{\hat{n}}})^{2}+{K_{b}\over 2}(\hat{\bf
z}\times{\bm{\nabla}}\times\delta{\bf{\hat{n}}})^{2}\bigg{]},$

where $B=2c|\psi_{0}|^{2}q_{0}^{2}$ is the smectic compression modulus. I
observe that the fluctuation mode
${\bm{\nabla}}_{\perp}u+\delta{\bf{\hat{n}}}_{\perp}$ is “massive” and leads
to an emergent Anderson-Higgs-like mechanism, a hallmark of gauge theories. As
a consequence, after a simple Gaussian integration over
$\delta{\bf{\hat{n}}}_{\perp}$, one finds that at long length scales,
$\delta{\bf{\hat{n}}}_{\perp}$ fluctuations are constrained to follow
${\bm{\nabla}}_{\perp}u$, corresponding to locking of the director $\hat{\bf
n}$ to the smectic layer normal. The low-energy elastic smectic Hamiltonian is
then obtained by the replacement

$\delta{\bf{\hat{n}}}_{\perp}\rightarrow-{\bm{\nabla}}_{\perp}u\;,$ (7)

everywhere in Eq.(6) and Frank energy Eq.(4). Valid in the long wavelength
limit and provided dislocations are confined, one thus obtain a nonlinear
elastic Goldstone-mode Hamiltonian of the smectic phase,

$\displaystyle H_{sm}[{u}]$ $\displaystyle=$
$\displaystyle\int\\!d^{d}x\bigg{[}{K\over
2}(\nabla^{2}_{\perp}{u})^{2}+{B\over
2}(\partial_{z}{u}-{\frac{1}{2}}({\bm{\nabla}}u)^{2})^{2}\bigg{]}\;,$ (8)

where $K=K_{s}$ is the splay modulus.ChaikinLubensky ; GP

I observe that the harmonic elasticity of the smectic – $m=d-1$-Lifshitz
modelmLifshitz  – is highly anisotropic (at harmonic level with scaling $z\sim
x_{\perp}^{2}$), displaying compression modulus $B$ along ${\bf q}_{0}$ and
higher-order curvature (Laplacian) modulus transverse to ${\bf q}_{0}$ , with
the “tension” (transverse gradient) modulus vanishing exactly. This is ensured
by the underlying rotational symmetry, as discussed in the Introduction to
critical phases. As I will show in forthcoming sections, as a consequence,
smectic fluctuations are highly enhanced (relative to e.g., its XY model
counterpart), resulting in importance of nonlinear Goldstone-modes’ elasticity
for $d\leq 3$, that I therefore retained in (8). I also note that the
compressional modulus $B$ multiplies a square of the nonlinear strain tensor,
whose precise form is determined by the underlying invariance under arbitrary
large rotation of smectic layers,

$\displaystyle u_{0}({\bf r})=z(1-\cos\theta)+x\sin\theta,$ (9)

and maintained under coarse-graining RG. I conclude this subsection by also
noting that the aforementioned connection to the Anderson-Higgs mechanism
elucidates why a smectic state is characterized by only a single Goldstone
mode $u$, rather than three, despite partially breaking two rotational and one
translational symmetries. In contrast, a charged superconductor is well-known
to have all its Goldstone modes “eaten” – fully gapped out by the Anderson-
Higgs mechanism.

####  II.2.2 Smectic from an isotropic fluid

I now derive the nonlinear smectic elasticity in a more basic, complementary
way, starting instead with an isotropic fluid state.GP  I begin with a generic
energy functional that captures system’s tendency to develop a unidirectional
wave at wavevector ${\bf q}_{0}$, with an arbitrary direction, and magnitude
fixed at $q_{0}$,

${\mathcal{H}}_{sm}={\frac{1}{2}}J\left[(\nabla^{2}\rho)^{2}-2q_{0}^{2}(\nabla\rho)^{2}\right]+{\frac{1}{2}}t\rho^{2}-w\rho^{3}+v\rho^{4}+\ldots,$
(10)

where $J,q_{0},t,w,v$ are parameters of the isotropic fluid phase. Clearly,
the first term is engineered so that dominant fluctuations and condensation
are on a spherical surface at a nonzero wavevector with a magnitude $q_{0}$.
Thus I focus on the density at a wavevector ${\bf q}$, that for now is
unrelated to $q_{0}$

$\rho({\bf x})={\text{Re}}\left[\rho_{q}({\bf x})e^{i{\bf q}\cdot{\bf
x}}\right],$ (11)

where $\rho_{q}({\bf x})$ is a complex scalar. Without loss of generality,
based on the discussion of the previous subsection, the order parameter
$\rho_{q}({\bf x})=|\rho_{q}|e^{-iqu}$ is taken to have a (constant) magnitude
$|\rho_{q}|$ and a phase $qu({\bf x})$. Clearly $u({\bf x})$ is just a phonon
displacement along ${\bf q}$. Substituting this expression for $\rho({\bf x})$
and its gradients into ${\mathcal{H}}_{sm}$ one finds

$\displaystyle{\mathcal{H}}_{sm}$ $\displaystyle=$ $\displaystyle
J\rho_{0}^{2}\left[\frac{1}{4}q^{2}(\nabla^{2}u)^{2}+\bigg{(}q{\bf
q}\cdot\nabla u-{\frac{1}{2}}q^{2}(\nabla
u)^{2}\bigg{)}^{2}+4(q^{2}-q_{0}^{2})\bigg{(}q{\bf q}\cdot\nabla
u-{\frac{1}{2}}q^{2}(\nabla u)^{2}\bigg{)}\right]+\ldots,$ (12)

where constant parts as well as fast oscillating pieces were dropped as they
average away after spatial integration of the above energy density. Firstly, I
observe that (as discussed on general grounds above) in a harmonic part linear
gradient elasticity in $u$ only appears for gradients along ${\bf q}$, namely
${\bf q}\cdot\nabla$, with elasticity transverse to ${\bf q}$ starting with a
Laplacian form. Secondly, the elastic energy density is an expansion in a
rotationally-invariant strain tensor combination

$u_{qq}=\hat{{\bf q}}\cdot\nabla u-{\frac{1}{2}}(\nabla u)^{2},$ (13)

whose nonlinearities in $u$ ensure that it is fully rotationally invariant
even for large rotations. To see this (picking $\hat{\bf q}={\bf{\hat{z}}}$)
note that a rigid (distortion-free) rotation of ${\bf q}$
($q_{0}{\bf{\hat{z}}}\rightarrow{\bf
q}=q_{0}(\cos\theta{\bf{\hat{z}}}+\sin\theta{\bf{\hat{x}}}$), can be
interpreted as a spatially linear “distortion” $u_{0}({\bf
x})=z(1-\cos\theta)+x\sin\theta$, for which nonlinear strain $u_{qq}$ vanishes
identically, thus, as required, corresponds to a vanishing energy. Thirdly,
the last term in (12) vanishes for $|{\bf q}|$ picked to equal $q_{0}$,
corresponding to energetically preferred choice of modulation wavevector.

Looking ahead, as one includes effects of fluctuations, the “bare” condition
$q=q_{0}$ will need to be adjusted so as to eliminate the fluctuation-
generated linear term in $u_{qq}$ order by order, which amounts to an
expansion in the nonlinear strain $u_{qq}$ around the correct (fluctuation-
corrected) ground state. Finally, I note that the relation between the
curvature modulus $K$ of Laplacian (first) term and the bulk modulus $B$
gradient (second) term is not generic and can be relaxed to have distinct
elastic constants, as can be seen if higher order gradient terms are included
in the original energy density, Eq. (10).

Choosing the coordinate system such that $\hat{\bf z}$ is aligned along ${\bf
q}$, one finds that for $q=q_{0}$, the Goldstone-mode Hamiltonian (12) reduces
to the standard smectic elastic energy density, (8) derived from the de Gennes
model above. deGennes ; ChaikinLubensky .

####  II.2.3 Smectic as cholesteric and helical fluids

Another beautiful example of a system that realizes a uniaxial periodic
critical state is a cholesteric phase of chiral liquid crystals, illustrated
in Fig.2(b). It is in fact the first liquid crystal phase discovered in
cholesterol benzoate by Reinitzer in 1886.deGennes ; ChaikinLubensky  Such
orientational (as opposed to mass) density wave state is ubiquitous in nature
and is equivalent to the helical co-planar spin-spiral state that is found in
non-centrosymmetric magnets like MnSi, FeSi and many othersMnSi  (neglecting
crystal-field pinning that is always present due to spin-orbit interactions
and explicit breaking of rotational symmetry.).

Following analysis of Radzihovsky and LubenskycholestericLR , below I discuss
the low-energy nonlinear elasticity of a cholesteric, on length scales longer
than its period, $a=2\pi/q_{0}$. As first proposed by de Gennes, the latter is
expected, based on symmetry and an explicit harmonic derivation by
LubenskycholestericLubensky ; deGennes , to be identical to that of a smectic,
i.e., that of a one-dimensional crystalline order, that spontaneously breaks
underlying translational and rotational symmetry, with constant-orientation
“layers” transverse to the ordering wavevector, ${{\bf q}}_{0}$. Although the
underlying chirality has only subtle surface effects within the cholesteric
state, its effects become important in the nature of the phase transition in
and out of the cholesteric critical state.

The simplest model of a chiral nematic liquid crystal (a cholesteric), is
captured by the chiral Frank elastic free energy density,

$\displaystyle{\mathcal{H}}^{*}_{F}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}K_{s}(\nabla\cdot{\hat{n}})^{2}+{\frac{1}{2}}K_{b}({\hat{n}}\times\nabla\times{\hat{n}})^{2}+{\frac{1}{2}}K_{t}({\hat{n}}\cdot\nabla\times{\hat{n}}+q_{0})^{2}.$
(14)

The broken chiral symmetry allows a chiral $q_{0}$ term, that leads to a twist
of the nematic structure into a helix with a pitch $2\pi/q_{0}$ along a
spontaneously chosen axis.

In a simplified isotropic limit of $K_{s}=K_{b}=K_{t}$, the chiral Frank
energy density is then given by

$\displaystyle{\mathcal{H}}^{*}_{F}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}K\left[(\partial_{i}{\hat{n}}_{j})^{2}+2q_{0}{\hat{n}}\cdot\nabla\times{\hat{n}}+q_{0}^{2}\right],$
(15) $\displaystyle=$
$\displaystyle{\frac{1}{2}}K\left(\partial_{i}{\hat{n}}_{j}+q_{0}\epsilon_{ijk}{\hat{n}}_{k}\right)^{2}-{\frac{1}{2}}Kq_{0}^{2},$
(16)

where after integration by parts I utilized the identity

$\displaystyle(\partial_{i}{\hat{n}}_{j})^{2}$ $\displaystyle=$
$\displaystyle(\nabla\cdot{\hat{n}})^{2}+(\nabla\times{\hat{n}})^{2}+\nabla\cdot\big{[}({\hat{n}}\cdot\nabla){\hat{n}}-{\hat{n}}\nabla\cdot{\hat{n}}\big{]}.$
(17)

In the absence of topological defects, neglecting the boundary term,
${\mathcal{H}}^{*}_{F}$ is a sum of squares and is therefore minimized by a
twist-only cholesteric state

$\displaystyle{\hat{n}}({\bf r})$ $\displaystyle=$ $\displaystyle{\hat{\bf
e}}_{10}\cos({\bf q}\cdot{\bf r})+{\hat{\bf e}}_{20}\sin({\bf q}\cdot{\bf
r}),$ (18)

where ${\hat{\bf e}}_{10},{\hat{\bf e}}_{20},{\hat{\bf e}}_{30}\equiv{\hat{\bf
e}}_{10}\times{\hat{\bf e}}_{20}$ form an orthonormal triad, with a constant
twist

${\hat{n}}\cdot\nabla\times{\hat{n}}=-q_{0}.$ (19)

I now derive the low-energy Goldstone-mode elasticity about the cholesteric
helical ground state, that can be parameterized according to

$\displaystyle{\hat{n}}({\bf r})$ $\displaystyle=$ $\displaystyle{\hat{\bf
e}}_{1}({\bf r})\cos({\bf q}\cdot{\bf r}+\chi({\bf r}))+{\hat{\bf e}}_{2}({\bf
r})\sin({\bf q}\cdot{\bf r}+\chi({\bf r})),$ (20)

where fluctuations are captured by the spatially dependent orthonormal triad
${\hat{\bf e}}_{1}({\bf r}),{\hat{\bf e}}_{2}({\bf r}),{\hat{\bf e}}_{3}({\bf
r})$ and the helical phase

$\chi({\bf r})=-q_{0}u({\bf r}),$ (21)

that corresponds to the phonon field $u({\bf r})$ of the constant-orientation
layers.

The helical state breaks a group of three dimensional translations and
rotations $G=T_{x,y,z}\times O(3)$ of the isotropic fluid down to
$H=T_{x}\times T_{y}\times U(1)$ (latter
$U(1)=\text{diagonal}[T_{z},O_{z}(2)]$). Thus, since dim$[G/H=O(3)]=3$, one
may expect three independent Goldstone modes $\chi({\bf r})$ and ${\hat{\bf
e}}_{3}({\bf r})$, corresponding to three degrees of freedom of the
orthonormal triad; the azimuthal angle $\phi({\bf r})$ defining the
orientation of the ${\hat{\bf e}}_{1,2}$ around ${\hat{\bf e}}_{3}$ is not
independent of $\chi({\bf r})$ as it can be absorbed into it. The low-energy
coset space is isomorphic to $S^{1}\times S^{2}$, a ball (radius $\pi$) of the
group manifold of SO(3). However, despite of this standard counting, as I will
show below and is expected on general grounds anticipated by de Gennes, the
cholesteric state is characterized by a single smectic-like Goldstone mode
$u({\bf r})$.

To this end, substituting the form for ${\bf{\hat{n}}}({\bf r})$ from Eq. (20)
into the Frank energy of the chiral nematic (14), and defining effective
connection gauge fields, ${\bf a}$ and ${\bf c}_{1,2}$

$\displaystyle\partial_{i}{\hat{\bf e}}_{1j}$ $\displaystyle=$ $\displaystyle
a_{i}{\hat{\bf e}}_{2j}+c_{1i}{\hat{\bf e}}_{3j},$ (22a)
$\displaystyle\partial_{i}{\hat{\bf e}}_{2j}$ $\displaystyle=$ $\displaystyle-
a_{i}{\hat{\bf e}}_{2j}+c_{2i}{\hat{\bf e}}_{3j},$ (22b) $\displaystyle
a_{i}={\hat{\bf e}}_{2}\cdot\partial_{i}{\hat{\bf e}}_{1},\ \ c_{1i}$
$\displaystyle=$ $\displaystyle-{\hat{\bf e}}_{1}\cdot\partial_{i}{\hat{\bf
e}}_{3},\ \ c_{2i}=-{\hat{\bf e}}_{2}\cdot\partial_{i}{\hat{\bf e}}_{3},$
(22c)

one finds,

$\displaystyle{\mathcal{H}}^{*}_{F}$ $\displaystyle=$
$\displaystyle\frac{K}{2}(\nabla\chi+{\bf a}+{\bf q}-q_{0}{\hat{\bf
e}}_{3})^{2}+\frac{K}{4}({\bf c}_{1}+q_{0}{\hat{\bf
e}}_{2})^{2}+\frac{K}{4}({\bf c}_{2}-q_{0}{\hat{\bf e}}_{1})^{2}.$ (23)

In above, I again neglected spatially oscillating and constant terms.

Taking ${\bf q}=q_{0}{\bf{\hat{z}}}$, with ${\bf{\hat{z}}}$ defining the
helical axis (distinct from the normal to the helical plane, ${\hat{\bf
e}}_{3}$) and noting the compatibility condition on effective flux or its
equivalent vanishing of the Pontryagin density

$\displaystyle\nabla\times{\bf a}=\epsilon_{ij}{\hat{\bf
e}}_{3}\cdot\partial_{i}{\hat{\bf e}}_{3}\times\partial_{j}{\hat{\bf
e}}_{3}=0,$ (24)

required by well-defined cholesteric layers, i.e., in the absence of
dislocations and disclinations in the layer structure, allows one to take

$\displaystyle{\bf a}={\bm{\nabla}}\phi.$ (25)

Under this condition $\phi$ can be eliminated in favor of $\chi$, i.e.,
$\chi+\phi\rightarrow\chi$ and in the laboratory coordinate system
${\bf{\hat{x}}},{\bf{\hat{y}}},{\bf{\hat{z}}}$, the fluctuations are
characterized by the local helical frame described by $\chi$ and ${\hat{\bf
e}}_{3}$, with

$\displaystyle{\hat{\bf e}}_{3}$ $\displaystyle=$ $\displaystyle{\hat{\bf
e}}_{3\perp}+{\bf{\hat{z}}}\sqrt{1-{\hat{\bf e}}_{3\perp}^{2}}\approx{\hat{\bf
e}}_{3\perp}+{\bf{\hat{z}}}(1-{\frac{1}{2}}{\hat{\bf e}}_{3\perp}^{2}).$ (26)

I thus obtain

$\displaystyle{\mathcal{H}}^{*}_{F}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}(\nabla_{\perp}\chi-q_{0}{\hat{\bf
e}}_{3\perp})^{2}+{\frac{1}{2}}(\partial_{z}\chi+{\frac{1}{2}}q_{0}{\hat{\bf
e}}_{3\perp}^{2})^{2}+\frac{1}{4}({\bf c}_{1}^{2}+{\bf
c}_{2}^{2})+\frac{q_{0}}{2}({\bf c}_{1}\cdot{\hat{\bf e}}_{2}-{\bf
c}_{2}\cdot{\hat{\bf e}}_{1}).$ (27)

From the minimization of the first term (or equivalently integrating out the
independent ${\hat{\bf e}}_{3\perp}$ degree of freedom), I obtain an effective
constraint

$\nabla_{\perp}\chi=q_{0}{\hat{\bf e}}_{3\perp},$ (28)

that is an example of an emergent Higgs mechanism (akin to smectic liquid
crystals discussed above) locking the cholesteric layers with the molecular
frame orientation. With this constraint (valid at low energies) the effective
cholesteric Hamiltonian reduces to

$\displaystyle{\mathcal{H}}^{*}_{F}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}\big{[}\partial_{z}\chi+\frac{1}{2q_{0}}(\nabla_{\perp}\chi)^{2}\big{]}^{2}+\frac{1}{4q_{0}^{2}}({\hat{\bf
e}}_{\alpha\gamma}\cdot\partial^{\perp}_{\beta}\partial^{\perp}_{\gamma}\chi)^{2}+\frac{q_{0}}{2}({\hat{\bf
e}}_{1\alpha}{\hat{\bf
e}}_{2\beta}\partial_{\beta}\partial_{\alpha}\chi-{\hat{\bf
e}}_{2\alpha}{\hat{\bf
e}}_{1\beta}\partial_{\beta}\partial_{\alpha}\chi),\hskip 14.22636pt$ (29a)
$\displaystyle=$
$\displaystyle\frac{B}{2}\big{[}\partial_{z}u-{\frac{1}{2}}(\nabla
u)^{2}\big{]}^{2}+\frac{K}{2}(\nabla^{2}u)^{2}$ (29b)

where compressional modulus $B=Kq_{0}^{2}$, ${\hat{\bf e}}_{\alpha,i}{\hat{\bf
e}}_{\alpha,j}=\delta_{ij}$ and ${\hat{\bf e}}_{1,i}{\hat{\bf
e}}_{2,j}-{\hat{\bf e}}_{2,i}{\hat{\bf e}}_{i,j}={\hat{\bf
e}}_{3,k}\epsilon_{ijk}$ were used, and to eliminate the last term in the
first line above I used the condition of the single-valuedness of the phase
field $\chi$, i.e., well defined cholesteric layers with no dislocations.
However, this last condition is violated and thus distinguishes the
cholesteric from a smectic near a phase transition out of the cholesteric
state.

Thus, as advertised and expected on symmetry grounds one indeed
findscholestericLR , that cholesteric Goldstone-mode elasticity, even at the
nonlinear level is identical to that of a conventional smectic – at harmonic
level controlled by the $m=d-1$-Lifshitz modelmLifshitz . Thus smectic
critical state universal phenomenology, to be worked out below, will equally
apply to the cholesteric state.

####  II.2.4 Smectic as a helical bosonic superfluid

I next discuss a quantum realization of a smectic state, namely via
interacting superfluid bosons on a frustrated honeycomb
lattice.helicalBosonsHMR22  Other bosonic realizations also appear in p-wave
Feshbach-resonantLR_ChoiPRL  and Rashba spin-orbital coupled HuiZhai  bosons,
engineered in a controlled way in cold atom experimentsMonikaAidelsburger ;
HuiZhai  to condense at a nonzero momentum, ${\bf k}_{0}$.

I refer the reader to the original literature for these latter systems and
here focus on bosons on a frustrated honeycomb latice, modeled by a tight-
binding dispersion with nearest $t_{1}$ and next-nearest (antiferromagnetic)
$-t_{2}$ neighbor hopping amplitudes, with

$H=-t_{1}\sum_{\langle
ij\rangle}a^{\dagger}_{i,1}a_{j,2}+t_{2}\sum_{\langle\langle
ij\rangle\rangle}(a^{\dagger}_{i,1}a_{j,1}+a^{\dagger}_{i,2}a_{j,2})+h.c.+\frac{U}{2}\sum_{i}\sum_{s=1,2}n_{i,s}(n_{i,s}-1).$
(30)

For sufficiently frustrated hopping, with $t_{2}/t_{1}>1/6$ (that has been
engineered in an optical lattice through Floquet techniques MonikaAidelsburger
), above system displays a dispersion minimum at a nonzero momentum closed
contour, illustrated in Fig.5.

Figure 5: (a) A black circular curve is a nonzero momentum ${\bf k}_{0}$
minimum in the noninteracting microscopic dispersion $\epsilon^{-}_{\bf k}$
(around the $\Gamma$ point in the Brillouin zone, dashed blue curve) of bosons
on a frustrated honeycomb lattice, (30). (b) The dispersion of the interacting
helical superfluid, $E_{{\bf q}}$ around ${\bf q}={\bf k}-{\bf k}_{0}=0$, with
green (see inset) indicating quadratic $q_{\perp}^{2}$ form along $q_{\perp}$
and blue (see inset) $|q_{\parallel}|$ along the condensate momentum, ${\bf
k}_{0}$, consistent with symmetry-expected smectic form, given by (35). The
black-dashed contour indicates the degenerate minimum of the noninteracting
band $\epsilon^{-}_{\bf k}$.

Such nonzero dispersion, with a closed minimum contour is approximately
circular for $t_{2}/t_{1}\approx 1/6+\epsilon$ and can be faithfully modeled
by a continuum field theory, encoded in a noninteracting Hamiltonian
$H_{0}=\int_{\bf r}\hat{\Psi}_{\bf r}^{\dagger}\hat{\varepsilon}_{{\bf
r}}\hat{\Psi}_{\bf r}$ with,

$\hat{\varepsilon}_{{\bf r}}=J(-\nabla^{2}-{k}_{0}^{2})^{2}+\varepsilon_{0},$
(31)

with a quartic dispersion

$\varepsilon_{{\bf k}}=J({\bf k}^{2}-{k}_{0}^{2})^{2}+\varepsilon_{0},$ (32)

and minima lying on contour ${\bf k}^{2}={k}_{0}^{2}$. The corresponding
Euclidean (imaginary-time $\tau$) Lagrangian is given by,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\Psi^{\ast}\partial_{\tau}\Psi+J|\nabla^{2}\Psi|^{2}-2J{k}_{0}^{2}|\nabla\Psi|^{2}+(\tilde{\varepsilon}_{0}-\mu)|\Psi|^{2}+\frac{U}{2}|\Psi|^{4}.$
(33)

For $\mu>\epsilon_{0}$ bosons condense into a nonzero-momentum superfluid
state, a simplest version of which is a helical condensate at a single point
${\bf k}_{0}$ on the dispersion contour minimum – encoded in $\varepsilon_{\bf
k}$, (32). In the density-phase representation, the state is characterized by

$\Psi({\bf r})=\sqrt{n}e^{i{\bf k}_{0}\cdot{\bf
r}+i\phi}=\sqrt{n_{0}+\pi}e^{i{\bf k}_{0}\cdot{\bf r}+i\phi}$ (34)

with a nonzero condensate density (at mean-field level)
$n_{0}=(\mu-\varepsilon_{0})/U$ and momentum $k_{0}$, for
$\mu>\varepsilon_{0}$ (vanishing otherwise), and density and phase
fluctuations, $\pi$ and $\phi$, respectively. To quadratic order, the helical
superfluid is then described by a Goldstone-mode Lagrangian density, which,
after integrating out density field $\pi$ gives a zero-temperature quantum
“columnar” Lagrangian – $m=d-2$-Lifshitz model in $d$ space-time
dimensionsmLifshitz  – (to also appear in subsequent sections)

$\displaystyle\mathcal{L}_{\phi}\approx$
$\displaystyle~{}B_{\tau}(\partial_{\tau}\phi)^{2}+B(\partial_{\parallel}\phi)^{2}+K(\partial_{\perp}^{2}\phi)^{2}.$
(35)

At nonzero $T$, (35) reduces to the classical smectic ($m=d-1$-Lifshitz
modelmLifshitz ) Hamiltonian for $\phi$ discussed in previous subsections.

####  II.2.5 Smectic as a Fulde-Ferrell-Larkin-Ovchinnikov superconductor

A singlet superconductor, frustrated by a depairing Zeeman field has been
predicted by Fulde and FerrellFF  and by Larkin and OvchinnikovLO  (FFLO) to
pair at a nonzero momentum, thereby exhibiting a periodic modulation of the
Cooper-pair amplitude – a pair-density wave (PDW)ffloLR ; PDWaop . Such FFLO
state, in the absence of deleterious orbital field and pinning lattice effects
was proposedSRprl ; SRaop ; SRreview  to be realizable in spin imbalanced
(polarized) Feshbach-resonant paired atomic superfluids. It is now believed to
have been observed in an array of decoupled one-dimensional traps, with recent
progress toward a higher dimensional realization of the FFLO superfluid.Hulet

While microscopic analysis is a bit involvedSRprl ; SRaop ; ffloLR , its
upshot is a derivation of an effective Landau theory of the form (33) for the
pairing amplitude, with a harmonic disperson $\varepsilon_{k}$, for
intermediate Zeeman field (chemical potential for spin imbalance)
$h_{c1}<h<h_{c2}$, displaying a minimum on nonzero momentum closed contour,
akin to that of the helical superfluid (33), above. Fulde-Ferrell and Larkin-
Ovchinnikov are two simplest states that minimize this Lagrangian. The FF
state is characterized by a pairing amplitude, FF

$\displaystyle\Psi_{FF}({\bf r})=\Psi_{q_{0}}e^{i{\bf q}_{0}\cdot{\bf
r}+i\phi},$ (36)

that is a plane-wave with the momentum ${\bf q}_{0}$ and a single Goldstone
mode $\phi$, a local superconducting phase. The state carries a nonzero,
uniform spontaneously-directed supercurrent

$\displaystyle{\bf j}_{FF}=\frac{1}{m}|\Psi_{q_{0}}|^{2}({\bf
q}_{0}+\nabla\phi),$ (37)

and thereby breaks time-reversal and rotational symmetries, chosen
spontaneously along ${\bf q}_{0}$, as well as the global gauge symmetry,
corresponding to the total atom conservation. The low-energy Lagrangian for
$\phi({\bf r},\tau)$ takes the smectic (columnar at $T=0$) form, (35).

The LO stateLO  is a time-reversal symmetric counterpart, that is a
superposition of $\pm{\bf q}_{0}$ paired condensates, given by,

$\displaystyle\Psi_{LO}({\bf r})$ $\displaystyle=$ $\displaystyle\Psi_{+}({\bf
r})e^{i{\bf q}\cdot{\bf r}}+\Psi_{-}({\bf r})e^{-i{\bf q}\cdot{\bf r}},$ (38a)
$\displaystyle=$ $\displaystyle
2|\Psi_{q_{0}}|e^{i{\frac{1}{2}}(\phi_{+}+\phi_{-})}\cos\big{[}{\bf
q}_{0}\cdot{\bf r}+{\frac{1}{2}}(\phi_{+}-\phi_{-})\big{]},\ \ \ \ \ \ \ $
(38b) $\displaystyle=$ $\displaystyle 2|\Psi_{q_{0}}|e^{i\phi}\cos\big{[}{\bf
q}_{0}\cdot{\bf r}+\theta\big{]}.$ (38c)

It is characterized by the superfluid phase $\phi$ and the phonon
$u=\theta/q_{0}$ Goldstone modes, with a combination of the XY and smectic
sectors in a classical Hamiltonian,

$\displaystyle{\mathcal{H}}_{LO}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}K(\nabla^{2}u)^{2}+{\frac{1}{2}}B\big{(}\partial_{\parallel}u-\frac{1}{2}(\nabla
u)^{2}\big{)}^{2}+\frac{1}{2}\rho_{s}^{\parallel}(\partial_{\parallel}\phi)^{2}+\frac{1}{2}\rho_{s}^{\perp}(\nabla_{\perp}\phi)^{2},$
(39)

whose zero-temperature quantum counterpart is a combination of XY and
“columnar” – $m=d-2$-Lifshitz model in $d$ space-time dimensionsmLifshitz
forms. At nonzero temperature one thus expects FF and LO states to display
critical smectic phenomenology that I discuss in the next section.

####  II.2.6 Smectic as quantum Hall stripes

One other prominent example of a quantum smectic has been argued
theoreticallyFogler ; Moessner ; MacDonaldFisherSm ; FradkinKivelsonPRB  to be
realized in half-filled large $N$ Landau level regime and observed
experimentallyEisensteinSm ; Du99  at such filling $\nu=N+1/2$ through
appearance of a highly anisotropic transport below 100 milli-Kelvin. At mean-
field level the state is described as a periodic array of alternative integer
filling $N$ and $N+1$ quantum Hall stripes, predicted to be the exact ground
state in the $N\gg 1$ limit. Beyond Hartree-Fock theoryFogler ; Moessner ,
fluctuations of such a striped state are well described by chiral edge
Luttinger liquids, corresponding to inter- (positions) and intra- (shape)
stripe phonons of this quantum Hall smectic. Much of the analysis discussed
below, importantly beyond its harmonic fluctuations, will also apply to this
state at nonzero temperature leading to a critical smectic state. At zero
temperature, it is described by “cholesteric”-like, i.e., $m=d-2$-Lifshitz
model in $d$ space-time dimensionsmLifshitz .

###  II.3 Nonzero $T$ Gaussian fluctuation in a harmonic smectic

To assess the extent of thermal fluctuations of the smectic Goldstone mode
$u({\bf r})$ I first analyze them within a harmonic approximation, neglecting
elastic nonlinearities in $H_{sm}$. ChaikinLubensky ; Caille  In terms of the
Fourier modes $u_{\bf k}$, the Hamiltonian decouples, reducing to

$\displaystyle
H_{sm}={\frac{1}{2}}\int\frac{d^{d}k}{(2\pi)^{d}}\left(Kk_{\perp}^{4}+Bk_{z}^{2}\right)|u_{\bf
k}|^{2},$ (40)

thus allowing a straightforward computation of phonon correlation functions
via standard Gaussian integrals or equivalently via equipartition. This gives
mean-squared fluctuations

$\displaystyle\langle u^{2}\rangle_{0}^{T}$ $\displaystyle=$
$\displaystyle\int^{\Lambda_{\perp}}_{L_{\perp}^{-1}}\frac{d^{d}k}{(2\pi)^{d}}\frac{T}{Bk_{z}^{2}+Kk_{\perp}^{4}}\approx\left\\{\begin{array}[]{ll}\frac{T}{2\sqrt{BK}}C_{d-1}L_{\perp}^{3-d},&d<3,\\\
\frac{T}{4\pi\sqrt{BK}}\ln q_{0}L_{\perp},&d=3,\\\ \end{array}\right.$ (43)

where I defined a constant
$C_{d}=S_{d}/(2\pi)^{d}=2\pi^{d/2}/[(2\pi)^{d}\Gamma(d/2)]$, with $S_{d}$ a
surface area of a $d$-dimensional sphere, and introduced an infrared cutoff by
considering a system of finite extent $L_{\perp}\times L_{z}$, with $L_{z}$
the length of the system along the ordering ($z$) axis and $L_{\perp}$
transverse to $z$. Unless it has a huge aspect ratio, such that $L_{z}\sim
L_{\perp}^{2}/\lambda>>L_{\perp}$, any large system
($L_{\perp},L_{z}>>\lambda$) will have $\lambda L_{z}\ll L_{\perp}^{2}$.

The key observation here is that the smectic phonons exhibit fluctuations that
diverge, growing logarithmically in 3d and linearly in 2d with system size
$L_{\perp}$; for $d>3$ fluctuations are bounded. Thus, as a consequence of
their “soft” elasticity (a vanishing $({\bm{\nabla}}_{\perp}u)^{2}$ modulus)
3d smectics are akin to 2d XY systems, such as superfluid films and two-
dimensional crystals, Landau ; Peierls ; MerminWagner ; Hohenberg ; Coleman ,
exhibit a power-law order in three dimensions.

The expression for the mean-squared phonon fluctuations in (43) leads the
emergence of important crossover length scales $\xi_{\perp},\xi_{z}$, related
by

$\displaystyle\xi_{\perp}$ $\displaystyle=$
$\displaystyle(\xi_{z}\sqrt{K/B})^{1/2}\equiv\sqrt{\xi_{z}\lambda},$ (44)

that characterize the finite-temperature smectic state. These are defined as
scales $L_{\perp},L_{z}$ at which phonon fluctuations are large, comparable to
the smectic period $a=2\pi/q_{0}$. Namely, setting

$\langle u^{2}\rangle_{0}^{T}\approx a^{2}$ (45)

in Eq. (43) one finds

$\displaystyle\xi_{\perp}$ $\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{a^{2}\sqrt{BK}}{T}\sim\frac{K}{Tq_{0}},&d=2,\\\
ae^{4\pi a^{2}\sqrt{BK}/T}\sim ae^{\frac{cK}{Tq_{0}}},&d=3,\\\
\end{array}\right.$ (48)

where in the second form of the above expressions I took the simplest
approximation for the smectic anisotropy length $\lambda=\sqrt{K/B}$ to be
$\lambda=a\sim 1/q_{0}$, and introduced $O(1)$ Lindemann constant
$c$,Lindemann  that depends on the somewhat arbitrary definition of “large”
phonon root-mean-squared fluctuations.

The smectic connected correlation function

$C_{u}({\bf x}_{\perp},z)=\langle\left[u({{\bf x}_{\perp}},z)-u({\bf
0},0)\right]^{2}\rangle_{0}\;.$ (49)

is also straightforwardly worked out, in 3d giving the logarithmic Caillé
formCaille

$\displaystyle C^{3d}_{u}({\bf x}_{\perp},z)$ $\displaystyle=$ $\displaystyle
2T\int{d^{2}{k_{\perp}}dk_{z}\over(2\pi)^{3}}{1-e^{i{\bf k}\cdot{{\bf
x}}}\over Kk_{\perp}^{4}+Bk_{z}^{2}}\equiv{T\over
2\pi\sqrt{KB}}\;g^{3d}_{T}\left({z\lambda\over x_{\perp}^{2}},{x_{\perp}\over
a}\right)\;$ (50a) $\displaystyle=$ $\displaystyle{T\over
2\pi\sqrt{KB}}\left[\ln\left({x_{\perp}\over
a}\right)-\frac{1}{2}{\text{Ei}}\left({-x_{\perp}^{2}\over
4\lambda|z|}\right)\right],\ \ \ \ \ \ \ \ \ $ (50b) $\displaystyle\approx$
$\displaystyle{T\over
2\pi\sqrt{KB}}\left\\{\begin{array}[]{lr}\ln\left({x_{\perp}\over
a}\right),&x_{\perp}\gg\sqrt{\lambda|z|}\;,\\\ \ln\left({4\lambda z\over
a^{2}}\right),&x_{\perp}\ll\sqrt{\lambda|z|}\;,\\\ \end{array}\right.$ (50e)

where $\text{Ei}(x)$ is the exponential-integral function. As indicated in the
last form, in the asymptotic limits of $x_{\perp}\gg\sqrt{\lambda z}$ and
$x_{\perp}\ll\sqrt{\lambda z}$ above 3d correlation function reduces to a
logarithmic growth with $x_{\perp}$ and $z$, respectively.

In 2d one instead findsTonerNelsonSm

$\displaystyle C^{2d}_{u}(x,z)$ $\displaystyle=$ $\displaystyle
2T\int{d{k_{x}}dk_{z}\over(2\pi)^{2}}{1-e^{i{{\bf k}}\cdot{{\bf x}}}\over
Kk_{x}^{4}+Bk_{z}^{2}}\equiv{T\over
2\pi\sqrt{KB}}\;g^{2d}_{T}\left({z\lambda\over x^{2}},{x\over a}\right)\;,$
(51a) $\displaystyle=$ $\displaystyle{2T\over
B}\bigg{[}\left(\frac{|z|}{4\pi\lambda}\right)^{1/2}e^{-x^{2}/(4\lambda|z|)}+\frac{|x|}{4\lambda}\mbox{erf}\big{(}\frac{|x|}{\sqrt{4\lambda|z|}}\big{)}\bigg{]}\
\ \ \ \ \ \ $ (51b) $\displaystyle\approx$ $\displaystyle{2T\over
B}\left\\{\begin{array}[]{lr}\left(\frac{|z|}{4\pi\lambda}\right)^{1/2},&x\ll\sqrt{\lambda|z|}\;,\\\
\frac{|x|}{4\lambda},&x\gg\sqrt{\lambda|z|}\;,\\\ \end{array}\right.$ (51e)

where $\text{erf}(x)$ is the Error function.

As a consequence of above divergent phonon fluctuations, the smectic density
wave order parameter (11) vanishes in thermodynamic limit

$\displaystyle\langle\rho({\bf x})\rangle_{0}$ $\displaystyle=$ $\displaystyle
2|\rho_{q}|\langle\cos\big{[}{\bf q}_{0}\cdot{\bf x}-qu({\bf
x})\big{]}\rangle_{0},$ (52a) $\displaystyle=$ $\displaystyle
2|\rho_{q}|e^{-{\frac{1}{2}}q_{0}^{2}\langle u^{2}\rangle_{0}}\cos\big{(}{\bf
q}_{0}\cdot{\bf x}),$ (52b) $\displaystyle=$ $\displaystyle
2\tilde{\rho}_{q}(L_{\perp})\cos\big{(}{\bf q}_{0}\cdot{\bf x}),$ (52c)

with the thermally suppressed order parameter amplitude given by

$\displaystyle\tilde{\rho}_{q}(L_{\perp})$ $\displaystyle=$
$\displaystyle|\rho_{q}|\left\\{\begin{array}[]{ll}e^{-L_{\perp}/\xi_{\perp}},&d=2,\\\
\left(\frac{a}{L_{\perp}}\right)^{\eta/2},&d=3,\\\ \end{array}\right.$ (53c)
$\displaystyle\rightarrow$ $\displaystyle 0,\ \ \mbox{for
$L_{\perp}\rightarrow\infty$},$ (53d)

where I used results for the phonon and phase fluctuations, (43), and defined
the Caillé exponent

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{q_{0}^{2}T}{8\pi\sqrt{BK}}.$ (54)

Thus, in qualitative contrast to its mean-field cartoon, at long scales
(longer than $\xi_{\perp,z}$) the smectic state is actually characterized by a
uniform mass density.

Since the average density is actually uniform, a better characterization of
the smectic state is through the structure function, $S({\bf q})$, a Fourier
transform of the density correlation function, that in 3d is given by

$\displaystyle S({\bf q})$ $\displaystyle=$ $\displaystyle\int
d^{3}x\langle\delta\rho({\bf x})\delta\rho(0)\rangle e^{-i{\bf q}\cdot{\bf
x}},$ (55a) $\displaystyle\approx$
$\displaystyle{\frac{1}{2}}\sum_{q_{n}}|\rho_{q_{n}}|^{2}\int_{{\bf x}}\langle
e^{-iq_{n}(u({\bf x})-u(0))}\rangle_{0}e^{-i({\bf q}-q_{n}{\hat{z}})\cdot{\bf
x}},\ \ \ \ \ \ \ \ \ $ (55b) $\displaystyle\approx$
$\displaystyle{\frac{1}{2}}\sum_{n}\frac{|\rho_{q_{n}}|^{2}}{|q_{z}-nq_{0}|^{2-n^{2}\eta}},\
\ \mbox{for $d=3$},$ (55c)

where I approximated phase and phonon fluctuations by Gaussian statistics (in
3d valid up to weak logarithmic correctionsGP ). Thus as anticipated one finds
that the logarithmically divergent 3d phonon fluctuations lead to a structure
function, with highly anisotropic ($q_{z}\sim q_{\perp}^{2}/\lambda$) quasi-
Bragg peaks replacing the true ($\delta$-function) Bragg peaks characteristic
of a true long-range periodic order.deGennes ; ChaikinLubensky ; SqSmExp

In two dimensions, smectic order is even more strongly suppressed by thermal
fluctuations. The linear growth of the 2d phonon fluctuations leads to
exponentially short-ranged correlations of the density, expected to result in
dislocation unbinding at any nonzero temperature, thereby completely
destroying smectic state in 2d.TonerNelsonSm

###  II.4 Nonlinear elasticity: beyond Gaussian fluctuations

####  II.4.1 Perturbation theory

As is clear from the analysis of the previous subsection, the restoration of
the translational symmetry (a vanishing $\langle\rho_{q}({\bf x})\rangle$,
etc.) by thermal fluctuations is a robust prediction of the quadratic theory,
that cannot be overturned by the left-out nonlinearities. However, as
discussed in the Introduction and illustrated instability of the smectic
Gaussian fixed point (in $d<3$) in Fig.3, the asymptotic long-scale form of
the correlation functions computed within the harmonic approximation only
extends out to the nonlinear length scales $\xi_{\perp,z}^{NL}$. On longer
scales, the divergently large smectic phonon fluctuations invalidate the
neglect of phonon nonlinearities,

$\displaystyle{\mathcal{H}}_{\text{nonlinear}}$ $\displaystyle=$
$\displaystyle-{\frac{1}{2}}B(\partial_{z}u)(\nabla u)^{2}+\frac{1}{8}B(\nabla
u)^{4}.\ \ \ \ $ (56)

These will necessarily qualitatively modify predictions (50e), (51e), and (55)
on scales longer than the crossover scales $\xi^{NL}_{\perp,z}$, that I
compute next.

Figure 6: Feynman graph that renormalizes the elastic moduli $K$, $B$ of the
smectic state.

To see this, one uses a perturbative expansion in the nonlinear operators (56)
to assess the size of their contribution to e.g., the free energy. Following
the standard field-theoretic RG analysis, pioneered by Wilson and
FisherepsilonWilsonFisher  (but here applied to the critical smectic phase,
rather than to a critical point) these can be accounted for as corrections to
the compressional $B$ and bend $K$ elastic moduli, with the leading softening
(negative) contribution to $\delta B$, summarized graphically in Fig.6, and
given by

$\displaystyle\delta B$ $\displaystyle=$
$\displaystyle-{\frac{1}{2}}TB^{2}\int_{\bf q}q_{\perp}^{4}G_{u}({\bf
q})^{2}\;,$ $\displaystyle\approx$
$\displaystyle-{\frac{1}{2}}TB^{2}\int_{-\infty}^{\infty}{dq_{z}\over
2\pi}\int_{L^{-1}_{\perp}}{d^{d-1}q_{\perp}\over(2\pi)^{d-1}}{q_{\perp}^{4}\over(Kq_{\perp}^{4}+Bq_{z}^{2})^{2}}\;,$
$\displaystyle\approx$ $\displaystyle-{1\over 8}{C_{d-1}T\over
3-d}\;\left({B\over K^{3}}\right)^{1/2}L_{\perp}^{3-d}B\;.$ (57b)

In above, I used the smectic correlator, $G_{u}({\bf q})$, focused on $d\leq
3$ (which allowed the dropping of the uv-cutoff ($\Lambda$) dependent part
that vanishes for $\Lambda\rightarrow\infty$), and cutoff the divergent
contribution of the long wavelength modes via the infra-red cutoff
$q_{\perp}>1/L_{\perp}$ by considering a system of a finite extent
$L_{\perp}$. Similar analysis gives a positive fluctuation correction to the
bend modulus $K$, stiffening the undulation mode.ffloLR

Clearly the anharmonicity becomes important when the fluctuation corrections
to the elastic constants (e.g., $\delta B$ above) become comparable to its
bare microscopic value. As the analysis below demonstrates, the divergence of
this correction as $L_{\perp}\rightarrow\infty$ signals the breakdown of the
conventional harmonic elastic theory on length scales longer than a crossover
scale

$\displaystyle\xi^{NL}_{\perp}$ $\displaystyle\approx$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{1}{T}\left(\frac{K^{3}}{B}\right)^{1/2},&d=2,\\\
ae^{\frac{c}{T}\left(\frac{K^{3}}{B}\right)^{1/2}},&d=3,\end{array}\right.$
(60)

which I define here as the value of $L_{\perp}$ at which $|\delta
B(\xi^{NL}_{\perp})|=B$. Within the approximation of the smectic screening
length $\lambda=a$, these nonlinear crossover lengths reduce to the phonon
disordering lengths (44), (48), defined by the Lindemann-like
criterion.Lindemann  Clearly, on scales longer than $\xi^{NL}_{\perp,z}$ (also
a crossover scale from Gaussian to smectic fixed point in Fig.1) the
perturbative contributions of nonlinearities dominate and therefore cannot be
neglected. Their contribution are thus expected to qualitatively modify the
harmonic predictions of the previous subsection.

####  II.4.2 Renormalization group analysis in $d=3-\epsilon$ dimensions

To describe the physics beyond the crossover scales, $\xi^{NL}_{\perp,z}$ –
i.e., to make sense of the infra-red divergent perturbation theory found in
Eq.57b – requires a renormalization group analysis.epsilonWilsonFisher ;
MEFrmp74 ; WilsonKogutPR  This was first performed in the context of
conventional 3d smectic liquid crystals and Lifshitz points in a seminal work
by Grinstein and Pelcovits (GP)GP . Below, I complement GP’s treatment with
Wilson’s momentum-shell renormalization group (RG) analysis, extending it to
an arbitrary dimension $d$, so as to connect to the behavior in 2d, that has
an exact solutionGW .

To this end I integrate (perturbatively in ${\mathcal{H}}_{\text{nonlinear}}$)
short-scale Goldstone modes in an infinitesimal cylindrical shell of
wavevectors, $\Lambda e^{-\delta\ell}<q_{\perp}<\Lambda$ and
$-\infty<q_{z}<\infty$ ($\delta\ell\ll 1$ is infinitesimal). The leading
perturbative momentum-shell coarse-graining contributions come from terms
found in direct perturbation theory above, but with the system size
divergences controlled by the infinitesimal momentum shell. The thermodynamic
averages can then be equivalently carried out with an effective coarse-grained
Hamiltonian of the form (8), with momentum-dependent couplings obtained from
integrating up their infinitesimal corrections, obtained via momentum-shell
RG. For smectic moduli $B$ and $K$ this gives,

$\displaystyle\delta B$ $\displaystyle\approx$
$\displaystyle-\frac{1}{8}gB\delta\ell,\ \ \ \delta
K\approx\frac{1}{16}gK\delta\ell,$ (61)

where the dimensionless coupling is given by

$\displaystyle g$ $\displaystyle=$ $\displaystyle
C_{d-1}\Lambda_{\perp}^{d-3}T\left({B\over
K^{3}}\right)^{1/2}\approx\frac{T}{2\pi}\left({B\over K^{3}}\right)^{1/2}\;,$
(62)

and in the second form I approximated $g$ by its value in 3d. Eqs.(61) show
that $B$ is softened and $K$ is stiffened by the nonlinearities in the
presence of thermal fluctuations, making the system effectively more
isotropic, as one may expect on general physical grounds.

For convenience I then anisotropically rescale the lengths and the remaining
long wavelength part of the fields $u^{<}({\bf r})$ according to
$r_{\perp}=r_{\perp}^{\prime}e^{\delta\ell}$,
$z=z^{\prime}e^{\omega\delta\ell}$ and $u^{<}({\bf
r})=e^{\phi\delta\ell}u^{\prime}({\bf r^{\prime}})$, so as to restore the
ultraviolet cutoff $\Lambda_{\perp}e^{-\delta\ell}$ back up to
$\Lambda_{\perp}$. The underlying rotational invariance ensures that the
nonlinear fluctuation corrections preserve the rotationally invariant strain
operator $\big{(}\partial_{z}u-{\frac{1}{2}}({\nabla}_{\perp}u)^{2}\big{)}$,
renormalizing it as a whole. It is therefore convenient (but not necessary) to
choose the dimensional rescaling that also preserves this form. It is easy to
see that this choice leads to

$\phi=2-\omega\;.$ (63)

The leading (one-loop) corrections to the effective coarse-grained and
rescaled free energy functional can then be summarized by differential RG
flows

$\displaystyle\frac{dB(\ell)}{d\ell}$ $\displaystyle=$
$\displaystyle(d+3-3\omega-{1\over 8}g(\ell))B(\ell)\;,$ (64a)
$\displaystyle\frac{dK(\ell)}{d\ell}$ $\displaystyle=$
$\displaystyle(d-1-\omega+{1\over 16}g(\ell))K(\ell)\;.$ (64b)

From these one readily obtains the flow of the dimensionless coupling
$g(\ell)$

$\displaystyle\frac{dg(\ell)}{d\ell}$ $\displaystyle=$
$\displaystyle(3-d)g-\frac{5}{32}g^{2}\;,$ (65)

whose flow for $d<3$ away from the $g=0$ Gaussian fixed point encodes the
long-scale divergences found in the direct perturbation theory above. This
resembles the Wilson-Fisher flow at the critical point just below
$d=4$epsilonWilsonFisher , but, as discussed in the Introduction requires no
fine tuning. As summarized in Fig.3 for $d<3$ the flow terminates at a nonzero
fixed-point coupling $g_{*}=\frac{32}{5}\epsilon$ (with $\epsilon\equiv 3-d$),
that determines the nontrivial long-scale behavior of the smectic critical
phase (see below). As with treatments of critical pointsepsilonWilsonFisher ;
ChaikinLubensky , but here extending over the whole smectic phase, the RG
procedure is quantitatively justified by the proximity to $d=3$, i.e.,
controlled by the smallness of $\epsilon$.

One can now use a standard matching calculation to determine the long-scale
asymptotic form of the correlation functions on scales beyond
$\xi^{NL}_{\perp,z}$. Namely, applying above coarse-graining RG analysis to a
computation of correlation functions allows one to relate its form at long
length scales of interest (that, because of infrared divergences is impossible
to compute via a direct perturbation theory in nonlinearities) to its
counterpart at short scales, evaluated with coarse-grained couplings,
$B(\ell)$, $K(\ell)$,…. In contrast to the former, the latter is readily
computed via a perturbation theory, that, because of shortness of the length
scale is convergent. The result of this matching calculation to lowest order
gives correlation functions from an effective Gaussian theory,

$\displaystyle G_{u}({\bf k})$ $\displaystyle\approx$
$\displaystyle\frac{T}{B({\bf k})k_{z}^{2}+K({\bf k})k_{\perp}^{4}},$ (66)

with moduli $B({\bf k})$ and $K({\bf k})$ that are singularly wavevector-
dependent, determined by the solutions $B(\ell)$ and $K(\ell)$ of the RG flow
equations (64a) and (64b), with initial conditions given by the microscopic
values $B$ and $K$.

2d analysis: In $d=2$, at long scales $g(\ell)$ flows to a nontrivial infrared
stable fixed point $g_{*}=32/5$, and the matching analysis predicts
correlation functions characterized by anisotropic wavevector-dependent moduli

$\displaystyle K({\bf k})$ $\displaystyle=$ $\displaystyle
K\left(k_{\perp}\xi^{NL}_{\perp}\right)^{-\eta_{K}}f_{K}(k_{z}\xi^{NL}_{z}/(k_{\perp}\xi^{NL}_{\perp})^{\zeta})\;,\
\ \ \ \ \ \ \ \ $ $\displaystyle\sim$ $\displaystyle k_{\perp}^{-\eta_{K}},$
$\displaystyle B({\bf k})$ $\displaystyle=$ $\displaystyle
B\left(k_{\perp}\xi^{NL}_{\perp}\right)^{\eta_{B}}f_{B}(k_{z}\xi^{NL}_{z}/(k_{\perp}\xi^{NL}_{\perp})^{\zeta})\;,$
$\displaystyle\sim$ $\displaystyle k_{\perp}^{\eta_{B}}.$

Thus, on scales longer than $\xi^{NL}_{\perp,z}$ these qualitatively modify
the real-space correlation function asymptotics of the harmonic analysis in
the previous subsection. In Eqs.(67) the universal anomalous exponents are
given by

$\displaystyle\eta_{B}$ $\displaystyle=$ $\displaystyle{1\over 8}g_{*}={4\over
5}\;\epsilon\;,$ (68a) $\displaystyle\approx$ $\displaystyle{4\over
5}\;,\;\;\;\mbox{for}\;d=2\;,$ $\displaystyle\eta_{K}$ $\displaystyle=$
$\displaystyle{1\over 16}g_{*}={2\over 5}\;\epsilon\;,$ (68b)
$\displaystyle\approx$ $\displaystyle{2\over 5}\;,\;\;\;\mbox{for}\;d=2\;,$

determining the $z-{\bf x}_{\perp}$ anisotropy exponent via (66) to be

$\displaystyle\zeta$ $\displaystyle\equiv$ $\displaystyle
2-(\eta_{B}+\eta_{K})/2\;,$ (69a) $\displaystyle=$
$\displaystyle\frac{7}{5},\;\mbox{for}\;d=2\;,$ (69b)

as expected reduced by thermal fluctuations down from its harmonic value of
$2$. The ${\bf k}_{\perp}-k_{z}$ dependence of $B({\bf k}),K({\bf k})$ is
determined by universal scaling functions, $f_{B}(x),f_{K}(x)$ that I will not
compute here. The underlying rotational invariance gives an exact relation
between the two anomalous $\eta_{B,K}$ exponents

$\displaystyle 3-d$ $\displaystyle=$ $\displaystyle{\eta_{B}\over 2}+{3\over
2}\eta_{K}\;,$ (70a) $\displaystyle 1$ $\displaystyle=$
$\displaystyle{\eta_{B}\over 2}+{3\over 2}\eta_{K}\;,\ \ \mbox{for $d=2$},$
(70b)

which is obviously satisfied by the anomalous exponents, Eqs.(68b),(68a),
computed here to first order in $\epsilon=3-d$.

Thus, as advertised, I find that at nonzero temperature, a 2d smectic state is
highly nontrivial and qualitatively distinct from its mean-field perfectly
periodic form. In addition to a vanishing density modulation and associated
fluctuation-restored translational symmetry, it is characterized by universal
nonlocal length-scale dependent moduli, Eq. (67). Consequently its Goldstone
mode theory and the associated correlations are not describable by a local
harmonic field theory, that is an analytic expansion in local field operators.
Instead, in 2d, on length scales beyond $\xi^{NL}_{\perp,z}$ (but shorter than
the expected dislocation unbinding length $\xi_{disl}$) thermal fluctuations
and correlations of this smectic critical phase are controlled by a nontrivial
fixed point, illustrated in Fig. 3, characterized by universal anomalous
exponents $\eta_{K,B}$ and scaling functions $f_{B,K}(x)$, defined above.

Above I obtained this nontrivial structure from an RG analysis and estimated
these exponents within a controlled but approximate $\epsilon$-expansion.
Remarkably, in 2d an exact solution of this problem was discovered by
Golubovic and WangGW . It predicts critical phenomenology in a qualitatively
agreement with the RG predictions above, and gives exact exponents

$\displaystyle\eta_{B}^{2d}$ $\displaystyle=$ $\displaystyle 1/2,\ \
\eta_{K}^{2d}=1/2,\ \ \zeta^{2d}=3/2,$ (71)

derived by maping onto the 1+1d KPZ equation.KPZ

3d analysis: In $d=3$, the nonlinear coupling $g(\ell)$ is marginally
irrelevant, flowing to $0$ at long scales. Despite this, as discovered by
Larkin and Khmel’nitskii and by Michael Fisher, et al., in the context of the
Ising modelLarkinKhm69 ; MEFmarginal72 ; MEFrmp74 , the marginal flow to the
Gaussian fixed point is sufficiently slow (logarithmic in lengths) that (as
usual at a marginal dimensionChaikinLubensky ) its power-law in $\ell$
dependence leads to a universal, asymptotically exact logarithmic wavevector
dependence found in a smectic by Grinstein and PelcovitsGP ,

$\displaystyle K({{\bf k}_{\perp},k_{z}=0})$ $\displaystyle\sim$
$\displaystyle K|1+\frac{5g}{64\pi}\ln(1/k_{\perp}a)|^{2/5}\;,$ (72a)
$\displaystyle B({{\bf k}_{\perp}=0,k_{z}})$ $\displaystyle\sim$
$\displaystyle B|1+\frac{5g}{128\pi}\ln(\lambda/k_{z}a^{2})|^{-4/5}.\ \ \ \ \
\ \ \ \ \ $ (72b)

This translates into the smectic order parameter correlations given by

$\displaystyle\langle\rho^{*}_{q}({\bf x})\rho_{q}(0)\rangle$
$\displaystyle\sim$ $\displaystyle e^{-c_{1}(\ln z)^{6/5}}\cos(q_{0}z),$ (73)

with $2/5,4/5$, and $6/5$ exact universal exponents and $c_{1}$ a nonuniversal
constantGP . Although these 3d anomalous effects are less dramatic and likely
to be difficult to observe in practice, theoretically they are quite
significant as they represent a qualitative breakdown of the mean-field and
harmonic descriptions, that respectively ignore interactions and thermal
fluctuations.

I conclude this section by noting that all of the above analysis is predicated
the validity of the purely elastic model, Eq. (8), that neglects topological
defects, such as dislocations. If these unbind (as they undoubtedly do in 2d
at any nonzero temperatureTonerNelsonSm ), then above prediction only hold on
scales shorter than the separation $\xi_{disl}$ between these defects.

##  III Biaxially periodic critical states: spontaneous line-crystals

Another class of critical states are those partially crystalized along two of
the three axes. They can be dubbed as “spontaneous line-crystals” as they are
a periodic 2d crystalline array of 1d liquids, oriented along a spontaneously
chosen axis, illustrated in Fig.2(c), and in this sense dual to a uniaxially
periodic critical state – a 1d periodic array of 2d liquids – the
$m=d-1$-Lifshitz modelmLifshitz , discussed in Sec.II. As I discuss below, its
key feature is rotational symmetry-enforced vanishing of the tilt modulus –
the $m=1$-Lifshitz modelmLifshitz , that leads to its enhanced critical
fluctuations.

###  III.1 Columnar liquid crystal

One example of a spontaneous line-crystal is a ubiquitous “columnar” liquid
crystal. It emerges from a nematic fluid composed of a high aspect ratio disk-
shaped constituents, illustrated in Fig. 2(c).deGennes  In this phase disk-
shaped molecules stack into one-dimensional fluid columns along a spontaneous
nematic axis, that freeze into a 2d crystal (typically triangular lattice),
retaining fluid order along the columns. Such discotic liquid crystal, forming
a 2d crystal of 1d fluid columns compliments the smectic state of 1d periodic
array of 2d fluids, discussed in earlier sections.

Simple analysis similar to that of a smectic, shows that for $d<5/2$ the
columnar liquid crystal ($m=1$-Lifshitz modelmLifshitz ) is a critical phase,
though given this dimensional constraint not a very practical one for
experimental realization. However, although I do not pursue the subject here,
I note that far stronger distortion effects of quenched disorder (e.g.,
columnar state in aerogelRTaerogelPRL ; RTaerogelPRB ; BRTCaerogelScience )
lead to a critical columnar glass for $d<7/2$, characterized by an infrared
attractive fixed point controlled by disorder.SRTcolumnar

The columnar state is characterized by a two-component phonon field ${\bf
u}=(u_{x},u_{y})$, that are two Goldstone modes associated with translational
symmetry breaking in the plane transverse to the columns. Analogously to a
smectic, the Goldstone modes associated with the rotational symmetry are
gapped out by the emergent Higgs mechanism.cholestericLR

Although one can derive the Goldstone mode elasticity for the columnar state
by following the approach analogous to that of a smectic, here I will simply
write it down based on symmetry and experience with the smectic. To this end I
note that the columnar state exhibits two types of rigidities, the in-plane
(transverse to the columnar axis ${\hat{z}}$) crystalline elasticity and
bending elasticity of the columns. The former is characterized by shear and
bulk elasticity of a 2d crystal. The latter is captured by the higher-
derivative curvature (rather than tension) filament elasticity. Together these
give the energy density

$\displaystyle{\mathcal{H}}_{col}=\frac{1}{2}\kappa(\partial_{z}^{2}{\bf
u})^{2}+\frac{\lambda}{2}\,u_{\alpha\alpha}^{2}+\mu\,u_{\alpha\beta}^{2},$
(74)

where $\kappa$ is the curvature modulus, $\mu$ and $\lambda$ are Lamé elastic
moduli of 2d triangular latticeChaikinLubensky , and (with the summation
convention over repeated indices) the strain tensor is given by

$\displaystyle
u_{\alpha\beta}=\frac{1}{2}\,\Big{(}\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}-\partial_{\gamma}u_{\alpha}\partial_{\gamma}u_{\beta}\Big{)}\approx\frac{1}{2}\,\Big{(}\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}-\partial_{z}u_{\alpha}\partial_{z}u_{\beta}\Big{)}.$
(75)

The two phonons, ${\bf u}(x,y,z)=(u_{x},u_{y})$ are fields in a three-
dimensional space, and in the second approximate form above I only kept the
most important strain nonlinearities.

As in the discussion of the smectic elasticity, the columnar nonlinear
elasticity (74) is strictly constrained by symmetry. The linear $z$ derivative
in the harmonic terms is forbidden by the rotational invariance about the $x$
axes (broken spontaneously), at infinitesimal level corresponding to ${\bf
u}\rightarrow{\bf u}+\theta z{\bf{\hat{y}}}$. This “softness” with respect to
$\partial_{z}{\bf u}$ modes then requires one to keep corresponding
nonlinearities in the in-plane strain tensor, $u_{\alpha\beta}$ above, the
form of which is dictated by the in-plane rotational invariance.

As in a smectic, here too the analysis of quadratic phonon fluctuations leads
to power-law divergence for $d<5/2$, requiring inclusion of elastic
nonlinearities in (74) via (75). Account of these via an RG analysis very
similar to that of the smectic in the previous section, leads to a nontrivial
infrared stable fixed point controlled by $\epsilon=5/2-d$, that characterizes
the universal properties of the resulting columnar critical phase (as in
Fig.3). For technical details, I refer the reader to the original
literature.SRTcolumnar

###  III.2 Spontaneous vortex lattice in magnetic superconductors

Another putative realization of a biaxially periodic (line-crystal) critical
state is a “spontaneous vortex lattice”’ in a ferromagnetic supercondutor
(FS). Rare-earth borocarbide materials are believed to be examples of the
latter, exhibiting a rich phase diagram that includes superconductivity,
antiferromagnetism, ferromagnetism and spiral magnetic order.PhysicsToday ;
experiments ; Varma  In particular, there is now ample experimental evidence
that, at low temperatures, superconductivity and ferromagnetism competitively
coexist in ErNi2B2C compounds, and perhaps in more recently discovered high
temperature superconductor Sr2Y Ru1-xCuxO6.

For sufficiently strong ferromagnetism, such FS’s have been predictedVarma  to
exhibit a spontaneous vortex (SV) state driven by the spontaneous
magnetization, rather than by an external magnetic field ${\bf H}$. It is
clear based purely on symmetry arguments that for ${\bf H}=0$ and in the
absence of atomic crystal anisotropy, the elastic properties of the resulting
SV solid differ dramatically and qualitatively from those of a conventional
Abrikosov lattice. The key underlying difference is the aforementioned
vanishing of the tilt modulus (74), which is guaranteed by the underlying
rotational invariance. Although this invariance is broken by the
magnetization, ${\bf M}$, the tilt modulus remains zero because this breaking
is spontaneous (neglecting the ever-present crystal field anisotropy). This
contrasts strongly with a conventional vortex solid, where the rotational
symmetry is explicitly broken by the applied field ${\bf H}$ and lattice
anisotropy.

Consequently, the elasticity of the SV crystal is identical to that of the
columnar phase, described by(74) together with (75), neglecting crystalline
anisotropy. In the presence of purely thermal fluctuations, elastic
nonlinearities remain irrelevant for such 3d SV state, since $d=3>5/2$.
However, as discovered and explored in Refs. magneticSCprl, ; magneticSCprb, ,
in the presence of ever-present quenched disorder, SV is a disorder- (rather
than temperature-) controlled critical glass state, treatable via RG
perturbative in $\epsilon=7/2-d$, with many interesting properties, e.g., the
unusually singular $B(H)$ relation.

##  IV Polymerized membrane

A fluctuating, tensionless and therefore curvature-(as opposed to tension-)
controlled membrane is a strongly fluctuating system. Focus on such membranes
was originally stimulated in part by ubiquitous biophysical realizations
(e.g., liposomes, cellular membranes, cytoskeleton of a red blood cell,
etc)JWSmembranes , but more recently found an ideal realization in fluctuating
graphene sheets.graphene  The latter, polymerized (solid) membrane,
characterized by a nonzero in-plane shear rigidity, is particularly
fascinating case of a critical phase for its internal dimension $D<4$.

Figure 7: Illustration of a phase diagram of a polymerized membrane,
controlled by temperature and in-plane anisotropy encoded in reduced
temperatures $\tau_{x}$ and $\tau_{y}$. An isotropic membrane is described by
$\tau_{x}=\tau_{y}\equiv\tau$, exhibiting temperature $\tau-$ driven
transition along path $P_{2}$ from a crumpled phase (with vanishing tangent
$\vec{t}_{x,y}=0$ order parameters) to a “flat” phase a phase with
$\vec{t}_{x,y}\neq 0$. Flat phase spontaneously breaks the rotational $O(3)$
symmetry of the embedding space and is a critical phase that exhibits rich
universal power-law phenomenology, controlled by a nontrivial infrared stable
fixed point. A polymerized membrane with in-plane anisotropy is predicted to
undergo a two-stage transition along $P_{1}$ from the crumpled-to-tubule phase
(latter characterized by only one of the two tangent order parameters
nonzero), followed by tubule-to-flat phase transitionRTtubulePRL ; RTtubulePRE
, later observed in simulations.tubuleSimulationsBowick

As was first argued by Nelson and PelitiNelsonPeliti , in addition to the
rotationally invariant “crumpled” phase of linear polymers and liquid
membranes, polymerized membranes exhibit a finite temperature “flat” phase. As
I will discuss below, this too is a critical phase characterized by universal
power-law correlations (e.g., membrane roughness) and anomalous elasticity,
with the state’s very existence in two-dimensional membranes being a beautiful
illustration of a phenomena of order-from-disorder.

Figure 8: Illustration of (a) a graphene sheet as a modern ideal realization
of a thermally fluctuating polymerized membrane, predicted to exhibit rich
anomalous elasticity as a critical phase, (b) flat phase of a polymerized
membrane defining its normals, Monge gauge embedding and the definition of the
roughness exponent, $\zeta$.

I note in passing that, as illustrated in Fig.7, a non-self-avoiding –
“phantom” anisotropic polymerized membrane was predictedRTtubulePRL ;
RTtubulePRE  and observed in simulationstubuleSimulationsBowick  to exhibit an
intermediate tubule phase, crumpled along one dimension and extended along the
other, that, despite its one-dimensionality, is stable to thermal
fluctuations.

A detailed derivation of the flat-phase Goldstone-modes elasticity is
availableNelsonPeliti ; JWSmembranes  starting from the Landau theory of the
crumpled phase in terms of the tangent vectors ${\bm{\nabla}}{\vec{r}}({\bf
x})$, where ${\vec{r}}({\bf x})$ is the embedding of the $D$-dimensional
membrane (parameterized by ${\bf x}$) in the $d$-dimensional space,
${\vec{r}}$ (with $D=2$ and $d=3$ the physical case). The flat phase
parameterized by

${\vec{r}}({\bf x})=(t_{0}{\bf x}+{\bf u}({\bf x}),{\vec{h}}({\bf x})),$ (76)

in the Monge gauge, spontaneously breaks the $O(d)$ symmetry of the crumpled
phase down to $O(D)$, where $t_{0}$ scale factor is the effective order
parameter of the flat phase, ${\bf u}({\bf x})$ is the $D$-component in-plane
phonon, and ${\vec{h}}({\bf x})$ is the “height function” describing the
membrane’s transverse undulation into the embedding space; we have implicitly
generalized to arbitrary co-dimension $d_{c}=d-D$.

The resulting energy functional is a sum of the bending and the in-plane
elastic contributions:

$H_{flat}[\vec{h},{\bf u}]=\int d^{D}x\left[{\kappa\over
2}(\nabla^{2}\vec{h})^{2}+\mu u_{\alpha\beta}^{2}+{\lambda\over
2}u_{\alpha\alpha}^{2}\right]\;,$ (77)

where the nonlinear strain tensor is

$\displaystyle u_{\alpha\beta}$ $\displaystyle=$
$\displaystyle{\frac{1}{2}}(\partial_{\alpha}{\vec{r}}\cdot\partial_{\beta}{\vec{r}}-\delta_{\alpha\beta})\approx{1\over
2}({\partial_{\alpha}u_{\beta}}+{\partial_{\beta}u_{\alpha}}+{\partial_{\alpha}{\vec{h}}}\cdot{\partial_{\beta}{\vec{h}}}),$
(78)

defined as the deviation of the embedding-induced metric $g_{\alpha\beta}$
from the flat metric, and in the second form I neglected elastic
nonlinearities that are subdominant at long scales. A crucial feature of
$H_{flat}[\vec{h},{\bf u}]$ (77) (as with other critical phases) is the
higher-order curvature elasticity of ${\vec{h}}$, with the surface tension
modulus for $({\bm{\nabla}}{\vec{h}})^{2}$ vanishing exactly, enforced by the
underlying rotational invariance (of the embedding space) and tension-free
membrane.

One can also integrate out the noncritical in-plane phonon field ${\bf u}$,
that appears only harmonically, to obtain a convenient equivalent form, purely
in terms of ${\vec{h}}$ NelsonPeliti ; AronovitzLubensky ; LRphdThesis ;
LRmembraneAOP

$H_{flat}[\vec{h}]=\int d^{D}x[{\kappa\over
2}(\nabla^{2}\vec{h})^{2}+{1\over{4d_{c}}}(\partial_{\alpha}\vec{h}\cdot\partial_{\beta}\vec{h})R_{\alpha\beta,\gamma\delta}(\partial_{\gamma}\vec{h}\cdot\partial_{\delta}\vec{h})]$
(79)

where for convenience, I rescaled Lamé coefficients so that the quartic
coupling is of order $1/d_{c}$. The four-point coupling fourth-rank tensor is
given by

$R_{\alpha\beta,\gamma\delta}={K-2\mu\over
2(D-1)}P^{T}_{\alpha\beta}P^{T}_{\gamma\delta}+{\mu\over
2}\left(P^{T}_{\alpha\gamma}P^{T}_{\beta\delta}+P^{T}_{\alpha\delta}P^{T}_{\beta\gamma}\right)\;,$
(80)

where $P^{T}_{\alpha\beta}=\delta_{\alpha\beta}-q_{\alpha}q_{\beta}/q^{2}$ is
a transverse (to ${\bf q}$) projection operator. The convenience of this
decomposition is that $K=2\mu(2\mu+D\lambda)/(2\mu+\lambda)$ and $\mu$ moduli
renormalize independently and multiplicatively.LeDoussalRadzihovskyPRL ;
LRmembraneAOP

I note that this last $h$-only elastic form indeed reflects the general
discussion of critical phases in the Introduction, namely that they are
described by an energy functional (of $\phi^{4}$ form with
$\vec{\phi}_{\alpha}\equiv\partial_{\alpha}{\vec{h}}$) enforced to be critical
(i.e., missing the $\phi^{2}$ term) by the underlying rotational symmetry, as
illustrated in Fig.3.

Applying Wilson-Fisher momentum-shell RG analysis to membrane’s elastic
nonlinearitiesNelsonPeliti ; AronovitzLubensky ; DG ; LeDoussalRadzihovskyPRL
; LRmembraneAOP  one sees that for $D<4$ (and arbitrary embedding dimension
$d$) the membrane exhibits height undulations that diverge with its extent
$L$. As for smectics and other critical phases discussed earlier, one can make
sense of the associated divergent perturbation theory in elastic
nonlinearities by performing an RG analysis controlled by
$\epsilon=4-D$AronovitzLubensky  or equivalently $1/d_{c}$ (large embedding
dimension, that is the analog of $1/N$ large $N$ expansion of the $O(N)$
model’s Wilson-Fisher critical point)DG . The RG approach for arbitrary $d$
gives flows of the dimensionless nonlinear coupling constants,
$\hat{\mu}(\ell)=\mu/\kappa^{2},\hat{\lambda}(\ell)=\lambda/\kappa^{2}$, after
integrating out modes in a momentum shell $\Lambda e^{-\delta\ell}<q<\Lambda$
(as for a smectic in previous section), and rescaling for convenience.

The dimensionless couplings flow to a tuning-free infrared attractive
Aronovitz-Lubensky (AL) fixed point,NelsonPeliti ; AronovitzLubensky ; DG ;
LeDoussalRadzihovskyPRL , with two other (in addition to the Gaussian) fixed
points that describe a marginally mechanically unstable membrane (of physical
relevance to nematic elastomers of the next section). The resulting globally
infrared stable AL fixed point controls the properties of the highly
nontrivial “flat” phase, that is critical and power-law rough. A complementary
approach of Le Doussal and RadzihovskyLeDoussalRadzihovskyPRL  is via a
solution of self-consistent integral equations (SCSA) that build on the large
embedding $1/d_{c}$ expansion, with virtue of being exact in three limits: (i)
$D\rightarrow 4$ ($\epsilon$ expansion) for arbitrary $d_{c}$, (ii)
$d_{c}\rightarrow\infty$ ($1/d_{c}$ expansion) and (iii) $d_{c}=0$ for
arbitrary $D$, thereby providing best quantitative predictions, as judged by
numerics and experiments.

The “flat” critical phase is characterized by universal anomalous elasticity
with length scale-dependent elastic moduli

$\displaystyle\kappa(q)\sim q^{-\eta},\ \ \mu(q)\sim\lambda(q)\sim
q^{\eta_{u}},$ (81)

where for physical membranes $\eta\approx 0.82$ and the underlying embedding-
space rotational invariance imposes exact relations NelsonPeliti ;
AronovitzLubensky ; DG ; LeDoussalRadzihovskyPRL ,

$\displaystyle\eta_{u}=4-D-2\eta_{\kappa},\ \ \ \zeta=(4-D-\eta)/2.$ (82)

These indicate that at long scales, $q\rightarrow 0$, thermal fluctuations
stiffen the effective bending modulus $\kappa(q)$ and soften membrane’s in-
plane moduli, $\mu(q),\lambda(q)$.

Based on these, the SCSALeDoussalRadzihovskyPRL  predicts a universal and
negative Poisson ratio,

$\mathop{{\rm Lim}}\limits_{q\rightarrow 0}\;\;\sigma\equiv{\lambda(q)\over
2\mu(q)+(D-1)\lambda(q)}=-{1\over 3}\;,\ \text{for}\ D=2,$ (83)

that measures the ratio of in-plane compression of a membrane along an axis
transverse to its strained direction. Its negative value indicates that such
membrane actually expands transversely as it is being stretched. This
approximate value of $-1/3$ compares extremely well with largest
simulationszetaBowick  and higher order computations. This amazing
fluctuation-driven anomalous elasticity phenomenology, uncovered using
systematic Wilson-Fisher RG, can be understood qualitatively by playing around
with a roughened piece of paper.

Finally I conclude by emphasizing that the universal anomalous elasticity is
at the heart of stability of the 2D “flat” phase. To see this, observe that in
contrast to the unstable Gaussian fixed point (describing fluctuations of a
microscopic membrane), the “flat” critical phase at the AL fixed point, with a
renormalized, momentum-dependent $\kappa(q)$, is characterized by root-mean-
squared height undulations

$\displaystyle h_{rms}$ $\displaystyle=$ $\displaystyle\sqrt{\langle
h^{2}({\bf
x})\rangle}=\left[\int_{L^{-1}}\frac{d^{D}q}{(2\pi)^{d}}\frac{T}{\kappa(q)q^{4}}\right]^{1/2}\sim
L^{\zeta},$ (84)

with the roughness exponent $\zeta<1$. Its best estimate for $D=2,d=3$
LeDoussalRadzihovskyPRL  is

$\zeta=0.59,$

and has also been computed within $\epsilon$ and $1/d_{c}$
expansionsAronovitzLubensky ; DG ; LeDoussalRadzihovskyPRL ; epsHigherOrder .
I note that despite unbounded power-law height undulations (84), the
rotational $O(d)$ symmetry of the embedding space remains broken since
$L^{\zeta}\ll L\rightarrow\infty$, despite two-dimensionality ($D=2$) of the
polymerized membrane field theory. It thus circumvents a naive application of
the Hohenberg-Mermin-Wagner-Coleman theoremHohenberg ; MerminWagner ; Coleman
. The rigorous form of the latter does not a priori apply to the higher-
derivative polymerized membrane field theory, and thus does not forbid above
findings, e.g., stability of the 2D flat phase. More physically, as can be
seen from (84), theorem’s common heuristic application fails precisely because
the state is controlled by a non-Gaussian fixed point with anomalous,
momentum-dependent elastic moduli, specifically with $\eta>0$. As a result
$h_{rms}(L)/L\sim L^{-\eta/2}\rightarrow 0$ indicates that the “flat” phase is
stabilized (for nonzero positive $\eta>0$ generated by thermal fluctuations)
by the very fluctuations that attempt to destabilize it, a phenomenon known as
order-from-disorder.

Finally, I note that random heterogeneity also ubiquitously appears in most
membrane realizations (see e.g., work on graphenegrapheneMceuen ) and leads to
a disorder-driven critical membrane phase, studied extensively starting with
the work of Nelson and RadzihovskyRadzihovskyNelsonEuro ; RadzihovskyNelsonPRA
, with many that followedMorseLubensky ; LRflatglassPRB ; LRmembraneAOP .

##  V Nematic elastomer

A final example of a critical phase that I will briefly discuss is a nematic
elastomer, namely rubber – a randomly crosslinked polymer network composed of
mesogenic groups, illustrated in Fig.9. It exhibits a spontaneous transition
to a nematic phase, thereby driving an accompanying spontaneous uniaxial
distortion of the elastic matrix, illustrated in Fig.9.bookWarner

Figure 9: Spontaneous uniaxial distortion of nematic elastomer driven through
the isotropic-nematic transition.

Even in the absence of fluctuations, bulk nematic elastomers were predictedGL
and later observed to display an array of fascinating phenomenabookWarner ;
reviewNE , the most striking of which is the vanishing of stress for a range
of strain, applied transversely to the spontaneous nematic axis. This striking
softness is generic, stemming from the spontaneous orientational symmetry
breaking by the nematic state, accompanied by the nemato-elastic Goldstone
mode, that leads to the observed soft distortion and strain-induced director
reorientationGL ; LMRX , illustrated in Fig.10. This unique elastic phenomenon
is captured even at the harmonic approximation of the fully nonlinear uniaxial
Hamiltonian in Eq.(85),

$\displaystyle{\mathcal{H}}_{elast}$ $\displaystyle=$
$\displaystyle\frac{1}{2}B_{z}\,w_{zz}^{2}+\lambda_{z\perp}\,w_{zz}\,w_{\alpha\alpha}+\frac{1}{2}\,\lambda\,w_{\alpha\alpha}^{2}+\mu\,w_{\alpha\beta}^{2}+\frac{K}{2}\,(\nabla_{\perp}^{2}u_{z})^{2},$
(85)

where akin to earlier examples the components of the (rescaled) effective
nonlinear strain tensor $\underline{\underline{{w}}}$ are given by

$\displaystyle w_{zz}$ $\displaystyle=$
$\displaystyle\partial_{z}u_{z}+\frac{1}{2}\,(\nabla_{\perp}u_{z})^{2},$ (86)
$\displaystyle w_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\left(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}-\partial_{\alpha}u_{z}\partial_{\beta}u_{z}\right).$
(87)

The underlying rotational symmetry of the parent isotropic phase guarantees a
vanishing of one of the five elastic constantsGL ; LMRX , $\mu_{\perp z}$
(i.e., the $w_{z\alpha}^{2}$ is missing from (85)), that usually characterizes
harmonic deformations of a three-dimensional uniaxial solidLandau , here with
the uniaxial axis ${\hat{z}}$ chosen spontaneously. Interestingly, examining
(87), $w_{zz}$ and $w_{\alpha\beta}$ have structure of, respectively, a
uniaxially (e.g., smectic) and biaxially (e.g., columnar) periodic critical
state. Thus, nematic elastomer is a three dimensional amorphous solid that is
an amalgam of a smectic and columnar liquid crystals.

Figure 10: (a) Simultaneous reorientation of the nematic director and of the
uniaxial distortion is a low-energy nemato-elastic Goldstone mode of an ideal
elastomer, that is responsible for its softness and (b) its resulting
idealized flat (vanishing stress) stress-strain curve for a range of strains
$0<\varepsilon<\varepsilon_{c}$. LMRX

Given this softness of the harmonic elasticity associated with symmetry-
imposed vanishing of the $\mu_{z\perp}$ modulus, thermal and heterogeneity-
driven fluctuations are divergent in 3d.XingRadzAOP ; NEelSL  Concomitantly,
the elastic terms $({\bm{\nabla}}_{\perp}u_{z})^{2}$ and $(\partial_{z}{\bf
u}_{\perp})^{2}$ associated with transverse distortions are missing from the
quadratic part of the elastic energy. Thus, as in other critical matter
discussed in earlier sections, one expects qualitative importance of above
elastic nonlinearities (87) in the presence of thermal fluctuations and
network heterogeneity.XingRadzAOP ; NEelSL

Similar to their effects in smectic, columnar liquid crystals, polymerized
membranes, and other critical matter discussed earlier, in bulk elastomers
thermal fluctuations (and network heterogeneity) leads to anomalous elasticity
controlled by a tuning-free infrared attractive fixed point, with universal
length-scale dependent elastic moduli, XingRadzAOP ; NEelSL

$\displaystyle K_{eff}$ $\displaystyle\sim$ $\displaystyle L^{\eta},\ \
\mu_{eff}\sim L^{-\eta_{u}}.$ (88)

The resulting critical state can be shownXingRadzAOP  to be characterized by a
universal non-Hookean stress-strain relation

$\sigma_{zz}\sim(\varepsilon_{zz})^{\delta}$ (89)

and a negative Poisson ratio for extension $\varepsilon_{xx}>0$ transverse to
the nematic axis

$\displaystyle\varepsilon_{yy}=\frac{5}{7}\varepsilon_{xx},\ \ \
\varepsilon_{zz}=-\frac{12}{7}\varepsilon_{xx}.$ (90)

While considerable progress has been made in understanding these fascinating
materials, many questions, particularly associated with network heterogeneity
remain open.

##  VI Summary and conclusions

In this chapter, I discussed a novel class of ordered states of matter that I
dub “critical matter”. The common crucial feature of these exotic states is
that, as a result of the underlying symmetry broken in the ordered state, a
subset of elastic moduli of the associated Goldstone modes vanish identically.
As a result, for sufficiently low dimensions, such system exhibit harmonic
fluctuations that are divergent in the thermodynamic limit, and require
treatment of Goldstone modes’ nonlinearities throughout the ordered phase (not
just near a critical point). Treating these within renormalization group
analysis, leads to an infrared stable fixed point that controls the resulting
highly nontrivial, strongly interacting, and universal ordered state. The
latter exhibits properties akin to that of a critical point, but extending
over the entire phase, which I therefore naturally refer to as a “critical
phase”.

As prominent examples of such systems, I have discussed smectics (with many
soft and hard matter realizations, from paired superfluids (FFLO), spin-orbit
coupled and frustrated bosons to quantum Hall stripes), cholesterics, helical
magnets, columnar phases, putative spontaneous vortex lattices, polymerized
membranes and nematic elastomers. While significant progress has been made in
characterizing these systems, discovery of new systems (e.g., as quantum
examples in physically accessible dimensions), understanding of effects of
random heterogeneity and dynamics remain challenging open problems.

I finish by noting that there are also critical phases (many appearing as
counter-parts of the ones discussed in this chapter), that I have not
discussed here, where fluctuations are driven by quenched disorder, rather
than by temperature or quantum fluctuations. These exhibit critical states of
glassy Goldstone modes and generically appear in dimensions higher than their
thermal counterpartsRTaerogelPRL ; RTaerogelPRB ; BRTCaerogelScience ;
SRTcolumnar ; magneticSCprl ; magneticSCprb ; RadzihovskyNelsonEuro ;
RadzihovskyNelsonPRA ; LRflatglassPRB ; MorseLubensky ; LRmembraneAOP ;
LRbuckling ; XingRadzAOP , and are thus of even stronger experimental
relevance.

##  VII Acknowledgments

The material presented in these lectures is based on research done with a
number of wonderful colleagues and friends, most notably David Nelson, John
Toner, Pierre Le Doussal, Tom Lubensky, and Xiangjun Xing. I am indebted to
them for much of my insight into the material presented here. This work was
supported by the National Science Foundation through grants DMR-1001240 and
DMR-0969083 as well by the Simons Investigator award from the Simons
Foundation.

On a personal note, I have met and discussed physics with Michael Fisher at
conferences, most often at the wonderful Rutgers Stat Mech meetings. I recall
fondly Michael always sitting in a front row, sometimes flanked by Daniel and
Matthew, strikingly paying close attention and asking probing questions even
in the 3 mins student talks, and as attentively as in invited talks by
distinguished senior faculty. I recall him always being kind and supportive in
my own career. Although I have never had the pleasure of working with Michael,
I have greatly benefited from his insight through his many seminal papers that
influenced my own RG work (some described in this chapter). But perhaps more
importantly, I indirectly learned physics from Michael through David Nelson,
Daniel and Matthew, all of whom have strongly shaped my style, taste and
approach to physics. And for that I am eternally grateful.

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