Lifshitz gauge duality Leo Radzihovsky Department of Physics and Center for Theory of Quantum Matter

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Lifshitz gauge duality

Leo Radzihovsky

Department of Physics and Center for Theory of Quantum Matter

University of Colorado, Boulder, CO 80309∗

(Dated: October 10, 2022)

Motivated by a variety of realizations of the compact Lifshitz model I derive its fractonic gauge

dual. The resulting U(1) vector gauge theory eﬃciently and robustly encodes the restricted mobility

of its dipole conserving charged matter and the corresponding topological vortex defects. The gauge

theory provides a transparent formulation of the three phases of the Lifshitz model and gives a ﬁeld

theoretic formulation of the associated two-stage Higgs transitions.

Introduction and motivation. Recently, there have

been much interest in systems with ﬁne-tuned generalized

global symmetries and their fractonic gauge duals.[1] One

of the simplest is the Lifshitz model (and its m-Lifshitz

generalization[2]), that describes a diverse number of

physical systems. Its classical realizations date back a

half century in studies of Goldstone modes of cholesteric,

smectic, and columnar liquid crystals, tensionless mem-

branes and nematic elastomers[3–10], and many other

soft-matter phases exhibiting rich phenomenology [11,

12].

Quantum realizations of the Lifshitz model in-

clude Hall striped states of a two-dimensional elec-

tron gas at half-ﬁlled high Landau levels[13–18],

striped spin and charge states of weakly doped cor-

related quantum magnets[19, 20], critical theory of

the RVB - VBS transition[21–23], ferromagnetic tran-

sition in one-dimensional spin-orbit-coupled metals[24],

the putative Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)

paired superﬂuid[25, 26] in imbalanced degenerate

atomic gases[27, 28], and spin-orbit coupled Bose

condensates.[29, 30], as well helical states of bosons or

spins on a frustrated lattice[31].

The most notable feature of the 3d classical and 2+1d

quantum Lifshitz model is its enlarged “tilt” or dipo-

lar symmetry and the concomitant logarithmic “rough-

ness”, φ2

rms ∼log Lof its Goldstone mode φ(akin to

the XY model in two dimensions), that leads to its

power-law correlated, quasi-long-range ordered state for

the matter ﬁeld eiφ. Depending on the nature of its

physical realization, the enlarged symmetry may result

from ﬁne-tuning to a critical point – as e.g., in RVB -

VBS [21–23], paramagnetic-ferromagnetic in spin-orbit-

coupled metals[24] and a membrane buckling[8] phase

transitions, or is dictated by an underlying symmetry

– as e.g., “target-space” rotational invariance of smectic,

columnar, cholesteric, and tensionless membrane ordered

phases.[3–7, 28, 32] In these realizations the nonlinearities

(elastic in the context of smectics and membrane states)

become relevant for d<dc(dc= 3 and dc= 5/2 for

the classical smectic[28, 32] and columnar states[33, 34],

respectively), leading to universal “critical phases”.

Dipolar symmetry and fractonic order. In addition to

above examples, Lifshitz model also naturally arises as

the Goldstone-mode (superﬂuid phase, φ) ﬁeld theory of

the ordered state of interacting bosons with additional

dipole-charge conservation, explored in great detail in

Ref. 35. At harmonic level the symmetry is equivalent

to the aforementioned target-space rotational invariance

of a 3d smectic. [36, 37] Our interest in the Lifshitz model

is also motivated by the recent observation that general-

ized quantum elastic systems, e.g., 2+1d conventional

and Wigner crystals, supersolids, smectics and vortex

crystals, under elasticity - gauge duality[38–40] map onto

generalized “fractonic” gauge theories[41, 42], that ex-

hibit charged matter with restricted mobility.[36, 37, 43–

47, 49]

For concreteness, in what follows, when discussing

phases, transitions and topological defects, I will use the

language of bosons, ψ∼eiφ in the dipolar Bose-Hubbard

model.[35] The boson and dipole number conserving sym-

metry,

φ→φ+α+β·x(1)

is encoded in the high derivative “elasticity” , forbid-

ding lowest order gradient of the compact superﬂuid

phase φ(with only dipole-conserving hopping, e.g.,

ψ†

x+δψxψxψ†

x−δ∼d†

x,δψxψ†

x−δ+h.c.). The symmetry

parameters, α, βare zero modes that may be constrained

by system’s boundary conditions. The generalized Lif-

shitz model is a minimal such continuum ﬁeld theory,

with a Euclidean Lagrangian density,

L=1

2κ(∂τφ)2+1

2Kijkl(∂i∂jφ)(∂k∂lφ)2,(2)

where κis the compressibility, tensor Kijkl encodes lat-

tice hopping anisotropy and τis the 0-th imaginary

time component of xµ. I note that, in striking con-

trast to the rotational invariance of the closely-related

smectic and other Lifshitz systems discussed above, here,

the more stringent dipolar symmetry forbids all relevant

nonlinearities.[32] It thus protects the ﬁxed line of the

noncompact Lifshitz model (2).

In 2+1d the model (2) is generically expected to un-

dergo a two-stage disordering transition. In the famil-

iar context of smectic liquid crystals (with φxthe com-

pact phonon ﬁeld) it corresponds to a transition from a

arXiv:2210.03127v1 [cond-mat.str-el] 6 Oct 2022

smectic state (a periodic array of stripes, that sponta-

neously breaks rotational and translational symmetries,

with (quasi-) long-range ordered (ψx∼eiφx)dx,δk∼

eiδk·∇φx≡eiθkﬁeld), through the translationally-

invariant nematic ﬂuid (that breaks rotational C2sym-

metry) to a fully disordered isotropic and translation-

ally invariant ﬂuid[36, 37] (with the 3d classical analogue

studied for many decades[11, 12]). However, the critical

nature of the nematic-smectic transitions, even in the 3d

classical case[52] remains an open problem. Here, I uti-

lize duality to provide a gauge theory formulation of the

2+1d Lifshitz model, allowing a transparent characteri-

zation of its phases and a ﬁeld theoretic analysis of the

corresponding Higgs transitions.

Phases of Lifshitz model. To this end, as was intro-

duced in Refs. 36, 37, 49, for a vector gauge theory formu-

lation of fractons, it is convenient to reformulate the Lif-

shitz model in terms of coupled XY models for the atom

(ψx∼eiφx) and dipole (dx,δk=ψ†

xψx+δk∼eiδk·∇φx≡

eiθk) superﬂuid phases, φand (θ)k=θk, with a La-

grangian density,[50]

L=1

2κ(∂τφ)2+1

2g(∇φ+θ)2+1

2I(∂τθ)2

2Kijkl(∂iθj)(∂kθl).(3)

At low energies the gcoupling in Lenforces ∇φ≈ −θ

(i.e., φ≈φ0−θ·r) and thus reduces L(3) to the stan-

dard form in (2), with corrections that are subdominant

at low energies. This form of Lifshitz Lagrangian (3) dis-

plays a gauge-like coupling between atoms and dipoles,

that thereby underlies a nontrivially intertwined atom-

dipole (and corresponding vortices) dynamics of the Lif-

shitz ﬂuid and its aforementioned phase transitions.[51]

Before turning to a detailed analysis, (3) already re-

veals the structure of the phases of the Lifshitz model:

(i) In the absence of vortices, i.e., a fully Bose-condensed

state of atoms and dipoles, BECad is characterized by

single-valued φand θphases. The state is gapless and is

well-described by a Gaussian ﬁxed line of standard Lif-

shitz form (2), with a dynamical exponent z= 2. For

constant θ, the BECad state is orientationally ordered,

atomic condensate at momentum θakin to a Fulde-

Ferrell[25–28], a spin-orbit coupled condensate[29, 30]

and a helical state of frustrated bosons[31]. However,

I expect it to be challenging to probe this momentum,

since in the bulk it can be gauged away.[53] Given the

resemblance of (3) to the Abelian-Higgs model (with a

non-gauge invariant “Maxwell” sector for θ, character-

ized by Kijkl), I expect the BECad - BECdtransition

to be in a generalized normal-superconductor universal-

ity class. This is expected due to a nontrivial gauge-like

coupling between the dipolar and atomic condensates,

whose consequences we will also see in the dual gauge

theory formulation descussed below.

(ii) Increasing ﬂuctuations (at zero temperature done by

increasing boson interaction relative to dipole hopping),

drives a proliferation of vortices in the atomic phase

φMott-insulating atoms, with dipoles remaining Bose-

condensed in BECd, and in the case of an underlying

isotropic system spontaneously breaks rotational sym-

metry by the choice of θ. With this ∇φbecomes an

independent vector ﬁeld (with both transverse and lon-

gitudinal components) that can thus be safely integrated

out, leading to a z= 1 XY-like Lagrangian density for

the dipolar Goldstone mode ,

LBECd=1

2I(∂τθ)2+1

2Kijkl(∂iθj)(∂kθl).(4)

(iii) Increasing interaction further then proliferates vor-

tices in θ, leading to a fully Mott-insulating phase, MI of

atoms and dipoles.

The shortcoming of the continuum form (3) of the com-

pact Lifshitz model, L, is that compactness (i.e., vor-

tex degrees of freedom) of the Goldstone modes φand

θis not manifest.[61] To remedy this, I make the corre-

sponding vortex degrees of freedom explicit by allowing

nonsingle-valued conﬁgurations of φand θ. Namely, I

“gauge” Lin (3), with atomic (a-) and dipolar (d-) vor-

tices, respectively represented by ﬂuxes of the associated

gauge ﬁelds,[62]

L=LM[˜aµ,˜

Aµ] + 1

2κ(∂τφ−˜a0)2+1

2g(∇φ−˜

a+θ)2

2I(∂τθ−˜

A0)2+1

2Kijkl(∂iθj−˜

Aij )(∂kθl−˜

Akl),

=LM[˜aµ,˜

Aµ] + κ

2(∂µφ−˜aµ+θµ)2+K

2(∂µθ−˜

Aµ)2.

(5)

with the corresponding discrete vortex (dual) 3-currents

given by

jµ=µνγ ∂ν˜aγ−µνk ˜

Aνk ≡(∂×˜a)µ−˜

A∗

µ,

pZτp

npˆvµ(τp)δ3(xν−xν

p(τp)),(6)

Jµ=µνγ ∂ν˜

Aγ≡(∂×˜

A)µ,

pZτp

Npˆ

Vµ(τp)δ3(xν−xν

p(τp)),(7)

θµ= (0, θi), ˜aµ= (˜a0,˜ai), ( ˜

Aµ)k=˜

Aµk = ( ˜

A0k,˜

Aik),

(np,Np) (boson, dipole) p-th vortex integer windings,

and (ˆvµ(τp),ˆ

Vµ(τp)) unit 3-velocities of their correspond-

ing world-lines. Throughout, to emphasize the structure

of the expressions I use a short-hand notation:(i) bold-

faced for Roman ﬂavor kand spatial i, j indices, and,

(ii) where obvious, omit the space-time Greek indices, as

deﬁned below. In the last line in (5), for transparency

of analysis I took Kijkl =Kδikδjl and rescaled coordi-

nates so that g=κand I=K, i.e., chose the speeds of

“sound” to be 1; in an isotropic lattice-free system Kijkl

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LifshitzgaugedualityLeoRadzihovskyDepartmentofPhysicsandCenterforTheoryofQuantumMatterUniversityofColorado,Boulder,CO80309(Dated:October10,2022)MotivatedbyavarietyofrealizationsofthecompactLifshitzmodelIderiveitsfractonicgaugedual.TheresultingU(1)vectorgaugetheoryecientlyandrobustlyencodestherestr...

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