Symmetry-Enforced Fractonicity and Two-Dimensional Quantum Crystal Melting

PHYSICAL REVIEW B 100, 045119 (2019)

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Symmetry-enforced fractonicity and two-dimensional quantum crystal melting

Ajesh Kumar and Andrew C. Potter Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA

(Received 5 May 2019; published 15 July 2019)

Fractons are particles that cannot move in one or more directions without paying energy proportional to their displacement. Here we introduce the concept of symmetry-enforced fractonicity, in which particles are fractons in the presence of a global symmetry, but are free to move in its absence. A simple example is dislocation defects in a two-dimensional crystal, which are restricted to move only along their Burgers vector due to particle number conservation. Utilizing a recently developed dual rank-2 tensor gauge description of elasticity, we show that accounting for the symmetry-enforced one-dimensional nature of dislocation motion dramatically alters the structure of quantum crystal melting phase transitions. We show that, at zero temperature, sufficiently strong quantum fluctuations of the crystal lattice favor the formation of a supersolid phase that spontaneously breaks the symmetry enforcing fractonicity of defects. The defects can then condense to drive the crystal into a supernematic phase via a phase transition in the (2 + 1)-dimensional XY universality class to drive a melting phase transition of the crystal to a nematic phase. This scenario contrasts the standard Halperin-Nelson scenario for thermal melting of two-dimensional solids in which dislocations can proliferate via a single continuous thermal phase transition. We comment on the application of these results to other scenarios such as vortex lattice melting at a magnetic field induced superconductor-insulator transition, and quantum melting of charge-density waves of stripes in a metal.

DOI: 10.1103/PhysRevB.100.045119

I. INTRODUCTION a putative zero-temperature quantum dislocation condensation transition must occur via the virtual (tunneling) dynamics of Condensation of topological defects plays a crucial role dislocations. Here the analogy between vortices and disloca- in the thermal and quantum melting of symmetry breaking tions breaks down, due to strong symmetry constraints that re- orders in low dimensional systems. Famous examples include strict the dynamics of dislocations. Namely, while dislocations the destruction of two-dimensional (2D) superfluids or easy- may freely move (or “glide”) along their Burgers vector, they plane magnets by vortex proliferation (the BKT transition [1–3]). The critical properties of these transitions are captured cannot “climb” perpendicular to this direction without adding by the 2D XY universality class at finite temperature, and or removing particles from the system. This so-called “glide extend to a related zero-temperature quantum phase transition constraint” has previously been noted in the literature [16–18], that is in the three-dimensional (3D) XY universality class, but, as we will argue, its consequences for quantum crystal with the extra dimension encoding the quantum dynamics and melting have not been fully appreciated. Specifically, in an fluctuations of the vortices. insulating crystal made of particles with a conserved number, The melting of 2D crystalline solids provides another clas- changing the particle number costs a nonzero amount of sic example of destruction of order by topological defects. At energy, and cannot occur at zero temperature. Similarly, since nonzero temperature, 2D crystals exhibit dislocation defects, the condensation of dislocations also requires a condensation whose thermal proliferation can drive a continuous melting of these symmetry-forbidden climb events, producing a quan- transition into a nematic or hexatic phase in which continuous tum superposition of states with different particle numbers, translation symmetry is restored, but rotational symmetry i.e., a dislocation condensate, is necessarily accompanied by breaking persists. As described by textbook Halperin-Nelson superfluid order. theory [4–6], this thermal dislocation-induced melting is es- In other words, a direct quantum melting transition via dis- sentially the same as the vortex condensation in a superfluid, location condensation would necessarily involve simultaneous with the minor distinction that the dislocations come in two restoration of translation symmetry and breaking of particle- flavors distinguished by different Burgers vectors. In the number conservation. In the conventional Landau paradigm spirit of the vortex melting of superfluids, an analogous zero- of phase transitions, a direct transition between two unrelated temperature quantum melting transition in the 3D XY univer- symmetry breaking patterns is generically not possible, and sality class was hypothesized, and invoked in various theories requires either fine tuning or the emergence of exotic decon- of melting of electronic crystals such as charge- and spin- fined particles and gauge fields [19,20]. Therefore, conven- density waves [7,8], and stripes in high-temperature cuprate tional wisdom would dictate that the climb constraint forbids a superconducting compounds [9–14] and neutral atomic continuous (second order) phase transition driven by quantum crystals [15]. dislocation condensation. However, whereas the thermal 2D solid melting transition To address this point, we construct an effective field theory occurs via the entropic proliferation of defect configurations, of dislocations, utilizing a dual description of the elastic

fluctuations of the crystal [21], which was recently and el- temperature phase diagram in Sec. III D, and in particular, egantly reformulated as a higher rank gauge theory [22]. In recover the Halperin-Nelson melting scenario. the latter formulation, the dual gauge charges are disclina- A dual gauge-theory of elasticity was first formulated in tions that cannot move without exciting a finite number of works by Zaanen et al. [21]reviewedinRef.[26], which then additional excitations per unit displacement [23,24], and thus studied quantum melting transitions of a 2D crystal. These at zero temperature are immobile objects, dubbed “fractons” works consider the glide constraint, and also identify that a [25]. dislocation condensate has coexisting nematic and superfluid The gauge charges of this theory are disclinations (ori- order. However, these works primarily neglect the role of entational defects), and are completely immobile fractonic excitations with nonzero particle number (interstitials and objects. The dislocations of the crystal appear in this dual vacancies), and their role in the origins of the superfluid. description as dipoles of the gauge charge, and are not inher- Moreover, they posit that a 2D crystal can melt via a single ently fractonic. Namely, in the absence of any symmetries, continuous transition to a smectic superfluid in the 3D XY they can move in any direction without producing additional universality class. In contrast, we explicitly incorporate inter- excitations. However, we will show that the glide constraint stitials and their coupling to the dislocation motion, and find dictates that, in the presence of U (1) particle-number conser- that the onset of a superfluid of interstitials and vacancies, vation symmetry, the dislocations cannot hop perpendicular and hence the resulting alleviation of the symmetry-enforced to their Burgers vector without producing excitations with fractonicity of the dislocations, is crucial for the disordering a net U (1) charge. In a charge insulating crystal, these are of crystalline order, and argue that the quantum nematic phase gapped and hence the symmetry forces the dislocations to be is likely preceded by an intermediate supersolid phase. 1D subdimensional fracton particles. We dub this phenomena: There are also previous works studying thermal vortex- symmetry-enforced fractonicity. line lattice melting in 3D [17,27]. Building on the work of This symmetry-enforced fractonicity rules out the exis- Marchetti and Nelson [27], where they describe the melting tence of an insulating nematic phase, as condensation of dis- by an unbinding transition of dislocation loops, Marchetti and locations inevitably produces condensation of particle num- Radzihovsky [17] incorporate the coupling of the climb mo- ber, i.e., producing a superfluid. Moreover, it implies that a tion of dislocation loops out of their glide plane to interstitials conventional, continuous quantum phase transition between and vacancies. They also argue that melting of a vortex solid a charge-insulating crystal and any noninsulating nematic to a vortex hexatic phase would require an intermediate vortex or hexatic phase is fundamentally forbidden. Instead, we supersolid phase if the transitions are continuous, in agree- show that, rather than directly melting the crystal, quantum ment with our discussion on (2 + 1)-dimensional quantum fluctuations of the crystal can instead drive condensation of vortex lattice melting in Sec. IV. the underlying particles to produce a supersolid phase with Looking beyond elasticity, our definition of symmetry- coexisting superfluid and crystalline orders. In this supersolid, enforced fractonicity is conceptually related to recently pro- the superfluid condensate alleviates the symmetry-enforced posed ideas of fractal-symmetry protected topological phases fractonicity, and frees the dislocations to move in any direc- [28]. In contrast, here we will consider only ordinary global tion. As quantum fluctuations of the crystal order are further symmetries, rather than a more exotic infinite number of increased, it is then possible to follow a quantum analog of the symmetries defined on different fractal-geometry subsystems. Halperin-Nelson theory, in which the supersolid directly transitions into a supernematic or superhexatic, before ultimately II. SYMMETRY-ENFORCED FRACTONICITY condensing the disclination defects to completely restore the translation symmetry and arrive at a superfluid phase. We first investigate the quantum melting of a 2D solid After describing this sequence of transitions using the formed by a crystal of bosonic atoms with a conserved dual higher-rank gauge theory language, we next explore number. To formulate an effective field theory description the consequences for these ideas to other types of crystals. of symmetry-enforced fractonicity of dislocations, we begin We first adapt these ideas to the quantum melting of vortex with the standard theory of elastic fluctuations of a crystal lattices in superfluids or superconductors. Finally, we explore in terms of a displacement field ui(r,τ) that describes the the consequences of symmetry-enforced fractonicity in the displacement the i ∈{x, y} direction of the atom located from quantum melting of 2D charge-density waves or “stripes” in a its equilibrium position r, at imaginary time τ (this Euclidean metal, where our analysis suggests that quantum fluctuations time coordinate is related to the real time by the usual Wick of stripe dislocations can favor pairing and superconductivity. rotation: τ = it). Here we adopt a continuum description, ob- Before embarking on the main subject of this paper, we tained by coarse graining on a length scale that includes many briefly comment on the relation of our results to those pre- unit cells of the underlying crystal lattice. In this limit we may viously obtained in the literature. Having mapped (2 + 1)- approximate the discrete atomic density by continuous density dimensional elasticity theory to a higher rank tensor gauge and current fields: theory, Pretko and Radzihovsky [22] describe the Halperin- − ρ ρ − ∇ · i 0(1 u) 2 Nelson sequence of melting transitions in the gauge theory jμ(x) = ≈ (−i) + O(∇ u). (1) j ∂τ u language, and further predict the possibility of a supersolid i phase at zero temperature, which agrees with our analysis. τ Here xμ = ( ) is the Euclidean space-time coordinate, and We work out the zero temperature phase diagram of the dual r gauge theory, and show that quantum melting occurs through ρ0 is the average density of particles in the crystal. an intermediate supersolid phase. We also discuss the finite For smooth elastic fluctuations of atomic displacements

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glide along their Burgers vector, and become subdimensional fractonic objects that can move only along one-dimensional submanifolds of the 2D system. As remarked above, in order to disorder the crystal in a continuous quantum phase transition to a nematic or hexatic phase, one must dynamically condense the dislocations—a - + process that is hampered by their symmetry-enforced fractonic nature. The dislocation motion is further inhibited by the long-range interactions between dislocations, mediated by elastic deformations of the crystal and grow logarithmi- cally in the distance between dislocations. In fact, for purely one-dimensional (1D) systems, it is impossible for particles with such long-range interactions to condense [29], but rather they always develop an interaction induced mass or form an + - incompressible crystal. This observation naturally raises the suspicion that it may be fundamentally impossible for the dislocations to condense ++2- - + in a continuous quantum phase transition. However, the dislocation system is not directly equivalent to a decoupled set - + - + - + of pure 1D systems, and requires further analysis. Namely, while each dislocation is confined to move along a 1D line, dislocations along different 1D lines interact via elastic fluc- - + tuations of the crystal. To analyze the resulting system, we develop an effective field theory for dislocations built on previously developed dual theories of elasticity [22,26]. In FIG. 1. Dislocations motion. (a) The dislocation shown with Burgers vector b is described by a dipole of gauge charges with the following section we briefly review the elegant rank-2 dipole moment perpendicular to b, in the dual gauge theory. (b) and gauge theory formulation of the dual elasticity [22], and then (c) The climb (glide) motion of the dislocation corresponds to the incorporate dislocation fields in a manner consistent with the motion of the dipole parallel (perpendicular) to its dipole moment, global number or charge conservation of the underlying atoms and requires the addition of a diagonal (off-diagonal) quadrupole forming the crystal. moment. A. Review: Rank-2 dual gauge theory description of elasticity

s To analyze the possibility of a quantum melting transition u , these currents satisfy the continuity equation: ∂μ jμ = s s driven by condensation of fractonic dislocations, we employ i[∂τ (∇ · u ) − ∇ · (∂τ u )] = 0. a dual rank-2 gauge theory description of the elastic fluctu- Dislocations arise as singular configurations with non- ations developed in Ref. [22], building on previous work of trivial circulation of u: ∂iu jdi = 2πb j, where b j is the jth component of the dislocation’s Burgers vector, and the Zaanen et al. [21]. We will then generalize these theories integral is taken along any path that encircles the dislocation. to include gauge-charged (disclinations) and gauge-dipolar Equivalently, we may characterize the μ ∈{τ,x, y} compo- (dislocations) matter. nent of the dislocation current with Burgers vector component The construction of the dual theory starts from the contin- i of unit length by uum description of elastic fluctuations of the atomic displacement vector field ui(x) with Lagrangian density: d μνλ J = ∂ν ∂λu , μ,i ( i ) (2) 1 2 2π L = ∂τ + ∂ ∂ , el. 2 [( ui ) Cijkl iu j kul ] (4) where μνλ is the fully antisymmetric unit tensor with space- time indices. where Cijkl is the rank-4 elasticity tensor, which is symmetric In the presence of a generic dislocation motion, the particle in arguments ij and kl. Following standard duality trans- π number current is actually not conserved. Instead, one finds formation, we introduce Hubbard-Stratonovich fields i (the lattice momentum) and σij, where σij (the symmetric stress ∂ = πρ d , μ jμ 2 0 ijJi, j (3) tensor): where ij is the antisymmetric unit tensor with spatial indices. 1 1 = τ 2 −1 σ σ + π π + σ ∂ − π ∂ . Equivalently, to move a dislocation by one lattice spacing S d d x Cijkl ij kl i i i ij iu j i i τ ui perpendicular to its Burgers vector, one must add or remove 2 2 a unit of charge to the system, as illustrated in Fig. 1.Ina (5) charge-insulating crystal (i.e., one which is not a supersolid or a more exotic compressible quantum liquid state), changing Separating the displacement field into u = us + ud , a smooth the charge density requires adding energy, and hence each (s) part and a singular defect (d) part that accounts for topo- climbing step requires producing gapped charge excitations, logical defects, and integrating out the smooth field constrains preventing the climb motion. Instead, dislocations may only the fields πi and σij to obey the momentum-conservation

045119-3 AJESH KUMAR AND ANDREW C. POTTER PHYSICAL REVIEW B 100, 045119 (2019) continuity equation illustrated in Fig. 1. The dipole moment is perpendicular to the Burgers vector of the corresponding dislocation, and hence we ∂τ π − ∂ σ = 0. (6) i j ij obtain the 1D particle property of dislocations. An intriguing analogy to electrodynamics arises upon rewrit- ing the stress tensor in terms of a rank-2 electric field, and the B. Effective theory of dislocations momentum in terms of its corresponding magnetic field bi: Having reviewed the dual higher-rank gauge theory de- = ˜ −1 σ , eij ik jlCklmn mn scription of phonon fluctuations, we now turn to developing bi = ijπ j, (7) an effective field theory for topological defects in the elastic medium, i.e., to incorporate matter into the dual gauge theory. where ij is the unit antisymmetric tensor with two spatial in- To this end, we will need to identify the leading relevant oper- ˜ = dices, and Cijkl ia jb kc ldCabcd . In terms of these variables ators describing the dynamics of dislocations that are invariant the constraint equation becomes a Faraday-type “law” under the rank-2 gauge structure, and carefully consider their

∂τ bi + jkC˜iklm∂ jelm = 0. (8) transformation properties under the global U (1) symmetry associated with the conserved number of the underlying parti- This constraint can be solved, at the expense of introducing cles forming the crystal. a gauge redundancy, by expressing eij and bi in terms of a symmetric rank-2 tensor gauge field aij and a scalar potential 1. Gauge charges (disclinations) a0, such that Eq. (8) is automatically satisfied: −1 The fundamental gauge charges of the dual elastic theory e = C˜ (∂τ a + ∂ ∂ a ), ij ijkl kl k l 0 are disclinations (defects in the orientational order of the ψ bi =− jk∂ jaki. (9) crystal). We can introduce a field c for these gauge-charged fields, which transforms under the rank-2 gauge transforma- The physical fields are invariant under gauge transformations tions [Eq. (10)] as of the form iξ (r,τ ) ψc(r,τ) → e ψc(r,τ). (15) aij → aij + ∂i∂ jξ, a0 → a0 + ∂τ ξ. (10)

Writing the action (5) in terms of eij and bi,gives As previously remarked, the higher-rank gauge structure implies that both the gauge-charge and gauge-dipole moment 2 1 1 S = dτd x eijC˜ijklekl + bibi − ρca0 − Jijaij , (11) of the defect matter fields are conserved. Consequently, all 2 2 gauge invariant operators have vanishing charge and dipole where ρc is the gauge-charge (disclination) density, and Jij moments. The lowest-order moment operators are those that is the tensor of disclination currents [30], and a0 acts as add gauge quadrupoles of the matter fields, such as a Lagrange multiplier to enforce the following Gauss’s law ,τ = ψ ψ† + ψ + + ψ† + constraint: Qxy(r ) c(r) c (r dxˆ) c[r d(ˆx yˆ)] c (r dyˆ) x+d y+d −i dx1 dy1 axy(x1,y1,τ ) ∂i∂ jeij = ρc. (12) × e x y , ,τ = ψ + ψ† 2ψ − Using the definition of the disclination density in the above Qxx(r ) c(r dxˆ)[ c (r)] c(r dxˆ) + + constraint, one obtains a mapping between the electric field x d x x1 −i dx1 − dx2 axx (x2,y,τ ) tensor and the strain tensor: × e x x x1 , ,τ = ψ + ψ† 2ψ − = 1 ∂ + ∂ . Qyy(r ) c(r dyˆ)[ c (r)] c(r dyˆ) eij 2 ik jl ( kul l uk ) (13) + + y d y y1 −i dy1 − dy2 ayy (x,y2,τ ) In addition to conservation of the total charge, the Gauss’s × e y y y1 , (16) law constraint (12) also implies conservation of the total dipole moment, which has the striking consequence that the where d is the lattice spacing. charges in the theory are immobile fractons [22,30,31]. The In a system where the number of particles that form the result is a dual tensor gauge theory coupled to fractonic matter. crystal is a conserved quantity, it is crucial to consider the The dipoles, in this case, are bound pairs of disclinations, physical charge quantum numbers of these quadrupolar op- which are dislocations. erators, i.e., their transformation properties under the U (1) The U (1) symmetry-enforced fractonic nature of disloca- symmetry associated with the particle number conservation. tions is captured by the following conservation law, obtained From the previous section we have seen that that dislocation using the Gauss’s law (12): climb by one lattice spacing requires adding or removing one particle. In the gauge theory, a dislocation is a dipolar 2 2 d x(ρx − 2eii ) = const. (14) composite of two opposite gauge charges that are displaced by distance d equal to a lattice spacing. The climb motion Using the duality mapping (13), one obtains eii ≈ n, where then corresponds to changing moving the gauge-dipole one n is the change in the particle density. Therefore, in a charge lattice spacing along its dipole moment. This is precisely the insulating crystal, the sum of the diagonal components of effect of the quadrupolar operators Qxx and Qyy [see Fig. 1(b)]. the quadrupole moment is conserved, forbidding the motion Therefore, under a U (1) rotation that transforms unit- of dipoles along the direction of their dipole moment, as charged operators by χ, these gauge-quadrupole operators

045119-4 SYMMETRY-ENFORCED FRACTONICITY AND … PHYSICAL REVIEW B 100, 045119 (2019) must likewise transform: climb operator adds a particle to the system and will likewise

−→ −iχ , acquire a phase under this U (1) rotation. In the gauge descrip- Qxx/yy e Qxx/yy U (1)χ tion, this dislocation climb operator, hops a dipolar particle −→ . by a unit lattice spacing along its dipole moment, and must Qxy Qxy (17) U (1)χ therefore transform as Using Eq. (16), this transformation then implies that the † id a d O , (r) =ψ (r + d,τ)e i ij j ψ (r,τ) disclination field transforms as climb d d d −→ −iχ . ψ ,τ −→ ir2χ/2d2 ψ ,τ . e Oclimb,d (23) c(r ) e c (r ) (18) U (1)χ U (1)χ In a charge-conserving system, such charged operators Therefore the operators Qxx and Qyy cannot appear alone in the action of any U (1)-symmetric theory. Instead, the minimal cannot appear on their own in the effective theory, but rather action for the gauge charges (disclinations) in a number- can only enter in charge-neutral composites (perhaps in- conserving system is volving other charged fields). Assembling these ingredients, the minimal form of the low-energy effective dislocation L = ψ† ∂ − ψ + λ ,τ + |ψ |2 , c c (i τ a0 ) c xyQxy(r ) Vc( c ) (19) field theory compatible with gauge invariance and charge where V is a potential for the disclinations, and λ is a conservation is c xy nonuniversal parameter. † 1 ⊥ 2 2 L = ψ Dτ ψ + d D ψ + V (|ψ | ) dis. d d 2m ij j d d d d d 2. Gauge dipoles (dislocations) + Ld , (24) Though disclinations are the elementary gauge-charged oct. objects, in a crystalline solid phase, they are not only fully where the d-sum ranges over different elemental lattice vec- Ld immobile, but also linearly confined [30]. Consequently, they tors, and oct. hops a diagonal gauge quadrupole along d,or do not play a role in the low-energy physics of a continu- in other words, adds the corresponding gauge octupole, and ous melting transition. In contrast, dislocations have a much ⊥ d = δ − ˆ ˆ weaker logarithmic-in-distance interaction (like vortices in a ij ( ij did j ) (25) superfluid), which makes them important near the thermal is a projection onto the direction perpendicular to the gauge- melting transition. In the dual description, dislocations are dipole moment. Crucially, the spatial derivative term omits dipolar composites of the fundamental gauge charges. Let gradients along the gauge-dipole direction (enforced by ⊥ us denote the dislocation field with gauge-dipole moment the projector d ), which would correspond to the U (1)- ψ d by d, which corresponds to the field of a dislocation forbidden climb motion, and V is a (generally nonlinear) = d with Burgers vector bi ijd j. Since the Burgers vector is effective Ginzburg-Landau-type potential for the dislocations. quantized to an integer multiple of lattice spacings, the ele- Ld However, oct. corresponds to a pair of dislocation climb mentary dislocation dipole moment |d| is equal to the lattice motions opposite to each other, or equivalently, the hopping spacing d. of a crystal particle, and therefore, respects the global U (1) Viewing the dislocation field as a tightly bound pair of op- symmetry. On the lattice, it takes the following form: posite unit gauge charges separated by unit distance in the ith Ld = † + + direction, one can deduce its gauge-transformation properties oct. thop Oclimb,d(r d )Oclimb,d(r) H.c. from those of the elementary charge fields, Eq. (10): d i[ξ (r+d,τ )−ξ (r,τ )] id·∇ξ (r,τ ) ψd(r,τ) → e ψd(r,τ) ≈ e ψd(r,τ), In the continuum limit, this gives second derivative terms, as shown below in Eq. (27). (20) For large quantum fluctuations of the atomic positions, where the last line holds in the continuum limit where we dislocations become important and Vd can develop a minimum coarse grain our distance scale on lengths much bigger than at a nonzero value of |ρd |. It is then useful to decompose the the underlying lattice spacing. dislocation field into amplitude and phase components: Gauge-invariant operators that move gauge-dipolar parti- √ φ ψ = ρ ei d . (26) cles (dislocations) from space-time point x = (r,τ) → x = d d (r,τ ) are Wilson lines of the form Integrating out massive fluctuations in the amplitude produces x † i d a d +d·∇a d ρ , = ψ x ( i ij j 0 0 )ψ . d 2 Wd(x x) d (x )e d(x) (21) L = (∂τ φ − d ·∇a ) dis. 2 d 0 Taking the limit of infinitesimal length Wilson lines, we see d that gauge invariant terms in a continuum dislocation field 1 ⊥ 2 + d (∂ φ − d a ) theory will involve covariant derivatives of the form 2m ij j d k kj d ψ = ∂ − ψ , Di d ( i idja ji) d 2 + thop (dîd ∂i∂ jφd − did ∂kaijdˆj ) , (27) ψ = ∂ − · ∇ ψ . j k Dτ d ( τ id a0 ) d (22) d To examine the particle-number quantum numbers of the where the absence of a linear time-derivative term is guar- climb operator, consider a U (1) transformation that rotates anteed by the zero net density of dislocations in the system unit-charged fields by phase χ. We know that the dislocation (due to inversion symmetry, which implies a particle-hole

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superfluid order parameter sf: 1 r u L = † ∂ + |∇ |2 + | |2 + | |4, sf sfi τ sf sf sf sf (29) 2ms 2 4! where, in a mean-field treatment, r > 0 would correspond to an insulator, and r < 0 to an ordered superfluid. −→ iχ Noting that under U (1) phase rotations, sf e sf,we U (1)χ see that the minimal gauge invariant and number conserving coupling between the the dislocation climb operators and the superfluid order parameter takes the form

Lsf-dis. = γsf Oclimb,d + H.c., (30) d which describes a process in which the dislocation climbs by removing an atom from the superfluid condensate to conserve the total particle number. We will first examine the backaction FIG. 2. Schematic phase diagram. At zero temperature, the frac- of the dislocation fields on the superfluid action, where we will tonic nature of the topological defects of the crystal leads to a de- see that the quantum fluctuations of the crystal can actually parture from the classical Halperin-Nelson thermal melting scenario. favor the formation of superfluid order. Subsequently, we will We obtain a sequence of disordering phase transitions preceded analyze the effective action for dislocations in the resulting by the onset of superfluidity of interstitials and vacancies, marked supersolid phase. in blue. At finite temperature, thermal fluctuations cause the long- range order of the superfluid to change to quasi-long-range order, and subsequently, to short-range order via a BKT transition at the A. Crystal fluctuation induced supersolidity boundary of the blue region. Consider the above theory when the dislocations are gapped, but close to a continuous condensation transition. symmetry for dislocations). For future reference, we note that Integrating out the dislocation fields, produces a renormal- → = − the transformation of the dislocation phase field φd under ized effective potential for the superfluid: r reff r 1 2 3 † γ , U (1) rotations by angle χ is 2 d x Oclimb,d(x)Oclimb d(0) . Since the dislocations are massive, the two point functions of the climb operator decay φ ,τ −→ φ ,τ + d·r χ. d(r ) d (r ) 2 (28) U (1)χ d exponentially producing − /ξ Having constructed effective field theories for the topolog- e x d r = r − Cγ 2 d3x , (31) ical defects of the crystal, we can now study the melting phase eff xp transitions driven by their condensation. First, we consider the theory of dislocations (27) and study their condensation while where C is a constant and p is the exponent at the critical ∼ξ 3−p disclinations remain gapped. For an ordinary, unconstrained point. Close to the critical point, the above integral is d , (2 + 1)-dimensional XY model (where the dislocations have which diverges at the critical point for p 3, and is finite an isotropic kinetic energy), there would be two distinct for p > 3. Hence, for p 3, r necessarily gets renormalized ξ phases. For small ρd , the phase fluctuations would be large, to a negative value at a sufficiently large d and hence the producing a gapped state with massive dislocations. Whereas, system develops superfluid order before the condensation of for large ρd , the phase fluctuations become stiff, producing the dislocations. In this case, the assumption of a single dis- long-range phase order. We will now see that the presence of location condensation transition fails, and instead is preceded the dynamical glide constraint alters this scenario by requiring by a supersolid phase. the onset of a superfluid of vacancies and interstitials in order If, instead, p > 3, r only gets shifted by a finite constant for the dislocation condensation to occur. at the critical point, and in particular, can still remain posi- tive, meaning that the system remains charge insulating. In the dislocation condensed phase (nematic phase), the climb III. QUANTUM MELTING VIA A SUPERSOLID ∂ φ − operator ∼eidi i d diaijd j is long-range ordered, and there- In this section we show that quantum fluctuations in the fore, the system has coexisting superfluid order. This leads crystal can actually favor the formation of an intermediate to the possibility of a continuous phase transition at which supersolid phase characterized by coexisting superfluid and the U (1) number conservation symmetry is spontaneously crystalline orders. In this supersolid phase, the particle num- broken and the broken translation symmetry of the crystal is ber symmetry is spontaneously broken, freeing the disloca- simultaneously restored. Such a scenario requires fine tuning tions and enabling them to both glide and climb. This enables in the Ginzburg-Landau-Wilson theory of phase transitions, a cascade of continuous quantum disordering transitions from where the generic possibilities are either a first order phase solid to supersolid, to supernematic, and finally to an isotropic transition or a pair of separate second order phase transitions superfluid, as shown in the schematic phase diagram in Fig. 2. corresponding to the two independent symmetries. This sug- To construct an effective field theory description for this gests that a conventional scenario for the quantum melting scenario, we first introduce a Ginzburg-Landau action for the transition would have p < 3. We note in passing, that such

045119-6 SYMMETRY-ENFORCED FRACTONICITY AND … PHYSICAL REVIEW B 100, 045119 (2019) a Landau forbidden transition is a characteristic feature of de- field description in Ref. [21]. However, we reformulate these confined critical points, which exhibit emergent fractionalized results in the rank-2 gauge description. excitations and gauge degrees of freedom at criticality [19,20]. The supersolid phase contains three types of Goldstone Whether such an exotic scenario could arise for quantum modes: a superfluid phase mode associated with the bro- crystal melting is intriguing, but we leave the computation of ken particle-number conservation, and two acoustic phonon the exponent p for future work (e.g., this question could be branches associated with the broken x- and y-translation sym- addressed by quantum Monte Carlo sampling of dislocation metries. In the dual description, these two phonon branches worldlines), and subsequently focus on the more conventional correspond to the two photon branches of the rank-2 gauge scenario of p < 3, where crystal melting can continuously field. Whereas, the superfluid phase mode remains essentially occur only via an intermediate superfluid region. unchanged across the supersolid to supernematic transition, In such a fluctuation-induced superfluid with large but the elastic Goldstone modes are altered. Namely, the su- finite ξd , the dislocations are still massive and the crystalline pernematic breaks only a single spatial symmetry (rotation), order remains intact. Therefore, the resulting phase is a su- and hence we expect only a single elastic Goldstone mode, persolid with both spontaneous breaking of particle number associated with long-wavelength fluctuations of the nematic conservation and translation symmetries.√ Denoting the phase director. In the dual theory, this arises via a Higgs mechanism iϕs of the ordered superfluid as ϕs (sf ∼ ρse ), the effective in which one of the two photon branches acquires a mass due action of the supersolid phase is to coupling with the dislocation (gauge-dipole) condensate. L = L + L + L + L , To see how this arises, we examine the low-energy effective ss el. sf dis. sf-dis. theory of the supernematic phase, starting from Eq. (32). To ρ 1 2 s 2 simplify the analysis, let us specialize to the square lattice, L = + ˜ , L = ∂μϕ , el. 2 bi eijCijklekl sf ( s ) 2 choose a diagonal elastic tensor Cij,kl = cδikδ jl, and rescale ρ d 2 aij and C to absorb factors of the gauge-dipole moment Ldis. = (∂τ φd − di∂ia0 ) 2 d, choose ρ = ρ , and pick units in which m = 1. While d d d d these choices simplify the analysis, they will not change the ρd ⊥ − cos d (∂ φ − d a )d , general structure of universal features (such as number and 2m ij il j d k kj l d d character of Goldstone modes). We will choose to work in the

axial gauge, where a = 0. Since vortices in the dislocation L =−ρ cos(d ∂ φ − d a d − ϕ ), (32) 0 sf-dis. d i i d i ij j s phase are suppressed by the dislocation condensate, we can d expand the cosine terms in Eq. (32) to quadratic order in their √ ρ = γ ρ arguments. The resulting action reads where d s, and the last term represents the superfluid assisted climb motion of the dislocations, and for simplicity ρ 1 ρ of notation we have used a Lorentz invariant action for the L = s ∂ ϕ 2 + 2 + 2 + d ∂ φ 2 sn ( μ s ) eij bi ( τ i ) superfluid phase fluctuations. Here we have not included the 2 2 2 ρ particle hopping term Ld ,inL , as it is less relevant d −1 2 oct. dis. + (∂iφ j − aij − δijd ϕs ) . (33) compared to the superfluid-climb coupling term. 2 In the absence of singular vortex configurations in the Since the interesting change in the collective mode struc- superfluid phase, we may simply absorb the smooth phase ture occurs in the elastic sector, let us freeze the superfluid field ϕ into the dual rank-2 gauge field a by a suitable s ij phase by fixing ϕ = 0 and examine the remaining equations gauge transformation. The resulting action has anisotropic, s of motion for a and φ (written in terms of a real time but nonzero kinetic energy for dislocations in both directions, coordinate t =−iτ): i.e., the superfluid alleviates the glide constraint and converts dislocations from fractons to ordinary mobile defects. We ∂2a = ∂2a − ∂ ∂ a + ρ (∂ φ − a ), refer to the dislocations as defects rather than particles, since t xx y xx x y xy d x x xx ∂2 = ∂2 − ∂ ∂ + ρ ∂ φ − , they still have logarithmic in distance interactions due to t ayy x ayy x yaxy d ( y y ayy ) the elastic fluctuations, so strictly speaking they are weakly ∂2 =∇2 − ∂ ∂ + confined. 2 t axy axy x y(axx ayy) + ρd (∂xφy + ∂yφx − 2axy),

B. Supersolid to supernematic transition ∂2 −∇2 φ =−∂ . t i ja ji (34) With the symmetry-enforced fracton constraint lifted, the crystalline order of the supersolid phase may melt by a Consider excitations with frequency ω and wave-vector continuous quantum phase transitions in which dislocations q = qxxˆ. In the absence of a dislocation condensate ρd = 0, condense. This transition falls in the 3D XY universality class, there would be two propagating photon modes, with either and produces a supernematic phase with both coexisting su- ayy = 0 (a longitudinal phonon) or axy = 0 (a transverse perﬂuid and orientational-symmetry breaking (but translation phonon). With nonzero ρd , the equation of motion for ayy ω2 − 2 + ρ = symmetry preserving) nematic order. We next examine the becomes [ (qx d )]ayy 0, i.e., the photon branch low-energy Goldstone mode excitations of the supernematic corresponding to the longitudinal phonon of the supersolid phase. The results we obtain agree, where they coincide, with acquires a Higgs mass. the well-known properties of general nematic superﬂuids, and In contrast, the off-diagonal components of axy and the have been previously obtained for the rank-1 U (1) gauge dislocation phase φy are coupled, as described by their mo-

045119-7 AJESH KUMAR AND ANDREW C. POTTER PHYSICAL REVIEW B 100, 045119 (2019) mentum space equations of motion: approximately constant, which we can set to 0, and the gauge ω2 − 2 φ = , charge-dipole coupling term becomes effectively Lc−d ≈ qx y iqxaxy √ −γc−d ρd ρc cos (d · ∇φc ). This takes the form of an ω2 − 2 =−ρ φ − . d 2 qx axy d (iqx y 2axy) (35) effective “‘hopping” -type kinetic energy for the disclinations. Solving these coupled equations, one finds two branches Hence we see that the disclinations, which were immobile 2+ ρ ± 2+ ρ 2 3qx 2 d (qx 2 d ) fractons in the crystal, may now hop by absorbing dislocations of modes with dispersions: ω± = .Theω+ 4 from the dislocation condensate. branch is gapped for all momenta, but the ω− branch contains gapless Goldstone modes with dispersion: ω = √1 |q |. To analyze the properties of this in the dual gauge de- 2 x scription, let us write down the full effective action for elastic We next verify that this Goldstone mode indeed corre- fluctuations, defects, and superfluid degrees of freedom in the sponds to fluctuations of the rotation breaking order. First phase-only approximation: note that the fluctuations of a and φ do not produce any xy y ρ ρ compression of the crystal atoms. Namely, the local change c 2 c −2 2 Lc ≈ (∂τ φc − a0 ) + (∂i∂ jφc − aij − d ϕsδij) , −∇ · u 2 2mc in density of the crystal is , in terms of the displace- ∼ ω + ment fields, or equivalently i eii i (axx ayy), which ρd ρd ⊥ L ≈ ∂ φ − ·∇ 2 + d ∂ φ vanishes for this gapless mode. Second, we can compute dis. ( τ d d a0 ) ( j d 2 2m ij θ = 1 ∂ u d d the distortion of the local bond-angle b 2 ij i j, associated with this Goldstone mode. To translate this quantity − 2 + ˆ ·∇φ − ˆ − −1ϕ 2 , into the dual gauge field variables, we again decompose the dkakj) (d d diaijd j d s ) (s) displacement field into smooth and defect parts: θb = θ + b 1 1 ρ θ (d) = 1 ∂ (s) + (d) L = 2 + ˜ + s ∂ ϕ 2. b 2 ij i(u j u j ). Note that the elastic contributions ss. b eCe ( μ s ) (37) (s) 1 2 2 2 to the bond angle can be written as ∂τ θ = ∂ b , and the b 2 i i L ∼ ρd (r−r) Writing the dislocation-disclination coupling as c−d θ (d) = 2 i ij j dislocation contribution is b d r | − |2 . Writing the − φ − · ∇φ r r d cos ( d d c ), we see that in the phase where both ρd = ρ ∂ φ dislocation density i d ( t i ) (in the axial gauge with dislocations and disclinations are condensed, these terms lock = ∼ a0 0), and inserting the above solution to q xˆ Goldstone d · ∇φc = φd, forcing the two components of the dislocation mode, one finds that the Fourier components of the bond angle phase fields φ to be gradients of a single scalar φ . ρ i c θ ∼ i d associated with the elastic Goldstone mode are b ω axy. In this case, we can perform a gauge transformation aij → Together, these observations confirm that the elastic Gold- aij + ∂i∂ jφc and a0 → a0 + ∂τ φc to remove the disclination stone mode of the supernematic phase is indeed a rotational phase field from the action, analogous to the familiar unitary mode as expected. gauge for superconductors. The resulting action in this unitary gauge is

C. Supernematic to superfluid transition −1 −1 ρ ρcm + ρd m L + L ≈ c a2 + c d (a − d−2ϕ δ )2, In the supernematic phase the disclinations are no longer c d 2 0 2 ij s ij immobile fractons. Instead, the disclinations can freely hop without creating further excitations, and have only logarithmic (38) in distance interactions, and thus are weakly confined (since so that all the components of the rank-2 gauge structure the diagonal components of the rank-2 gauge field, which me- acquire Higgs masses that lock a0 = axy = 0 and axx = ayy = −2 diated linear-in-distance interactions in the crystalline phases ϕsd . The only remaining excitation is the superfluid phase [30] acquire a Higgs mass). As the quantum fluctuations in the mode, signaling that the crystalline order has been completely nematic order are further increased, these disclination defects disordered by the defect condensation, restoring full trans- can become important at low energies and eventually can lation and rotation symmetry, and resulting in an isotropic condense to destroy the nematic order and restore rotational superfluid. We summarize the sequence of quantum melting invariance. phase transitions in Table I. To describe the disclination condensation process in the effective dual elastic theory, we should consider the D. Finite temperature crossover disclination field ψc with action Eq. (19). Note that disclination/antidisclination pairs displaced by a single lattice At zero temperature we have seen that the melting of a spacing are gauge dipoles, and can freely convert into dis- crystal to a nematic phase is accompanied by a superfluid locations, which have the same gauge-dipolar structure. This of vacancies and interstitials, where the dislocations lose coupling is described by a term their subdimensional property. However, one must recover the classical Halperin-Nelson scenario at finite temperature L ψ ,ψ =−γ ψ ,τ ψ † + ,τ c−d [ c d ] c−d d(r ) c (r d ) where there is a continuous phase transition from the crystal d to the nematic phase. To examine how this works, we must × ψ (r,τ) + H.c. (36) consider the effects of thermal fluctuations. c A detailed discussion of the finite temperature physics of In the supernematic phase, the dislocations are con- fractonic matter was presented in Ref. [32]. The basic point is densed, i.e., effectively ψd = 0. To describe this, it that fractons are only prevented from moving at zero temper- √ φ ψ = ρ i c is useful√ to introduce phase variables c ce and ature, where each forbidden move requires exciting additional iφ ψd = ρd e d . In the dislocation condensed phase, φd is gapped fracton excitations. However, nonzero temperature

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TABLE I. Summary of the zero-temperature phases, gapless modes propagating along the x direction, and the nature of the topological defects. The photon modes ayy and axy acquire Higgs masses when the dislocations and the disclinations are condensed, respectively, and ϕs is the Goldstone mode corresponding to the condensation of the bosons that constitute the lattice.

Phase Gapless modes Dislocations Disclinations

Solid axy, ayy 1D fractons, gapped 0D fractons, gapped Supersolid axy, ayy,ϕs unconstrained, gapped 0D fractons, gapped Supernematic axy,ϕs unconstrained, condensed unconstrained, gapped Superfluid ϕs unconstrained, condensed unconstrained, condensed excites a finite density of fractons, which can be absorbed or field), respectively. Again denoting the displacement field emitted to allow the forbidden motion. of the vortex positions by u, the low-energy effective field Here a similar scenario holds for symmetry-enforced frac- theory for this state can be constructed by performing standard tons, with the distinction that it is a thermally excited gas boson-vortex duality, in terms of a vortex field ψv minimally of mobile charged particles that liberate the dislocations at coupled to an emergent U (1) gauge field αμ whose flux is 2π nonzero temperature, as opposed to a thermal self-liberation times the density of the particles forming the superfluid. The of inherently fractonic particles. Specifically, for energy gap resulting effective theory is to charged-particle excitations of the crystal, nonzero tem- L = L ψ ,α + L + L α, , perature will produce a thermal gas of noncrystalline parti- v[ v ] el.[u] α[ A] cles of density ρ ∼ e− /T . Dislocations can then climb by T 1 2 2 L = ψ¯ (−i∂τ −μ−α )ψ + |(∇−iα)ψ | +V (|ψ | ), absorbing or emitting particles into this thermal gas. Since v v 0 v 2m v v thermally excited particles are required to assist these climb- v i 1 direction “hops” processes, their amplitude will be propor- L = ∂τ + ∂ ∂ + α ∇ · + α · ∂τ , el. ijui u j 2Cijkl iu j kul 0 u u tional to ρT , and hence will also display the same activated 2 2 temperature dependence. A second distinction from the 3D (μνλ∂ν αλ) iμνλαμ∂ν Aλ Lα = + , (39) thermal liberation of intrinsic fractons described in Ref. [32], 4κ2 2π is that thermal screening of 3D fractons results in weaker power-law interactions, and whereas the dislocations continue where μ is a chemical potential for the vortices, produced by to exhibit logarithmic confinement, up until a BKT transition, the external time-reversal breaking field that induced the vor- as described by the Halperin-Nelson theory. tex lattice, mv is the effective mass of the vortex excitations, We note that, for temperatures T , there will be and V (···) is an effective potential for the vortex excitations very few excited charged particles to assist the dislocation (additional vortices beyond those already present in the vortex climb, and the dynamics of the dislocations will be highly lattice are gapped excitations of the vortex liquid). In the last anisotropic, such that the climb motion is thermally frozen line, κ is the gauge coupling of the emergent gauge field out in the limit of T → 0. For T , the anisotropy is less α (here, for simplicity, we have written a Lorentz invariant pronounced, and the classical Halperin-Nelson scenario can form, though such a symmetry will not naturally be present take over. in the theory, deviations from this form will not alter the Finally, a possible alternative is that of a single first order subsequent discussion). For vortex lattices in a superfluid phase transition between the solid and liquid (at nonzero or superconductor respectively, A is either: (i) an external temperature) or between the solid and superfluid at zero (i.e., nondynamical or “background”) electromagnetic poten- temperature. tial field (for the superfluid) or (ii) the vector potential of a fluctuating magnetic field (for the superconductor).

IV. VORTEX LATTICE MELTING A. Phonons and dual elasticity theory The above discussion focused on the case where the The absence of time-reversal symmetry dramatically alters underlying crystal arose from bosonic atoms with a con- the phonon spectrum of the vortex solid compared to an ordi- served number. However, these principles have implications nary crystal. Formally, this enables the single time derivative for other types of crystals. As an example, we next consider term in Lel., which is ordinarily forbidden, but now dominates the quantum melting of a 2D vortex lattice of a superfluid 2 the usual (∂τ u) form at low energies. Consequently, instead or superconductor, where the objects forming the crystal of distinct longitudinal and transverse branches of acoustic are collective topological defects of a different order. This phonons, the vortex lattice exhibits a single phonon mode scenario is closely related to the one discussed above through that is a mixture of compression and rotation, and which boson-vortex duality, though there will be some phenomeno- has a nonrelativistic ω ∼ q2 dispersion. We note that other logical differences due to the absence of time-reversal magnetoelastic systems, for example phonons of a skyrmion symmetry due to the magnetic field required to produce the crystal, are described by an identical magnetoelastic theory vortex lattice state. [33]: Starting from a superfluid (superconductor) state, a vortex lattice can be induced by externally breaking time-reversal i 1 L [u] = u ∂τ u + C ∂ u ∂ u . (40) symmetry by applying a net rotation (or external magnetic m.el. 2 ij i j 2 ijkl i j k l

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Starting from the time-reversal asymmetric elasticity ac- not mean that they are free to move. To see this, note that tion, we can derive a dual field theory for the vortex crystal the flux of the dual gauge field α is locked to the particle phonons, following analogous steps to those of Ref. [22], number: ∇ × α = 2πρ, so that the vortices are forced to which has been independently obtained by Zhai et al. [34]. admit a finite gauge-flux density. To accommodate this gauge We first introduce Hubbard-Stratonovich fields πi and σij to flux, the vortex condensate cannot be spatially homogeneous, produce an action that is linear in u: but must itself have a lattice of dual-vortex defects. In the i original particle language this dual-vortex lattice is simply L [u,π,σ] = π ∂τ π + iπ ∂τ u m.el. 2 ij i j i i a crystal of the particles. As we have seen in the previous 1 section, dislocations of this crystal are symmetry-enforced + σ −1 σ − σ ∂ . ijCijkl kl i ij iu j (41) fractons that cannot climb, due to particle number conser- 2 vation. Hence, by condensing the vortices, we have gone The first term marks the chief departure from the time-reversal from a vortex lattice whose dislocations cannot climb due to invariant action. In particular, varying with respect to πi,we the topological vortex-number conservation, to an ordinary see that πi ≈ iju j. This implies that the original translation insulating crystal whose dislocations cannot climb due to symmetry of the underlying elastic action: ui → ui + δi,for particle-number conservation. any constant vector δi, will require invariance under πi → πi + ijδ j. Next, decomposing the displacement field into smooth (s) V. CHARGE-DENSITY WAVE MELTING IN A METAL and singular defect (d) parts: u = us + ud , and integrating out Before concluding, we turn our attention to electronic the smooth fluctuations enforces the constraint of Eq. (6)of 2D crystals such as charge-density waves (CDW), and stripe the main text, which can again be solved by introducing a phases of high-temperature cuprate superconductors [35–37], rank-2 gauge structure identical to that explained in Ref. [22] or other strongly correlated compounds [38]; the related situ- and above. ation where the 2D crystal forms in an electronic system due The resulting dual elasticity action reads to interactions. i 1 Electron-electron interactions in a metal can produce spon- Ldual[a] = biij∂τ b j + eijC˜ijklekl. (42) taneous charge-density wave (CDW) order at wave-vector Q 2 2 that may or may not be commensurate with the underlying The b2 term, present in the time-reversal symmetric case, is ionic lattice, and which are sometimes accompanied by spin- notably absent. Instead, the dual magnetic field term contains density wave (SDW) order. A controlled theory of the onset a linear time derivative of b, such that the dual-photon aij ex- CDW or SDW remains challenging, since the critical fluctua- hibits the ω ∼ q2 dispersion expected of the magnetophonons. tions of the DW order are strongly coupled to the continuum In fact, the above-noted πi → πi + ijδ j invariance of the of gapless particle-hole excitations of the Fermi surface action, which was a consequence of the underlying translation [39–41]. We will not attempt to address this challenging symmetry, manifests as a bi → bi − δi invariance of the dual situation, and instead examine the constraints placed by the action. Therefore, translational invariance forbids any terms symmetry-enforced fractonic nature of its dislocation defects, involving b without derivatives, preventing the nominally and comment on open issues for future work. more relevant b2 terms from appearing, and enforcing the Suppose we can tune a parameter in the system, such as correct quadratic phonon (dual-photon) dispersion. doping or pressure, that tends to destroy the CDW order. Then, as the quantum fluctuations in the CDW order increase, they B. Constraints on melting transitions will favor dislocation motion. However, as the glide constraint dictates, dislocations cannot climb without absorbing elec- The key feature of the above description is that the vortex trons from the Fermi surface into the CDW order. α number becomes the charge of the dual gauge field μ and is Naively, the minimal such process would be for a dis- hence conserved. Physically, this vortex number conservation location to climb enough to absorb a single electron from arises since no local fluctuation of the superconductor can near the Fermi surface. This process would be forbidden in change the global vorticity. In the dual elasticity theory, this a time-reversal invariant system, since the electron carries a again means that the dislocation climb terms must couple to a spin-1/2. It could conceivably occur in a spin-density wave α field with compensating gauge-charge under , such as system where time-reversal symmetry is absent. However,

∂ φ − 2 ∂ φ − 2 L = ei(d x x d axx ) + ei(d y y d ayy ) ψ + H.c., (43) in the above field theory formulation the dislocation climb climb v operator is a bosonic field, and hence cannot couple directly where is some nonuniversal coupling constant. Hence, to a single electron creation or annihilation operator (which as with the previous analysis, it is not possible to directly would correspond to an unphysical nonlocal conversion of condense the dislocations without condensing the vortices, to fermions to bosons). It remains an open question whether this destroy the underlying superfluid or superconducting state. statistical obstruction is fundamental, or whether it is possible Simply condensing the vortices on top of the background to modify the dislocation description in such a way to convert vortex lattice would produce an insulating state, still with the climb operator to a fermion object. translation symmetry breaking crystalline order. The disloca- To simplify our discussion, we will instead focus on a tions of this insulating crystal state that descends from the time-reversal invariant CDW state, in which case dislocations phase-disordered vortex lattice are no longer constrained to can only climb by adding or removing spin-singlet pairs glide by the vortex-number conservation. However, this does of electrons. We can then revisit the above analysis of the

045119-10 SYMMETRY-ENFORCED FRACTONICITY AND … PHYSICAL REVIEW B 100, 045119 (2019) previous sections, but replacing the role of the bosonic super- However, at this point we do not have a concrete scenario ﬂuid order parameter sf by a Cooper pairing ﬁeld Cp which for such an exotic transition. Even if such a phase transition is a charge-2e, spin-singlet operator, producing a coupling of existed, its critical exponents would be dramatically different the form than the conventional (2 + 1)-dimensional XY universality

∂ φ − 2 ∂ φ − 2 class conjectured in previous works [21,26]. L = 2i(d x x d axx ) + 2i(d y y d ayy ) + , climb e e Cp c.c. It would also be potentially interesting to investigate the

iq·r applicability of the concept of global symmetry-enforced (r,τ) = e , (τ )c + / ,↑c− + / ,↓. (44) Cp k q k q 2 k q 2 fractonicity in other contexts, including (3 + 1)-dimensional k crystals [43], or true fracton “topological phases” in which the Here the pairing parameter k,q can encode any symmetry subdimensional objects are deconfined particles with possible allowed pairing (e.g., s wave, d wave, etc.). anyonic behavior and exponential in system-size ground- Then, by analogy to the bosonic quantum melting story state degeneracy [44–47], rather than confined defects of a presented above, we see that the quantum fluctuations in symmetry-breaking phase. the CDW metal favor the formation of a superconducting Note added. Recently we became aware of a related work electron-pair condensate in order to alleviate the glide con- by M. Pretko and L. Radzihovsky [48] on closely related straint and allow the crystal defects to climb, following subject matter. which, they could condense to melt the CDW metal to a nematic metal (note that more detailed microscopic input is ACKNOWLEDGMENTS needed to decide which pairing symmetry would be most favorable) [42]. This scenario is reminiscent of the notion of We thank M. Pretko and P. T. Dumitrescu for insight- competing orders [37] in high-temperature superconductors, ful conversations. This work was supported by NSF DMR- and the symmetry-enforced fracton concepts could potentially 1653007. provide a useful new perspective on these complex materials. APPENDIX: COUPLING OF STRIPE DISLOCATIONS TO A VI. DISCUSSION FERMI SURFACE OF ELECTRONIC EXCITATIONS To summarize, we have introduced the notion of In this Appendix we consider possible couplings of symmetry-enforced fractonicity, where in the presence of a phonons and dislocations of an incommensurate CDW with global symmetry, certain point particles or defects cannot the reconstructed Fermi surface. We will find that the crystal move along certain directions without exciting gapped ex- excitations have only irrelevant couplings to the low-energy citations that carry nontrivial quantum numbers under that particle-hole continuum of electronic excitations, and hence symmetry. We have shown that dislocations of (2 + 1)- can be neglected in the analysis of dislocation condensation. dimensional insulating solids are a simple example of this Since the fermionic fluctuations have vanishing rank-2 concept. Moreover, our analysis shows that the symmetry- gauge charge, they cannot couple directly to the dual gauge enforced subdimensional nature of dislocations dramatically field aij, but rather only to its field strengths bi and eij. These alters the critical properties of quantum melting transitions couplings contain derivatives that make them irrelevant at from what was conjectured in previous literature [21,26]. low energies. Intuitively, this reflects the familiar fact that the Namely, by developing a dual higher-rank gauge theory dual photons described Goldstone modes of broken translation description of these symmetry-enforced fractonic defects, we symmetry, which quite generally decouple from other low- have shown that they can condense to drive a continuous energy excitations [49]. quantum phase transition from a crystal to a nematic phase The low energy effective field theory of the Fermi surface is only when the subdimensional constraint on them is lifted through the onset of a superfluid of vacancies and interstitials. L = dϕψ†[−iv (ϕ) · ∇]ψ , (A1) In retrospect, hints of this result were evident from more gen- ϕ F ϕ eral considerations. Namely, the symmetry-enforced fractonic ϕ nature of dislocations mean that any dislocation condensate where denotes the angle in momentum space (here we must also spontaneously break the U (1) symmetry associated assume a single sheet for the reconstructed Fermi sur- v with number conservation. Therefore, a direct transition from face for notational simplicity), F is the Fermi velocity normal to the Fermi surface at angle ϕ, and ψϕ (r) = crystal to nematic must simultaneously restore translation ϕ + kF ( ) −ik (ϕ)·r dk(ϕ)e F ψk ϕ is the low-energy fermion field symmetry and break number conservation. In a conventional kF (ϕ)− ( ) Ginzburg-Landau-Wilson framework, it is generically not near the Fermi surface at angle ϕ, where is a UV cutoff that possible to go between two phases with completely different is less than the Fermi wavelength. symmetry breaking patterns without fine tuning. Instead, the The low-energy modes of the Fermi surface are shape transition is generically first order, or splits into a pair of fluctuations described by operators separate second order transitions. In other contexts, more † exotic deconfined critical points can arise, in which such a M[ f ] = dϕ f (ϕ)ψϕ ψϕ, (A2) Landau-forbidden change of symmetry can occur in a direct continuous quantum phase transition [19]. It would be intrigu- where f is some real-valued function of the Fermi-surface ing to consider whether a possible deconfined critical scenario angle ϕ. The minimal coupling between the dislocation cur- could apply to quantum melting of a solid to a supernematic. rents and the low-energy modes of the Fermi surface would

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λ d ∼λ2ω |θ ω, |2 be of the form Jμ,iM[ f ], where M[ f ] has the right spatial tions of the form q i( q) . These interactions are symmetries to couple to the Jμ,i. In a Hertz-Millis type less relevant than the long-range dislocation interactions in- analysis [50,51], integrating out the electronic ﬂuctuations duced by the gauge coupling g, and are not expected to ∼λ2 |ω| | d ,ω |2 would produce a Landau damping term q Jμ,i(q ) . alter the analysis presented in the main text for a bosonic d ∼ ∂ θ crystal. If we write Jμ,i iμν ν i this term only produces interac-

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