Condensation of Lattice Defects and Melting Transitions in Quantum Hall phases

Gil Young Cho,1 Onkar Parrikar,1 Yizhi You,1 Robert G. Leigh,1 and Taylor L. Hughes1

1Department of Physics, Institute for Condensed Matter Theory,University of Illinois, 1110 W. Green St., Urbana IL 61801-3080, U.S.A.

(Dated: November 15, 2018)

Motivated by recent progress in understanding the interplay between lattice and electronic topo-logical phases, we consider quantum-melting transitions of weak quantum liquid crystals, a crystaland a nematic phase, in which electrons form a quantum Hall state. In certain classes of Chern bandinsulators and quantum Hall phases, it has been previously demonstrated that there are topologicalChern-Simons terms such as a Hall viscosity term and a gravitational Chern-Simons term for locallattice deformations. The Chern-Simons terms can induce anyonic statistics for the topological lat-tice defects and furthermore dress the defects with certain symmetry quantum numbers. On theother hand, the melting transitions of such liquid-crystalline orders are driven by the condensationof lattice defects. Based on these observations, we show how the topological terms can change thenature of the proximate disordered phases of the quantum liquid crystalline phases. We derive andstudy the effective dual field theories for the liquid crystalline phases with the geometric Chern-Simons terms, and carefully examine the symmetry quantum numbers and statistics of defects. Weshow that a crystal may go through a continuous phase transition into another crystal with thenew discrete translational symmetries because the dislocation, the topological defect in the crystal,carries non-zero crystal momentum due to the Hall viscosity term. For the nematic phase, the discli-nation will condense at the phase transition to the isotropic phase, and we show that the isotropicphase may support a deconfined fractionally charged excitation due to the Wen-Zee term, and thusthe isotropic phase and the nematic phase have different electromagnetic Hall responses.

I. INTRODUCTION

When a two-dimensional electron gas is subject to astrong magnetic field, it is well-known that the electronicenergy spectrum consists of highly degenerate harmonicoscillator levels called Landau levels. In a partially-filledLandau level, the determination of the true ground stateout of a massively degenerate set of levels is dominatedby the repulsive interaction between the electrons sincethe kinetic energy is quenched. Because of the interac-tions, and the flat single-particle energy spectrum, onemight naively expect for the electrons to form a crys-talline state and break spatial translation symmetries.Surprisingly, the electrons often form an isotropic liquid-like state with a quantized electronic Hall conductancewhen certain conditions (e.g. particular fillings of theLandau levels) are met1–7. For example, when the low-est Landau level is 1

3 -filled, the electrons form an incom-pressible fractional quantum Hall liquid described by theLaughlin wavefunction8, and exhibit a fractionally quan-

tized Hall conductance σxy = e2

3h .

In addition to the isotropic liquid phases there are alsomany types of phases in the quantum Hall regime whicheither exhibit spatial order or have a quantum Hall liq-uid coexisting with spatial order9–19. The conventionalspatial order can be classified by its broken spatial sym-metries (e.g., in 2D there are two continuous translationalsymmetries and one continuous rotational symmetry thatcan be broken). A solid crystal phase is a phase withbroken translational and rotational symmetries, e.g., aWigner crystal phase. Crystal phases can melt to restorefull translational symmetry through the condensation ofdislocations (N.B. we will use “melting transition” in-

terchangeably with phase transition)20–23. The resultingintermediate phase with restored translation symmetry,but broken rotation symmetry, is called a nematic phase.A nematic phase can successively melt and restore its ro-tational symmetry by the condensation of disclinations.After this it becomes an isotropic liquid phase with full2D spatial symmetry.

We are interested in cases where a quantum Hall fluidremains incompressible but is subjected to coexistingspatial quantum order9–11,13,18. For example, spatial or-der on top of the incompressible quantum Hall fluid isexpected9,18 to emerge if the filling of the Landau levelis slightly away from the ‘magic’ filling and the dilutegas of quasi-hole and quasi-electron excitations over the‘magic’ filling forms a liquid-crystalline or crystalline or-der. In principle, a sufficiently large quadrupolar inter-action between the electrons (which is the Lz = ±2 partof a rotationally symmetric interaction e.g., a Coulombinteraction) may induce liquid-crystalline order24 whilethe Hall fluid remains incompressible. In other recent re-sults it has been predicted that fractional Chern insulatorstates with higher Chern numbers can exhibit topologi-cal nematic phases25,26 though we will not consider thissystem directly. Additionally, we note that we will notaddress the perhaps more familiar examples of compress-ible quantum Hall smectic and nematic states16,17,19.

More generally, we focus on electrons in quantum Hallphases exposed to phase transitions in ‘weak’ quantumlattices27–30. We use the term ‘weak’ lattice to distin-guish it from the conventional lattice formed by the ionsin the solid which will not exhibit a quantum meltingtransition. There are two possibilities to generate weaklattices in quantum Hall systems : (i) through spatial

arX

iv:1

407.

5637

v2 [

cond

-mat

.str

-el]

26

Jul 2

014

2

order parameters of the electrons or quasiparticles ex-hibited in crystal or liquid crystalline phases or (ii) inrotating optical lattice systems31–33 where the potentialwells can be adjusted to be weak and the structure canbe tuned by hand. These weak lattices share the samesymmetry with the classical ionic lattices and will havethe same type of topological defects (dislocations anddisclinations), but they have the advantage of the pos-sibility of exhibiting melting transitions. The quantumlattice and its melting transition have also been shown tohost other types of interesting physical phenomena suchas linearized Einstein-Cartan gravity27,29. We note thatthroughout this work we assume that the gapped elec-tronic band structure (the Landau level cyclotron gap ormany-body fractional quantum Hall gap) remains gappedand incompressible during the melting transitions of thebackground weak lattice.

In such phases, it is natural to expect an interesting in-terplay between the geometric defects of the spatial orderand the quantum Hall fluid. The connection between ge-ometric degrees of freedom and the responses of quantumHall phases has been intensely studied over the last fiveyears based on older developments in the works of Wenand Zee34 and Avron et. al.35. For our work the relevantresults are the predictions that the quantum Hall stateshave Hall-like responses to geometric deformations34–43.In these works it was shown that the quantum Hall liq-uid responds to perturbations of the local geometry. TheWen-Zee term and gravitational Chern-Simons term en-code the charge and angular momentum response of thequantum Hall state in the presence of curvature, includ-ing disclinations when rotation symmetry is broken byspatial ordering. On the other hand, the Hall viscosityindicates how the fluid will respond to strain-rates (ve-locity gradients) and, when coupled to a weak lattice, theresponse to dislocations.

Similar to the pure charge response of a quantum Hallinsulator, the geometric responses also have a Chern-Simons form. As such, the corresponding Chern-Simonsterms in the hydrodynamic description will attach vari-ous quantum numbers to topological defects of the un-derlying spatial order, i.e., dislocations and disclinations.These quantum numbers can affect the statistics and thetypes of continuous phase transitions produced by con-densation of these defects. Thus, the quantum-meltingtransitions of a weak lattice controlled by the propertiesof the lattice defects are the main focus of this paper.We will separately consider a weak crystal and a ne-matic phase that coexists with a liquid-like quantum Hallstate. We will show that the geometric Chern-Simonsterms35–38,41–43 can change the statistics and quantumnumbers of the topological lattice defects responsible forthe melting transitions of the spatially ordered phases.We base our description of the relevant transitions on thedual theories of the relevant elastic theories28,30. This de-scription contains the low-energy physics of the spatiallyordered phases and can characterize the nature of the de-fects that lead to proximate disordered phases. We care-

fully examine when a continuous melting transition viacondensation of the defects is available. When a contin-uous transition is available, we show that the topologicalorder as well as the symmetries can change at a singletransition. Furthermore, the topological defects becomedressed with additional quantum numbers through theChern-Simons terms. For example, a dislocation in thecrystal phase is dressed with crystal momentum41,42, andthis implies that a crystal melts into another crystal withonly discrete translational symmetry, rather than a ne-matic phase which has continuous translation symmetry.

The rest of the paper is organized as follows. In Sec-tion II, we begin with the introduction to the dualitytransformation of a conventional bosonic theory coupledto a Chern-Simons term. In Section II A, we consider thecoexistence of a crystal phase and a quantum Hall stateand develop a dual theory of the crystal. We will showthat dislocations, i.e., the topological defect of the trans-lation symmetry of the crystal, is bound to momentumin the presence of a quantum Hall state, similar to theresponse in the Chern insulator discussed in Refs. 41 and42. We carefully examine the quantum numbers of thedislocation and study the nature of the disordered phaseof the crystal. In section II B, we study the dual theory ofa nematic phase coexisting with a quantum Hall phaseand study the statistics and the electric charge of thedisclination. We make some additional comments aboutpossible melting transitions of the nematic phase and dis-cuss some complications of the description of a completetheory of the nematic disordering transition that will bedeferred to future work.

II. QUANTUM HALL STATES WITH LATTICEDEFECTS

As mentioned in the introduction, we will consider spa-tially ordered phases coexisting with quantum Hall elec-tronic states. Our goal is to study the properties of thetopological defects in a broken symmetry phase, includ-ing the types of phase transitions mediated by their con-densation. Formally, the broken symmetries will be rep-resented by their corresponding bosonic order parame-ters, and the order parameters will couple to the elec-tronic degrees of freedom. After integrating out thegapped electronic degrees of freedom, one obtains an ef-fective bosonic theory. For spatial symmetries the orderparameters typically couple to the electrons as an effec-tive spatial vielbein eai or metric gij ≡ eai ebjδab field.

In our cases of interest it has been shown that after in-tegrating out the fermions one finds Chern-Simons typeterms of the geometric fields that couple to the currentsof the topological defects35–38,41–43. The Chern-Simonsterms could induce statistical transmutation of the latticedefects43 and/or dress the topological defects with cer-tain quantum numbers41,42. Both of these effects play animportant role in determining the available phase tran-sitions. If the defects undergo statistical transmutation,

3

then they can carry anyonic statistical angles. The any-onic statistics will have an important impact on the melt-ing transitions of the spatial order parameter, e.g., theorder parameter cannot melt if the defects are not at theright “filling” in analogy with the anyonic quasi-particlecondensation transitions in quantum Hall states. Themelting of the spatial order itself occurs due to the pro-liferation/condensation of the appropriate topological de-fect. Thus, if the topological defects are dressed with ad-ditional quantum numbers (charge, spin, etc.), the defectcondensate phase (i.e., the disordered phase) will breakthe symmetries associated with the attached quantumnumbers. We will now discuss several interesting typesof defect-mediated phase transitions, but we will first be-gin with reviewing the conventional theory of a bosonicorder parameter coupled to a Chern-Simons term.

To understand the effects of the Chern-Simons terms,and to introduce the standard duality transformationon which we will heavily rely in subsequent sections,we briefly review the theory of a conventional AbelianChern-Simons term coupled to a boson field Ψ which ischarged under a non-compact U(1) gauge field Aµ

44–48:

LCS =1

4πkεµνλαµ∂ναλ + |(∂µ − iαµ − iAµ)Ψ|2

+r

2|Ψ|2 + Veff (|Ψ|) +

1

4e2(εµνλ∂νAλ)2 + · · · (1)

From this action one can determine that the bosonic fieldΨ is bound to k flux quanta of the gauge field αµ due tothe presence of the Chern-Simons term (∼ εα∂α). Thus,Ψ experiences statistical transmutation, i.e., if k ∈ 2Z, Ψremains bosonic, otherwise, it can be fermionic or any-onic. The flux attachment, and its average field approx-imation, are expected to be reliable when the theory(1) describes the transition between the incompressiblestates of Ψ such as a Mott insulator and a quantum Hallstate. For now let us consider the case when the com-posite operator Ψ (bound to k fluxes of αµ) is bosonic(k ∈ 2Z). This allows us to study the transition betweena Mott insulator and a fractional quantum Hall state atfilling ν = 1

k .Then the theory in Eq. 1 has two phases depending

on the sign of r (the mass of the boson field Ψ). Whenr > 0, Ψ is gapped and does not condense, thus it is thedisordered phase of Ψ. In this case we can integrate outΨ and we find a gapless photon emerging from the non-compact U(1) gauge field Aµ. Because the gauge fieldAµ is non-compact, it is not confining and the photon isa stable gapless excitation. The effective action in thiscase is simply the Maxwell action:

L(r>0) =1

4e2(εµνλ∂νAλ)2 + · · · (2)

This is the Mott insulator phase of the boson Ψ;the gauge field αµ is not in the “flux-condensed”phase4,5,46–48 and the Chern-Simons term for αµ in (1)can be safely ignored. This is clearly seen in the dualaction of (1) which we will derive further below.

On the other hand, if r < 0 then Ψ will Bose condenseand gain an expectation value. When Ψ condenses, theaction will develop a term ∼ |Ψ|2|αµ +Aµ|2, i.e., there isa Higgs mass for the combination αµ + Aµ. In the low-energy theory, this effectively sets αµ = −Aµ, and wefind that the ordered phase of Ψ is described by a singleChern-Simons theory:

L(r<0) =1

4πkεµνλAµ∂νAλ + · · · (3)

In this phase, there is no gapless photon from Aµ as thereis a topological mass gap from the Chern-Simons term.

Furthermore we can show that this phase r < 0 sup-ports fractional excitations. To exhibit this explicitly,we will use a dual theory. First we rewrite the bosonfield in terms of its phase and modulus Ψ =

√ρ0 exp(iφ).

The phase field φ is compact over a range of 2π, i.e.,φ ∼ φ + 2π, which allows for topological defect configu-rations of vortices where∮

d~x · ∇φ = 2πZ. (4)

The kinetic term for the original bosonic field Ψ from Eq.1 can be rewritten in terms of the modulus and the phasevariables

1

2|(∂µ − iαµ − iAµ)Ψ| = ρ0

2|∂µφ− αµ −Aµ|2

→K2µ

2ρ0−Kµ(∂µφ− αµ −Aµ) (5)

where we have introduced the bosonic Hubbard-Stratonovich vector field Kµ in the above equation andassumed that ρ0 is a constant (the fluctuation associatedwith the modulus is massive and can be safely integratedout in the low-energy theory).

The phase variable φ can be separated into a smooth,single-valued part φsm and the singular part φvor, i.e.,φ = φsm + φvor. The singular part of the phase variableencodes the vortex configurations of Ψ via εij∂i∂jφvor 6=0. The smooth part φsm is non-compact and can beintegrated out to generate an equation of motion for Kµ:

Kµ∂µφsm → ∂µKµ = 0, Kµ ≡1

2πεµνλ∂νaλ. (6)

By identifying 12π ε

µνλ∂ν∂λφvor = Jµvor as the quantized

vortex current (∫J0d2x ∈ Z), we end up with the dual

theory of the theory in Eq. 1:

LCS =1

4πkεµνλαµ∂ναλ −

1

2πεµνλaµ∂ν(αλ +Aλ)− aµJµvor

+1

4g2(εµνλ∂νaλ)2 +

1

4e2(εµνλ∂νAλ)2 + · · · (7)

where g2 = 2π2ρ0. The theory is quadratic in αµ so wecan integrate it out and further simplify the low-energytheory in the long-wave length limit:

LCS =k

4πεµνλaµ∂νaλ − aµJµvor −

1

2πεµνλaµ∂νAλ

+1

4e2(εµνλ∂νAλ)2 + · · · (8)

4

To complete the duality transformation we need to ele-vate the vortex source current Jµvor to the level of a fluctu-ating quantum field by introducing the soft-spin variableΦ:

Jµvor ∼ iΦ∗(∂µΦ)− i(∂µΦ)∗Φ. (9)

Then we also need to supplement the theory with modelkinetic and vortex-interaction terms to obtain:

LCS =k

4πεµνλaµ∂νaλ −

1

2πεµνλaµ∂νAλ +

1

2|(∂µ − iaµ)Φ|2

+M2

2|Φ|2 + Veff (|Φ|) +

1

4e2(εµνλ∂νAλ)2 + · · ·

(10)

which is the result we have been seeking. At this point,we can identify Φ as the vortex of the original bosonΨ in (1) and it is anyonic with the statistical angle π

kfrom (10). Thus the ordered phase Ψ with the effectiveresponse (3) is a fractionalized phase with the anyon Φ.

With Eq. 10 in hand we can try to understand thetwo different phases of the theory in Eq. 1 from thedual action. The order-disorder transition of the originalboson Ψ from Eq. 1 should occur when Φ condenses.In the ordered phase of the Φ field, Φ condenses and aµdevelops a Higgs term and becomes massive. This resultsin an effective theory:

L =1

4e2(εµνλ∂νAλ)2 + · · · (11)

at low energy. This is the same action that was found inthe disordered phase of Ψ. Thus we identify the orderedphase of Φ with the disordered phase of Ψ. Here, one canquestion how the field Φ can condense because the fieldΦ is anyonic with the Chern-Simons term of aµ. In thedual theory description of the transition, we have treatedΦ as a bosonic field by ignoring the fluctuations of aµ atthe transition of Φ. This was first used by Haldane7 andalso used by Wen1 in the construction of the hierarchyfractional quantum Hall states. We note that one canalso proceed with a Φ with an anyonic statistical angleto obtain the same hierarchy states.

On the other hand, if Φ does not condense and is mas-sive, then we can integrate out the Φ field to find

L =k

4πεµνλaµ∂νaλ −

1

2πεµνλaµ∂νAλ + · · ·

→ 1

4πkεµνλAµ∂νAλ + · · · (12)

where we have also integrated out aµ to obtain the secondline. This is precisely the same effective theory (3) ap-pearing in the ordered phase of Ψ. Hence we identify thedisordered phase of Φ as the ordered phase of Ψ. Thus,we have completed the dual description.

Let us comment on the above dual description of thetransition. First of all, exactly the same form of thetheory46 applies to the transition between quantum Hall

states with different topological order. Second, the tran-sition is continuous46,47 in the large-N limit (where Nis the number of flavors of the matter field coupled tothe Chern-Simons gauge field) despite the presence ofthe Chern-Simons term, and hence is assumed to remaincontinuous in the small-N limit.

Now that we have completed our review we will showthat a similar field theory appears in the melting transi-tion of a weak lattice with which a quantum Hall phasecoexists. The melting transition is described by the con-densation of the topological defects of the symmetry-broken phase. As in the transition between quantum Hallstates, we will see that the geometric Chern-Simons termsrestrict the operators that can condense at the transition.Furthermore, the defects are dressed with certain symme-try quantum numbers, just as the charge Chern-Simonsterm in (1) binds flux to a charge. Thus, after the con-densation transition, the symmetries associated with theattached quantum numbers appear as the broken symme-tries in the ordered phase of the defects. Furthermore,because the melting transition is equivalent to the tran-sition between fractional quantum Hall states, we willsee that the topological order as well as the symmetrychanges at a single transition.

We will explain these ideas carefully as we discuss dis-location mediated melting of a weak lattice coexistingwith a quantum Hall phase, and then move on to discussthe disordering of a nematic phase.

A. Case I. Dislocation Driven Transitions

We will first consider the melting transition of a weakcrystal lattice with no coexisting quantum Hall phaseand later include the effects of a quantum Hall fluidthat is coupled to the weak lattice. A crystal phase isa phase with broken translational and rotational symme-tries. The low-energy theory is described by the localdisplacement vector ua(~x, t), a = 1, 2, and the quantizedfluctuations of the displacement ua(~x, t) are the phononsof the crystal. The phonons are described by the La-grangian

L =1

2g2

∑a=1,2

(∂µua(~x))2. (13)

There are two branches of phonons in two spatial di-mensions (one longitudinal, one transverse) because thephonons are the Goldstone modes of the two broken con-tinuous translational symmetries. A rotational Gold-stone mode is absent from the low-energy physics, asit has been suggested30 that the rotational symmetry isconfined in the crystal phase. Rotations of the spatialframe act to rotate the ua amongst themselves. Sincethere is no time-like component of ua, the low energytheory will not have local Lorentz invariance, and therewill be only two essentially Abelian currents involved inthe low energy physics.

5

In the crystal there are two types of topological defects:dislocations and disclinations28,42. A dislocation is a tor-sional defect responsible for the restoration of the trans-lational symmetry, whereas a disclination corresponds tothe curvature defect responsible for the restoration of therotational symmetry. The condensation of dislocationsusually induces a transition toward a nematic phase inwhich the translational symmetries are restored. As thedisclination is gapped both in the crystal and in the ne-matic phase, one can ignore the disclination29 in the de-scription of the initial transition between the crystal andthe nematic phase. We will first study this transition.

To introduce a field theory for the transition, we iden-tify the dislocation in terms of the displacement vector~u = (u1, u2) via ∮

dxj∂j~u = ~b (14)

where ~b = (b1, b2) is the Burgers’ vector of the disloca-tions contained within the integration path. For conve-nience, we restrict ourselves to underlying spatial orderwhich is a C2 symmetric rectangular lattice generated bythe two unit vectors (l1, 0) and (0, l2) (we will further re-strict to C4 symmetry later when convenient). Then the

Burgers’ vector ~b = (b1, b2) can be always representedas n1(l1, 0) + n2(0, l2) with (n1, n2) ∈ Z2. The Burgers’vector represents two quantized “fluxes” carried by thedislocation. The lattice thus sets a compactness rela-tion for the displacement vector ua ∼ ua + la. Hence wesplit the displacement vector ~u(~x, t) into the smooth part~usm(~x, t) and the singular part ~udis(~x, t) which containsthe information about the dislocation configuration via(14).

Now that we have identified the relevant topologicaldefect we can dualize the theory (13). The discussionhere will parallel Ref. 28 and the previous subsection.We begin with the Hubbard-Stratonovich transformationof the elastic phonon Lagrangian

L =1

2g2

∑a=1,2

(∂µua(~x))2 →∑a=1,2

(g2

2(P aµ )2 − P aµ∂µua

)

→∑a=1,2

(g2

2l2a(εµνλ∂νKaλ)2 − 1

laεµνλKaµ∂ν∂λuadis

)

=∑a=1,2

(g2

2l2a(εµνλ∂νKaλ)2 −Kaµjaµ

). (15)

In the above equation, we have integrated out the smoothpart ~usm ∈ R2 to generate an equation of motion ∂µP

aµ =

0 =⇒ P aµ ≡ εµνλ∂νKaλ/la for a = 1, 2. We also have

identified the quantized dislocation current (∫d2xja0 ∈

Z2):

jaµ =1

laεµνλ∂ν∂λu

adis, a = 1, 2. (16)

We will now suppose that the dynamics of the systemcan be modeled by the possible condensation of elemen-tary dislocation fields. For a particular dislocation, such

a field would carry a ‘charge’ corresponding to the Burg-ers’ vector of the dislocation. We suppose further thatthe dynamics is dominated by a pair of such fields corre-sponding to Burgers’ vectors along the lattice basis direc-tions. (Dislocations with other Burgers’ vectors might bethought of as bound states of our elementary fields butwe assume that these do not affect the vacuum structurein a significant way.) The dislocation currents jaµ canthen be represented in terms of these quantum fields Φa:

jaµ ∼ i(Φa)∗∂µΦa − i(∂µΦa)∗Φa, a = 1, 2. (17)

The dislocation fields Φa are the analogue of the vortexfields described earlier, and as implied by the form of thecurrent, carry the ‘translational winding’ associated withthe dislocation.

This completes the dual transformation of the elastictheory for the crystal (13)

L =∑a=1,2

[g2

2l2(εµνλ∂νKaλ)2 +

1

2|(∂µ − iKaµ)Φa|2

]+r

2|Φ1|2 +

r

2|Φ2|2 + Veff (|Φ1|, |Φ2|) + · · · (18)

In the dual theory (18), we have imposed C4 rotationsymmetry for convenience here, and thus the mass gapsfor Φ1 and Φ2 are the same (furthermore, this fixes l1 =l2 = l). If one further relaxes this C4 symmetry, then themass terms for Φ1 and Φ2 do not need to be the same.With C4 symmetry, there are two phases depending onthe sign of r in (18).

For r > 0, the dislocations are massive and can be inte-grated out. In the low-energy and long wavelength limit,we find that there are two branches of the photons de-scribed by the two non-compact gauge fieldsKaµ, a = 1, 2.This is the dual description of the two branches of thephonons in the original elastic theory (13). Thus we iden-tify the disordered phase of both Φa, a = 1, 2 as the crys-tal phase. In the crystal, the dislocations are minimallycoupled to the gauge fields and thus are logarithmicallyconfined.

For r < 0, the dislocations Φa, a = 1, 2 condense andHiggs the gauge fields Kaµ, a = 1, 2. Because the disloca-tion condensate will restore the continuous translationalsymmetry, the ordered phase of the dislocations is thephase with translational symmetry. However, the discli-nations are still gapped in this phase and thus the rota-tional symmetry is still broken. We identify this phaseas the nematic phase.

Now let us consider the impact of the electronic quan-tum Hall state coexisting with the crystal. We still as-sume that disclinations are gapped, and we are interestedin the regime where the low-energy physics is describedby the phonons and the dislocations. For the topologicalChern insulator49 it was shown that the response of theChern insulator state coupled to the deformation vector~u(~x, t) can be derived42, and it is the Hall viscosity term

6

representing the response to torsion:

LζH =∑a

ζH2εµνλeaµ∂νea,λ

= −∑a

ζH2εµνλ∂µu

a∂ν∂λua

= −∑a

ζH2εµνλ∂µu

adis∂ν∂λu

adis. (19)

We have used the fact that the vielbein eaµ and the de-formation vector ua are related to each other by eaµ =

δaµ + ∂µua in linearized elasticity theory42. This Hall vis-

cosity term encapsulates the response of the insulator todislocations. Namely, to each dislocation there is an as-sociated localized momentum density that is attached tothe defect. While this was only shown for the Chern in-sulator, we will present arguments here that it shouldbe a generic feature of quantum Hall states coupled todislocation defects.

In a quantum Hall fluid, electrons in the bulk movechirally along equipotential lines of an electrostatic po-tential; this is a consequence of the Lorentz force. Thespatial order that coexists with the Hall fluid will onlycouple through such an electrostatic potential. Whena dislocation is created in the crystal or charge densitywave order over the quantum Hall fluid, an equipotentialline will surround the dislocation (see Fig. 1c, for ex-ample) and this implies that there should be an electroncurrent around the dislocation. For one-component Hallfluids the electric current is proportional to the local mo-mentum density, and thus there is non-zero momentumdensity localized at the dislocation in the direction of theBurgers’ vector. This is the same phenomenon predictedby Eq. 19.

To be more explicit, let us consider a quantum Hallfluid coexisting with a uni-directional charge densitywave ρB(x, y) with periodicity a0 >> `B (here `B is themagnetic length) is imposed as in Fig. 1a:

ρB(x, y) = ρ0∑n∈Z

δ(x− na0). (20)

We consider the case that the charge density wave gen-erates a short-ranged electrostatic potential VB(x, y) forsimplicity. The range of the potential we expect to belsc . `B . With this assumption the background electro-static potential VB(x, y) due to this charge distributioncan be modeled as follows:

(∂2x + ∂2y −1

l2sc)VB(x, y) = −ρB(x, y). (21)

The background electric field due to the density wave

is ~E = −~∇VB . We introduce a dislocation with theBurgers’ vector (a0, 0) into the charge density wave. Asdemonstrated in Fig. 1c, the dislocation can be modeledby removing charge along a half-line

ρdis(x, y) = −ρ0δ(x)θ(−y). (22)

x

= 𝒂𝒂𝟎𝟎

x

𝝆𝝆(𝒙𝒙)

𝒂𝒂𝟎𝟎

(a) (b)

(c)

x

=

x

𝑦𝑦 = 0

𝑥𝑥, 𝑦𝑦 = (0,0)

FIG. 1. Charge density wave over a quantum Hall fluid. (a)Charge density wave with periodicity a0 along x axis. (b)The charge density distribution along x-axis. The straightlines represent the position of the extra charge due to thedensity wave. (c) A dislocation with the Burgers’ vector(a0, 0) is introduced at the origin (0, 0) by removing charge.The charge distribution with the dislocation is ρ(x, y) =ρ0

∑n∈Z δ(x−na0)− ρ0θ(−y)δ(x). The position of the dislo-

cation is identified with the red circle. From this charge dis-tribution, it is straightforward to see that there is no net mo-mentum current along y-axis. The red arrows on the equipo-tential line represent the current which is proportional to themomentum density.

x (b)

R

(a)

𝑦𝑦′(𝑠𝑠1)

𝑦𝑦′(𝑠𝑠2) 𝑦𝑦′(𝑠𝑠3)

x

FIG. 2. Deformation of the Volterra cut for a dislocation. (a)We introduce a local frame (x′, y′) at each point where x′ ∝d~f(s)ds

where ~f(s) parametrizes the location of the removedcharge density. We introduce y′ by requiring y′ · x′ = 0. (b)

The curvature κ(s) of ~f(s) at s ∈ [0, 1] is 1R(s)

where R(s) is

the radius of the osculating circle at s.

7

The total charge distribution with the dislocation overthe Hall fluid is given by ρ(x, y) = ρB(x, y) + ρdis(x, y).The electrostatic potential VB(x, y) will be modified ac-cordingly as VB → VB + Vdis where Vdis is the potentialinduced by a dislocation ρdis(x, y). These quantities sat-isfy

(∂2x + ∂2y −1

l2sc)(VB + Vdis) = −ρB(x, y)− ρdis(x, y)

→ (∂2x + ∂2y −1

l2sc)Vdis(x, y) = −ρdis(x, y). (23)

Because ρdis(x, y) = ρdis(−x, y) then Vdis(x, y) =Vdis(−x, y), and thus there is no net electric field alongthe x-axis around the dislocation. This translates as nonet electric current along the y-axis due to the Lorentzforce. On the other hand, there will be a non-zero netelectric field along the y-axis at the tip of the disloca-tion as illustrated in Fig.1c, and thus there is a localizedmomentum density at the dislocation in the direction ofthe Burgers’ vector. The momentum density should belocalized around the dislocation within a circle of radius∼ lsc . `B .

In this argument we have used the fact that the“string” attached to the point-like dislocation is astraight line. This is essentially a gauge choice, and wewould like to see that this result is more general, and onlydepends on the point-like nature of the dislocation, notthe attached string. So, with the result of this simple casein hand, we can consider a slightly more complicated caseas in Fig. 2 in which the string attached to the disloca-tion is curved, but with a curvature κ << 1/lsc. Becausethe electrostatic potential should be screened on a lengthscale larger than lsc ∼ `B , we can concentrate only on theregion very close to the region of the removed charge (i.e.,the string). This region is the area in which the current islocalized and is much smaller than 1/κ. For convenience,we parametrize the position of the removed charge (the

string) by ~f(s), s ∈ [0, 1]. Then we can introduce a local

frame x′ and y′ by x′(s) ∝ d~fds (the tangent vector to the

line ~f) and y′(s) ⊥ x′(s). Because we are interested inthe region near the string within the length scale muchsmaller than the inverse of the curvature, we effectivelyhave the same problem in the previous case (c.f. Fig. 1c)by ignoring any terms of O(`Bκ):

(∂2x′ + ∂2y′ −1

l2sc)Vdis(x

′, y′) = −ρdis(x′, y′). (24)

The important thing here is that the electron current isso tightly bound near the dislocation that it does notsee the curvature, i.e., it cannot “see” far enough awayto distinguish the bending string from a straight line.Thus, ρdis(x

′, y′) = ρdis(x′,−y′), and we again have

Vdis(x′, y′) = Vdis(x

′,−y′) and thus there is no net mo-mentum along x′-axis. On the other hand, it is apparentthat there is a localized momentum density at the tip ofthe dislocation along the direction of the Burger’s vector

since at the tip, the local frame returns to just the ini-tial x, y axes. We note that effectively the assumptionabout the curvature is a restriction on the rate at whichthe curve bends so that when the string is thickened byincluding the region where the current is localized, thenthe thickened string will never intersect itself, i.e., differ-ent equipotential lines will not cross. It is possible thiscondition could be relaxed by microscopically resolvingwhat occurs at the intersection between different equipo-tential lines, but we do not have an argument for thatcase.

From our arguments above, we claim that a dislocationcoupled to a quantum Hall fluid traps momentum density.Hence, we will model the contribution of the quantumHall fluid to the elastic action as

LζH = −∑a

ζH2εµνλ∂µu

adis∂ν∂λu

adis. (25)

There are two caveats about this expression. First, ourarguments have shown that there should be a momen-tum bound to a dislocation but not what the magnitudeof the momentum is in terms of the Burgers’ vector. ForChern insulators, the proportionality between momen-tum and Burgers’ vector is just the Hall viscosity, andwe are assuming the same here. If it is not the Hall vis-cosity, it will generically be a response coefficient that isodd under time reversal symmetry, has the units of Hallviscosity, and is tunable by the magnetic field. Theseare the only properties on which our analysis relies so forsimplicity we will take the coefficient to simply be theHall viscosity. Second, we have not proven for the Hallfluid that this should take a relativistic form as it does inthe Chern insulator, however introducing different coef-ficients for the various terms will not qualitatively affectour results so we will take the simplest case where theyare all equal and we recover the relativistic form. We canproceed to invert this Hall viscosity term by introducingan auxiliary field Gaµ, a = 1, 2 to write the Chern-Simonsterm coupled to the dislocation

LζH = −∑a

ζH2εµνλ∂µu

adis∂ν∂λu

adis

= −∑a

ζH l2

2εµνλ∂µ

(uadisl

)∂ν∂λ

(uadisl

)=∑a

[1

2ζH l2εµνλGaµ∂νGaλ − Gaµεµνλ

(∂ν∂λu

adis

l

)]=∑a

[1

2ζH l2εµνλGaµ∂νGaλ − Gaµjaµ

](26)

where we have re-scaled by the lattice constant l of theunderlying weak square lattice. It is apparent from thisresult that the statistics of a dislocation depends on theHall viscosity ζH . If we drag a dislocation around an-other dislocation it is translated by the Burgers’ vectorba. It thus picks up a relative phase exp(ipab

a) wherepa is the momentum carried by the dislocation traveling

8

around the loop. Since the amount of momentum carriedby the dislocation is proportional to ζH l and the lengthof the Burgers’ vector is an integer in units of l then thestatistical phase depends on ζH l

2. In quantum Hall flu-ids, or a Chern insulator, the Hall viscosity ζH can becomputed in linear response theory35–38,41–43.

ζH =1

8π`2B(27)

for some length scale `B set by the electronic structure inthe Hall fluid/Chern insulator (note we have set ~ = 1.)In the quantum Hall fluid `B is the magnetic length, andin a Chern insulator it is a scale set by the bulk insulatinggap. We will introduce a dimensionless number keff =ζH l

2

2π to rewrite Eq. (26) in a more conventional form:

LζH =∑a

[1

4πkeffεµνλGaµ∂νGaλ − Gaµjaµ

]. (28)

Now we can introduce the soft-spin field Φa to describethe dislocation, and we end up with the dual theory cou-pled with the Chern-Simons theory emergent from theHall viscosity term.

L =∑a=1,2

[g2

2l2(εµνλ∂νKaλ)2 +

1

2|(∂µ − iKaµ − iGaµ)Φa|2

]+∑a

[1

4πkeffεµνλGaµ∂νGaλ

]+r

2|Φ1|2 +

r

2|Φ2|2 + · · ·

(29)

This is formally equivalent to the theory that we havestudied in (1). However there is a subtle difference in thatkeff is not necessarily quantized. One might worry thatnon-quantized keff implies that the gauge theory has ir-rational topological degeneracy keff on T 2. However wesee that the dislocation Φa is logarithmically confined dueto the gapless photon Kaµ, and thus there is no topologi-cal degeneracy associated with keff in the crystal phase.Equivalently, the Wilson loop operators that transformfrom one ground state to a degenerate ground state costenergy and will split the ground-state degeneracy.

We note that the dislocation field Φa is bosonic if thelattice constant l and the magnetic length `B have thecorrect ratio. In the context of the quantum Hall effect(1), this is equivalent to saying that we should have thecorrect filling factor k ∈ 2Z for the electrons to have atransition to a stable fractional quantum Hall state atfilling ν = 1

k . When we are at the correct filling factorkeff for the dislocations, we may have a transition fromthe crystal phase to another phase via the condensationof the dislocations. The condition to have a bosonic ex-citation can be derived:

keff ∈ 2Z↔ `B =l

4π√

2N, N ∈ Z. (30)

We have noted that the Hall viscosity term (25) bindscrystal momentum to a dislocation. This is analogous to

the fact that the Chern-Simons term in (1) binds electriccharge to the flux of the gauge field. It can be shownthat the dislocation of the “flux” (nx, ny) ∈ Z2 carriesthe momentum ~p41,42 (“charge” bound to “flux”):

~p =2πkeff

l(nx, ny) . (31)

Because the momentum is carried by the dislocation,a condensate of dislocations will break the continuoustranslational symmetry with a period (the size of the unitcell) set by the momentum (31). This should be sharplydistinguished from the conventional melting transition ofthe crystal in which the dislocation condensation fully re-stores the translational symmetry. We should stress thatthe original discrete symmetry generated by the two unitvectors (l, 0) and (0, l) is a subset of the new discretetranslational symmetry set by (31). Nevertheless, it isremarkable that we cannot restore the continuous trans-lational symmetry by condensation of the dislocations.

Furthermore, we also notice that the dual theory (29)is equivalent to the theory (1) of the transition betweena Mott insulator and a fractional quantum Hall state.Let us consider the case of keff = 2N ,N ∈ Z to de-scribe a continuous transition from a disordered phase ofthe dislocations to the condensed phase of the disloca-tions. If we assume that dislocations in both directionscondense simultaneously then the condensate in (29) pro-duces a Higgs mass for the combination Gaµ+Kaµ and setsGaµ ≈ −Kaµ (for a = 1, 2) in the low energy regime. Thenwe see that Aaµ receives a Chern-Simons term which wasoriginally in terms of Kaµ:

L =1

4πkeffεµνλGaµ∂νGaλ + · · · (32)

The Chern-Simons term then generates the mass gap forthe photon emergent from the gauge field Gaµ. Addition-ally, it is straightforward to see that the ordered phaseof the dislocations supports a deconfined fractional exci-tation which is the vortex of Φa. Because the disloca-tion Φa is the order parameter for the new crystallineorder with the lattice period dictated by (31), we ex-pect that the vortex of Φa should be again the disloca-tion in the new crystal phase. This phase correspondsto the “torsion-condensed” phase, which is analoguousto the “flux-condensed” phase,i.e., a fractional quantumHall phase. In the fractional quantum Hall phase, theexcitations are the (magnetic) flux, and thus the excita-tion (which is the vortex of Φa) in this torsion-condensedphase is the source of the torsion, or equivalently a dis-location.

In summary, we find that the ordered phase of the dis-locations is again a crystal phase which breaks continuoustranslational symmetry. Furthermore it has topologicalorder with deconfined dislocations excitations. The dis-ordered phase of the dislocation is trivially the originalcrystal phase.

9

Explicit Example: We now consider a specific exam-ple to illustrate the contents of the field theory we devel-oped. Let

lx = ly = 4π`B√

2. (33)

From Eq. 30 this implies that we are at the correct “fill-ing” (in analogy with the quantum Hall states) to havea bosonic dislocation in the excitation spectrum, i.e.,

kxeff = kyeff = 2. (34)

Hence the minimal dislocations, with the “fluxes”(nx, ny) = (1, 0) and (0, 1), are bosonic.

Now, consider the theory where the (1, 0) dislocationcondenses. This dislocation carries a crystal momentum

~p =

(2πkxefflx

, 0

)=

(2π × 2

4π`B√

2, 0

)=

(1

`B√

2, 0

).

(35)

The momentum sets a real space vector ~R such that ~p ·~R = 2π, which defines a new unit cell in the orderedphase of Φx.

~R =

(2π

1/`B√

2, 0

)=(

2π`B√

2, 0)

=

(lx2, 0

). (36)

Notice that we would usually expect for the condensationof the dislocation (1, 0) to restore the translational sym-metry completely along the x-direction in the absence ofthe Hall viscosity term. However, we see that we cannotrestore the continuous translational symmetry by con-densation of the dislocation but end up with anothercrystal with half the period.

B. Case II. Disclination Driven Transitions

Now we consider a background nematic phase withcontinuous translation symmetry and broken rotationalsymmetry. Let us review the relevant properties of thenematic phase and disclinations. The continuous rotationsymmetry of two-dimensional Euclidean space is brokendown to a discrete subgroup Cn, or possibly completelybroken. We will focus on the case with a discrete ro-tational symmetry Cn, n ∈ 2, 3, 4, 6. For example, a tri-angular lattice can melt into a nematic phase with C6

rotational symmetry, namely a hexatic state20–23. Oneconsequence of the broken continuous rotational symme-try is the presence of a single Goldstone mode. An-other consequence is the existence of a topological de-fect, namely a disclination. Ultimately the prolifera-tion of disclinations is responsible for the restorationof the continuous rotational symmetry to return to anisotropic liquid state. In the presence of Cn rotationalsymmetry a disclination carries a quantized Frank angleθF = 2mπ

n ,m = 1, 2, · · ·n−1. Generally the disclinationsare coupled with the Goldstone mode and are logarithmi-cally confined in the nematic phase. A disclination with

the Frank angle θF ∈ [0, 2π] can be modeled as a metrictensor for a point-source of curvature50,51

ds2 = dr2 + (1 +θF2π

)2r2dφ2. (37)

The spin connection in this geometry has a non-zero com-ponent

ωφr =

(1 +

θF2π

)dφ. (38)

Thus, a disclination is a point source of curvature (asopposed to torsion) with an amount of curvature that isprecisely θF .

While the nematic phase can support curvature de-fects, the torsion in a nematic phase will generically van-ish. To understand the vanishing torsion in the nematicphase, we start by considering the crystal phase prox-imate to the nematic phase. In the crystal phase, thetorsion T a present in the system is proportional to thedensity of Burgers’ vector ba (i.e., the dislocation den-sity)27–30. In a conventional crystal to nematic transitionthe dislocations are trivially bosonic and may condenseto drive a transition toward the nematic phase. In fact,the nematic phase is the phase that the minimal disloca-tions with Burgers’ vectors along the axes with discretetranslation symmetry both condense. Hence, the densityof Burgers’ vector in the nematic phase is zero, and thusthe torsion vanishes. We can also conclude that torsion isstill zero even in the presence of disclinations (which bydefinition carry a non-zero Frank angle, but can possiblycarry a Burgers’ vector), because the torsion induced bythe disclination can be screened by the dislocation con-densate. This is true when the nematic is not coupled toan electronic system with a weak topological index of theelectronic band structure52,53.

To describe the nematic melting transition we intro-duce an angular variable θ ∼ θ + 2π

n = θsm + θvor. Inthis splitting θsm describes the Goldstone mode and θvordescribes the disclination in the nematic phase30 throughthe Lagrangian

L =1

2g2(∂µθ)

2. (39)

Using the results from previous sections we use a similarmethod to dualize this elastic theory to obtain an effec-tive theory for the disclination θvor coupled to a non-compact U(1) gauge field aµ:

L =g2

8π2n2(εµνλ∂νaλ)2 − εµνλaµ

n

2π∂ν∂λθvor

=g2

8π2n2(εµνλ∂νaλ)2 − aµJµ, Jµ =

n

2πεµνλ∂ν∂λθvor.

(40)

The non-compact gauge field aµ is a dual description ofthe Goldstone mode. Here the disclination current withthe minimal Frank angle θF = 2π

n is quantized,∫J0 ∈ Z.

10

We introduce a soft-spin variable Φ to complete the dualtransformation of the elastic theory (39).

L =g2

8π2n2(εµνλ∂νaλ)2 +

1

2|(∂µ − iaµ)Φ|2

+r

2|Φ|2 + Veff (|Φ|) + · · · (41)

Now, one can easily find the nature of the two phasesseparated by condensation of the disclination field Φ.r > 0 corresponds to the nematic phase in which the low-energy physics is described by a photon emerging fromaµ. The photon is the dual description of the Goldstonemode of the broken rotational symmetry in the nematicphase. For r < 0, the disclination condenses and the ro-tational symmetry is restored. This phase is the isotropicphase with the full continuous symmetry of two dimen-sional space.

Armed with this knowledge, we now turn on a mag-netic field to form the electronic integer quantum Hallstate over the nematic phase of the quantum lattice. Forconcreteness, we consider the electrons in a uniform mag-netic field in which only the lowest Landau level is filled.We are interested in the linear response of electrons to thegeometric perturbations which generate curvature with-out torsion. The electronic degrees of freedom can beintegrated out and we end up with the effective theoryconsisting of the electromagnetic Chern-Simons term, theWen-Zee term2,34, and the gravitational Chern-Simonsterm42,43:

L =1

4πεµνλAemµ ∂νA

emλ +

1

4πεµνλAemµ ∂νωλ

+1

24πεµνλωµ∂νωλ. (42)

We note that usually in 2+1-d the spin connection isa matrix of one-forms ωabµ dx

µ where a, b = 0, 1, 2. Forthe non-relativistic quantum Hall effect the only impor-tant component is the rotational piece ωxyµ . In Eq. 42ωµ ≡ ωxyµ . As we can see from this action, the elec-trons are sensitive to curvature, however, to proceed,we must assume that the electrons are sensitive to thesources of curvature supplied by disclinations in the ne-matic and that disclinations in the liquid crystal can bemodeled as shown in the metric above. We expect thatthis assumption will only hold when the electrons are ex-tremely strongly coupled to the nematic. It will almostcertainly fail when the electrons are only weakly coupledbecause it is known that the electrons will not even seea quantized curvature from nematic disclinations13. Tobe precise, the assumption we make is that curvature∂µων − ∂νωµ = (∂µ∂ν − ∂ν∂µ)θvor where θvor is the non-single-valued disclination field.

We are now prepared to study the properties of thedisclination-driven transition. First we rewrite the Wen-Zee term and the gravitational Chern-Simons term (42)

in terms of the disclination field θvor

L =1

4πεµνλAemµ ∂ν∂λθvor +

1

24πεµνλ∂µθvor∂ν∂λθvor

=1

2nAemµ Jµ − 6n2

4πεµνλΩµ∂νΩλ −

n

2πεµνλΩµ∂ν∂λθvor

= (1

2nAemµ − Ωµ)Jµ − 6n2

4πεµνλΩµ∂νΩλ. (43)

We have inverted the gravitational Chern-Simons termby introducing an auxiliary field Ωµ and used the quan-tized disclination current Jµ ∈ Z with the smallest Frankangle 2π

n . We can see that the disclination with the small-

est Frank angle traps an electric charge 12n which is con-

sistent with the result of Ref. 43. It is the identificationof a quantized disclination current where we have had touse the strong-coupling assumption mentioned above.

From the theory (43), we notice that the disclinationis bound with a flux 2π

6n2 and has an anyonic statisti-cal angle π

6n2 . To discuss the condensed phase of thedisclinations, we first perform the flux-smearing approx-imation1–5 and thus the disclinations will form integerquantum Hall states with the filling νeff = 6n2 ∈ 2Z.This state can be thought as 3n2 copies of the ν = 2bosonic integer quantum Hall state54–56 which has beendiscussed in the context of the bosonic symmetry pro-tected topological phases. The excitation in this state isa bosonic excitation54–56 with the electromagnetic charge12n (this excitation was the disclination in the nematicphase and thus inherits the electric charge of the discli-nation). This implies that the electromagnetic Hall effectwill be increased by

δσxy = 6n2 × (1

2n)2e2

h=

3e2

2h(44)

in the ordered phase of the disclination. Furthermore,there will be no Goldstone mode associated with the bro-ken rotational symmetry in the condensed phase of thedisclination and thus the resulting phase will be an in-compressible quantum Hall state. Here we have consid-ered the case where only the lowest Landau level is fullyfilled, but it is straightforward to generalize our calcula-tion to the other fillings.

As there is experimental evidence for a nematic frac-tional quantum Hall state57, some of these predictionsmay carry over to that system, though the connectionis not obvious. Away from the strong-coupling limit wewould expect the curvature induced by the disclinationto be non-quantized, but it still determines the statisticalangle and the charge of the disclination. A shot-noise ex-periment58 or tunneling experiments might be applicableto detect the (non-quantized) statistics and charge of thedisclination. This non-quantized statistics of the discli-nation could be clearly contrasted with the quasi-holeand quasi-electron excitations which alway carry quan-tized statistics in the quantum Hall states to distinguishthe contributions.

Before finishing this section, we comment on a rele-vant issue for the case with electrons in a Chern Insula-

11

tor studied in Refs. 41 and 42. In these references, theauthors found geometric response terms including a grav-itational Chern-Simons term in terms of the non-abelianspin connection wab42:

L =1

96πεµνλ

[wµ

ab∂νwλ

ba +

2

3wµ

abwν

bcwλ

ca

]. (45)

Their calculations also show torsional response terms,but no clear candidate for a Wen-Zee term. For the cur-vature response, a disclination corresponds to a flux inwxyµ and traps a spin density Sz = ψ†σzψ of the elec-

trons ψ = (ψ↑, ψ↓)T in the Chern insulator. With this in

mind, it is not difficult to see that the condensation ofdisclinations will induce a spin/magnetic ordering in Szand hence the isotropic phase is a magnetized phase withnon-zero 〈Sz〉. It would be interesting to explore this infuture work, as well as the unresolved issue of how toproperly take an Abelian projection onto the wxyµ sectorof Eq. 45 to recover the gravitational Chern-Simons termwith proper normalization as written in Eq. 42.

III. CONCLUSION

In this paper, we discussed the proximate disorderedphases and the continuous melting transitions driven bycondensation of topological defects of crystals and liq-uid crystals coupled to a quantum Hall state. By de-veloping the dual descriptions for the spatial order, the

topological defects are shown to carry anyonic statisti-cal angles and attached quantum numbers due to theChern-Simons terms of the geometric deformations. Be-cause the Chern-Simons terms attach the fluxes to thedefects, the melting transitions are equivalent to a tran-sition between different quantum Hall states. Thus thetopological order as well as the symmetry may change ata single continuous transition. We also carefully examinethe quantum numbers of the lattice defects and identifythe proximate disordered phase of the liquid crystals. Inthe case of the crystal, we showed that the dislocationcarries crystal momentum and thus the disordered phaseof the crystal is again a crystal with the broken transla-tional symmetry. We also studied the nematic phase andfound that in cases where the electrons are strongly cou-pled to the nematic, the disclination carries fractionalelectric charge and an anyonic statistical angle. Whenthe disclinations condense, the isotropic phase supportsa deconfined bosonic excitation with fractional electriccharge. Thus the isotropic phase and the nematic phasehave different electromagnetic Hall effects.

ACKNOWLEDGMENTS

Authors thank A. Beekman, M. Stone, Z. Nussinov andE. Fradkin for helpful discussions. The authors acknowl-edge support from the NSF grant number DMR-1064319(G.Y.C), NSF CAREER DMR-1351895 (TLH), and U.S.DOE contract DE-FG02-13ER42001.

1 X.-G. Wen, Quantum field theory of many-body systems:from the origin of sound to an origin of light and electrons(Oxford University Press Oxford, 2004).

2 X.-G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992).3 X. G. Wen, Adv. Phys. 44, 405 (1995).4 S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev.

Lett. 62, 82 (1989).5 A. Lopez and E. Fradkin, Phys. Rev. B 44, 5246 (1991).6 J. K. Jain, Phys. Rev. Lett. 63, 199 (1989).7 F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983).8 R. B. Laughlin, Physical Review Letters 50, 1395 (1983).9 L. Balents, EPL (Europhysics Letters) 33, 291 (1996).

10 Z. Tesanovic, F. Axel, and B.I. Halperin, Physical ReviewB 39, 8525 (1989).

11 G. Murthy, Phys. Rev. Lett. 85, 1954 (2000).12 M. Mulligan, C. Nayak, and S. Kachru, Physical Review

B 84, 195124 (2011).13 J. Maciejko, B. Hsu, S.A. Kivelson, Y.J. Park, and

S.L. Sondhi, Phys. Rev. B 88, 125137 (2013).14 S. Kivelson, C. Kallin, D. P. Arovas, and J. R. Schrieffer,

Physical review letters 56, 873 (1986).15 S. Kivelson, C. Kallin, D. P. Arovas, and J. R. Schrieffer,

Phys. Rev. B 36, 1620 (1987).16 E. Fradkin and S. A. Kivelson, Phys. Rev. B 59, 8065

(1999).

17 M. M. Fogler, International Journal of Modern Physics B16, 2924 (2002).

18 S.-Y. Lee, V. W. Scarola, and J.K. Jain, Phys. Rev. B 66,085336 (2002).

19 L. Radzihovsky and A. T. Dorsey, Phys. Rev. Lett. 88,216802 (2002).

20 J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181(1973).

21 B.I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41, 121(1978).

22 D. R. Nelson and B.I. Halperin, Phys. Rev. B 19, 2457(1979).

23 A.P. Young, Phys. Rev. B 19, 1855 (1979).24 Y. You, G. Y. Cho, and E. Fradkin, to appear soon.25 M. Barkeshli and X.-L. Qi, Phys. Rev. X 2, 031013 (2012).26 Y.-H. Wu, J. Jain, and K. Sun, arXiv preprint

arXiv:1309.1698 (2013).27 H. Kleinert and J. Zaanen, Physics Letters A 324, 361

(2004).28 J. Zaanen, Z. Nussinov, and S. Mukhin, Annals of Physics

310, 181 (2004).29 J. Zaanen and A. Beekman, Annals of Physics 327, 1146

(2012).30 A. J. Beekman, K. Wu, V. Cvetkovic, and J. Zaanen,

Phys. Rev. B 88, 024121 (2013).

12

31 A. S. Sørensen, E. Demler, and M. D. Lukin, Phys. Rev.Lett. 94, 086803 (2005).

32 M. Hafezi, A. S. Sørensen, E. Demler, and M. D. Lukin,Phys. Rev. A 76, 023613 (2007).

33 M. A. Baranov, K. Osterloh, and M. Lewenstein, Phys.Rev. Lett. 94, 070404 (2005).

34 X. G. Wen and A. Zee, Phys. Rev. Lett. 69, 953 (1992).35 J. E. Avron, R. Seiler, and P. G. Zograf, Phys. Rev. Lett.

75, 697 (1995).36 N. Read, Phys. Rev. B 79, 045308 (2009).37 N. Read and E. H. Rezayi, Phys. Rev. B 84, 085316 (2011).38 C. Hoyos and D. T. Son, Phys. Rev. Lett. 108, 066805

(2012).39 B. Bradlyn, M. Goldstein, and N. Read, Phys. Rev. B 86,

245309 (2012).40 F. Haldane, arXiv preprint arXiv:0906.1854 (2009).41 T. L. Hughes, R. G. Leigh, and E. Fradkin, Phys. Rev.

Lett. 107, 075502 (2011).42 T. L. Hughes, R. G. Leigh, and O. Parrikar, Phys. Rev.

D 88, 025040 (2013).43 A. G. Abanov and A. Gromov, ArXiv e-prints (2014),

arXiv:1401.3703 [cond-mat.str-el].44 X.-G. Wen, Phys. Rev. Lett. 84, 3950 (2000).45 D.-H. Lee and C. L. Kane, Phys. Rev. Lett. 64, 1313

(1990).

46 X.-G. Wen and Y.-S. Wu, Phys. Rev. Lett. 70, 1501 (1993).47 W. Chen, M. P. A. Fisher, and Y.-S. Wu, Phys. Rev. B

48, 13749 (1993).48 L. Pryadko and S.-C. Zhang, Phys. Rev. Lett. 73, 3282

(1994).49 F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).50 M. A. Vozmediano, F. de Juan, and A. Cortijo, in Journal

of Physics: Conference Series, Vol. 129 (IOP Publishing,2008) p. 012001.

51 C. Furtado, F. Moraes, and A. de M Carvalho, PhysicsLetters A 372, 5368 (2008).

52 J. C. Y. Teo and T. L. Hughes, Phys. Rev. Lett. 111,047006 (2013).

53 S. Gopalakrishnan, J. C. Y. Teo, and T. L. Hughes, Phys.Rev. Lett. 111, 025304 (2013).

54 T. Senthil and M. Levin, Phys. Rev. Lett. 110, 046801(2013).

55 Y.-M. Lu and D.-H. Lee, ArXiv e-prints 1210.0909 (2012),arXiv:1210.0909 [cond-mat.str-el].

56 Y.-M. Lu and A. Vishwanath, Phys. Rev. B 86, 125119(2012).

57 J. Xia, J. Eisenstein, L. Pfeiffer, and K. West, NaturePhysics 7, 845 (2011).

58 E.-A. Kim, M. Lawler, S. Vishveshwara, and E. Fradkin,Physical review letters 95, 176402 (2005).