# Fractonic gauge theory of smectics

Zhengzheng Zhai and Leo Radzihovsky  Department of Physics and Center for
Theory of Quantum Matter
University of Colorado, Boulder, CO 80309<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by striped correlated quantum matter, and the recently developed
duality between elasticity of a two-dimensional (2D) crystal and a gauge
theory, we derive a dual coupled U(1) vector gauge theory for a two-
dimensional (2D) quantum smectic, where the disclination is mapped onto the
fractonic charge, that we demonstrate can only move transversely to smectic
layers. This smectic gauge theory dual also emerges from a gauge dual of a
quantum crystal after a Higgs transition corresponding to a single flavor of
its dipole condensation, an anisotropic quantum melting via dislocation
proliferation. A condensation of the second flavor of dislocations is
described by another Higgs transition describing the smectic-to-nematic
melting. We also utilize the electrostatic limit of this duality to formulate
a melting of a 2D classical smectic in terms of a higher derivative sine-
Gordon model, demonstrating its instability to a nematic at any nonzero
temperature. Generalizing this classical duality to a 3D smectic, gives
formulation of a 3D nematic-to-smectic transition in terms of an anisotropic
Abelian-Higgs model.

##  I Introduction

###  I.1 Motivation and background

A smectic state of matter, is a liquid crystal phase that partially breaks
rotational and translational symmetries spontaneously, exhibiting a periodic
layered order. Classical smectics form in systems of rod-like constituents
(molecules like 5CB) ProstDeGennes  and are driven by anisotropic entropic
(exclusion volume) interactions. In striking contrast, quantum smectic states
appear even in systems of isotropic point-like constituents as a result of
frustrated competition between kinetic energy and interactions. In cold atom
systems, quantum smectics may be realized in a putative Fulde-Ferrell-Larkin-
Ovchinnikov paired superfluids in imbalanced degenerate atomic gases FF ; LO ;
LR_VishwanathPRL ; LRpra  and in spin-orbit coupled Bose condensates HuiZhai ;
LR_ChoiPRL . Quantum smectics are also a natural explanation for a striking
resistive anisotropy observed in quantum Hall systems at half-filled high
Landau levels, and for “striped” spin and charge states of weakly doped
correlated quantum magnets EisensteinQSm ; Fogler ; Moessner ; FisherMacdonald
; LR_Dorsey .

A two-dimensional (2D) smectic can emerge from partial, anisotropic
meltingHalperinOstlund  of a crystal, with only one species of dislocations
unbinding, such that only one direction of translational symmetry is restored,
in a Kosterlitz-Thouless (KT)-like KT  phase transition. However, a 2D smectic
is unstable to thermal fluctuations, and is always driven into a nematic fluid
at any nonzero temperature HalperinOstlund ; TonerNelson . In contrast, a
(2+1)D quantum smectic at zero temperature, is a stable state of matter, whose
studies have been limited to a simplest harmonic description, with effects of
topological defects and of elastic nonlinearities neglected beyond qualitative
discussions (for an exception see Ref. [LR_VishwanathPRL, ; LRpra, ]). This,
together with ubiquitous putative realizations provides a strong motivation
for the present detailed work, a brief preview of which appeared in a recent
publication.Smecticgauge

A complementary motivaton for our study is its relation to a new class of
topological quantum states of matter – dubbed “fractons” – discovered in
theoretical exactly solvable models.Chamon05 ; Bravyi11 ; Haah11 ;
Castelnovo12 ; Yoshida13 ; Bravyi13 ; Vijay15 ; Vijay16  These feature a
number of fascinating properties that are believed to lie beyond a
conventional quantum field theoretic description.QiAOP2020  The most striking
of these are quasiparticles with robust (not just fine-tuned or symmetry
imposed) restrictions on their mobility, such as an immobile fracton, and its
subdimensional multipoles. Although experimental realizations have been sorely
lacking, these theoretical models are intensely studied, motivated by their
promise for a robust quantum memory and fundamental interest in a new class of
topological quantum liquidsHNreview ; AbhinavPremReview .

Following this gapped class of lattice qubit models, fracton-like phenomena
were also uncovered in gapless symmetric tensor gauge theories, encoded in a
generalized Gauss law, that conserves charge multipoles and thereby constrains
mobility of chargesPretko1703 ; Pretko1707 . Contemporaneously, a similarity
of the constrained dynamics of disclination and dislocation defects in a
crystal was conjectured to be dual to charges and dipoles of a gauge
theoryPretkoLRdualityPRL2018 . Utilizing a generalizaton of the familiar XY-
to-gauge theory (boson-vortex) dualitydasgupta ; fisher , Pretko and
Radzihovsky formalized this relation through a duality mapping (explored in
other contexts by Zaanen and company Zaanen2017 ) between a quantum 2D crystal
elasticity and a symmetric tensor gauge theory. Under this mapping the stress
tensor $\sigma_{ij}$ and momentum vector $\pi_{i}$ fields map onto the
electric tensor $E_{ij}$ and magnetic vector $B_{i}$ fields, respectively,
with Newton’s law (conservaton of momentum) corresponding to Faraday’s law of
the tensor gauge theory.

This established fracton-elasticity duality allows for numerous predictions
for phases and phase transitions of the fracton system, based on the extensive
understanding of 2D crystal and their decendent states. For example, different
phases of the scalar fracton model - fracton insulator, dipole condensate and
fracton condensate, can be regarded as gauge theory counterparts to the
“commensurate” and “incommensurate” (supersolid)
crystalsPretkoLRsymmetryEnrichedPRL2018 ; Kumar19 , hexatic, and isotropic
fluid phases of the elasticity theory. The associated finite-temperature
dipole-unbinding transition and fracton charge unbinding transition correspond
to the classical two-stage melting transitions, i.e., crystal-to-hexatic and
hexatic-to-liquid transition respectively.

A complementary and physically more transparent formulation of elasticity-to-
fractonic coupled vector gauge theory was recently
presentedRadzihovskyHermeleVectorGaugePRL2020 . The resulting dual coupled
vector gauge theory involves three U(1) vector gauge fields ${\bf A}_{k}$
(with $k=x,y$ denoting flavors) and ${\bf a}$, and their canonically conjugate
electric fields ${\bf E}_{k}$ and ${\bf e}$, that encode coupled Goldstone
modes, the phonons $u_{k}$ and the local bond angle $\theta$. Building on the
treatment of the quantum crystalRadzihovskyHermeleVectorGaugePRL2020  and a
recent analysis of the quantum smecticSmecticgauge , we derive and explore
extensively the coupled vector gauge theory duality to study the (2+1)D
smectic and its quantum phase transitions to a crystal and a nematic,
formulated in terms of an array of Higgs transitions. We also utilize it to
study 2D and 3D classical smectic and the corresponding classical nematic-to-
smectic phase transitions.

###  I.2 Summary of Results

In this paper, we develop and explore in detail a dual coupled U(1) vector
gauge theory for a 2D quantum smectic, building on a recent study of a 2D
quantum crystalRadzihovskyHermeleVectorGaugePRL2020  and a smecticSmecticgauge
by one of the authors. The dual description we derive is formulated in terms
of two coupled U(1) vector gauge theories, with electric fields $\hat{{\bf
E}}$ and $\hat{{\bf e}}$, and canonically conjugate vector potentials
$\hat{{\bf A}}$ and $\hat{{\bf a}}$, sourced by dipole and charge current
densities, $n_{b},{\bf j}_{b}$ (dislocations) and $n_{s},{\bf j}_{s}$
(disclinations), respectively. The corresponding dual Hamiltonian density is
given by

$\begin{split}\tilde{\mathcal{H}}_{\text{sm}}=&\frac{1}{2}\chi\hat{\bf
E}^{2}+\frac{1}{2}\left({\bm{\nabla}}\times\hat{\bf
A}\right)^{2}+\frac{1}{2}K\hat{\bf e}^{2}\\\
&+\frac{1}{2}\left({\bm{\nabla}}\times\hat{{\bf a}}+\hat{\bf x}\times\hat{\bf
A}\right)^{2}-\hat{\bf A}\cdot{\bf j}_{b}-\hat{\bf a}\cdot{\bf
j}_{s},\end{split}$ (1)

supplemented by the generalized Gauss laws,

$\displaystyle{\bm{\nabla}}\cdot\hat{{\bf E}}$ $\displaystyle=$ $\displaystyle
n_{b}+\hat{{\bf e}}\cdot\hat{\bf x},$ (2)
$\displaystyle{\bm{\nabla}}\cdot\hat{{\bf e}}$ $\displaystyle=$ $\displaystyle
n_{s}.$ (3)

We demonstrate that the $n_{s}$ charges (disclinations in the smectic) of this
dual gauge theory are subdimensional “lineon”, mobile only transverse to
smectic layers (that we take to be along $\hat{\bf x}$), enforced by
generalized gauge invariance and associated continuity equation,

$\displaystyle\partial_{t}n_{b}+{\bm{\nabla}}\cdot{\bf j}_{b}$
$\displaystyle=$ $\displaystyle-\hat{\bf x}\cdot{\bf j}.$ (4)

In contrast the dipoles (dislocations) exhibit a finite but highly
anisotropically mobility.

Motivated to also understand the quantum crystal-smectic transition, we derive
the smectic gauge dual and transition to it by utilizing gauge dual of the
quantum crystalRadzihovskyHermeleVectorGaugePRL2020  and condensing one flavor
of dipoles (dislocations). The associated Higgs transition gaps out the
corresponding flavor of the gauge fields ${\bf A}_{k}$, and leads to a dual
quantum smectic Lagrangian, that matches exactly the description obtained
through direct duality of smectic elasticity,

$\tilde{\mathcal{L}}_{\text{sm}}=\frac{1}{2}|\left(\partial_{\mu}+ipA_{\mu}\right)\psi_{x}|^{2}-V\left(|\psi_{x}|\right)+\mathcal{L}_{\text{M}}^{\text{sm}},$
(5)

where, $\mathcal{L}_{\text{M}}^{\text{sm}}$ is the Maxwell sector from
Equation (1), with $V\left(|\psi_{x}|\right)$ a U(1)-invariant Landau
potential for $x$-flavor dipoles, $\psi_{x}$. The flow chart in Fig. 1
summarizes these two routes to the dual gauge theory of a quantum smectic.

Figure 1: Quantum crystal-smectic duality relations and the associated quantum
melting transition.

In this formulation $\psi_{x}=0$ is the Coulomb phase corresponds to the
quantum smectic phase, and the $\psi_{x}\neq 0$ Higgs phase, gives a
condensation of unbound $\hat{\bf x}-$dipoles ($\hat{\bf y}-$dislocations)
that gaps out the gauge field $A_{\mu}$, which drives a Higgs transition to a
quantum nematic. The quantum melting transitions and the corresponding phases
are illustrated in Fig. 2.

Figure 2: Illustration of quantum melting of a 2D crystal into a smectic,
followed by smectic-to-nematic melting, respectively driven by a condensation
of $\hat{\bf x}$-dislocations and of $\hat{\bf y}$-dislocations, tuned by
applied shear stress, $\sigma_{xx}$.

We also explore the classical limit of this duality, and formulate the 2D
smectic-to-nematic melting and a subsequent nematic-to-isotropic fluid
transition in terms of a higher-derivative sine-Gordon model,

$\begin{split}\tilde{\mathcal{H}}_{\text{sm}}=&\frac{1}{2}\chi(\partial_{x}^{2}\alpha)^{2}+\frac{1}{2}K^{-1}(\partial_{y}\alpha)^{2}-g_{b}\cos(b\partial_{x}\alpha)\\\
&-g_{s}\cos(2\pi\alpha).\end{split}$ (6)

The first two terms capture the elasticity of a 2D smectic, and the two cosine
correspond to dislocations and disclinations, tuned by the corresponding
fugacities $g_{b,s}$. We demonstrate that in 2D the dislocations are always
relevant, corresponding to the instability of a 2D smectic to a nematic at any
nonzero temperatureNelsonToner . The resulting sine-Gordon model in $\alpha$
then captures the the nematic-to-isotropic fluid transition, illustrated in
Fig. 3.

Figure 3: A phase diagram illustrating an instability of 2D smectic to
arbitrary weak thermal fluctuations that drive it into a nematic at any
nonzero temperature. The nematic phase transition into an isotropic fluid is
through a KT disclination unbinding transition $T_{c}$.

We also derive a classical dual gauge theory for a 3D smectic, that captures
its finite-temperature melting into a nematic through a dual normal-
superconductor transition with an higher-derivative Maxwell sector, equivalent
to Toner’s original treatment of the nematic-to-smectic-A transition.Toner82

###  I.3 Outline

The rest of this paper is organized as follows. In Sec. II, after briefly
introducing the elasticity theory of a smectic phase and its topological
defects, we map a two-dimensional quantum smectic to a dual coupled U(1)
vector gauge theory, and use it to demonstrate that its charges
(disclinations) exhibit subdimensional constrained mobility. In Sec. III,
starting with the coupled U(1) vector gauge theory for a quantum crystal, and
“softening” it into a generalized Abelian-Higgs model, we rederive the dual
gauge theory of a quantum smectic through a Higgs transition of one flavor of
its dipoles. Furthermore, we derive an equivalent low-energy tensor gauge-
theory description. In Sec. IV, we explore the classical analogue of these
dualities and associated phase transitions, and formulate a higher derivative
sine-Gordon model, capturing classical thermal smectic melting transitions. We
use it to demonstrate that indeed a 2D smectic is unstable and driven into a
nematic at any nonzero temperature. We also generalize this discussion to a 3D
classical smectic, and reformulate the 3D nematic to smectic-A transition
mediated by unbinding of dislocation loops in terms of a higher-derivative
classial normal-superconductor transition. We conclude in Sec. V with a
summary of our results and discussion of potential utility of our work.

##  II Smectic and its duality

###  II.1  Classical smectic

Ideal smectics are equidistantly layered structures, with a well-defined
interlayer spacing $d$, which can be determined through diffraction
experiments. With the layers correlations are liquid-like and exhibit crystal-
like periodic modulation transverse to the layers, with corresponding density
given by,

$\rho\left({\bf r}\right)=\rho_{0}+\left(\psi e^{{\bf q}_{0}\cdot{\bf
r}}+h.c.\right),$ (7)

where, ${\bf q}_{0}=\frac{2\pi}{d}\hat{\bf z}$ is the modulation wavevector,
and $\psi$ its amplitude, that is the order parameter that distinguishes the
smectic phase from the nematic phase.

The deformation of a smectic can be described by its layer displacement field
$u({\bf r})$. As the system is invariant under uniform translations, the
elastic energy should be expressed purely in terms of derivatives of $u$.
Furthermore, the first-order derivatives along the layer,
${\bm{\nabla}}_{\perp}u$, corresponding to merely a uniform rotation of the
layers, must cost no energy. Thus, to harmonic order, only the curvature of
the layers, ${\bm{\nabla}}_{\perp}^{2}u$, can enter the quadratic part of the
elastic energy functional. This point can be seen more explicitly by the
following argument. The order parameter $\psi\left({\bf r}\right)$, describing
phonon fluctuations, can be represented as

$\psi\left({\bf r}\right)=|\psi|e^{-iq_{0}u({\bf r})},$ (8)

and the locations of the layer planes can be determined as the constant phase
of the molecular density wave,

$\phi\left({\bf r}\right)\equiv{\bf q}_{0}\cdot{\bf r}-q_{0}u\left({\bf
r}\right)=2\pi n,n=0,\pm 1,\pm 2,...$ (9)

The layers local unit-normal is given by,

$\begin{split}{\bf
N}=&\frac{{\bm{\nabla}}\phi}{|{\bm{\nabla}}\phi|}=\frac{\left(-{\bm{\nabla}}_{\perp}u,1-\nabla_{\parallel}u\right)}{\sqrt{1+\left({\bm{\nabla}}u\right)^{2}}}\\\
=&\left(-\partial_{x}u,-\partial_{y}u,1\right)+O[\left({\bm{\nabla}}u\right)^{2}].\end{split}$
(10)

The first-order derivative,
${\bm{\nabla}}_{\perp}u\approx\left(\partial_{x}u,\partial_{y}u,0\right)\approx\hat{\bf
z}-{\bf N}$, therefore, corresponds to a rigid rotation of the layers around
an axis along the layer plane, and does not contribute to the elastic energy,
and, $\nabla_{\parallel}u=\partial_{z}u$, in 3D.

Figure 4: Elastic deformations of a smectic. (a) Smectic with equilibrium
layer spacing: $q_{0}=\frac{2\pi}{d},u=0$. (b) A uniformly translated smectic
with the same energy as that in (a): $u=u_{0}$. (c) A rotated smectic with the
same energy to harmonic order as that in (a): $u=\theta x$. (d) A smectic with
compressed layers with energy increased relative that in (a):
$q_{0}^{\prime}=\frac{2\pi}{d^{\prime}},u=\left(1-\frac{d}{d^{\prime}}\right)z$.
[Figures adapted from Reference Chaikin2000, .]

Consistent with this, the continuum elastic Hamiltonian density for a
D-dimensional smectic is given by a well-known expression,

$\mathcal{H}_{\text{sm}}=\frac{1}{2}\chi\left(\nabla_{\parallel}u\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}_{\perp}^{2}u\right)^{2},$
(11)

a Landau-Peierls elastic energy Landau37 ; Peierls36  for a one-dimensional
solid, where $\chi$ is inverse of the compressional modulus, and $K$ is the
bend modulus.

Note that (10) is only correct up to $O[({\bm{\nabla}}u)^{2}]$ for small
rotations. For any finite rotations $\theta$,

$\nabla_{\parallel}u=1-\cos\theta,\;\;\nabla_{\perp}u=-\sin\theta,$ (12)

and the nonlinear strain,
$\nabla_{\parallel}u-\frac{1}{2}({\bm{\nabla}}u)^{2}$, can be straighforwardly
seen to be independent of the rotation angle $\theta$. Thus, the rotationally
invariant energy density is given by,

$\mathcal{H}_{\text{sm}}=\frac{1}{2}\chi\left[\nabla_{\parallel}u-\frac{1}{2}({\bm{\nabla}}u)^{2}\right]^{2}+\frac{1}{2}K\left({\bm{\nabla}}_{\perp}^{2}u\right)^{2},$
(13)

which introduces non-linear elasticity into $\mathcal{H}_{\text{sm}}$, that
for $d\leq 3$ leads to a nontrivial anomalous smectic elasticity Grinstein82 ;
LR2011PRE . However, because the focus of our work is on a quantum smectic,
these elastic nonlinearities remain irrelevant in (2+1)D and will thus be
neglected in the rest of the manuscript.

For the smectic-A phase, the local normal field (layer orientation) ${\bf N}$
and the director field ${\bf n}=\hat{\bf z}+\delta{\bf n}$ are aligned in
equilibrium. Thus,

${\bm{\nabla}}_{\perp}u=-\delta{\bf
n},\;\;{\bm{\nabla}}_{\perp}^{2}u=-{\bm{\nabla}}_{\perp}\delta{\bf
n}=-{\bm{\nabla}}{\bf n},$ (14)

and the elastic energy, ignoring nonlinearities, can be represented as

$\mathcal{H}_{\text{sm}}=\frac{1}{2}\chi\left({\bm{\nabla}}u+\delta{\bf
n}\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}{\bf n}\right)^{2}.$ (15)

For a 2D smectic, with the layers along $\hat{\bf x}$ (with layer normal along
$\hat{\bf y}$), the elastic Hamiltonian density in Equation (11) reduces to

$\mathcal{H}_{\text{sm}}=\frac{1}{2}\chi\left(\partial_{y}u\right)^{2}+\frac{1}{2}K\left(\partial_{x}^{2}u\right)^{2},$
(16)

where the layer displacement $u$ is along the $\hat{\bf y}$ axis, and the
layer orientation (director field) is, ${\bf n}=-\hat{\bf
x}\sin\theta+\hat{\bf y}\cos\theta=\hat{\bf y}+\delta{\bf n}$.

Another way to obtain smectic elasticity is to start out with a elasticity of
a 2D crystal and allow nonsingle-valued displacement field $u_{x}$, with
${\bm{\nabla}}u_{x}={\bf v}_{x}$, accounting for a plasma of unbound
dislocations with Burgers vector along the $\hat{\bf x}-$directed smectic
layers, where ${\bf v}_{x}$ is an arbitrary vector with
$\epsilon_{ij}\partial_{i}v_{jx}=b_{x}$. Integrating over strain tensor field
${\bf v}_{x}$, leads to the smectic harmonic elasticity,

$\begin{split}\mathcal{H}_{\text{sm}}=&\mathcal{H}_{\text{cr+disl.}}=\mu
u_{ij}^{2}+\frac{1}{2}\lambda u_{ii}^{2}\\\
=&\mu\left(u_{xx}^{2}+u_{yy}^{2}+2u_{xy}^{2}\right)+\frac{1}{2}\lambda\left(u_{xx}+u_{yy}\right)^{2}+\frac{1}{2}E_{c}b_{x}^{2}\\\
=&\mu\left[v_{xx}^{2}+\left(\partial_{y}u_{y}\right)^{2}+\frac{1}{2}\left(\partial_{x}u_{y}+v_{yx}\right)^{2}\right]\\\
&+\frac{1}{2}\lambda\left(v_{xx}+u_{yy}\right)^{2}+\frac{1}{2}E_{c}\left(\partial_{x}v_{yx}-\partial_{y}v_{xx}\right)^{2}\\\
=&\frac{1}{2}\chi\left(\partial_{y}u_{y}\right)^{2}+\frac{1}{2}K\left(\partial_{x}^{2}u_{y}\right)^{2},\end{split}$
(17)

where compressional modulus is $\chi=4\mu(\mu+\lambda)/(2\mu+\lambda)$, bend
modulus $K=E_{c}$, and higher derivative terms are neglected after integrating
out the ${\bf v}_{x}$ field in the last step.

Equivalently, the smectic elasticity can be formulated in terms of the
orientational (nematic) angle degree of freedom $\theta$, which corresponds to
the orientation of the layers, with the elastic Hamiltonian density given by,

$\mathcal{H}_{\text{sm}}=\frac{1}{2}\chi\left({\bm{\nabla}}u-\theta\hat{\bf
x}\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}\theta\right)^{2}.$ (18)

At low energies set by $\chi$, this Hamiltonian reduces to the conventional
form (16) after Higgs’ing out the bond angle $\theta$, locking
$\partial_{x}u=\theta$.

###  II.2 Two-dimensional quantum smectic

Classical elastic Hamiltonian in (18) is easily generalized to a quantum
smectic by elevating $u$ and $\theta$ to operators, and adding canonically
conjugate linear and angular momenta operators, $\hat{\pi}$ and $\hat{L}$,
respectively. This gives,

$\hat{\mathcal{H}}_{\text{sm}}=\frac{1}{2}\hat{\bf\pi}^{2}+\frac{1}{2}\hat{L}^{2}+\frac{1}{2}\chi\left({\bm{\nabla}}\hat{u}-\hat{\theta}\hat{\bf
x}\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}\hat{\theta}\right)^{2},$ (19)

for bosonic smectic supplemented with canonical commutation relations
($\hbar=1$),

$\displaystyle\left[\hat{u}({\bf r}),\hat{\pi}({\bf r}^{\prime})\right]$
$\displaystyle=$ $\displaystyle i\delta^{2}\left({\bf r}-{\bf
r}^{\prime}\right),$ (20a) $\displaystyle\left[\hat{\theta}({\bf
r}),\hat{L}({\bf r}^{\prime})\right]$ $\displaystyle=$ $\displaystyle
i\delta^{2}\left({\bf r}-{\bf r}^{\prime}\right).$ (20b)

It is convenient to work with a path-integral formulation where quantum nature
of these fields is accounted for by functional integration in phase-space of
these fields. We consider the evolution operator for the quantum smectic,

$U(u^{\prime},\theta^{\prime},u,\theta;t)=\langle
u^{\prime},\theta^{\prime}|e^{-i\int_{{\bf
r}}\hat{\mathcal{H}}_{\text{sm}}t}|u,\theta\rangle,$ (21)

and rewrite it in phase-space functional integral formulation as,

$\begin{split}U&=\int\left[du\right]\left[d\pi\right]\left[d\theta\right]\left[dL\right]e^{iS_{\text{sm}}}\\\
&=\int\left[du\right]\left[d\pi\right]\left[d\theta\right]\left[dL\right]e^{i\int
dt\int_{{\bf r}}\mathcal{L}_{\text{sm}}},\end{split}$ (22)

with the corresponding Lagrangian density given by,

$\begin{split}\mathcal{L}_{\text{sm}}=&\pi\partial_{t}u+L\partial_{t}\theta-\frac{1}{2}\pi^{2}-\frac{1}{2}L^{2}+\frac{1}{2}\chi^{-1}{\bf\sigma}^{2}+\frac{1}{2}K^{-1}{{\bf
j}}^{2}\\\ &-{\bf\sigma}\cdot\left({\bm{\nabla}}u-\theta\hat{\bf
x}\right)-{{\bf j}}\cdot{\bm{\nabla}}\theta.\end{split}$ (23)

Figure 5: Topological defects in a 2D smectic. $\hat{{\bf{n}}}$ is the Frank
director, and ${\bf b}$ is the Burger’s vector. (a) A $-\frac{1}{2}$
disclination, i.e., the director $\hat{{\bf{n}}}$ changes $-\pi$ around a
closed loop counterclockwise. (b) A $+\frac{1}{2}$ disclination, i.e., the
director $\hat{{\bf{n}}}$ changes $+\pi$ around a closed loop
counterclockwise. (c) A single positive dislocation, i.e., the layer
displacemnt $u$ changes $+d$ around a closed circuit counterclockwise, which
can be regarded as a tightly bound dipole of two, opposite disclinations with
charges $\pm\frac{1}{2}$.

In addition to the single-valued (smooth) Goldstone mode degrees of freedom,
$\tilde{\theta}$ and $\tilde{u}$, we must also include topological defects –
disclinations and dislocations, captured by including a nonsingle-valued
component of the bond angle field $\theta^{s}$ and of the phonon distortion
field $u^{s}$, respectively. In the smectic, a disclination at a point ${\bf
r}_{0}$, is defined by a nonzero closed line-integral of the gradient of the
bond angle around ${\bf r}_{0}$, $\oint_{{\bf r}_{0}}d\theta=2\pi n_{s}$, or
equivalently in a differential form,

$\hat{\bf z}\cdot{\bm{\nabla}}\times{\bm{\nabla}}\theta^{s}=2\pi
n_{s}\delta^{2}({\bf r}-{\bf r}_{0})\equiv n_{s}({\bf r}),$ (24)

measuring the deficit/surplus bond angle. $n_{s}({\bf r})$ is the disclination
charge density. A disclination with charge $n_{s}=-\frac{1}{2}$, and a
disclination with charge $n_{s}=+\frac{1}{2}$ are illustrated in Fig. 5(a, b).

A dislocation at ${\bf r}_{0}$ with a Burgers charge $n_{b}$ (that is an
integer multiples of the elementary layer spacing), is defined by a closed
line-integral, $\oint_{{\bf r}_{0}}du=n_{b}$, or equivalently in the
differential form,

$\hat{\bf z}\cdot{\bm{\nabla}}\times{\bm{\nabla}}u^{s}=n_{b}\delta^{2}({\bf
r}-{\bf r}_{0})\equiv n_{b}({\bf r}),$ (25)

where $n_{b}({\bf r})$ is the Burgers charge density. A dislocation in the
smectic is shown in Fig. 5(c), which can be regarded as a tightly bound pair
of $+\frac{1}{2}$ and $-\frac{1}{2}$ disclinations.

In anticipation of our more rigorous duality derivation, already here we can
argue for the subdimensional nature of the disclination dynamics in a smectic.
Consider a pair of oppositely charged $\pm\frac{1}{2}$ disclinations separated
along $\hat{x}$ axis, as shown in Fig. 6. The separation between the pair is
large such that we can regard them as isolated disclinations. Moving the ‘$+$’
disclination along the layers (i.e., $\hat{\bf x}$ axis) by two layer
spacings, requires the introduction of four extra half-layers of molecules,
which is a highly non-local process in terms of atoms quantum dynamics, and is
therefore not allowed. While moving the ‘$+$’ disclination transversely (i.e.,
in $\hat{y}$ direction) preserves the strength of the dislocation, and thus,
is allowed dynamically, even though there is an energy cost for seperating the
‘$+$’ and ‘$-$’ disclination pair that make up the dislocation. Similar
analysis applies to the ‘$-$’ disclination. Thus, we conclude that
disclinations can only move transversely to the layers, i.e., they manifests
the subdimensional lineon dynamics.

To include the topological defects in the complete description of the smectic,
we decompose the distortion field $u$ and the bond angle $\theta$ into the
smooth elastic and nonsingle-valued components,

$u=\tilde{u}+u^{s},\ \ \theta=\tilde{\theta}+\theta^{s}.$ (26)

Integrating out the single-valued parts $\tilde{u}$ and $\tilde{\theta}$ out
of the total generating function,

$U=\int\left[d\tilde{u}\right][d\tilde{\theta}]\left[du^{s}\right]\left[d\pi\right]\left[d\theta^{s}\right]\left[dL\right]e^{i\int
dt\int_{{\bf r}}\mathcal{L}_{\text{sm}}},$ (27)

leads to,

$U=\int\left[du^{s}\right]\left[d\pi\right]\left[d\theta^{s}\right]\left[dL\right]e^{i\int
dt\int_{{\bf r}}\mathcal{L}_{\text{sm}}},$ (28)

where the new Lagrangian density is given in terms of only nonsingle
components,

$\begin{split}\mathcal{L}_{\text{sm}}=&\pi\partial_{t}u^{s}+L\partial_{t}\theta^{s}-\frac{1}{2}\pi^{2}-\frac{1}{2}L^{2}+\frac{1}{2}\chi^{-1}{\bf\sigma}^{2}\\\
&-{\bf\sigma}\cdot\left({\bm{\nabla}}u^{s}-\theta^{s}\hat{\bf
x}\right)+\frac{1}{2}K^{-1}{{\bf j}}^{2}-{{\bf
j}}\cdot{\bm{\nabla}}\theta^{s},\end{split}$ (29)

with two enforced constraints,

$\displaystyle\partial_{t}\pi-{\bm{\nabla}}\cdot{\bf\sigma}\equiv\partial_{\mu}J_{\mu}=0,$
(30) $\displaystyle\partial_{t}L-{\bm{\nabla}}\cdot{\bf j}-\sigma_{x}=0,$ (31)

where we implicitly introduced currents
$J_{\mu}=\left(\pi,-\sigma_{i}\right)$.

Figure 6: Restricted along-layers mobility of disclinations in a 2D quantum
smectic. Consider a pair of disclinations seperated by five layer spacings.
(a) Moving the ‘$+$’ disclination along the layers by two layer spacings
requires an introduction of four half-layers of underlying bosons, which is a
non-local process, and therefore is not allowed. (b) Moving the ‘$+$’
disclination transverse to the layers by two layer spacings preserves the
strength of the dislocation, and thus, is dynamically allowed.

###  II.3 Quantum smectic-gauge theory duality

The momentum continuity Equation (30) can be solved in terms of gauge
potential fields, $A_{\mu}$, with

$J_{\mu}=\epsilon_{\mu\nu\gamma}\partial_{\nu}A_{\gamma}=\left(\epsilon_{ij}\partial_{i}A_{j},\epsilon_{i\nu\gamma}\partial_{\nu}A_{\gamma}\right),$
(32)

such that

$\displaystyle\pi=\epsilon_{ij}\partial_{i}A_{j}=\hat{\bf
z}\cdot\left({\bm{\nabla}}\times{\bf A}\right),$ (33)
$\displaystyle\sigma_{i}=-\epsilon_{i\nu\gamma}\partial_{\nu}A_{\gamma}=\epsilon_{ij}(\partial_{t}A_{j}-\partial_{j}A_{0}),$
(34)

and Equation (31) transforms into

$\partial_{t}\left(L-\epsilon_{xj}A_{j}\right)-\partial_{i}\left(j_{i}-\epsilon_{xi}A_{0}\right)\equiv\partial_{\mu}\tilde{j}_{\mu}=0,$
(35)

with
$\tilde{j}_{\mu}=\left(L-\epsilon_{xj}A_{j},-j_{i}+\epsilon_{xi}A_{0}\right)$,
which is then solved by introducing another vector gauge potential $a_{\mu}$,

$\tilde{j}_{\mu}=\epsilon_{\mu\nu\gamma}\partial_{\nu}a_{\gamma},$ (36)

such that

$\displaystyle L$ $\displaystyle=$
$\displaystyle\epsilon_{ij}\partial_{i}a_{j}+\epsilon_{xj}A_{j}=\hat{\bf
z}\cdot\left({\bm{\nabla}}\times{\bf a}+\hat{\bf x}\times{\bf A}\right),$ (37)
$\displaystyle j_{i}$ $\displaystyle=$
$\displaystyle\epsilon_{ij}(\partial_{t}a_{j}-\partial_{j}a_{0})+\epsilon_{xi}A_{0}.$
(38)

Substituting the solutions of $\pi,\sigma_{i},L$,and $j_{i}$ in terms of the
gauge fields into the Lagrangian density, leads to the dual Lagrangian density

$\tilde{\mathcal{L}}_{\text{sm}}=\mathcal{L}_{\text{M}}^{\text{sm}}+{\bf
A}\cdot{\bf j}^{b}+A_{0}n_{b}+{\bf a}\cdot{\bf j}^{s}+a_{0}n_{s},$ (39)

where $\mathcal{L}_{\text{M}}$ is the Maxwell part, given by

$\begin{split}\mathcal{L}_{\text{M}}^{\text{sm}}=&\frac{1}{2}\chi^{-1}\left(\partial_{t}{\bf
A}-{\bm{\nabla}}A_{0}\right)^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
A}\right)^{2}\\\ &+\frac{1}{2}K^{-1}\left(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}-A_{0}\hat{\bf
x}\right)^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf a}+\hat{\bf
x}\times{\bf A}\right)^{2},\end{split}$ (40)

and the charge contributions are obtained by integrating by parts and defining
the dislocation and disclination charge and current densities as,

$\displaystyle n_{b}$ $\displaystyle=$
$\displaystyle\epsilon_{ij}\partial_{i}\partial_{j}u,$ (41a) $\displaystyle
n_{s}$ $\displaystyle=$
$\displaystyle\epsilon_{ij}\partial_{i}\partial_{j}\theta,$ (41b)
$\displaystyle j^{b}_{i}$ $\displaystyle=$
$\displaystyle\epsilon_{ij}\left(\partial_{j}\partial_{t}u-\partial_{t}\partial_{j}u\right),$
(41c) $\displaystyle j^{s}_{i}$ $\displaystyle=$
$\displaystyle\epsilon_{ij}\left(\partial_{j}\partial_{t}\theta-\partial_{t}\partial_{j}\theta\right).$
(41d)

Introducing Hubbard-Stratonovich fields ${\bf E}$ and ${\bf e}$, the
Lagrangian density transforms into

$\begin{split}\tilde{\mathcal{L}}_{\text{sm}}=&-{\bf
E}\cdot\left(\partial_{t}{\bf A}-{\bm{\nabla}}A_{0}\right)-\frac{1}{2}\chi{\bf
E}^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf A}\right)^{2}-\frac{1}{2}K{\bf
e}^{2}\\\ &-{\bf e}\cdot\left(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}-A_{0}\hat{\bf
x}\right)-\frac{1}{2}\left({\bm{\nabla}}\times{\bf a}+\hat{\bf x}\times{\bf
A}\right)^{2}\\\ &+{\bf A}\cdot{\bf j}^{b}+A_{0}n_{b}+{\bf a}\cdot{\bf
j}^{s}+a_{0}n_{s}.\end{split}$ (42)

Integrating over $A_{0}$ and $a_{0}$ gives the Gauss law, leaving the standard
Lagrangian form, $\tilde{\mathcal{L}}_{\text{sm}}=-{\bf
E}\cdot\partial_{t}{\bf A}-{\bf e}\cdot\partial_{t}{\bf
a}-\tilde{\mathcal{H}}_{\text{sm}}$, from which we can read off the dual
Hamiltonian density, that is given by,

$\begin{split}\tilde{\mathcal{H}}_{\text{sm}}=&\frac{1}{2}\chi{\hat{\bf
E}}^{2}+\frac{1}{2}\left({\bm{\nabla}}\times{\hat{\bf
A}}\right)^{2}+\frac{1}{2}K{\hat{\bf e}}^{2}\\\
&+\frac{1}{2}\left({\bm{\nabla}}\times{\hat{{\bf a}}}+\hat{\bf
x}\times{\hat{\bf A}}\right)^{2}-{\hat{\bf A}}\cdot{\bf j}^{b}-\hat{\bf
a}\cdot{\bf j}^{s},\end{split}$ (43)

supplemented by the generalized Gauss laws

$\displaystyle{\bm{\nabla}}\cdot\hat{\bf E}$ $\displaystyle=$ $\displaystyle
n_{b}+\hat{\bf e}\cdot\hat{\bf x},$ (44)
$\displaystyle{\bm{\nabla}}\cdot\hat{\bf e}$ $\displaystyle=$ $\displaystyle
n_{s},$ (45)

where, ${\bf E}$ and ${\bf e}$, are independent electric fields, canonically
conjugate to the corresponding vector potentials, ${\bf A}$ and ${\bf a}$,
respectively.

The above Hamiltonian must be invariant under the gauge transformations:

$\displaystyle(A_{0},A_{i})\to
A_{\mu}^{\prime}=\left(A_{0}+\partial_{t}\chi,(A_{i}+\partial_{i}\chi)\right),$
(46a) $\displaystyle(a_{0},a_{i})\to
a_{\mu}^{\prime}=\left(a_{0}+\partial_{t}\phi,(a_{i}+\partial_{i}\phi+\hat{\bf
x}_{i}\chi)\right).\;\;\;\;\;\;\;$ (46b)

Requiring the source term to preserve this gauge invariance, we obtain coupled
continuity equations for charges (disclinations) and dipoles (dislocations),
satisfying,

$\displaystyle\partial_{t}n_{s}+{\bm{\nabla}}\cdot{\bf j}_{s}$
$\displaystyle=$ $\displaystyle 0,$ (47)
$\displaystyle\partial_{t}n_{b}+{\bm{\nabla}}\cdot{\bf j}_{b}$
$\displaystyle=$ $\displaystyle-j_{s}^{x}.$ (48)

We observe that the dipole (dislocation) continuity equation is violated by a
nonzero charge (disclination) current $j_{s}^{x}$ in the $\hat{\bf x}$ (along
the layers) direction. Thus, in the absence of gapped $\hat{\bf x}-$dipoles
($\hat{\bf y}$-dislocations), we find that $j_{s}^{x}=0$, i.e., motion of
isolated fracton charges (disclinations) is restricted to be transverse to the
smectic layers, as moving along the layers requires $\hat{\bf x}$-dipoles
($\hat{\bf y}$-dislocations) that are gapped in the smectic. Therefore, the
fractons (disclinations) exhibit subdimensional lineon dynamics, as argued in
Section II.2.

##  III Higgs transition of quantum crystal-to-smectic melting

As discussed in the Introduction, a smectic can emerge from anisotropic
melting HalperinOstlund  of a crystal, understood in terms of a Kosterlitz-
Thouless (KT)-like, KT  single-species dislocation unbinding transition. This
classical partial melting transition of an anistropic solid was studied by
Halperin and Ostlund HalperinOstlund , and we have used this fact in Equation
(17) to derive the smectic harmonic elasticity by including $\hat{{\bf
x}}-$orientated dislocations in a classical crystal. Although such a 2D
classical smectic is unstable to thermal fluctuations, driven into a nematic
fluid at any nonzero temperature HalperinOstlund ; TonerNelson  (see Section
IV), a (2+1)D quantum smectic at zero temperature is a stable state of matter.
In this section, we demonstrate that similarly, a quantum smectic can also
emerge from partial, anisotropic quantum melting of a crystal.

The Hamiltonian density of a 2D quantum crystal is,

$\begin{split}\mathcal{H}_{\text{cr}}=&\frac{1}{2}C_{ij,kl}(\partial_{i}\hat{u}_{j}-\hat{\theta}\epsilon_{ij})(\partial_{k}\hat{u}_{l}-\hat{\theta}\epsilon_{kl})+\frac{1}{2}K({\bm{\nabla}}\hat{\theta})^{2}\\\
&+\frac{1}{2}\hat{\bf\pi}^{2}+\frac{1}{2}\hat{L}^{2},\end{split}$ (49)

where, $\hat{\bf u}$ is the phonon field operator, $\hat{\theta}$ is the
orientational bond-angle field operator, and, $\hat{\pi}$ and $\hat{L}$ are
their corresponding canonically conjugate momentum respectively. $C_{ij,kl}$
is the elastic constant tensor, which takes the form,
$C_{ij,kl}=\lambda\delta_{ij}\delta_{kl}+2\mu\delta_{ik}\delta_{jl}$, for an
isotropic hexagonal lattice, characterized by two independent Lamé
coefficients, $\lambda$ and $\mu$. Working in the path-integral formulation
with the field operators replaced by corresponding classical fields, the
action is

$\displaystyle S_{\text{cr}}=$ $\displaystyle\int dt\int_{\bf
r}\frac{1}{2}\left[(\partial_{t}{\bf
u})^{2}+(\partial_{t}\theta)^{2}-K({\bm{\nabla}}\theta)^{2}\right.$
$\displaystyle\left.-C_{ij,kl}(\partial_{i}u_{j}-\theta\epsilon_{ij})(\partial_{k}u_{\ell}-\theta\epsilon_{kl})\right].$
(50)

Now, as the $\hat{\bf x}-$dislocations condensed, we include dislocations with
Burgers vector along the layers ($\hat{\bf x}$) by replacing
$\partial_{i}u_{x}=v_{ix}$, where $v_{i}$ is an arbitrary vector field with
$\epsilon_{ij}\partial_{i}v_{jx}=b_{x}$. Then, the Lagrangian density

$\displaystyle\mathcal{L}_{\text{cr+disl.}}=$
$\displaystyle\frac{1}{2}\left[(\partial_{t}{\bf
u})^{2}+(\partial_{t}\theta)^{2}-K({\bm{\nabla}}\theta)^{2}-\lambda\left({\bm{\nabla}}\cdot{\bf
u}\right)^{2}\right.$ (51)
$\displaystyle\left.-2\mu(\partial_{i}u_{j}-\theta\epsilon_{ij})^{2}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[(\partial_{t}{\bf
u})^{2}+(\partial_{t}\theta)^{2}-K({\bm{\nabla}}\theta)^{2}-\lambda(v_{xx}+\partial_{y}u_{y})^{2}\right.$
$\displaystyle\left.-2\mu\left(v_{xx}^{2}+(\partial_{y}u_{y})^{2}+(v_{yx}-\theta)^{2}+(\partial_{x}u_{y}-\theta)^{2}\right)\right.$
$\displaystyle\left.-\frac{1}{2}E_{c}(\partial_{x}v_{yx}-\partial_{y}v_{xx})^{2}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[(\partial_{t}{\bf
u})^{2}+(\partial_{t}\theta)^{2}-K({\bm{\nabla}}\theta)^{2}-\chi_{1}(\partial_{y}u_{y})^{2}\right.$
$\displaystyle\left.-\chi_{2}(\partial_{x}u_{y}-\theta)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[(\partial_{t}u_{y})^{2}+(\partial_{t}\theta)^{2}-\chi\left({\bm{\nabla}}u_{y}-\theta\hat{\bf
x}\right)^{2}-K({\bm{\nabla}}\theta)^{2}\right],$

where, in the third line, we have integrated out $v_{ix}$ and $u_{x}$, and
have defined $\chi_{1}=4\mu(\mu+\lambda)/(2\mu+\lambda)$ and $\chi_{2}=2\mu$,
which are taken to be equal in the last line, $\chi_{1}=\chi_{2}=\chi$, for
simplicity, corresponding to the case when $\lambda\to 0$. Then, we arrive at
the Lagrangian density of a smectic, starting with that of a quantum crystal,

$\mathcal{L}_{\text{sm}}=\mathcal{L}_{\text{cr+disl.}},$ (52)

as summarized by the flow chart in Fig. 1.

Motivated by this possibility of partial quantum melting of a crystal into a
smectic and the subsequent melting into a quantum nematic, we will explore its
dual for in this section. Below, we will also derive a dual gauge theory of a
quantum smectic, through a Higgs transition from a dual gauge theory of an
incommensurate quantum crystal (supersolid) by condensing one flavor of
dipoles, and thereby Higgs’ing out a flavor component of the gauge fields. As
required by consistency, we indeed find that the resulting quantum smectic
dual is in full agreement with a direct duality derived in Section II.

###  III.1 Soft-spin descriptions of quantum crystal and quantum smectic

####  III.1.1 Quantum crystal

The dual coupled U(1) vector gauge theory for a quantum crystal was first
derived in Ref. [RadzihovskyHermeleVectorGaugePRL2020, ], characterized by the
Lagrangian density

$\begin{split}\tilde{\mathcal{L}}_{\text{cr}}=&\mathcal{L}_{\text{M}}^{\text{cr}}+{\bf
A}_{k}\cdot{\bf J}_{k}+A_{0k}n_{k}^{b}+{\bf a}\cdot{\bf
j}^{s}+a_{0}n_{s},\end{split}$ (53)

where $k=(x,y)$ indexes different flavors, ${\bf A}_{k}$, ${\bf a}$ gauge
fields capture the $k=x,y$ phonons and bond orientational order respectively.
The Maxwell part, $\mathcal{L}_{\text{M}}^{\text{cr}}$, is given by

$\begin{split}\mathcal{L}_{\text{M}}^{\text{cr}}=&\frac{1}{2}\chi^{-1}\left(\partial_{t}{\bf
A}_{k}-{\bm{\nabla}}A_{0k}\right)^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
A}_{k}\right)^{2}\\\ +&\frac{1}{2}K^{-1}(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}-A_{0k}\hat{\bf
e}_{k})^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf a}-\hat{\bf z}\times{\bf
A}_{k}\right)^{2}.\end{split}$ (54)

To access descendant phases and corresponding quantum phase transitions, we
need to treat dislocation and disclination defects as dynamical charges.
Following a standard analysis and focusing on dislocations (dipoles) for the
moment, we introduce the dynamical field $\psi_{k}({\bf
r},t)=\sqrt{\rho_{k}}e^{i\varphi_{k}}$ for each gauge-charged dipole species
${\bf p}_{k}$, and add corresponding kinetic energies,
$\frac{\rho_{k}}{2}\left(\partial_{t}\varphi_{k}+pA_{0k}\right)^{2}$, where we
have integrated out the massive magnitude fluctuations to focus on the low-
energy phase fluctuations only.

Figure 7: Dislocation climb via vacancies diffusion. An edge dislocation moves
out of the slip plane onto a parallel plane directly above or below the slip
plane. This movement is termed nonconservative, as compared with conservative
movement (glides). (a) Diffusion of vacancy to edge dislocation. (b)
Dislocation climbs up one lattice spacing.

As discussed in the Introduction, in the Mott-insulating commensurate crystal
phase, the dipole ${\bf p}_{k}$ can only move in the direction perpendicular
to ${{\bf p}}_{k}$ while the along-dipole climb is forbidden due to the U(1)
particle-number conservation symmetry (“glide-constraint”). Thus, we have only
the glide motion, $\Pi_{ij}^{\perp{{\bf p}_{k}}}D_{j}\psi_{{\bf p}_{k}}$,
with, ${\bf D}={\bm{\nabla}}+ip_{k}{\bf A}_{k}$, the covariant spatial
derivative, and $\Pi_{ij}^{\perp{{\bf
p}_{k}}}=\delta_{ij}-\frac{p_{i,k}p_{j,k}}{p_{k}^{2}}$, the transverse
projection operator. However, the crystal also exhibits scalar non-topological
point defects, corresponding to deficiency and excess in atom density, which
permits the climb process of the dipoles (dislocations), as shown in Fig. 7.
Combined with the bosonic statistics of the underlying particles, the quantum
crystal can first develop into a super-solid phase (“incommensurate” crystal),
featuring both the crystalline and the superfluid orders. The condensation of
vacancies or interstitials in the super-solid phase, therefore, frees these
symmetry-forbidden climb constraintsMarchettiRadzihovsky;
PretkoLRsymmetryEnrichedPRL2018 ; Kumar19 , $\Pi_{ij}^{\parallel{{\bf
p}_{k}}}D_{j}\psi_{{\bf p}_{k}}$, where $\Pi_{ij}^{\parallel{{\bf
p}_{k}}}=\frac{p_{i,k}p_{j,k}}{p_{k}^{2}}$ is the longitudinal projection
operator. In Appendix A, we show in detail how the dislocation-superfluid
coupling alleviates the glide constraint and converts the dislocations
(dipoles) from subdimensional quasi-particles to ordinary mobile defects,
acted upon by the full spatial derivative, ${\bf D}\psi_{{\bf p}_{k}}$.

Figure 8: A disclination quadrupole, constructed as a bound pair of two equal
and opposite dislocations with Burgers vectors ${\bf b}$ and $-{\bf b}$,
carries a unit of vacancy number, as can be seen by the deficiency of a single
atom in the middle of the configuration. As the dislocations proliferate, the
condensation of pairs of opposite dislocations $\left({\bf b},-{\bf b}\right)$
much always accompany the condensation of vacancies/interstitials, which
correspond to terms of $\mathcal{L}_{\text{sf-dis}}$, the coupling between
dislocation climb operators and vacancy or interstitial condensate (see
Appendix A).

Introducing defect’s core energy, $E_{j_{\mu}^{b}}$, to account for lattice-
scale physics, and the dipole (dislocation) charge density and current density
on a discrete lattice is given as a sum of their discrete charges

$\displaystyle n^{b}_{k}({\bf r})$ $\displaystyle=$ $\displaystyle\sum_{{\bf
r}_{n}}d\cdot n^{b}_{k}({{\bf r}_{n}})\delta^{2}({\bf r}-{\bf r}_{n}),$ (55)
$\displaystyle{\bf j}^{b}_{k}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{{\bf r}_{n}}d\cdot{\bf j}^{b}_{k}({{\bf
r}_{n}})\delta^{2}({\bf r}-{\bf r}_{n}),$ (56)

where $d$ is the lattice spacing for a 2D crystal, which is also the
elementary charge of the gauge dipoles (unit of the dislocation charge), i.e.,
$|{\bf p}_{k}|=p=d$. The partition function is then given by

$\displaystyle Z$ $\displaystyle=\sum_{k}\int\prod_{{\bf
r}_{n}}dA_{\mu,k}({\bf
r}_{n})\sum_{j_{\mu,k}^{b}}\delta\left(\Delta_{\mu}j_{\mu,k}^{b}\right)\cdot
e^{-S_{\text{M}}^{\text{cr}}}$ (57) $\displaystyle\cdot e^{-\int dt\sum_{{\bf
r}_{n}}\sum_{k}\left[\right.\frac{m_{p}}{2}\left(\partial_{t}\varphi_{k}+pA_{0,k}\right)^{2}-d^{2}E_{j_{\mu,k}^{b}}|j_{\mu,k}^{b}({\bf
r}_{n})|^{2}+dA_{\mu,k}j_{\mu,k}^{b}\left.\right]}$
$\displaystyle\equiv\sum_{k}\int\prod_{{\bf r}_{n}}dA_{\mu,k}({\bf
r}_{n})d\varphi_{k}({\bf
r}_{n})\sum_{j_{\mu,k}^{b}}e^{-\tilde{S}_{\text{cr}}[A_{\mu,k},j_{\mu,k}^{b}]}$

with the action

$\begin{split}\tilde{S}_{\text{cr}}=&\int dt\sum_{{\bf
r}_{n}}\sum_{k}\bigg{[}\frac{m_{p}}{2}\left(\partial_{t}\varphi_{k}+pA_{0,k}\right)^{2}-\tilde{E}_{j_{\mu,k}^{b}}|j_{\mu,k}^{b}({\bf
r}_{n})|^{2}\\\
&+d\left(A_{\mu,k}+\frac{1}{d}\Delta_{\mu}\varphi_{k}\right)j_{\mu,k}^{b}({\bf
r}_{n})\bigg{]}+S_{\text{M}}^{\text{cr}},\end{split}$ (58)

where, $m_{p}$ is the effective inertia mass of the dipoles,
$\tilde{E}_{j_{\mu,k}^{b}}=d^{2}E_{j_{\mu,k}^{b}}$ in the discrete lattice,
and, $\Delta_{\mu}j_{\mu,k}^{b}=j^{b}_{\mu,k}({\bf
r}+{\bf\mu})-j^{b}_{\mu,k}({\bf r})$ and
$\Delta_{\mu}\varphi_{k}=\varphi_{k}({\bf r}+{\bf\mu})-\varphi_{k}({\bf r})$,
$\mu=x,y$, are the discrete lattice derivatives. Note the continuity equation
$\Delta_{\mu}j_{\mu,k}^{b}=0$ is automatically satisfied when we integrate out
$\varphi_{k}$, which is the phase of the $k-$flavor dipole field
$\psi_{k}=|\psi_{k}|e^{i\varphi_{k}}$. After tracing over the 3-currents
$j_{\mu,k}^{b}=(n^{b}_{k},{\bf j}^{b}_{k})$, we obtain

$\begin{split}\tilde{S}_{\text{cr}}=&\int dt\sum_{{\bf
r}_{n}}\sum_{k}\bigg{[}\frac{m_{p}}{2}\left(\partial_{t}\varphi_{k}+pA_{0,k}\right)^{2}\\\
&-g^{b}_{k}\cos\left(\Delta_{\mu}\varphi_{k}+pA_{\mu,k}\right)\bigg{]}+S_{\text{M}}^{\text{cr}},\end{split}$
(59)

where $g^{b}_{k}=2e^{-\tilde{E}_{j^{b}_{k}}}$, and we have approximated the
resulting Villain potential by its lowest harmonic.

In the continuum limit, we have

$\tilde{S}_{\text{cr}}=\int dt\int d^{2}r\tilde{\mathcal{L}}_{\text{cr}},$
(60)

with the Lagrangian density $\tilde{\mathcal{L}}_{\text{cr}}$ given by

$\displaystyle\tilde{\mathcal{L}}_{\text{cr}}=$
$\displaystyle\sum_{k}\left[\frac{\rho_{k}}{2}\left(\partial_{t}\varphi_{k}+pA_{0,k}\right)^{2}-\tilde{g}^{b}_{k}\cos\left({\bm{\nabla}}\varphi_{k}+p{\bf
A}_{k}\right)\right]$ (61) $\displaystyle+\mathcal{L}_{\text{M}}^{\text{cr}},$

where,
$\tilde{g}^{b}_{k}=\frac{g^{b}_{k}}{d^{2}}=\frac{2}{d^{2}}e^{-\tilde{E}_{j^{b}_{k}}}$.

We then turn to an equivalent “soft-spin" description by noting that this
ordered phase action emerges from a corresponding quantum Ginzburg-Landau
theory for the complex order parameter, $\psi_{k}=|\psi_{k}|e^{i\varphi_{k}}$,
and write $\tilde{\mathcal{L}}_{\text{cr}}$ as,

$\tilde{\mathcal{L}}_{\text{cr}}=\sum_{k}\frac{1}{2}|\left(\partial_{\mu}+ipA_{\mu,k}\right)\psi_{k}|^{2}-V\left(\\{|\psi_{k}|\\}\right)+\mathcal{L}_{\text{M}}^{\text{cr}},$
(62)

where, $\psi_{k}$ correspond to $\hat{\bf x}$\- and ${\bf y}$-oriented dipole
fields ($\hat{\bf y}-$ and $\hat{\bf x}-$dislocations) in a square lattice,
and $V\left(\\{|\psi_{k}|\\}\right)$, is the Landau U(1)-invariant potential
for the quantum crystal, the form of which controls the type and subsequence
of phase transitions (See Section III.2). It is straightforward to verify that
Equation (62) reduces to Equation (61) when the gapped Higgs-like magnitude
degrees of freedom, $|\psi_{k}|$, whose fluctuations are controlled by
$V\left(\\{|\psi_{k}|\\}\right)$, are integrated out, and therefore these two
Lagrangians are equivalent.

####  III.1.2 Quantum smectic

Similar analysis applies also for the quantum smectic. The condensation of
vacancies or interstitials in the super-smectic (“incommensurate" smectic)
phase also endows the full mobility of dipoles (dislocations) in the smectic.

Then, starting with $\tilde{\mathcal{L}}_{\text{sm}}$ given by Equation (39)
and elevating dislocation and disclination defects into dynamical charges, we
add corresponding kinetic energies,
$\frac{\rho}{2}\left(\partial_{t}\varphi_{x}+pA_{0}\right)^{2}$, and follow
the same procedure as what have been done for the quantum crystal above, which
leads to the effective Lagrangian density of the super-smectic in the
continuum given by,

$\displaystyle\tilde{\mathcal{L}}_{\text{sm}}=$
$\displaystyle\frac{\rho}{2}\left(\partial_{t}\varphi_{x}+pA_{0}\right)^{2}-\tilde{g}_{b}\cos\left({\bm{\nabla}}\varphi_{x}+p{\bf
A}\right)+\mathcal{L}_{\text{M}}^{\text{sm}}$ (63) $\displaystyle=$
$\displaystyle\frac{1}{2}|\left(\partial_{\mu}+ipA_{\mu}\right)\psi_{x}|^{2}-V\left(|\psi_{x}|\right)+\mathcal{L}_{\text{M}}^{\text{sm}},$

where for concreteness we have taken the layers to be along the $\hat{\bf x}$
axis, and replaced $\tilde{g}^{b}_{x}$ simply by $\tilde{g}_{b}$ for the
$\hat{\bf x}$-dipole ($\hat{\bf y}$-dislocation) fugacity, and $A_{\mu,x}$
simply by $A_{\mu}$. In the second form, we have written it as an equivalent
“soft-spin” description in terms of $\psi_{x}=|\psi_{x}|e^{i\varphi_{x}}$,
with the Landau U(1)-invariant potential, $V\left(|\psi_{x}|\right)$, which is
equivalent to the first form when the gapped Higgs magnitude degree of
freedom, $|\psi_{x}|$, whose fluctuations are controlled by
$V\left(|\psi_{x}|\right)$, is integrated out.

###  III.2 Crystal-to-smectic and smectic-to-nematic transitions

The quantum crystal phase can go through a fully isotropic melting transition,
mediated by dislocations, into a hexatic (or nematic) phase, or through a
multi-stage anisotropic transition, first partially melting into a smectic
phase, depending on the form of the Landau potential $V(\\{|\psi_{k}|\\})$. A
simple discussion based on Ginzburg-Landau theory of continuous phase
transitions is given in the following.

Considering a 2D square lattice, the U(1)-invariant Landau potential
$V(\\{|\psi_{k}|\\})$, satisfying the symmetries of the system, expanded to
fourth-order of the dipole fields $\psi_{k}$, is given by

$\begin{split}V&=\frac{\alpha}{2}\sum_{k}|\psi_{k}|^{2}+\frac{\beta}{4}\sum_{k}|\psi_{k}|^{4}+\frac{\beta^{\prime}}{2}|\psi_{x}|^{2}|\psi_{y}|^{2}\\\
&\equiv\frac{\alpha}{2}|\Psi|^{2}+\frac{\beta}{4}|\Psi|^{4}+\frac{1}{2}\left(\beta^{\prime}-\beta\right)|\psi_{x}|^{2}|\psi_{y}|^{2},\end{split}$
(64)

where, $\beta,\beta^{\prime}>0$, and we have defined the vector complex order
parameter $\Psi=(\psi_{x},\psi_{y})$. Note that by these two forms, we can
regard this potential as two identical complex Ising models coupled together,
or a complex XY model with the originally rotational symmetry broken down to a
rectangular one. As usual, at mean-field level the phase transition takes
place at $\alpha=0$, as $\alpha$ sign.

####  III.2.1 Crystal-to-nematic transition

We first consider the case of $\beta>\beta^{\prime}$, when it is energetically
favorable for both flavors of dipole fields to condense with the same
expectation value
$|\psi_{x}|=|\psi_{y}|=\sqrt{\frac{|\alpha|}{\beta+\beta^{\prime}}}$,
corresponding to the nematic phase. This crystal-to-nematic transition,
analogue of the crystal-to-hexatic melting in a 2D classical crystal, has been
discussed in Ref. [PretkoZhaiLRdualityPRB, ], formulated as a Ginzburg-Landau
theory of tensor superconductors, i.e., dipole fields coupled to the symmetric
tensor gauge field. However, as pointed out in the Introduction, that tensor-
only gauge theory formulation of the Mott-insulating “commensurate” crystal,
fails to capture the full dipole mobility endowed by the condensation of
vacancies and interstitials. Here, we take a complementary coupled vector
gauge theory descriptionRadzihovskyHermelePRL and condense both $\hat{\bf
x}$\- and $\hat{\bf y}$-dipoles, $\psi_{x}\neq 0,\psi_{y}\neq 0$. Via
Anderson-Higgs mechanism this gaps out all gauge field flavors,
$A_{\mu,k},k=x,y$. These can then be safely integrated out at low energies at
wavelengths longer than $\sqrt{\frac{2m_{p}}{\rho_{k}p^{2}}}$ (i.e., the
penetration length of $A_{\mu,k}$), thereby reducing the Maxwell Lagrangian of
the crystal to that of a form described by the rotational gauge field
$a_{\mu}$ only. With details relegated to Appendix B, the result is given by,

$\begin{split}\mathcal{L}_{\text{M}}^{\text{cr}}\left(A_{\mu,k}=0,a_{\mu}\right)\approx&\mathcal{L}_{\text{M}}^{\text{nm}}\left(a_{\mu}\right)\\\
=&\frac{1}{2}K^{-1}\left(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}\right)^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
a}\right)^{2},\end{split}$ (65)

which corresponds to setting, $A_{\mu,k}\approx 0$, to the lowest order, and
neglecting anisotropies of the resulting nematic state.

As demonstrated by foundational papers on duality dasgupta ; fisher , such
Abelian gauge theory is dual to a quantum XY model of the nematic given by,

$\mathcal{L}_{\text{nm}}=\frac{1}{2}\left(\partial_{t}\theta\right)^{2}-\frac{1}{2}K\left({\bm{\nabla}}\theta\right)^{2},$
(66)

where $\theta$ is the layer orientation, confirming a consistency with our
physical expectations.

####  III.2.2 Crystal-to-smectic and smectic-to-nematic transitions

We next consider the case of $\beta<\beta^{\prime}$, when it is favorable to
have only one flavor of dipole fields condensed, with $\psi_{x}=0$ and
$|\psi_{y}|=\sqrt{\frac{|\alpha|}{\beta}}$, or, $\psi_{y}=0$ and
$|\psi_{x}|=\sqrt{\frac{|\alpha|}{\beta}}$, corresponding to the crystal-to-
smectic phase transition, which restores translational symmetry in only one
direction. With our interest in the quantum smectic, in the following we will
thus focus on the $\beta<\beta^{\prime}$ case, and dualize the quantum smectic
alternatively, via Anderson-Higgs mechanism, gapping out one of the flavors of
the gauge fields in this smectic phase, formulated in a “soft-spin”
description of a 2D quantum crystal.

The crystal-smectic partial melting transition corresponds to condensation of
one flavor of the dipole fields, that according to Fig. 2, we take to be
$\hat{\bf y}$-dipoles, i.e. $\psi_{y}\neq 0$. Within this Higgs phase, the
corresponding $y$ flavor gauge field $A_{\mu,y}$ is gapped out, and can be
safely integrated out in the low-energy regime with wavelengths of excitations
much greater than $\sqrt{\frac{2m_{p}}{\rho_{y}p^{2}}}$ (i.e., the penetration
length of $A_{\mu,y}$ ). To lowest order, it corresponds to $A_{\mu,y}\approx
0$, reducing the crystal’s Maxwell Lagrangian to that of a smectic,

$\mathcal{L}_{\text{M}}^{\text{cr}}\left(A_{\mu,x},A_{\mu,y}=0,a_{\mu}\right)\approx\mathcal{L}_{\text{M}}^{\text{sm}}\left(A_{\mu,x},a_{\mu}\right).$
(67)

The detailed derivations are given in Appendix B. Subsequent melting with a
condensation of $\hat{\bf x}-$dipoles ($\hat{\bf y}-$dislocations) leads to a
single vector gauge theory for $a_{\mu}$, with both $A_{\mu,k}$ gapped out, in
the low-energy regime with wavelengths of excitations much greater than
$\sqrt{\frac{2m_{p}}{\rho_{x}p^{2}}}$, i.e.,

$\begin{split}\mathcal{L}_{\text{M}}^{\text{sm}}\left(A_{\mu,x}=0,a_{\mu}\right)\approx&\mathcal{L}_{\text{M}}^{\text{nm}}\left(a_{\mu}\right)\\\
=&\frac{1}{2}K^{-1}\left(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}\right)^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
a}\right)^{2},\end{split}$ (68)

to the lowest order. As expected, this corresponds to the dual of the quantum
XY model of the nematic state.

###  III.3 Vector to tensor gauge theory redux in low-energy limit

The Maxwell part of a crystal, given by Equation (54), is gauge invariant
under the following transformation,

$\begin{split}\left(A_{0k},{\bf
A}_{k}\right)&\to\left(A_{0k}+\partial_{t}\chi_{k},{\bf
A}_{k}+{\bm{\nabla}}\chi_{k}\right),\\\
\left(a_{0},a_{k}\right)&\to\left(a_{0}+\partial_{t}\phi,a_{k}+\partial_{k}\phi+\chi_{k}\right).\end{split}$
(69)

As demonstrated explicitly in Ref. [RadzihovskyHermeleVectorGaugePRL2020, ],
the enlarged gauge redundancy allows us to completely eliminate ${\bf a}$ by
choosing $\chi_{k}=a_{k}$, as a result of which, the term
$\frac{1}{2}\left({\bm{\nabla}}\times{\bf a}-\hat{\bf z}\times{\bf
A}_{k}\right)^{2}$ reduces to
$\frac{1}{2}\left(\epsilon_{ik}A_{ik}\right)^{2}$, thereby gapping out the
antisymmetric component $\epsilon_{ik}A_{ik}$ at energies well below this gap,
i.e., with length scales greater than $\sqrt{\chi}$. Furthermore, the electric
field term $\frac{1}{2}K^{-1}(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}-A_{0k}\hat{\bf e}_{k})^{2}$ reduces to
$\frac{1}{2}K^{-1}({\bm{\nabla}}a_{0}+A_{0k}\hat{\bf e}_{k})^{2}$ under this
transformation, enforcing $A_{0k}=-\partial_{k}a_{0}$ at low enough energies
with length scales greater than $\sqrt{\frac{K}{\chi}}$. Therefore, the dual
coupled U(1) vector gauge theory for a quantum crystal, reduces to the dual
tensor gauge theory in the low-energy limit, with,
$\mathcal{L}_{\text{M}}^{\text{cr}}$, reduces to that in the tensor gauge
theory, described by,

$\displaystyle\mathcal{L}_{\text{M}}^{\text{cr}}=$
$\displaystyle\frac{1}{2}\chi^{-1}\left(\partial_{t}A_{ik}+\partial_{i}\partial_{k}a_{0}\right)^{2}-\frac{1}{2}\left(\epsilon_{ji}\partial_{j}A_{ik}\right)^{2}$
(70) $\displaystyle=$ $\displaystyle-
E_{ik}\left(\partial_{t}A_{ik}+\partial_{i}\partial_{k}a_{0}\right)-\frac{1}{2}\chi
E_{ik}^{2}-\frac{1}{2}B_{k}B^{k},$

where $A_{ik}$, is a rank-2 symmetric tensor field, which corresponds to the
$i^{\text{th}}$ component of $k$-flavor vector gauge field ${\bf A}_{k}$ in
the coupled vector gauge theory, $a_{0}$ is a scalar field with
$\partial_{k}a_{0}$ corresponding to $A_{0k}$ in the vector gauge theory,
$B_{k}=\epsilon_{ji}\partial_{j}A_{ik}$, and $E_{ik}$ is the electric tensor
field canonically conjugate to $A_{ik}$.

In contrast, within the smectic Higgs phase, corresponding to the condensation
of $\hat{\bf y}-$dipoles ($\hat{\bf x}-$dislocations), we cannot eliminate
${\bf a}$ completely, since we have already made a gauge choice with
$\chi_{y}=\frac{1}{p}\varphi_{y}$,

$A_{iy}\to A_{iy}+\frac{1}{p}\partial_{i}\varphi_{y},\;A_{0y}\to
A_{0y}+\frac{1}{p}\partial_{t}\varphi_{y},\;a_{y}\to
a_{y}+\frac{1}{p}\varphi_{y},$ (71)

to absorb the phase $\varphi_{y}$ of the condensed dipole field
$\psi_{y}=\sqrt{\rho_{y}}e^{i\varphi_{y}}$ into the gauge fields $A_{\mu,y}$.
Integrating out the gapped gauge field components, $A_{iy}$ and
$\partial_{y}a_{0}$, reduces $\mathcal{L}_{\text{M}}^{\text{cr}}$, in a
condensate of $\hat{\bf y}-$dipoles, to that of a smectic,

$\displaystyle\mathcal{L}_{\text{M}}^{\text{cr}}=$
$\displaystyle\frac{1}{2}\chi^{-1}\left(\partial_{t}A_{ix}+\partial_{i}\partial_{x}a_{0}\right)^{2}-\frac{1}{2}\left(\epsilon_{ji}\partial_{j}A_{ix}\right)^{2}$
(72) $\displaystyle+\frac{1}{2}K^{-1}\left(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}-A_{0x}\hat{\bf
x}\right)^{2}+\frac{1}{2}\left(\epsilon_{ij}\partial_{i}a_{j}+A_{yx}\right)^{2}$
$\displaystyle=$ $\displaystyle-{\bf E}_{x}\cdot\left(\partial_{t}{\bf
A}_{x}-{\bm{\nabla}}A_{0x}\right)-\frac{1}{2}\chi{\bf
E}_{x}^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf A}_{x}\right)^{2}$
$\displaystyle-{\bf e}\cdot\left(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}-A_{0x}\hat{\bf x}\right)-\frac{1}{2}K{\bf e}^{2}$
$\displaystyle-\frac{1}{2}\left({\bm{\nabla}}\times{\bf a}+\hat{\bf
x}\times{\bf A}_{x}\right)^{2},$

which matches exactly with Equation (40) after setting
$A_{0x}=-\partial_{x}a_{0}$ and dropping the ‘x’ index.

We may also explore a 2D quantum smectic at low energies by similar analysis,
as what has been done for a crystal in Equation (70). In the smectic case,
choosing $\chi=a_{x}$ in the gauge transformation (46) allows us to eliminate
$a_{x}$ completely, as a result of which, the term
$\frac{1}{2}\left({\bm{\nabla}}\times{\bf a}+\hat{\bf x}\times{\bf
A}\right)^{2}$ of $\mathcal{L}_{\text{M}}^{\text{sm}}$ given by (40), reduces
to, $\frac{1}{2}\left(\partial_{x}a_{y}+A_{y}\right)^{2}$, thereby enforcing
$A_{y}=-\partial_{x}a_{y}$ at sufficiently low energies, with length scales
greater than $\sqrt{\chi}$. Furthermore, the term
$\frac{1}{2}K^{-1}\left(\partial_{t}{\bf a}-{\bm{\nabla}}a_{0}-A_{0}\hat{\bf
x}\right)^{2}$, reduces to
$\frac{1}{2}K^{-1}\left[\left(\partial_{t}a_{y}-\partial_{y}a_{0}\right)^{2}+\left(\partial_{x}a_{0}+A_{0}\right)^{2}\right]$
under this transformation, enforcing $A_{0}=-\partial_{x}a_{0}$ in low energy
regime with length scales greater than $\sqrt{\frac{K}{\chi}}$. Therefore, in
the low-energy limit, $\mathcal{L}_{\text{M}}^{\text{sm}}$ reduces to,

$\begin{split}\mathcal{L}_{\text{M}}^{\text{sm}}=&\frac{1}{2}\chi^{-1}\left(\partial_{t}A_{i}+\partial_{i}\partial_{x}a_{0}\right)^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
A}\right)^{2}\\\
&+\frac{1}{2}K^{-1}\left(\partial_{t}a_{y}-\partial_{y}a_{0}\right)^{2}\\\
=&-E_{ix}\left(\partial_{t}A_{i}+\partial_{x}\partial_{x}a_{0}\right)-\frac{1}{2}\chi
E_{ix}^{2}-\frac{1}{2}B_{x}^{2}\\\
&+\frac{1}{2}K^{-1}\left(\partial_{t}a_{y}-\partial_{y}a_{0}\right)^{2},\end{split}$
(73)

where $E_{ix}$ is a tensor gauge field, corrsponding to the $i^{\text{th}}$
component of the vector field ${\bf E}_{x}$.

Figure 9: Vector gauge theory of a 2D quantum crystal and smectic, in the low-
energy limit reduce to their corresponding tensor gauge theory forms.

##  IV Classical limit of smectic-gauge theory duality

###  IV.1 2D classical smectic duality

As a consistency check on our quantum dual theory for a 2D quantum smectic, we
anticipate that the classical smectic theory must emerge as the classical
limit of the above duality, as we demonstrate explicitly below.

The elasticity of a 2D classical smectic is given by the Hamiltonian density,

$\begin{split}\mathcal{H}_{\text{sm}}&=\frac{1}{2}\chi\left({\bm{\nabla}}u-\theta\hat{\bf
x}\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}\theta\right)^{2}\\\
&=\frac{1}{2}\chi^{-1}{\bf\sigma}^{2}-i{\bf\sigma}\cdot\left({\bm{\nabla}}u-\theta\hat{\bf
x}\right)+\frac{1}{2}K^{-1}{\bf j}^{2}-i{\bf
j}\cdot{\bm{\nabla}}\theta.\end{split}$ (74)

where we have introduced two Hubbard-Stratanovich fields ${\bf\sigma}$ and
${\bf j}$ to decouple the two elastic terms.

We derive its classical dual by integrating out the smooth part of $u$ and
$\theta$, which leads to the constraints:

$\begin{split}\partial_{i}\sigma_{i}&=0,\\\
\partial_{i}j_{i}+\sigma_{x}&=0.\end{split}$ (75)

The first equation can be solved in terms of a scalar potential $\phi$,
$\sigma_{i}=\epsilon_{ij}\partial_{j}\phi$, which inside the second constraint
gives,

$\partial_{i}(j_{i}+\epsilon_{xi}\phi)=0,$ (76)

that is then solved by introducing another potential $\alpha$,
$j_{i}=\epsilon_{ij}\partial_{j}\alpha-\epsilon_{xi}\phi$.

Substituting $\sigma_{i}$ and $j_{i}$ back into the original Hamiltonian, and
integrating by parts, lead to

$\begin{split}\tilde{\mathcal{H}}_{\text{sm}}=&\frac{1}{2}\chi^{-1}\left({\bm{\nabla}}\phi\right)^{2}-i\epsilon_{ij}\partial_{j}\phi\left(\partial_{i}u-\delta_{ix}\theta\right)\\\
&+\frac{1}{2}K^{-1}\left(\epsilon_{ij}\partial_{j}\alpha-\epsilon_{xi}\phi\right)^{2}-i\left(\epsilon_{ij}\partial_{j}\alpha-\epsilon_{xi}\phi\right)\partial_{i}\theta\\\
=&\frac{1}{2}\chi^{-1}\left({\bm{\nabla}}\phi\right)^{2}+i\phi\epsilon_{ij}\partial_{j}\left(\partial_{i}u-\delta_{ix}\theta\right)\\\
&+\frac{1}{2}K^{-1}\left({\bm{\nabla}}\alpha+\phi\hat{\bf
x}\right)^{2}+i\theta\epsilon_{ij}\partial_{i}\partial_{j}\alpha+i\phi\epsilon_{xi}\partial_{i}\theta\\\
=&\frac{1}{2}\chi^{-1}\left({\bm{\nabla}}\phi\right)^{2}+\frac{1}{2}K^{-1}\left({\bm{\nabla}}\alpha+\phi\hat{\bf
x}\right)^{2}-i\phi b-i\alpha s,\end{split}$ (77)

where we have defined $b=\epsilon_{ij}\partial_{i}\partial_{j}u$ and
$s=\epsilon_{ij}\partial_{i}\partial_{j}\theta$, as the dislocation and
disclination densities respectively.

Low-energy regime, $\frac{1}{2}K^{-1}\left({\bm{\nabla}}\alpha+\phi\hat{\bf
x}\right)^{2}\approx k_{B}T$, i.e., at length scales greater than
$\sqrt{Kk_{B}T}$, integrating over $\phi$ , to lowest-order, sets
$\phi=\partial_{x}\alpha$, and therefore, gives

$\tilde{\mathcal{H}}_{\text{sm}}=\frac{1}{2}\chi^{-1}\left(\partial_{x}^{2}\alpha\right)^{2}+\frac{1}{2}K^{-1}\left(\partial_{y}\alpha\right)^{2}-i\partial_{x}\alpha
b-i\alpha s,$ (78)

which, as expected, turns out to be the electrostatic limit of the quantum
smectic duality, i.e., Equation (39) and Equation (40), with $\phi=A_{0}$ and
$\alpha=a_{0}$.

Focusing on dislocations and neglecting the high energy disclination defects,
we can straightforwardly integrate out $\alpha({\bf r})$ in the partition
function, obtaining a dislocation Coulomb gas Hamiltonian

$H_{\text{b}}=\frac{1}{2}\int\frac{d^{2}q}{(2\pi)^{2}}b({\bf q})\tilde{K}({\bf
q})b(-{\bf q}),$ (79)

with,

$\tilde{K}({\bf
q})=\frac{d^{2}}{a^{2}}\frac{Kq_{x}^{2}}{q_{y}^{2}+\lambda^{2}q_{x}^{4}}+2E_{b},$
(80)

where, $E_{\text{b}}$ is the defect core energy, d is the layer spacing, and a
is the lattice spacing between atoms within the layers, and “penetration"
length $\lambda$ is defined as, $\lambda^{2}\equiv K/\chi$. Thus, the
dislocations Coulomb gas Hamiltonian in real space reduces to,

$H_{\text{b}}=\frac{1}{2}\int_{{\bf r}_{1},{\bf r}_{2}}U({\bf r}_{1}-{\bf
r}_{2})b({\bf r}_{1})b({\bf r}_{2})+E_{\text{b}}\int_{\bf r}b^{2}({\bf r}),$
(81)

where,

$U({\bf
r})=\frac{1}{4}\frac{d^{2}}{a^{2}}\chi\left(\frac{\lambda}{\pi|y|}\right)^{1/2}e^{-x^{2}/4\lambda|y|},$
(82)

as first found by Toner and Nelson in Ref. [TonerNelson, ]. The complete
Hamiltonian contains also a smooth phonon part, $H_{0}$, depending only the
smooth, single-valued part, $\tilde{u}$, of the displacement,
$u=\tilde{u}+u^{s}$, i.e.,

$H=H_{0}+H_{b}.$ (83)

with $H_{0}$ given by

$H_{0}=\frac{1}{2}\int
d^{2}r\left[\chi\left(\partial_{y}\tilde{u}\right)^{2}+K\left(\partial_{x}^{2}\tilde{u}\right)^{2}\right]$
(84)

With this Hamiltonian, we can study the effects of phonons and dislocations at
finite temperatures on translational and orientational orders.

Effect of phonon fluctuations on the translation order is expressed in terms
of correlations in the order parameter, $\psi\left({\bf
r}\right)=|\psi|e^{-iq_{0}u({\bf r})}$ as,

$\begin{split}\langle\psi({\bf r})\psi^{*}({\bf 0})\rangle&\sim\langle
e^{-iq_{0}\left[u({\bf r})-u({\bf 0})\right]}\rangle\\\
&=e^{-\frac{1}{2}q_{0}^{2}\langle[u({\bf r})-u({\bf 0})]^{2}\rangle}\\\
&\sim\left\\{\begin{array}[]{ll}\exp\left(-\frac{q_{0}^{2}k_{B}T}{\chi}\sqrt{\frac{|y|}{4\pi\lambda}}\right),\mbox{
for $x^{2}\ll\lambda y$},\\\
\exp\left(-\frac{q_{0}^{2}k_{B}T}{4\chi\lambda}|x|\right),\mbox{ for
$x^{2}\gg\lambda y$,}\end{array}\right.\end{split}$ (85)

where we have used the fact that

$\begin{split}\langle[u({\bf r})-u({\bf 0})]^{2}\rangle&=\int_{{\bf
q}}\int_{{\bf q}^{\prime}}\left(e^{i{\bf q}\cdot{\bf
r}}-1\right)\left(e^{i{\bf q}^{\prime}\cdot{\bf r}}-1\right)\langle u({\bf
q})u({\bf q}^{\prime})\rangle\\\ &=\int_{{\bf q}}\left(2-2\cos({\bf
q}\cdot{\bf
r})\right)\frac{k_{B}T}{\chi\left(q_{y}^{2}+\lambda^{2}q_{x}^{4}\right)}\\\
&\sim\left\\{\begin{array}[]{ll}\frac{k_{B}T}{\chi}\sqrt{\frac{|y|}{\pi\lambda}},\mbox{
for $x^{2}\ll\lambda y$},\\\ \frac{k_{B}T}{2\chi\lambda}|x|,\mbox{ for
$x^{2}\gg\lambda y$.}\end{array}\right.\end{split}$ (86)

Smectic layers orientational order is expressed in terms of correlations in
the nematic-like order parameter, ${\bf N}({\bf r})=\left(\cos\theta({\bf
r}),\sin\theta({\bf r})\right)$, as,

$\begin{split}\lim_{r\to\infty}\langle{\bf N}({\bf r})\cdot{\bf N}({\bf
0})\rangle&=\lim_{r\to\infty}\langle\cos\left[\theta({\bf r})-\theta({\bf
0})\right]\rangle\\\
&=\lim_{r\to\infty}e^{-\frac{1}{2}\langle\left[\theta({\bf r})-\theta({\bf
0})\right]^{2}\rangle}\\\ &=e^{-\frac{K_{B}T\Lambda}{\chi\lambda
a}}=\text{constant},\end{split}$ (87)

where we have used

$\begin{split}&\langle\left[\theta({\bf r})-\theta({\bf
0})\right]^{2}\rangle=\langle\left[\partial_{x}u({\bf r})-\partial_{x}u({\bf
0})\right]^{2}\rangle\\\ &=-\int_{{\bf q}}\int_{{\bf
q}^{\prime}}\left(e^{i{\bf q}\cdot{\bf r}}-1\right)\left(e^{i{\bf
q}^{\prime}\cdot{\bf r}}-1\right)q_{x}q_{x}^{\prime}\langle u({\bf q})u({\bf
q}^{\prime})\rangle\\\ &=\int_{{\bf q}}\left(2-2\cos({\bf q}\cdot{\bf
r})\right)\frac{q_{x}^{2}k_{B}T}{\chi q_{y}^{2}+Kq_{x}^{4}}\\\
&\sim\frac{2k_{B}T}{\chi\lambda a},\mbox{ for $r\to\infty$.}\end{split}$ (88)

with $\Lambda=\frac{2\pi}{a}$, a convenient cutoff. Therefore, in a 2D
classical smectic at nonzero temperature, the translational order is destroyed
by thermal phonon fluctuationsLandau37 ; Peierls36 ; TonerNelson , while the
orientational order persists even in the presence of thermally excited
phonons, destroyed only at higher temperatures by proliferation of
dislocations.

In presence of unbound dislocations, appearing at density, $n_{d}\sim
e^{-E_{c}/(k_{\text{B}}T)}$, in thermal equilibrium, the effective elasticity
in Debye-Huckel approximation, reduces to that of a nematic at scales greater
than, $\xi_{\text{D}}\sim e^{-E_{c}/(2k_{\text{B}}T)}$,

$F_{\text{N}}=\frac{1}{2}K(T)\int d^{2}r\left({\bm{\nabla}}\theta\right)^{2},$
(89)

and the correlations in orientational order become decay algebraically,

$\lim_{r\to\infty}\langle{\bf N}({\bf r})\cdot{\bf N}({\bf 0})\rangle\sim
r^{-\eta(T)},$ (90)

with, $\eta(T)=2k_{\text{B}}T/[\pi K(T)]$. Therefore, a 2D smectic is unstable
to thermal fluctuations, driven into a nematic fluid at any nonzero
temperatures. At high temperature, the nematic to isotropic liquid transition,
driven by unbinding of disclinations, is described by Kosterlitz and Thouless,
with Stein78 ,

$\eta\left(T_{c}^{-}\right)=\frac{1}{4},$ (91)

at the critical temperature $T_{c}$.

Motivated by our formulation of two-dimensional melting of a classical
crystal, via a dual theory in terms of a higher derivative vector sine-Gordon
model Zhai19 , we expect to find the analogous description for the 2D smectic.
To this end, we express the dislocation and disclination densities in terms of
a sum of their discrete charges as,

$\displaystyle\begin{split}b({\bf r})&=&\sum_{{\bf r}_{n}}b_{{\bf
r}_{n}}\delta^{2}({\bf r}-{\bf r}_{n}),\\\ s({\bf r})&=&\sum_{{\bf
r}_{n}}s_{{\bf r}_{n}}\delta^{2}({\bf r}-{\bf r}_{n}).\end{split}$ (92)

In terms of these discrete topological defect charges, the Hamiltonian is
given by

$\begin{split}H_{\text{sm}}=&\frac{1}{2}\int_{\bf
r}\left[\chi^{-1}(\partial_{x}^{2}\alpha)^{2}+K^{-1}(\partial_{y}\alpha)^{2}\right]\\\
&+\sum_{{\bf r}_{n}}\left[E_{b}b_{{\bf r}_{n}}^{2}+E_{s}{s_{{\bf
r}_{n}}}\right]-\sum_{{\bf r}_{n}}\left[i\partial_{x}\alpha b_{{\bf
r}_{n}}+i2\pi\alpha s_{{\bf r}_{n}}\right].\end{split}$ (93)

Following a standard analysis, summing over the charges, we obtain the dual
sine-Gordon Hamiltonian,

$\begin{split}\tilde{H}_{\text{sm}}=&\int_{\bf
r}\bigg{[}\frac{1}{2}\chi(\partial_{x}^{2}\alpha)^{2}+\frac{1}{2}K^{-1}(\partial_{y}\alpha)^{2}-g_{b}\cos(b\partial_{x}\alpha)\\\
&-g_{s}\cos(2\pi\alpha)\bigg{]}.\end{split}$ (94)

where $g_{b}=\frac{2}{a^{2}}e^{-a^{2}E_{b}}$ and
$g_{s}=\frac{2}{a^{2}}e^{-E_{s}}$, which provides a transparent description of
the continuous two-stage melting in terms of the renormalization-group
relevance of two cosine operators that control the sequential unbinding of
dislocations and disclinations, respectively corresponding to the smectic-to-
nematic and nematic-to-isotropic fluid transitions.

Figure 10: Different phases involved in the melting transitions of a
triangular crystal and a smectic in two dimensions. (a) shows a triangular
crystal at low temperatures, which first melts at $T_{\text{m}}$ via an
unbinding of dislocations into a hexatic, and the sixfold orientational order
in the hexatic phase is destroyed at a higher temperature $T_{\text{hex}}$ by
an unbinding of disclinations. (b) shows a zero-temperature smectic, which is
unstable and driven into a nematic at any nonzero temperatures. This nematic
phase transforms into an isotropic fluid via a further disclination unbinding
transition at $T_{c}$. [Figures adapted from Reference TonerNelson, .]

Because of the second-order Laplacian elasticity, standard analysis around the
Gaussian fixed line $g_{b}=g_{s}=0$ shows that, the mean-squared fluctuations
of $\alpha({\bf r})$ is,

$\langle[\alpha({\bf r})-\alpha({\bf
0})]^{2}\rangle\sim\left\\{\begin{array}[]{ll}\frac{k_{B}T}{\chi^{-1}}\sqrt{\frac{\lambda|y|}{\pi}},\mbox{
for $x^{2}\ll\lambda^{-1}y$},\\\ \frac{k_{B}T}{2\chi^{-1}}\lambda|x|,\mbox{
for $x^{2}\gg\lambda^{-1}y$,}\end{array}\right.$ (95)

which leads to an exponentially (as opposed to power-law in a conventional
sine-Gordon model) vanishing Debye-Waller factor, $\langle e^{i2\pi\alpha({\bf
r})}e^{-i2\pi\alpha({\bf 0})}\rangle=e^{-2\pi^{2}\langle[\alpha({\bf
r})-\alpha({\bf 0})]^{2}\rangle}$, and in turn to a strongly irrelevant
disclination cosine, $g_{s}$, that can therefore be neglected. In contrast,
mean-squared fluctuations of $\partial_{x}\alpha({\bf r})$ is,

$\langle\left[\partial_{x}\alpha({\bf r})-\partial_{x}\alpha({\bf
0})\right]^{2}\rangle\sim\text{constant},$ (96)

for large $r$, and orientational correlation therefore given by,

$\displaystyle\langle e^{ib\cdot\partial_{x}\alpha({\bf
r})}e^{-ib\cdot\partial_{x}\alpha({\bf 0})}\rangle$
$\displaystyle=e^{-\frac{1}{2}b^{2}\langle[\partial_{x}\alpha({\bf
r})-\partial_{x}\alpha({\bf 0})]^{2}\rangle}$ (97)
$\displaystyle\sim\text{constant}.$

This therefore leads to the conclusion that the dislocation cosine, $g_{b}$,
is always relevant. At sufficiently long scales, dislocation cosine in
Equation (94) reduces to a harmonic potential for $\partial_{x}\alpha$,
$-g_{b}\cos(b\partial_{x}\alpha)\simeq{\frac{1}{2}}g_{b}b^{2}(\partial_{x}\alpha)^{2}$.
The effective Hamiltonian is then given by

$\tilde{H}_{\text{nm}}\simeq\int_{\bf
r}\left[{\frac{1}{2}}g_{b}b^{2}(\partial_{x}\alpha)^{2}-g_{s}\cos(2\pi\alpha)\right],$
(98)

where we have neglected the $\chi^{-1}$ “curvature” elasticity relative to the
gradient one encoded in $g_{b}$, and restored the disclination cosine operator
$g_{s}\cos(2\pi\alpha)$. The resulting conventional sine-Gordon model in
$\alpha$ can then exhibit the second KT-like “roughenning” transition,
capturing the nematic-to-isotropic fluid transition, associated with the
unbinding of disclinations, with well-known standard KT phenomenology.

###  IV.2 3D classical smectic duality

Motivated by the correspondence of a $(2+1)$D quantum smectic and a 3D
classical smectic, and the extensively studied 3D nematic to smectic-A
transition deGennes72 ; HalperinMa ; Helfrich78 ; NelsonToner ; Lubensky81 ;
Grinstein86 ; Toner82 , we formulate a dual gauge theory of a 3D classical
smectic, akin to a mapping of a 3D classical XY model onto a classical charged
superconductor.dasgupta ; fisher

The elasticity of a 3D classical smectic with its layers along $xy$ plane, is
captured by the Hamiltonian density,

$\begin{split}\mathcal{H}_{\text{sm}}^{\text{3d}}&=\frac{1}{2}\chi\left({\bm{\nabla}}u+\delta{\bf
n}\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}\delta{\bf n}\right)^{2}\\\
&=\frac{1}{2\chi}{\bf\sigma}^{2}-i{\bf\sigma}\cdot\left({\bm{\nabla}}u+\delta{\bf
n}\right)-\frac{1}{2K}{\bf j}\frac{1}{{\bm{\nabla}}^{2}}{\bf j}-i{\bf
j}\cdot\delta{\bf{n}},\end{split}$ (99)

where $\delta{\bf{n}}={\bf{n}}-\hat{\bf z}=\left(\delta n_{x},\delta
n_{y},0\right)$ represents fluctuations in layer orientation, and we
introduced two Hubbard-Stratanovich fields ${\bf\sigma}$ and ${\bf j}$ to
decouple the two elastic terms. The Hamiltonian density in Eq. (99) is
equivalent to the standard smectic form,

$\mathcal{H}_{\text{sm}}^{\text{3d}}=\frac{1}{2}\chi\left(\nabla_{z}u\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}_{\perp}^{2}u\right)^{2},$
(100)

in the low-energy limit, where the orientational degree of freedom,
$\delta{{\bf{n}}}$, locks to the layer normals with
$\delta{\bf{n}}=-{\bm{\nabla}}_{\perp}u$.

Integrating out the smooth part of $u$ and $\delta{\bf{n}}$, leads to the
constraints:

$\begin{split}{\bm{\nabla}}\cdot{\bf\sigma}&=0,\\\ {\bf
j}+{\bf\sigma}^{\perp}&=0.\end{split}$ (101)

The first equation can be solved in terms of a vector potential ${\bf A}$,
$\sigma_{i}=\epsilon_{ijk}\partial_{j}A_{k},({\bf{\sigma}}={\bm{\nabla}}\times{\bf
A})$, which inside the second constraint gives,

$j_{i}+\epsilon_{ijk}^{\perp}\partial_{j}A_{k}=0,$ (102)

such that $j_{i}$ is solved as
$j_{i}=-\epsilon_{ijk}^{\perp}\partial_{j}A_{k}$.

Substituting $\sigma_{i}$ and $j_{ij}$ back into the original Hamiltonian, and
integrating by part, lead to

$\begin{split}\mathcal{H}_{\text{sm}}^{\text{3d}}=&\frac{1}{2\chi}\left({\bm{\nabla}}\times{\bf
A}\right)^{2}-i\epsilon_{ijk}\partial_{j}A_{k}\left(\partial_{i}u+\delta
n_{i}\right)\\\
-&\frac{1}{2K}\epsilon_{ijk}^{\perp}\partial_{j}A_{k}\frac{1}{{\bm{\nabla}}^{2}}\epsilon_{imn}^{\perp}\partial_{m}A_{n}+i\epsilon_{ijk}^{\perp}\partial_{j}A_{k}\delta
n_{i}\\\ =&\frac{1}{2\chi}\left({\bm{\nabla}}\times{\bf
A}\right)^{2}+iA_{k}\epsilon_{ijk}\partial_{j}\left(\partial_{i}u+\delta
n_{i}\right)\\\
-&\frac{1}{2K}\epsilon_{ijk}^{\perp}\partial_{j}A_{k}\frac{1}{{\bm{\nabla}}^{2}}\epsilon_{imn}^{\perp}\partial_{m}A_{n}-iA_{k}\epsilon_{ijk}^{\perp}\partial_{j}\delta
n_{i}\\\ =&\frac{1}{2\chi}\left({\bm{\nabla}}\times{\bf
A}\right)^{2}+iA_{k}\epsilon_{ijk}\partial_{j}\partial_{i}u\\\
-&\frac{1}{2K}\epsilon_{ijk}^{\perp}\partial_{j}A_{k}\frac{1}{{\bm{\nabla}}^{2}}\epsilon_{imn}^{\perp}\partial_{m}A_{n}+iA_{k}\left(\epsilon_{ijk}-\epsilon_{ijk}^{\perp}\right)\partial_{j}\delta
n_{i}\\\ =&\frac{1}{2\chi}\left({\bm{\nabla}}\times{\bf
A}\right)^{2}-\frac{1}{2K}\epsilon_{ijk}^{\perp}\partial_{j}A_{k}\frac{1}{{\bm{\nabla}}^{2}}\epsilon_{mnk}^{\perp}\partial_{m}A_{n}-iA_{k}b_{k},\end{split}$
(103)

where dislocation density is given by
$b_{k}=\epsilon_{ijk}\partial_{i}\partial_{j}u$.

Figure 11: Different types of dislocations in a 3D smectic. (a) A screw
dislocation with its line core tangent along $\hat{\bf z}$. (b) A positive
edge dislocation with its line core tangent along $\hat{\bf x}$ axis, i.e.,
perpendicular to $\hat{\bf z}$.

In momentum space, we have

$\begin{split}\mathcal{H}_{\text{sm}}^{\text{3d}}=&\frac{1}{2\chi}\left|{\bf
q}\times{\bf A}({\bf q})\right|^{2}+\frac{1}{2K}\frac{|\left({\bf q}\times{\bf
A}({\bf q})\right)_{\perp}|^{2}}{q^{2}}\\\ &-iA_{k}({\bf q})b_{k}(-{\bf q})\\\
=&\frac{1}{2\chi}\left|{\bf q}_{\perp}\times{\bf A}_{\perp}({\bf
q})\right|^{2}-i{\bf A}_{\perp}({\bf q}){\bf b}_{\perp}(-{\bf q})\\\
+&\frac{1}{2}\left(\frac{1}{\chi}+\frac{1}{Kq^{2}}\right)|\left(q_{z}{\bf
A}_{\perp}-A_{z}{\bf q}_{\perp}\right)|^{2}-iA_{z}({\bf q})b_{z}(-{\bf
q}),\end{split}$ (104)

Functionally integrating out $A_{z}$, leads to,

$\begin{split}\mathcal{H}_{\text{sm}}^{\text{3d}}=&\frac{1}{2\chi}\left[\left|{\bf
q}_{\perp}\times{\bf A}_{\perp}({\bf
q})\right|^{2}+\left(1+\frac{1}{\lambda^{2}q^{2}}\right)q_{z}^{2}|{\bf
A}_{\perp}|^{2}\right]\\\ &-i{\bf A}_{\perp}({\bf q}){\bf b}_{\perp}(-{\bf
q})+\frac{1}{2}\frac{\chi}{q_{\perp}^{2}\left[1+1/(\lambda^{2}q^{2})\right]}\left|b_{z}({\bf
q})\right|^{2}\\\
=&\frac{1}{2\chi}A^{\perp}_{i}\left[q_{\perp}^{2}P_{ij}^{\text{T}}+\left(q_{z}^{2}+\frac{q_{z}^{2}}{\lambda^{2}q^{2}}\right)\left(P_{ij}^{\text{L}}+P_{ij}^{\text{T}}\right)\right]A^{\perp}_{j}\\\
&-i{\bf A}_{\perp}({\bf q}){\bf b}_{\perp}(-{\bf q})+\frac{1}{2}\Gamma({\bf
q})\left|b_{z}({\bf q})\right|^{2}\\\ =&\frac{1}{2\chi}A_{i}^{\perp}({\bf
q})\left(q^{2}+\frac{q_{z}^{2}}{\lambda^{2}q^{2}}\right)P_{ij}^{\text{T}}A_{j}^{\perp}(-{\bf
q})\\\ &-i{\bf A}_{\perp}({\bf q})\cdot{\bf b}_{\perp}(-{\bf
q})+\frac{1}{2}\Gamma({\bf q})\left|b_{z}({\bf q})\right|^{2},\end{split}$
(105)

where we have chosen the Coulomb (transverse) gauge
${\bm{\nabla}}_{\perp}\cdot{\bf A}_{\perp}=0$, i.e., ${\bf q}_{\perp}\cdot{\bf
A}_{\perp}=0$, $P^{\text{T}}_{ij}=\delta_{ij}-q_{i}q_{j}/q^{2}$, and,
$P^{\text{L}}_{ij}=q_{i}q_{j}/q^{2}$, are transverse and longitudinal
projection operators respectively, $\Gamma({\bf q})\equiv\chi
q_{\perp}^{-2}\left[1+1/(\lambda^{2}q^{2})\right]^{-1}$ is the interaction
potential for screw dislocation $b_{z}$’s, and we have used the fact that,
$P_{ij}^{\text{L}}A_{j}^{\perp}=0$, in the Coulomb gauge. If we integrate out
${\bf A}_{\perp}$ further, we get the dislocation Coulomb gas model, given by,

$\begin{split}H_{b}^{\text{3d}}=&\frac{1}{2}\int_{{\bf
q}}\left(\frac{{K}q^{2}}{q_{z}^{2}+\lambda^{2}q^{4}}P_{ij}^{\perp}+2E_{b}^{i}\delta_{ij}\right)b_{i}({\bf
q})b_{j}(-{\bf q})\\\ &+\frac{1}{2}\int_{{\bf q}}\Gamma({\bf
q})\left|b_{z}({\bf q})\right|^{2}\\\ =&\frac{1}{2}\int_{{\bf
q}}\left(\frac{{K}\left[\hat{\bf z}\cdot{\bf q}\times{\bf b}({\bf
q})\right]^{2}}{q_{z}^{2}+\lambda^{2}q_{\perp}^{4}}+2E_{b}^{i}{\bf b}({\bf
q}){\bf b}(-{\bf q})\right),\end{split}$ (106)

where,
$P_{ij}^{\perp}=\left(\delta_{ij}-q_{i}^{\perp}q_{j}^{\perp}/q_{\perp}^{2}\right)(1-\delta_{iz}\delta_{jz})$,
is the transverse projection operator for edge dislocations, i.e. screw
dislocations $b_{z}-$components projected away, and $E_{b}^{i}$’s are core
energies of dislocations Toner82 . We note that $\Gamma({\bf q})$ is
approximately a constant at small $q$, where $q$ is smaller than $1/\lambda$,
and therefore contributes to the core energy of a screw dislocation, i.e.,
$E_{b}^{z}\to E_{b}^{z}+\Gamma({\bf q})$. Note that for simplicity, we have
assumed that the lattice spacing between atoms within the layers is equal to
the layer spacing, i.e., $a=d$, such that we have no factor like
‘$\frac{d^{2}}{a^{2}}$’ as in Equation (80) for the 2D case.

Interested in the nature of the nematic to smectic-A transition, TonerToner82
mapped a model of a smectic onto a Coulomb gas of dislocation loops, which he
then transformed into an anisotropic superconductor in a vector gauge field
${\bf A}$, and analyzed it with a momentum-shell renormalization group. In the
long-wavelength limit of $q\ll 1/\lambda$, indeed our model reduces to
Toner’s, with a generalization that screw dislocations in our model have a
finite interaction.

In analogy to what we have done for a 2D smectic, we transform the Coulomb gas
Hamiltonian, Equation (106) into a classical gauge theory. The partition
function for the dislocation-loop Coulomb gas on a lattice is given by,

$\begin{split}Z&=\int\prod_{{\bf r}_{n}}d{\bf A}({\bf r}_{n})\sum_{\\{b_{{\bf
r}_{n}}\\}}e^{-H_{A}\left[\\{{\bf A}({\bf r}_{n})\\}\right]+\sum_{{\bf
r}_{n}}ib_{{\bf r}_{n}}d\cdot{\bf A}({\bf r}_{n})}\\\ &\cdot e^{-\sum_{{\bf
r}_{n}}E_{b}^{i}d^{2}|b^{i}_{{\bf r}_{n}}|^{2}}\delta(\Delta\cdot{\bf b}_{{\bf
r}_{n}})\delta(A_{z})\delta(\Delta\cdot{\bf A}_{{\bf r}_{n}})\\\
&=\int\prod_{{\bf r}_{n}}d{\bf A}({\bf r}_{n})d\varphi({{\bf
r}_{n}})\delta(A_{z})\delta(\Delta\cdot{\bf A}_{{\bf r}_{n}})\\\ &\cdot
e^{-H_{A}\left[\\{{\bf A}({\bf r}_{n})\\}\right]+\sum_{{\bf r}_{n}}ib_{{\bf
r}_{n}}\cdot\left[d{\bf A}({\bf r}_{n})-\Delta\varphi({\bf
r}_{n}\right]-E_{b}^{i}d^{2}|b^{i}_{{\bf r}_{n}}|^{2}},\end{split}$ (107)

with,

$\begin{split}H_{A}\left[\\{{\bf A}({\bf
r}_{n})\\}\right]&=\frac{1}{2}\int\frac{d^{3}q}{(2\pi)^{3}}\left(\frac{q_{z}^{2}}{{K}q_{\perp}^{2}}+\frac{q_{\perp}^{2}}{{\chi}}\right)|{\bf
A}({\bf q})|^{2},\end{split}$ (108)

and we have introduced an auxiliary scalar field $\varphi({\bf r})$, such that
integrating out $\varphi$ recovers the constraint, $\Delta\cdot{\bf b}_{{\bf
r}_{n}}=0$.

After tracing over the dislocation charges ${\bf b}_{{\bf r}_{n}}$, we obtain

$\begin{split}\tilde{H}_{\text{sm}}^{\text{3d}}=H_{A}-g_{b_{i}}^{\text{3d}}\sum_{{\bf
r}_{n}}\cos\left[\Delta_{i}\varphi({\bf r}_{n})-d\cdot A_{i}({\bf
r}_{n})\right]\end{split}$ (109)

where $g_{b_{i}}^{\text{3d}}=2e^{-E_{b}^{i}d^{2}}$, and we have approximated
the resulting Villain potential by its lowest harmonic. In the continuum
limit, it becomes

$\begin{split}\tilde{H}_{\text{sm}}^{\text{3d}}&=H_{A}-\tilde{g}_{b_{i}}^{\text{3d}}\int
d^{3}r\cos\left(\partial_{i}\varphi-d\cdot A_{i}\right)\\\ &=H_{A}-\int
d^{3}r\left[\frac{1}{2}\left|({\bm{\nabla}}-id\cdot{\bf
A})\psi\right|^{2}+V(|\psi|)\right],\end{split}$ (110)

with,
$\tilde{g}_{b_{i}}^{\text{3d}}=\frac{g_{b_{i}}^{\text{3d}}}{a^{2}d}=\frac{2}{a^{2}d}e^{-E_{b}^{i}d^{2}}$,
and in the second form we have written it as an equivalent “soft-spin”
description in terms of $\psi=|\psi|e^{i\varphi}$, with the Landau
U(1)-invariant potential, $V\left(|\psi|\right)$, which is equivalent to the
first form below the energy scale of the gapped Higgs-like magnitude degree of
freedom, $|\psi|$. Thus, we reproduce Toner’s anisotropic superconductor
modelToner82  expected to have the same critical properties as the above
dislocation-loop Coulomb gas model.

##  V Summary and conclusion

In this paper, after a brief review of smectic elasticity, we developed a
coupled U(1) vector gauge theory for a two-dimensional quantum smectic, where
the phonons and orientational Goldstone modes map onto coupled gauge fields,
and topological defects correspond gauge charges and dipoles. We discovered
that charges (disclinations) exhibit subdimensional lineon dynamics,
restricted to move transverse to the layer. Motivated by the partial quantum
melting of a crystal into a smectic, and the subsequent smectic-to-nematic
transition, we reproduced the dual description of a quantum smectic by
condensing the one flavor species of dipoles within the generalized Abelian-
Higgs model of a 2D quantum crystal.

We also applied this duality to treat a classical smectic liquid crystal. To
this end, we formulated the smectic-to-nematic and nematic-to-isotropic fluid
transitions as a higher-derivative sine-Gordon model of a 2D classical
smectic. Motivated by the correspondence between a (2+1)D quantum system and a
3D classical system, we also derived a dual theory for a 3D classical smectic,
and reproduced smectic’s dislocation-loop Coulomb gas description for the
nematic-smectic transition, which we then mapped onto an anisotropic Abelian-
Higgs model.

We expect this fractonic gauge theory reformulation of smectics will be useful
for further detailed explorations, e.g., subjected to an external stress and
in presence of a substrate. We leave a study of the true critical behavior
(beyond mean-field) of the crystal-smectic and smectic-nematic transitions
using the dual gauge theory for future studies. The duality analysis in the
presence of elastic nonlinearties also remains a challenging open problem.

###### Acknowledgements.

We acknowledge earlier collaboration with Michael Prekto that motivated this
work. This work was supported by the Simons Investigator Award from the James
Simons Foundation and by the NSF MRSEC grant DMR-1420736.

##  Appendix A Crystal-to-supersolid transition

In the Mott-insulating “commensurate” crystal phase associated with the
particle-number conservation symmetry, the Lagrangian density of a square-
lattice quantum-crystalwith lattice spacing $d$, for the two dipole fields
$\psi_{\bf p}=\psi_{x},\psi_{y}$, corresponding to the two minimal dipole
species ${\bf p}_{x}=p\hat{\bf x},{\bf p}_{y}=p\hat{\bf y}$, plus the Maxwell
gauge field part, takes the form

$\begin{split}\mathcal{L}_{\text{dis}}=&\sum_{{\bf p}}i\psi^{\dagger}_{{\bf
p}}D_{0}\psi_{{\bf p}}-\frac{1}{2m_{p}}\sum_{{\bf p}}|\Pi_{ij}^{\perp{{\bf
p}}}D_{j}\psi_{{\bf p}}|^{2}-V(\\{\psi_{k}\\})\\\
&+\mathcal{L}_{\text{M}}^{\text{cr}},\end{split}$ (111)

where, $m_{p}$ is the effective mass of the dipole,
$D_{0}=\partial_{t}+ip_{k}A_{0k}$ and $D_{j}=\partial_{j}+ip_{k}A_{jk}$ (i.e.,
${\bf D}={\bm{\nabla}}+ip_{k}{\bf A}_{k}$) are the covariant
derivativesKumar19 , $V(\\{\psi_{k}\\})$ is the U(1)-invariant Ginzburg-Landau
potential, $\mathcal{L}_{\mathcal{\text{M}}}^{\text{cr}}$ the Lagrangian
density of the Maxwell part, and $\Pi_{ij}^{\perp{{\bf
p}}}=\delta_{ij}-\frac{p_{i}p_{j}}{p^{2}}$ is the projection operator since in
this Mott-insulating crystal phase, the dipole can only move in the direction
perpendicular to p while the along-dipole climbs are forbidden due to the U(1)
particle-number conservation symmetry (“ glide-constraint").

However, as discussed in the main text, the crystal also exhibits scalar non-
topological point defects, corresponding to deficiency and excess in atom
density, which permits the climb process of the dislocations (See Fig. 7).
Combined with the bosonic statistics of the underlying particles, the quantum
crystal can first develop into a super-solid phase (incommensurate crystal),
featuring both the crystalline order and the superfluid order. The
condensation of vacanies or interstitials in the super-solid phase, therefore,
frees these symmetry-forbidden climb events. Therefore, for a complete
description, we also need to add the superfluid part of the underlying bosonic
particles, $\mathcal{L}_{\text{sf}}$, and the minimal gauge-invariant coupling
between the dislocation climb operators and superfluid order parameter
,$\mathcal{L}_{\text{sf-dis}}$, into the full Lagrangian Kumar19 ,

$\displaystyle\mathcal{L}_{\text{sf}}=i\Psi^{\dagger}_{\text{sf}}\partial_{t}\Psi_{\text{sf}}-\frac{1}{2m_{s}}|{\bm{\nabla}}\Psi_{\text{sf}}|^{2}-\mu|\Psi_{\text{sf}}|^{2}-\frac{U}{2}|\Psi_{\text{sf}}|^{4},\;\;\;\;\;\;\;\;$
(112) $\displaystyle\mathcal{L}_{\text{sf-
dis}}=\gamma\Psi_{\text{sf}}\sum_{\bf p}\mathcal{O}_{\text{climb},{\bf
p}}+h.c.,$ (113)

where, $\Psi_{\text{sf}}$ is the superfluid order parameter,
$\mathcal{O}_{\text{climb},{\bf p}}=\psi_{\bf p}^{\dagger}({\bf r}+{\bf
p})e^{ip_{i}A_{ij}p_{j}}\psi_{\bf p}({\bf r})$ is the ${\bf p}$-dipole climb
operator, and $\gamma$ is the coupling constantKumar19 . Fig. 8 shows an
example of terms in $\mathcal{L}_{\text{sf-dis}}$.

The Mott insulator-to-superfluid transition, described by
$\mathcal{L}_{\text{sf}}$, occurs at the critical point $\mu=0$, with
$|\Psi_{\text{sf}}|=\sqrt{\frac{|\mu|}{U}}\equiv\sqrt{\rho_{s}^{0}}$. Writing
$\Psi_{\text{sf}}=\sqrt{\rho_{s}}e^{i\varphi_{s}}=\sqrt{\rho_{s}^{0}+\delta\rho_{s}}e^{i\varphi_{s}}$,
the superfluid part becomes.

$\mathcal{L}_{\text{sf}}=-\rho_{s}\partial_{t}\varphi_{s}-\frac{1}{2m_{s}}\left[\frac{({\bm{\nabla}}\rho_{s})^{2}}{4\rho_{s}}+\rho_{s}({\bm{\nabla}}\varphi_{s})^{2}\right]-\mu\rho_{s}-\frac{U}{2}\rho_{s}^{2}.$
(114)

Integrating out the massive magnitude fluctuations $\delta\rho_{s}$, leads to,

$\begin{split}\mathcal{L}_{\text{sf}}&=\frac{1}{2}\partial_{t}\varphi_{s}\frac{1}{U-1/(4m_{s}\rho_{s}^{0})\cdot{\bm{\nabla}}^{2}}\partial_{t}\varphi_{s}-\frac{\rho_{s}^{0}}{2m_{s}}({\bm{\nabla}}\varphi_{s})^{2}\\\
&\approx\frac{1}{2U}(\partial_{t}\varphi_{s})^{2}-\frac{\rho_{s}^{0}}{2m_{s}}({\bm{\nabla}}\varphi_{s})^{2}\\\
&\equiv\frac{\rho_{s}^{0}}{2m_{s}}(\partial_{\mu}\varphi_{s})^{2},\end{split}$
(115)

where, in the second line, we have assumed that $\varphi_{s}({\bf r})$ varies
slowly in space and droped the term,
$\frac{1}{4m_{s}\rho_{s}^{0}}{\bm{\nabla}}^{2}$, and in the last line, we have
written it as that of a sound mode, in a Lorentz-invariant form for
simplicity, with
$\partial_{\mu}\equiv\left(\frac{1}{c_{\text{ph}}}\partial_{t},{\bm{\nabla}}\right)$
and $c_{\text{ph}}=\sqrt{\frac{\rho_{s}^{0}U}{m_{s}}}$.

Combined, $\mathcal{L}_{\text{dis}}$, with, $\mathcal{L}_{\text{sf}}$ and
$\mathcal{L}_{\text{sf-dis}}$, the resulting phase is a supersolid with the
spontaneous breaking of both partice number conservation U(1) symmetry and
translational symmetry. Writing $\psi_{k}=\sqrt{\rho_{k}}e^{i\varphi_{k}}$,
and integrating out the massive magnitude fluctuations, the effective
Lagrangian density for the super-solid phase is given by

$\begin{split}\tilde{\mathcal{L}}_{\text{cr}}=&\frac{\rho_{k}}{2}\left(\partial_{t}\varphi_{k}+pA_{0k}\right)^{2}-\frac{\rho_{k}}{2m_{p}}\cos\left[\Pi_{ij}^{\perp
k}\left(\partial_{j}\varphi_{k}+pA_{jk}\right)\right]\\\
&-\rho_{p}^{\parallel}\cos\left[\Pi_{ij}^{\parallel
k}\left(\partial_{j}\varphi_{k}+pA_{jk}\right)+\varphi_{s}\right]+\mathcal{L}_{\text{M}}^{\text{cr}}+\mathcal{L}_{\text{sf}}.\end{split}$
(116)

Freezing the superfluid phase by fixing $\varphi_{s}=0$, and rescaling the
longitudinal and transverse gradients, we get (correct to quadratic order in
the argument of cosine terms),

$\begin{split}\tilde{\mathcal{L}}_{\text{cr}}=&\frac{\rho_{k}}{2}\left(\partial_{t}\varphi_{k}+pA_{0k}\right)^{2}-\frac{\rho_{k}}{2m_{p}}\cos\left({\bm{\nabla}}\varphi_{k}+p{\bf
A}_{k}\right)+\mathcal{L}_{\text{M}}^{\text{cr}}\\\
=&\frac{1}{2}|\left(\partial_{\mu}+ipA_{\mu,k}\right)\psi_{k}|^{2}-V\left(\\{\psi_{k}\\}\right)+\mathcal{L}_{\text{M}}^{\text{cr}},\end{split}$
(117)

where in the second form, we have written it as an equivalent “soft-spin"
description in terms of $\psi_{k}$, with the landau U(1)-invariant potential,
$V\left(\\{\psi_{k}\\}\right)$. Note that the fugacity
$e^{-\tilde{E}_{k}^{b}}$ and density $\rho_{k}$ are related by the relation:
$\frac{m_{p}}{d^{2}}e^{-\tilde{E}_{k}^{b}}=\rho_{k}$ to the lowest order, such
that $\frac{\rho_{k}}{m_{p}}=2\tilde{g}_{k}^{b}$. Therefore, Equation (117) is
in the same form as Equation (61).

Similar analysis applies for the quantum smectic phase with just one species
of dipoles, i.e., $\hat{\bf x}-$dipoles ($\hat{\bf y}-$dislocations), which
leads to the full mobility of dipoles (dislocations) in the smectic and the
effective Lagrangian density of the super-smectic phase given by,

$\displaystyle\tilde{\mathcal{L}}_{\text{sm}}=$
$\displaystyle\frac{\rho}{2}\left(\partial_{t}\varphi_{x}+pA_{0}\right)^{2}-\frac{\rho}{2m_{p}}\cos\left({\bm{\nabla}}\varphi_{x}+p{\bf
A}\right)+\mathcal{L}_{\text{M}}^{\text{sm}}$ (118) $\displaystyle=$
$\displaystyle\frac{1}{2}|\left(\partial_{\mu}+ipA_{\mu}\right)\psi_{x}|^{2}-V\left(|\psi_{x}|\right)+\mathcal{L}_{\text{M}}^{\text{sm}},$

which is in the same form as Equation (63).

##  Appendix B From Crystal to Smectic Dual

The crystal-smectic partial melting transition corresponds to condensation of
one flavor of the dipole fields, that according to Fig. 2, we take to be
$\hat{\bf y}$-dipoles, i.e. $\psi_{y}\neq 0$. In this $\hat{\bf y}$-dipole
condensate, the vortices in the phase field $\varphi_{y}$ are suppressed,
i.e., ${\bm{\nabla}}\varphi_{y}$ is small, and thus, we can expand the
corresponding cosine term in (117) to the quadratic order in its argument,
which leads to

$\begin{split}\tilde{\mathcal{L}}_{\text{cr}}=&\frac{\rho_{x}}{2}\left(\partial_{t}\varphi_{x}+pA_{0x}\right)^{2}-\frac{\rho_{x}}{2m_{p}}\cos\left({\bm{\nabla}}\varphi_{x}+p{\bf
A}_{x}\right)+\mathcal{L}_{\text{M}}^{\text{cr}}\\\
&+\frac{\rho_{y}}{2}\left(\partial_{t}\varphi_{y}+pA_{0y}\right)^{2}+\frac{1}{2}\cdot\frac{\rho_{y}}{2m_{p}}\left({\bm{\nabla}}\varphi_{y}+p{\bf
A}_{y}\right)^{2}.\end{split}$ (119)

With the gauge transformation by choosing $\chi_{y}\to\frac{1}{p}\varphi_{y}$
in Equation (46),

$A_{iy}\to A_{iy}+\frac{1}{p}\partial_{i}\varphi_{y},\;A_{0y}\to
A_{0y}+\frac{1}{p}\partial_{t}\varphi_{y},\;a_{y}\to
a_{y}+\frac{1}{p}\varphi_{y},$ (120)

we absorb the gradients of the phase $\varphi_{y}$ into the gauge fields
$A_{\mu,y}$, and the last two terms in Equation (B1) become quadratic terms of
$\hat{\bf y}$-flavor gauge fields, $A_{\mu,y}^{2}$, which make the original
massless modes $A_{\mu,y}$ become massive. And,
$\tilde{\mathcal{L}}_{\text{cr}}$ can be written as

$\begin{split}\tilde{\mathcal{L}}_{\text{cr}}=&\frac{\rho_{x}}{2}\left(\partial_{t}\varphi_{x}+pA_{0x}\right)^{2}-\frac{\rho_{x}}{2m_{p}}\cos\left({\bm{\nabla}}\varphi_{x}+p{\bf
A}_{x}\right)+\mathcal{L}_{\text{M}}^{\text{eff}},\end{split}$ (121)

where the effective Maxwell part $\mathcal{L}_{\text{M}}^{\text{eff}}$ in
terms of newly defined $A_{\mu,y}$ is given by

$\mathcal{L}_{\text{M}}^{\text{eff}}=\mathcal{L}_{\text{M}}^{\text{cr}}+\frac{1}{2}\rho_{y}p^{2}A_{0y}^{2}+\frac{1}{2}\cdot\frac{\rho_{y}p^{2}}{2m_{p}}{\bf
A}_{y}^{2}$ (122)

Therefore, within this condensate phase, the gauge field components
$A_{y,\mu}$ become gapped via the Anderson-Higgs mechanism by coupling to the
$\hat{\bf y}$-dipole ($\hat{\bf x}$-dislocation) condensate , and can be
safely integrated out in the low-energy regime with wavelengths of excitations
much greater than $\sqrt{\frac{2m_{p}}{\rho_{y}p^{2}}}$ (i.e., penetration
length of $A_{\mu,y}$) , which leads to

$\begin{split}\mathcal{L}_{\text{M}}^{\text{eff}}=&\frac{1}{2}\chi{\bf
E}_{x}^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
A}_{x}\right)^{2}+\frac{1}{2}K_{x}e_{x}^{2}+\frac{1}{2}K_{y}e_{y}^{2}\\\
&-\frac{1}{2}\left(\frac{\rho_{y}p^{2}}{\rho_{y}p^{2}-m_{p}}\right)\left(\epsilon_{ij}\partial_{i}a_{j}+\epsilon_{ix}A_{ix}\right)^{2},\end{split}$
(123)

where, the modified bend modulus ${\bf K}=(K_{x},K_{y})$, with
$K^{-1}_{x}=K^{-1}$ and $K_{y}^{-1}=K^{-1}-\frac{K^{-1}}{K\rho_{y}p^{2}+1}$,
becomes anisotropic in this smectic case. In the lowest order approximation,
making $\rho_{y}\to\infty$ (i.e., the condensate is very dense),
$K_{x}=K_{y}=K$, and $\mathcal{L}_{\text{M}}^{\text{eff}}$ reduces to the
Maxwell Lagrangian of the smectic case,

$\begin{split}\mathcal{L}_{\text{M}}^{\text{eff}}\approx&\frac{1}{2}\chi{\bf
E}_{x}^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
A}_{x}\right)^{2}+\frac{1}{2}K{\bf
e}^{2}-\frac{1}{2}\left(\epsilon_{ij}\partial_{i}a_{j}+\epsilon_{ix}A_{ix}\right)^{2}\\\
=&\mathcal{L}_{\text{M}}^{\text{sm}}.\end{split}$ (124)

Therefore, to lowest order, the crystal’s Maxwell Lagrangian reduces to that
of a smectic,

$\mathcal{L}_{\text{M}}^{\text{cr}}\left(A_{\mu,x},A_{\mu,y}=0,a_{\mu}\right)\approx\mathcal{L}_{\text{M}}^{\text{sm}}\left(A_{\mu,x},a_{\mu}\right),$
(125)

which simply corresponds to setting $A_{\mu,y}\approx 0$. And, by replacing
$\rho_{x}$ simply with $\rho$ in Equation (B3), the dual Lagrangian of the
crystal reduces to that of the smectic exactly, to the lowest order,

$\tilde{\mathcal{L}}_{\text{cr}}\left(A_{\mu,x},A_{\mu,y}=0,a_{\mu},\psi_{x},\psi_{y}=0\right)\approx\tilde{\mathcal{L}}_{\text{sm}}\left(A_{\mu,x},a_{\mu},\psi_{x}\right).$
(126)

Similarly analysis applies for the further melting with a condensation of the
other, $\hat{\bf x}-$dipoles ($\hat{\bf y}-$dislocations). Within this
$\psi_{x}\neq 0$ Higgs phase, corresponding to a condensation of unbound
$\hat{\bf x}-$dipoles ($\hat{\bf y}-$dislocations) in the smectic, the gauge
field components $A_{\mu,x}$ become gapped also, via coupling to the $\hat{\bf
x}-$dipole ($\hat{\bf y}-$dislocation) condensate. This can be seen easily by
making a further gauge transformation with $\chi_{x}\to\frac{1}{p}\varphi_{x}$
in Equation (46),

$A_{ix}\to A_{ix}+\frac{1}{p}\partial_{i}\varphi_{x},\;A_{0x}\to
A_{0x}+\frac{1}{p}\partial_{t}\varphi_{x},\;a_{x}\to
a_{x}+\frac{1}{p}\varphi_{x},$ (127)

to absorb the gradients of the phase $\varphi_{x}$ into the gauge fields
$A_{\mu,x}$, and expanding the corresponding cosine term in (A6) to the
quadratic order in its argument, which leads to,

$\begin{split}\tilde{\mathcal{L}}_{\text{sm}}=&\mathcal{L}_{\text{M}}^{\text{sm}}+\frac{1}{2}\rho_{x}p^{2}A_{0x}^{2}+\frac{1}{2}\cdot\frac{\rho_{x}p^{2}}{2m_{p}}{\bf
A}_{x}^{2}.\end{split}$ (128)

where the last two quadratic terms make the original massless modes
$A_{\mu,x}$ become massive now. Integrating out $A_{\mu,x}$ in the low-energy
regime with wavelengths of excitations much greater than
$\sqrt{\frac{2m_{p}}{\rho_{x}p^{2}}}$ (i.e., penetration length of
$A_{\mu,x}$), leads to,

$\begin{split}\tilde{\mathcal{L}}_{\text{sm}}=&\frac{1}{2}\left(K^{-1}-\frac{1}{\rho_{x}p^{2}+K^{-1}}\right)(\partial_{t}a_{x}-\partial_{x}a_{0})^{2}\\\
&+\frac{1}{2}K^{-1}\left(\partial_{t}a_{y}-\partial_{y}a_{0}\right)^{2}-\frac{1}{2}\left(1+\frac{2m_{p}}{\rho_{x}p^{2}}\right)\left({\bm{\nabla}}\times{\bf
a}\right)^{2}\\\ \approx&\frac{1}{2}K^{-1}\left(\partial_{t}{\bf
a}-{\bm{\nabla}}a_{0}\right)^{2}-\frac{1}{2}\left({\bm{\nabla}}\times{\bf
a}\right)^{2}\\\ =&\tilde{\mathcal{L}}_{\text{nm}},\end{split}$ (129)

in the lowest order approximation, making $\rho_{x}\to\infty$ (i.e., the
condensate is very dense), which is just the dual Lagrangian density of the
quantum xy model of a nematic. Therefore,

$\mathcal{L}_{\text{M}}^{\text{sm}}\left(A_{\mu,x}=0,a_{\mu}\right)\approx\mathcal{L}_{\text{M}}^{\text{nm}}\left(a_{\mu}\right),$
(130)

to the lowest order.

##  Appendix C 3D smectic elasticity

In the main text, we have given the elasticity of a d-dimensional smectic in
terms of the layer displacement $u$ only, given by Equation (11). For a 3D
smetic, we just set $d=3$. Here, we formulate the elasticity of a 3D smectic
in terms of the displacement $u$ and the Frank director ${\bf{n}}$
simultaneously. The elastic energy will not change, if all layers of molecules
are rotated together rigidly. However, there will be an energy cost if the
orientation directions of molecules, represented by Frank director ${\bf{n}}$,
are rotated away from their equilibrium local orientation, normal to the
layers. The elastic energy density of a 3D smectic, with its layers along $xy$
plane, is given by

$\begin{split}\mathcal{H}_{\text{sm}}^{\text{3d}}=&\frac{1}{2}\chi\left(\partial_{z}u\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}_{\perp}u+\delta{\bf{n}}\right)^{2}+\frac{1}{2}K_{1}\left({\bm{\nabla}}\cdot{\bf
n}\right)^{2}\\\
&+\frac{1}{2}K_{2}\left[{\bf{n}}\cdot({\bm{\nabla}}\times{\bf{n}})\right]^{2}+\frac{1}{2}K_{3}\left[{\bf{n}}\times({\bm{\nabla}}\times{\bf{n}})\right]^{2},\end{split}$
(131)

where $u$ is the layer displacement,
$\delta{\bf{n}}={\bf{n}}-\hat{z}=\left(\delta n_{x},\delta n_{y},0\right)$
represents the layer orientation degree of freedom, and the last three terms
represents the slay, twist and bend distortions of the director respectively,
with three independent, corresponding elastic constants $K_{1}$, $K_{2}$ and
$K_{3}$. To linear order in $\delta{\bf{n}}$,

$\begin{split}\mathcal{H}_{\text{sm}}^{\text{3d}}\simeq&\frac{1}{2}\chi\left(\partial_{z}u\right)^{2}+\frac{1}{2}K\left({\bm{\nabla}}_{\perp}u+\delta{\bf{n}}\right)^{2}+\frac{1}{2}K_{1}\left({\bm{\nabla}}\cdot{\bf
n}\right)^{2}\\\ &+\frac{1}{2}K_{2}\left[{\bf
z}\cdot({\bm{\nabla}}\times{\bf{n}})\right]^{2}+\frac{1}{2}K_{3}\left[{\bf
z}\times({\bm{\nabla}}\times{\bf{n}})\right]^{2}\\\ =&\frac{1}{2}\chi
q_{z}^{2}|u({\bf q})|^{2}+\frac{1}{2}K\left|{\bf q}_{\perp}u({\bf
q})+\delta{\bf{n}}({\bf q})\right|^{2}\\\ +&\frac{1}{2}K_{1}\delta
n_{i}P_{ij}^{\text{L}}\delta n_{j}+\frac{1}{2}K_{2}\delta
n_{i}P_{ij,z}^{\text{T}}\delta n_{j}+\frac{1}{2}K_{3}\delta
n_{i}P_{ij,\perp}^{\text{T}}\delta n_{j},\end{split}$ (132)

where, $P_{ij}^{\text{L}}=q_{i}q_{j}/q^{2}$ is the longitudinal projector,
$P_{ij,z}^{\text{T}}=\left(\delta_{ij}-q_{i}^{\perp}q_{j}^{\perp}/q^{2}\right)(1-\delta_{iz}\delta_{jz})$
and
$P_{ij,\perp}^{\text{T}}=\left(\delta_{ij}-q^{2}_{\perp}/q^{2}\right)\left(\delta_{ix}\delta_{jx}+\delta_{iy}\delta_{jy}\right)$
are the transverse-to-$\hat{\bf z}$ and transverse-to-layer projector,
respectively, and in the second line, we have transformed into the momentum
space. For long-wavelength limit, with the wave number
$q\ll\sqrt{\chi/K_{2}},\sqrt{\chi/K_{3}}$, we can integrate out the higher-
energy terms, which sets $\delta{\bf{n}}={\bm{\nabla}}_{\perp}u$ and reduces
$\mathcal{H}_{\text{sm}}^{\text{3d}}$ into the form,

$\begin{split}\mathcal{H}_{\text{sm}}^{\text{3d}}=&\frac{1}{2}\chi\left(\partial_{z}u\right)^{2}+\frac{1}{2}K_{1}\left({\bm{\nabla}}_{\perp}^{2}u\right)^{2},\end{split}$
(133)

matching exactly with the standard form of the elastic energy of a 3D smectic.
In Section IV, for a simple analysis with losing much qualitative physics, we
have set $K=\chi$, and made the isotropic elasticity approximation with
$K_{1}=K_{2}=K_{3}$, replacing them by $K$ for simplicity.

Below, we give a briefly discussion of dislocatins and their energies in a 3D
smectic, based on $\mathcal{H}_{\text{sm}}^{\text{3d}}$ given by Equation
(133). A more complete and detailed discussion based on
$\mathcal{H}_{\text{sm}}^{\text{3d}}$ given by Equation (131) in terms of both
$u$ and $\delta{\bf{n}}$, can be found in some standard textbooks, like that
by Chaikin and Lubensky. For a single positive screw dislocation with its line
core, located in the origin of $xy-$plane, in the $\hat{\bf z}$ direction, as
shown in Fig. 11(a) , from the Equation (9), that determines the positions of
the layer planes, we get the layer displacement given by,

$u_{\text{scrw}}({\bf r})=\frac{d}{2\pi}\tan^{-1}\frac{y}{x},$ (134)

taking place in the $xy$ plane only, and then,

$\displaystyle\partial_{z}u_{\text{screw}}$ $\displaystyle=0,$ (135)
$\displaystyle{\bm{\nabla}}_{\perp}u_{\text{screw}}$
$\displaystyle=\frac{d}{2\pi}\frac{-y\hat{\bf x}+x\hat{\bf
y}}{x^{2}+y^{2}}=-\frac{d}{2\pi}\frac{\hat{\bf\varphi}}{r_{\perp}},$

shuch that,

${\bm{\nabla}}_{\perp}^{2}u_{\text{screw}}=0,$ (136)

and therefore, the energy of a single screw dislocation in a smectic is $0$.

For a single positive edge dislocation with its line core perpendicular to the
$\hat{z}$ direction, say in the $\hat{x}$ direction, as shown in Fig. 11(b),
the layer displacement is then given by,

$u_{\text{edge}}({\bf r})=\frac{d}{2\pi}\tan^{-1}\frac{z}{y},$ (137)

and then,

$\displaystyle\partial_{z}u_{\text{edge}}$
$\displaystyle=\frac{y}{y^{2}+z^{2}},$ (138)
$\displaystyle{\bm{\nabla}}_{\perp}u_{\text{edge}}$
$\displaystyle=-\frac{z\hat{\bf y}}{y^{2}+z^{2}},$

such that,

${\bm{\nabla}}_{\perp}^{2}u_{\text{edge}}=\frac{2yz}{(y^{2}+z^{2})^{2}},$
(139)

and, the energy of a single edge dislocation in a smectic can be shown to be
divergent as the length scale of the system, after integrating (C3) over
space.

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