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Fractonic gauge theory of smectics

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


A. Motivation and background
A smectic state of matter, is a liquid crystal phase that partially breaks rotational and translational symmetries spontaneously, exhibiting a periodic layered order. Classical smectics form in systems of rod-like constituents (molecules like 5CB) 1,2 and are driven by anisotropic entropic (exclusion volume) interactions. In striking contrast, quantum smectic states appear even in systems of isotropic point-like constituents as a result of frustrated competition between kinetic energy and interactions. In cold atom systems, quantum smectics may be realized in a putative Fulde-Ferrell-Larkin-Ovchinnikov paired superfluids 3,4 in imbalanced degenerate atomic gases 5,6 and in spin-orbit coupled Bose condensates 7,8 . Quantum smectics are also a natural explanation for a striking resistive anisotropy observed in quantum Hall systems at half-filled high Landau levels 9-14 , and for "striped" spin and charge states of weakly doped correlated quantum magnets 15,16 .
A two-dimensional (2D) smectic can emerge from partial, anisotropic melting 17 of a crystal, with only one species of dislocations unbinding, such that only one direction of translational symmetry is restored, in a Kosterlitz-Thouless (KT)-like 18 phase transition. However, a 2D smectic is unstable to thermal fluctuations, and is always driven into a nematic fluid at any nonzero temperature 17,[19][20][21] . In contrast, a (2+1)D quantum smectic at zero temperature, is a stable state of matter, whose studies have been limited to a simplest harmonic description, with effects of topological defects and of elastic nonlinearities neglected beyond qualitative discussions (for an exception see Refs. 6,22,23). This, together with ubiquitous putative realizations provides a strong motivation for the present detailed work, a brief preview of which appeared in a recent publication. 24 A complementary motivation for our study is its rela-tion to a new class of topological quantum states of matter -dubbed "fractons" -discovered in theoretical exactly solvable models. [25][26][27][28][29][30][31][32] These feature a number of fascinating properties that are believed to lie beyond a conventional quantum field theoretic description. 33 The most striking of these are quasiparticles with robust (not just fine-tuned or symmetry imposed) restrictions on their mobility, such as an immobile fracton, and its subdimensional multipoles. Although experimental realizations have been sorely lacking, these theoretical models are intensely studied, motivated by their promise for a robust quantum memory 27 and fundamental interest in a new class of topological quantum liquids 34,35 .
Following this gapped class of lattice qubit models, fracton-like phenomena were also uncovered in gapless symmetric tensor gauge theories, encoded in a generalized Gauss law, that conserves charge multipoles and thereby constrains mobility of charges [36][37][38] . Contemporaneously, a similarity of the constrained dynamics of disclination and dislocation defects in a crystal was conjectured to be dual to charges and dipoles of a gauge theory 39 . Utilizing a generalization of the familiar XYto-gauge theory (boson-vortex) duality 40,41 , Pretko and Radzihovsky 42 formalized this relation through a duality mapping (explored in other contexts by Zaanen and company 43 ) between a quantum 2D crystal elasticity and a symmetric tensor gauge theory. Under this mapping the stress tensor σ ij and momentum vector π i fields map onto the electric tensor E ij and magnetic vector B i fields, respectively, with Newton's law (conservation of momentum) corresponding to Faraday's law of the tensor gauge theory. Relation of these tensor gauge theories to chiral topological elasticity was also explored in Ref. 44.
This established fracton-elasticity duality allows for numerous predictions for phases and phase transitions of the fracton system, based on the extensive understanding of 2D crystal and their descendent states. For example, different phases of the scalar fracton model -fracton insulator, dipole condensate and fracton condensate, can be regarded as gauge theory counterparts to the "commensurate" and "incommensurate" (supersolid) crystals [45][46][47] , hexatic, and isotropic fluid phases of the elasticity theory. The associated finite-temperature dipole-unbinding transition and fracton charge unbinding transition correspond to the classical two-stage melting transitions, i.e., crystal-to-hexatic and hexatic-to-liquid transition respectively.
A complementary and physically more transparent formulation of elasticity-to-fractonic coupled vector gauge theory was recently presented 48 . The resulting dual coupled vector gauge theory involves three U(1) vector gauge fields A k (with k = x, y denoting flavors) and a, and their canonically conjugate electric fields E k and e, that encode coupled Goldstone modes, the phonons u k and the local bond angle θ. Building on the treatment of the quantum crystal 48 and a recent analysis of the quantum smectic 24 , we derive and explore extensively the coupled vector gauge theory duality to study the (2+1)D smectic and its quantum phase transitions to a crystal and a nematic, formulated in terms of an array of Higgs transitions. We also utilize it to study 2D and 3D classical smectic and the corresponding classical nematic-tosmectic phase transitions [50][51][52][53][54][55][56] .

B. Summary of Results
In this paper, we develop and explore in detail a dual coupled U(1) vector gauge theory for a 2D quantum smectic, building on a recent study of a 2D quantum crystal 48 and a smectic 24 by one of the authors. The dual description we derive is formulated in terms of two coupled U(1) vector gauge theories, with electric fieldsÊ andê, and canonically conjugate vector potentialsÂ and a, sourced by dipole and charge current densities, n b , j b (dislocations) and n s , j s (disclinations), respectively. The corresponding dual Hamiltonian density is given bỹ supplemented by the generalized Gauss laws, We demonstrate that the n s charges (disclinations in the smectic) of this dual gauge theory are subdimensional "lineon", mobile only transverse to smectic layers (that we take to be alongx), enforced by generalized gauge invariance and associated continuity equation, In contrast the dipoles (dislocations) exhibit a finite but highly anisotropically mobility. Motivated to also understand the quantum crystalsmectic transition, we derive the smectic gauge dual and transition to it by utilizing gauge dual of the quantum crystal 48 and condensing one flavor of dipoles (dislocations). The associated Higgs transition gaps out the corresponding flavor of the gauge fields A k , and leads to a dual quantum smectic Lagrangian, that matches exactly the description obtained through direct duality of smectic elasticity, where, L sm M is the Maxwell sector from Equation (1), with V (|ψ x |) a U(1)-invariant Landau potential for x-flavor dipoles, ψ x . The flow chart in Fig. 1 summarizes these two routes to the dual gauge theory of a quantum smectic. In this formulation the ψ x = 0 Coulomb phase corresponds to the quantum smectic phase, and the ψ x = 0 Higgs phase gives a condensation of unboundx−dipoles (ŷ−dislocations) that gaps out the gauge field A µ , which drives a Higgs transition to a quantum nematic. The quantum melting transitions and the corresponding phases are illustrated in Fig. 2.
We also explore the classical limit of this duality, and formulate the 2D smectic-to-nematic melting and a sub-sequent nematic-to-isotropic fluid transition in terms of FIG. 2: Illustration of quantum melting of a 2D crystal into a smectic, followed by smectic-to-nematic melting, respectively driven by a condensation ofx-dislocations and ofŷ-dislocations, and tuned by applied shear stress, σxx. a higher-derivative sine-Gordon model, The first two terms capture the elasticity of a 2D smectic, and the two cosine correspond to dislocations and disclinations, tuned by the corresponding fugacities g b,s . We demonstrate that in 2D the dislocations are always relevant, corresponding to the instability of a 2D smectic to a nematic at any nonzero temperature 17,[19][20][21] . The resulting sine-Gordon model in α then captures the the nematic-to-isotropic fluid transition, illustrated in Fig. 3.

FIG. 3:
A phase diagram illustrating an instability of a 2D smectic to arbitrary weak thermal fluctuations that drive it into a nematic at any nonzero temperature. The nematic phase transition into an isotropic fluid is through a KT disclination unbinding transition Tc.
We also derive a classical dual gauge theory for a 3D smectic, that captures its finite-temperature melting into a nematic through a dual normal-superconductor transition with an higher-derivative Maxwell sector, equivalent to Toner's original treatment of the nematic-to-smectic-A transition. 56

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

II. SMECTIC AND ITS DUALITY
A. Classical smectic Ideal smectics are equidistantly layered structures, with a well-defined interlayer spacing d, which can be determined through diffraction experiments. With the layers correlations are liquid-like and exhibit crystal-like periodic modulation transverse to the layers, with corresponding density given by, where, q 0 = 2π dẑ is the modulation wavevector, and ψ its amplitude, that is the order parameter that distinguishes the smectic phase from the nematic phase.
The deformation of a smectic can be described by its layer displacement field u(r). As the system is invariant under uniform translations, the elastic energy should be expressed purely in terms of derivatives of u. Furthermore, the first-order derivatives along the layer, ∇ ⊥ u, corresponding to merely a uniform rotation of the layers, must cost no energy. Thus, to harmonic order, only the curvature of the layers, ∇ 2 ⊥ u, can enter the quadratic part of the elastic energy functional. This point can be seen more explicitly by the following argument. The order parameter ψ(r), describing phonon fluctuations, can be represented as and the locations of the layer planes can be determined as the constant phase of the molecular density wave, The layers local unit-normal is given by, The first-order derivative, ∇ ⊥ u ≈ (∂ x u, ∂ y u, 0) ≈ẑ − N, therefore, corresponds to a rigid rotation of the layers around an axis along the layer plane, and does not contribute to the elastic energy, and, ∇ u = ∂ z u, in 3D.
(See. Fig.4.) Consistent with this, the continuum elastic Hamiltonian density for a D-dimensional smectic is given by a well-known expression, and the nonlinear strain, ∇ u − 1 2 (∇u) 2 , can be straightforwardly seen to be independent of the rotation angle θ. Thus, the rotationally invariant energy density is given by, which introduces non-linear elasticity into H sm , that for d ≤ 3 leads to a nontrivial anomalous smectic elasticity 22,23 . However, because the focus of our work is on a quantum smectic, these elastic nonlinearities remain irrelevant in (2+1)D and will thus be neglected in the rest of the manuscript. For the smectic-A phase, the local normal field (layer orientation) N and the director field n =ẑ + δn are aligned in equilibrium. Thus, and the elastic energy, ignoring nonlinearities, can be represented as For a 2D smectic, with the layers alongx (with layer normal alongŷ), the elastic Hamiltonian density in Eq. (11) reduces to where the layer displacement u is along theŷ axis, and the layer orientation (director field) is, n = −x sin θ + y cos θ =ŷ + δn. Another way to obtain smectic elasticity is to start out with a elasticity of a 2D crystal and allow nonsinglevalued displacement field u x , with ∇u x = v x , accounting for a plasma of unbound dislocations with Burgers vector along thex−directed smectic layers, where v x is an arbitrary vector with ij ∂ i v jx = b x . Integrating over strain tensor field v x , leads to the smectic harmonic elasticity, where compressional modulus is χ = 4µ(µ + λ)/(2µ + λ), bend modulus K = E c , and higher derivative terms are neglected after integrating out the v x field in the last step. Equivalently, the smectic elasticity can be formulated in terms of the orientational (nematic) angle degree of freedom θ, which corresponds to the orientation of the layers, with the elastic Hamiltonian density given by, At low energies set by χ, this Hamiltonian reduces to the conventional form (16) after Higgs'ing out the bond angle θ, locking ∂ x u = θ.
It is convenient to work with a path-integral formulation where quantum nature of these fields is accounted for by functional integration in phase-space of these fields. We consider the evolution operator for the quantum smectic, and rewrite it in phase-space functional integral formulation as, with the corresponding Lagrangian density given by, In addition to the single-valued (smooth) Goldstone mode degrees of freedom,θ andũ, we must also include topological defects -disclinations and dislocations, captured by including a nonsingle-valued component of the bond angle field θ s and of the phonon distortion field u s , respectively. In the smectic, a disclination at a point r 0 , is defined by a nonzero closed line-integral of the gradient of the bond angle around r 0 , r0 dθ = 2πn s , or equivalently in a differential form, measuring the deficit/surplus bond angle. n s (r) is the disclination charge density. A disclination with charge n s = − 1 2 , and a disclination with charge n s = + 1 2 are illustrated in Fig. 5(a, b).
A dislocation at r 0 with a Burgers charge n b (that is an integer multiples of the elementary layer spacing), is defined by a closed line-integral, r0 du = n b , or equivalently in the differential form, where n b (r) is the Burgers charge density. A dislocation in the smectic is shown in Fig. 5(c), which can be regarded as a tightly bound pair of + 1 2 and − 1 2 disclinations.
In anticipation of our more rigorous duality derivation, already here we can argue for the subdimensional nature of the disclination dynamics in a smectic. Consider a pair of oppositely charged ± 1 2 disclinations separated alongx axis, as shown in Fig. 6. The separation between the pair is large such that we can regard them as isolated disclinations. Moving the '+' disclination along the layers (i.e.,x axis) by two layer spacings, requires the introduction of four extra half-layers of molecules, which is a highly nonlocal process in terms of atoms quantum dynamics, and is therefore not allowed. While moving the '+' disclination transversely (i.e., inŷ direction) preserves the strength of the dislocation, and thus, is allowed dynamically, even though there is an energy cost for separating the '+' and '−' disclination pair that make up the dislocation. Similar analysis applies to the '−' disclination. Thus, we conclude that disclinations can only move transversely to the layers, i.e., they manifests the subdimensional lineon dynamics.
To include the topological defects in the complete description of the smectic, we decompose the distortion field u and the bond angle θ into the smooth elastic and nonsingle-valued components, Integrating out the single-valued partsũ andθ out of the total generating function, leads to, where the new Lagrangian density is given in terms of only nonsingular components, with two enforced constraints, where we implicitly introduced currents J µ = (π, −σ i ). FIG. 6: Restricted along-layers mobility of disclinations in a 2D quantum smectic. Consider a pair of disclinations separated by five layer spacings. (a) Moving the '+' disclination along the layers by two layer spacings requires an introduction of four half-layers of underlying bosons, which is a non-local process, and therefore is not allowed. (b) Moving the '+' disclination transverse to the layers by two layer spacings preserves the strength of the dislocation, and thus, is dynamically allowed.

C. Quantum smectic-gauge theory duality
The momentum continuity equation (30) can be solved in terms of gauge potential fields, A µ , with such that and Eq.(31) transforms into , which is then solved by introducing another vector gauge potential a µ , such that Substituting the solutions of π, σ i , L,and j i in terms of the gauge fields into the Lagrangian density, leads to the dual Lagrangian densitỹ where L M is the Maxwell part, given by and the charge contributions are obtained by integrating by parts and defining the dislocation and disclination charge and current densities as, Introducing Hubbard-Stratonovich fields E and e, the Lagrangian density transforms intõ Integrating over A 0 and a 0 gives the Gauss law, leaving the standard Lagrangian form,L sm = −E·∂ t A−e·∂ t a− H sm , from which we can read off the dual Hamiltonian density, that is given by, supplemented by the generalized Gauss laws where, E and e, are independent electric fields, canonically conjugate to the corresponding vector potentials, A and a, respectively. The above Hamiltonian must be invariant under the gauge transformations: Requiring the source term to preserve this gauge invariance, we obtain coupled continuity equations for charges (disclinations) and dipoles (dislocations), satisfying, We observe that the dipole (dislocation) continuity equation is violated by a nonzero charge (disclination) current j x s in thex (along the layers) direction. Thus, in the absence of gappedx−dipoles (ŷ-dislocations), we find that j x s = 0, i.e., motion of isolated fracton charges (disclinations) is restricted to be transverse to the smectic layers, as moving along the layers requiresx-dipoles (ŷdislocations) that are gapped in the smectic. Therefore, the fractons (disclinations) exhibit subdimensional lineon dynamics, as argued in Section II B.

III. HIGGS TRANSITION OF QUANTUM CRYSTAL-TO-SMECTIC MELTING
As discussed in the Introduction, a smectic can emerge from anisotropic melting 17 of a crystal, understood in terms of a Kosterlitz-Thouless (KT)-like, 18 single-species dislocation unbinding transition. This classical partial melting transition of an anisotropic solid was studied by Halperin and Ostlund 17 , and we have used this fact in Eq. (17) to derive the smectic harmonic elasticity by includingx−orientated dislocations in a classical crystal. Although such a 2D classical smectic is unstable to thermal fluctuations, driven into a nematic fluid at any nonzero temperature 17,19-21 (see Section IV), a (2+1)D quantum smectic at zero temperature is a stable state of matter. In this section, we demonstrate that similarly, a quantum smectic can also emerge from partial, anisotropic quantum melting of a crystal.
The Hamiltonian density of a 2D quantum crystal is, where,û is the phonon field operator,θ is the orientational bond-angle field operator, and,π andL are their corresponding canonically conjugate momentum respectively. C ij,kl is the elastic constant tensor, which takes the form, C ij,kl = λδ ij δ kl + 2µδ ik δ jl , for an isotropic hexagonal lattice, characterized by two independent Lamé coefficients, λ and µ. Working in the pathintegral formulation with the field operators replaced by corresponding classical fields, the action is Now, as thex−dislocations condensed, we include dislocations with Burgers vector along the layers (x) by re- Then, the Lagrangian density where, in the third line, we have integrated out v ix and u x , and have defined χ 1 = 4µ(µ + λ)/(2µ + λ) and χ 2 = 2µ, which are taken to be equal in the last line, χ 1 = χ 2 = χ, for simplicity, corresponding to the case when λ → 0. Then, we arrive at the Lagrangian density of a smectic, starting with that of a quantum crystal, as summarized by the flow chart in Fig. 1. Motivated by this possibility of partial quantum melting of a crystal into a smectic and the subsequent melting into a quantum nematic, we will explore its dual in this section. Below, we will also derive a dual gauge theory of a quantum smectic, through a Higgs transition from a dual gauge theory of an incommensurate quantum crystal (supersolid) by condensing one flavor of dipoles, and thereby Higgs'ing out a flavor component of the gauge fields. As required by consistency, we indeed find that the resulting quantum smectic dual is in full agreement with a direct duality derived in Section II.
A. Soft-spin descriptions of quantum crystal and quantum smectic

Quantum crystal
The dual coupled U(1) vector gauge theory for a quantum crystal was first derived in Ref. [48], characterized by the Lagrangian densitỹ where k = (x, y) indexes different flavors, A k , a gauge fields capture the k = x, y phonons and bond orientational order respectively. The Maxwell part, L cr M , is given by To access descendant phases and corresponding quantum phase transitions, we need to treat dislocation and disclination defects as dynamical charges. Following a standard analysis and focusing on dislocations (dipoles) for the moment, we introduce the dynamical field ψ k (r, t) = √ ρ k e iϕ k for each gauge-charged dipole species p k , and add corresponding kinetic energies, ρ k 2 (∂ t ϕ k + pA 0k ) 2 , where we have integrated out the massive magnitude fluctuations to focus on the lowenergy phase fluctuations only. As discussed in the Introduction, in the Mottinsulating commensurate crystal phase, the dipole p k can only move in the direction perpendicular to p k while the along-dipole climb is forbidden due to the U(1) particle-number conservation symmetry ("glideconstraint"). Thus, we have only the glide motion, Π ⊥p k ij D j ψ p k , with, D = ∇ + ip k A k , the covariant spatial derivative, and Π ⊥p k , the transverse projection operator. However, the crystal also exhibits scalar non-topological point defects, corresponding to deficiency and excess in atom density, which permits the climb process of the dipoles (dislocations), as shown in Fig. 7. Combined with the bosonic statistics of the underlying particles, the quantum crystal can first develop into a super-solid phase ("incommensurate" crystal), featuring both the crystalline and the superfluid orders. The condensation of vacancies or interstitials in the supersolid phase (as illustrated in Fig.8 is a bound state of opposite charge dislocatons -a quadrupole), therefore, frees these symmetry-forbidden climb constraints [45][46][47]57 is the longitudinal projection operator. In Appendix A, we show in detail how the dislocation-superfluid coupling alleviates the glide constraint and converts the dislocations (dipoles) from subdimensional quasi-particles to ordinary mobile defects, acted upon by the full spatial derivative, Dψ p k . Introducing defect's core energy, E j b µ , to account for lattice-scale physics, and the dipole (dislocation) charge density and current density on a discrete lattice is given as a sum of their discrete charges where d is the lattice spacing for a 2D crystal, which is also the elementary charge of the gauge dipoles (unit of the dislocation charge), i.e., |p k | = p = d. The partition function is then given by with the actioñ where, m p is the effective inertia mass of the dipoles, µ,k (r) and ∆ µ ϕ k = ϕ k (r + µ) − ϕ k (r), µ = x, y, are the discrete lattice derivatives. Note the continuity equation ∆ µ j b µ,k = 0 is automatically satisfied when we integrate out ϕ k , which is the phase of the k−flavor dipole field ψ k = |ψ k |e iϕ k . After tracing over the 3-currents j b µ,k = (n b k , j b k ), we obtaiñ where k , and we have approximated the resulting Villain potential by its lowest harmonic.
In the continuum limit, we havẽ with the Lagrangian densityL cr given bỹ where,g b k . We then turn to an equivalent "soft-spin" description by noting that this ordered phase action emerges from a corresponding quantum Ginzburg-Landau theory for the complex order parameter, ψ k = |ψ k |e iϕ k , and writeL cr as, where, ψ k correspond tox-andŷ-oriented dipole fields (ŷ− andx−dislocations) in a square lattice, and V ({|ψ k |}), is the Landau U(1)-invariant potential for the quantum crystal, the form of which controls the type and subsequence of phase transitions (See Section III B).
It is straightforward to verify that Eq.(62) reduces to Eq.(61) when the gapped Higgs-like magnitude degrees of freedom, |ψ k |, whose fluctuations are controlled by V ({|ψ k |}), are integrated out, and therefore these two Lagrangians are equivalent.

Quantum smectic
Similar analysis applies also for the quantum smectic. The condensation of vacancies or interstitials in the super-smectic ("incommensurate" smectic) phase also endows the full mobility of dipoles (dislocations) in the smectic.
Then, starting withL sm given by Eq.(39) and elevating dislocation and disclination defects into dynamical charges, we add corresponding kinetic energies, ρ 2 (∂ t ϕ x + pA 0 ) 2 , and follow the same procedure as what have been done for the quantum crystal above, which leads to the effective Lagrangian density of the supersmectic in the continuum given by, where for concreteness we have taken the layers to be along thex axis, and replacedg b x simply byg b for thexdipole (ŷ-dislocation) fugacity, and A µ,x simply by A µ . In the second form, we have written it as an equivalent "soft-spin" description in terms of ψ x = |ψ x |e iϕx , with the Landau U(1)-invariant potential, V (|ψ x |), which is equivalent to the first form when the gapped Higgs magnitude degree of freedom, |ψ x |, whose fluctuations are controlled by V (|ψ x |), is integrated out.

B. Crystal-to-smectic and smectic-to-nematic transitions
The quantum crystal phase can go through a fully isotropic melting transition, mediated by dislocations, into a hexatic (or nematic) phase, or through a multistage anisotropic transition, first partially melting into a smectic phase, depending on the form of the Landau potential V ({|ψ k |}). A simple discussion based on Ginzburg-Landau theory of continuous phase transitions is given in the following.
Considering a 2D square lattice, the U(1)-invariant Landau potential V ({|ψ k |}), satisfying the symmetries of the system, expanded to fourth-order of the dipole fields ψ k , is given by where, β, β > 0, and we have defined the vector complex order parameter Ψ = (ψ x , ψ y ). Note that by these two forms, we can regard this potential as two identical complex Ising models coupled together, or a complex XY model with the originally rotational symmetry broken down to a rectangular one. As usual, at mean-field level the phase transition takes place at α = 0, as α changes its sign.

Crystal-to-nematic transition
We first consider the case of β > β , when it is energetically favorable for both flavors of dipole fields to condense with the same expectation value |ψ x | = |ψ y | = |α| β+β , corresponding to the nematic phase. This crystal-tonematic transition, analogue of the crystal-to-hexatic melting in a 2D classical crystal, has been discussed in Ref. 46, formulated as a Ginzburg-Landau theory of tensor superconductors, i.e., dipole fields coupled to the symmetric tensor gauge field. However, as pointed out in the Introduction, that tensor-only gauge theory formulation of the Mott-insulating "commensurate" crystal, fails to capture the full dipole mobility endowed by the condensation of vacancies and interstitials. Here, we take a complementary coupled vector gauge theory description 48 and condense bothx-andŷ-dipoles, ψ x = 0, ψ y = 0. Via Anderson-Higgs mechanism this gaps out all gauge field flavors, A µ,k , k = x, y. These can then be safely integrated out at low energies at wavelengths longer than 2mp ρ k p 2 (i.e., the penetration length of A µ,k ), thereby reducing the Maxwell Lagrangian of the crystal to that of a form described by the rotational gauge field a µ only. With details relegated to Appendix B, the result is given by, which corresponds to setting, A µ,k ≈ 0, to the lowest order, and neglecting anisotropies of the resulting nematic state.
As demonstrated by foundational papers on duality 40,41 , such Abelian gauge theory is dual to a quantum XY model of the nematic given by, where θ is the layer orientation, confirming a consistency with our physical expectations.

Crystal-to-smectic and smectic-to-nematic transitions
We next consider the case of β < β , when it is favorable to have only one flavor of dipole fields condensed, with ψ x = 0 and |ψ y | = |α| β , or, ψ y = 0 and |ψ x | = |α| β , corresponding to the crystal-to-smectic phase transition, which restores translational symmetry in only one direction. With our interest in the quantum smectic, in the following we will thus focus on the β < β case, and dualize the quantum smectic alternatively, via Anderson-Higgs mechanism, gapping out one of the flavors of the gauge fields in this smectic phase, formulated in a "soft-spin" description of a 2D quantum crystal.
The crystal-smectic partial melting transition corresponds to condensation of one flavor of the dipole fields, that according to Fig. 2, we take to beŷ-dipoles, i.e. ψ y = 0. Within this Higgs phase, the corresponding y flavor gauge field A µ,y is gapped out, and can be safely integrated out in the low-energy regime with wavelengths of excitations much greater than 2mp ρyp 2 (i.e., the penetration length of A µ,y ). To lowest order, it corresponds to A µ,y ≈ 0, reducing the crystal's Maxwell Lagrangian to that of a smectic, The detailed derivations are given in Appendix B. Subsequent melting with a condensation ofx−dipoles (ŷ−dislocations) leads to a single vector gauge theory for a µ , with both A µ,k gapped out, in the low-energy regime with wavelengths of excitations much greater than 2mp ρxp 2 , i.e., to the lowest order. As expected, this corresponds to the dual of the quantum XY model of the nematic state.
C. Vector to tensor gauge theory redux in low-energy limit The Maxwell part of a crystal, given by Eq. (54), is gauge invariant under the following transformation, As demonstrated explicitly in Ref. 48, the enlarged gauge redundancy allows us to completely eliminate a by choosing χ k = a k , as a result of which, the term 1 2 (∇ × a −ẑ × A k ) 2 reduces to 1 2 ( ik A ik ) 2 , thereby gapping out the antisymmetric component ik A ik at energies well below this gap, i.e., with length scales greater than √ χ. Furthermore, the electric field term 1 2 K −1 (∂ t a − ∇a 0 − A 0kêk ) 2 reduces to 1 2 K −1 (∇a 0 + A 0kêk ) 2 under this transformation, enforcing A 0k = −∂ k a 0 at low enough energies with length scales greater than K χ . Therefore, the dual coupled U(1) vector gauge theory for a quantum crystal, reduces to the dual tensor gauge theory in the low-energy limit, with, L cr M , reduces to that in the tensor gauge theory, described by, where A ik , is a rank-2 symmetric tensor field, which corresponds to the i th component of k-flavor vector gauge field A k in the coupled vector gauge theory, a 0 is a scalar field with ∂ k a 0 corresponding to A 0k in the vector gauge theory, B k = ji ∂ j A ik , and E ik is the electric tensor field canonically conjugate to A ik . In contrast, within the smectic Higgs phase, corresponding to the condensation ofŷ−dipoles (x−dislocations), we cannot eliminate a completely, since we have already made a gauge choice with χ y = 1 p ϕ y , to absorb the phase ϕ y of the condensed dipole field ψ y = √ ρ y e iϕy into the gauge fields A µ,y . Integrating out the gapped gauge field components, A iy and ∂ y a 0 , reduces L cr M , in a condensate ofŷ−dipoles, to that of a smectic, (72) which matches exactly with Eq.(40) after setting A 0x = −∂ x a 0 and dropping the 'x' index.
We may also explore a 2D quantum smectic at low energies by similar analysis, as what has been done for a crystal in Eq.(70). In the smectic case, choosing χ = a x in the gauge transformation (46) allows us to eliminate a x completely, as a result of which, the term 1 2 (∇ × a +x × A) 2 of L sm M given by (40), reduces to, 1 2 (∂ x a y + A y ) 2 , thereby enforcing A y = −∂ x a y at sufficiently low energies, with length scales greater than √ χ. Furthermore, the term 1 this transformation, enforcing A 0 = −∂ x a 0 in low energy regime with length scales greater than K χ . Therefore, in the low-energy limit, L sm M reduces to, where E ix is a tensor gauge field, corresponding to the i th component of the vector field E x . This vector-to tensorgauge theory reduction at low energies is illustrated in Fig.9.
FIG. 9: Vector gauge theory of a 2D quantum crystal and smectic, in the low-energy limit reduce to their corresponding tensor gauge theory forms.

IV. CLASSICAL LIMIT OF SMECTIC-GAUGE THEORY DUALITY
A. 2D classical smectic duality As a consistency check on our quantum dual theory for a 2D quantum smectic, we anticipate that the classical smectic theory must emerge as the classical limit of the above duality, as we demonstrate explicitly below.
The elasticity of a 2D classical smectic is given by the Hamiltonian density, where we have introduced two Hubbard-Stratonovich fields σ and j to decouple the two elastic terms.
We derive its classical dual by integrating out the smooth part of u and θ, which leads to the constraints: The first equation can be solved in terms of a scalar potential φ, σ i = ij ∂ j φ, which inside the second constraint gives, that is then solved by introducing another potential α, Substituting σ i and j i back into the original Hamilto-nian, and integrating by parts, lead tõ where we have defined b = ij ∂ i ∂ j u and s = ij ∂ i ∂ j θ, as the dislocation and disclination densities respectively. Low-energy regime, 1 2 K −1 (∇α + φx) 2 ≈ k B T , i.e., at length scales greater than √ Kk B T , integrating over φ , to lowest-order, sets φ = ∂ x α, and therefore, gives which, as expected, turns out to be the electrostatic limit of the quantum smectic duality, i.e., Eq.(39) and Eq. (40), with φ = A 0 and α = a 0 . Focusing on dislocations and neglecting the high energy disclination defects, we can straightforwardly integrate out α(r) in the partition function, obtaining a dislocation Coulomb gas Hamiltonian with,K where, E b is the defect core energy, d is the layer spacing, and a is the lattice spacing between atoms within the layers, and "penetration" length λ is defined as, λ 2 ≡ K/χ. Thus, the dislocations Coulomb gas Hamiltonian in real space reduces to, where, as first found by Toner and Nelson in Ref. 21. The complete Hamiltonian contains also a smooth phonon part, H 0 , depending only the smooth, single-valued part,ũ, of the displacement, u =ũ + u s , i.e., with H 0 given by With this Hamiltonian, we can study the effects of phonons and dislocations at finite temperatures on translational and orientational orders. Effect of phonon fluctuations on the translation order is expressed in terms of correlations in the order parameter, ψ (r) = |ψ|e −iq0u(r) as, where we have used the fact that Smectic layers orientational order is expressed in terms of correlations in the nematic-like order parameter, N(r) = (cos θ(r), sin θ(r)), as, where we have used with Λ = 2π a , a convenient cutoff. Therefore, in a 2D classical smectic at nonzero temperature, the translational order is destroyed by thermal phonon fluctuations [19][20][21] , while the orientational order persists even in the presence of thermally excited phonons, destroyed only at higher temperatures by proliferation of dislocations.
In presence of unbound dislocations, appearing at density, n d ∼ e −Ec/(kBT ) , in thermal equilibrium, the effective elasticity in Debye-Huckel approximation, reduces to that of a nematic at scales greater than, ξ D ∼ e −Ec/(2kBT ) , and the correlations in orientational order become decay algebraically, with, η(T ) = 2k B T /[πK(T )]. Therefore, a 2D smectic is unstable to thermal fluctuations, driven into a nematic fluid at any nonzero temperatures. At high temperature, the nematic to isotropic liquid transition, driven by unbinding of disclinations, is described by Kosterlitz and Thouless,with 18,49 , at the critical temperature T c . Motivated by our formulation of two-dimensional melting of a classical crystal 17,18,58-60 , via a dual theory in terms of a higher derivative vector sine-Gordon model 61 , we expect to find the analogous description for the 2D smectic. To this end, we express the dislocation and disclination densities in terms of a sum of their discrete charges as, b(r) = rn b rn δ 2 (r − r n ), s(r) = rn s rn δ 2 (r − r n ).

(92)
In terms of these discrete topological defect charges, the Hamiltonian is given by Following a standard analysis, summing over the charges, we obtain the dual sine-Gordon Hamiltonian, where g b = 2 a 2 e −a 2 E b and g s = 2 a 2 e −Es , which provides a transparent description of the continuous two-stage melting in terms of the renormalization-group relevance of two cosine operators that control the sequential unbinding of dislocations and disclinations, respectively corresponding to the smectic-to-nematic and nematic-to-isotropic fluid transitions. The resulting phase diagram is illustrated in Fig.10. Because of the second-order Laplacian elasticity, standard analysis around the Gaussian fixed line g b = g s = 0 shows that, the mean-squared fluctuations of α(r) is given by which leads to an exponentially (as opposed to power-law in a conventional sine-Gordon model) vanishing Debye-Waller factor, e i2πα(r) e −i2πα(0) = e −2π 2 [α(r)−α(0)] 2 , and in turn to a strongly irrelevant disclination cosine, g s , that can therefore be neglected. In contrast, meansquared fluctuations of ∂ x α(r) is, for large r, and orientational correlation therefore given by, This therefore leads to the conclusion that the dislocation cosine, g b , is always relevant. At sufficiently long scales, dislocation cosine in Eq.(94) reduces to a harmonic potential for ∂ x α, −g b cos(b∂ x α) The effective Hamiltonian is then given bỹ H nm where we have neglected the χ −1 "curvature" elasticity relative to the gradient one encoded in g b , and restored the disclination cosine operator g s cos(2πα). The resulting conventional sine-Gordon model in α can then exhibit the second KT-like"roughening"transition, capturing the nematic-to-isotropic fluid transition, associated with the unbinding of disclinations, with well-known standard KT phenomenology.

B. 3D classical smectic duality
Motivated by the correspondence of a (2+1)D quantum smectic and a 3D classical smectic, and the extensively studied 3D nematic to smectic-A transition 50-56 , we formulate a dual gauge theory of a 3D classical smectic, akin to a mapping of a 3D classical XY model onto a classical charged superconductor. 40,41 The elasticity of a 3D classical smectic with its layers along xy plane, is captured by the Hamiltonian density, where δn = n −ẑ = (δn x , δn y , 0) represents fluctuations in layer orientation, and we introduced two Hubbard-Stratonovich fields σ and j to decouple the two elastic terms. The Hamiltonian density in Eq. (99) is equivalent to the standard smectic form, in the low-energy limit, where the orientational degree of freedom, δn, locks to the layer normals with δn = −∇ ⊥ u.
Integrating out the smooth part of u and δn, leads to the constraints: The first equation can be solved in terms of a vector potential A, σ i = ijk ∂ j A k , (σ = ∇ × A), which inside the second constraint gives, such that j i is solved as Substituting σ i and j ij back into the original Hamiltonian, and integrating by part, lead to where dislocation density (see Fig.11) is given by In momentum space, we have Functionally integrating out A z , leads to, where we have chosen the Coulomb (transverse) gauge ∇ ⊥ · A ⊥ = 0, i.e., q ⊥ · A ⊥ = 0, P T ij = δ ij − q i q j /q 2 , and, P L ij = q i q j /q 2 , are transverse and longitudinal projection operators respectively, Γ(q) ≡ χq −2 ⊥ 1 + 1/(λ 2 q 2 ) −1 is the interaction potential for screw dislocation b z 's, and we have used the fact that, P L ij A ⊥ j = 0, in the Coulomb gauge. If we integrate out A ⊥ further, we get the dislocation Coulomb gas model, given by, where, , is the transverse projection operator for edge dislocations, i.e. screw dislocations b z −components projected away, and E i b 's are core energies of dislocations 56 . We note that Γ(q) is approximately a constant at small q, where q is smaller than 1/λ, and therefore contributes to the core energy of a screw dislocation, i.e., E z b → E z b + Γ(q). Note that for simplicity, we have assumed that the lattice spacing between atoms within the layers is equal to the layer spacing, i.e., a = d, such that we have no factor like ' d 2 a 2 ' as in Eq. (80) for the 2D case.
Interested in the nature of the nematic to smectic-A transition, Toner 56 mapped a model of a smectic onto a Coulomb gas of dislocation loops, which he then transformed into an anisotropic superconductor in a vector gauge field A, and analyzed it with a momentum-shell renormalization group. In the long-wavelength limit of q 1/λ, indeed our model reduces to Toner's, with a generalization that screw dislocations in our model have a finite interaction.
In analogy to what we have done for a 2D smectic, we transform the Coulomb gas Hamiltonian, Eq.(106) into a classical gauge theory. The partition function for the dislocation-loop Coulomb gas on a lattice is given by, with, and we have introduced an auxiliary scalar field ϕ(r), such that integrating out ϕ recovers the constraint, ∆ · b rn = 0. After tracing over the dislocation charges b rn , we obtaiñ where g 3d bi = 2e −E i b d 2 , and we have approximated the resulting Villain potential by its lowest harmonic. In the continuum limit, it becomes and in the second form we have written it as an equivalent "soft-spin" description in terms of ψ = |ψ|e iϕ , with the Landau U(1)invariant potential, V (|ψ|), which is equivalent to the first form below the energy scale of the gapped Higgs-like magnitude degree of freedom, |ψ|. Thus, we reproduce Toner's anisotropic superconductor model 56 expected to have the same critical properties as the above dislocationloop Coulomb gas model.

V. SUMMARY AND CONCLUSION
In this paper, after a brief review of smectic elasticity, we developed a coupled U(1) vector gauge theory for a two-dimensional quantum smectic, where the phonons and orientational Goldstone modes map onto coupled gauge fields, and topological defects correspond gauge charges and dipoles. We discovered that charges (disclinations) exhibit subdimensional lineon dynamics, restricted to move transverse to the layer. Motivated by the partial quantum melting of a crystal into a smectic, and the subsequent smectic-to-nematic transition, we reproduced the dual description of a quantum smectic by condensing the one flavor species of dipoles within the generalized Abelian-Higgs model of a 2D quantum crystal.
We also applied this duality to treat a classical smectic liquid crystal. To this end, we formulated the smecticto-nematic and nematic-to-isotropic fluid transitions as a higher-derivative sine-Gordon model of a 2D classical smectic. Motivated by the correspondence between a (2+1)D quantum system and a 3D classical system, we also derived a dual theory for a 3D classical smectic, and reproduced smectic's dislocation-loop Coulomb gas description for the nematic-smectic transition, which we then mapped onto an anisotropic Abelian-Higgs model.
We expect this fractonic gauge theory reformulation of smectics will be useful for further detailed explorations, e.g., subjected to an external stress and in presence of a substrate. We leave a study of the true critical behavior (beyond mean-field) of the crystal-smectic and smecticnematic transitions using the dual gauge theory for future studies. The duality analysis in the presence of elastic nonlinearities also remains a challenging open problem. part, takes the form where, m p is the effective mass of the dipole, D 0 = ∂ t + ip k A 0k and D j = ∂ j + ip k A jk (i.e., D = ∇ + ip k A k ) are the covariant derivatives 47 , V ({ψ k }) is the U(1)invariant Ginzburg-Landau potential, L cr M the Lagrangian density of the Maxwell part, and Π ⊥p ij = δ ij − pipj p 2 is the projection operator since in this Mott-insulating crystal phase, the dipole can only move in the direction perpendicular to p while the along-dipole climbs are forbidden due to the U(1) particle-number conservation symmetry (" glide-constraint").
However, as discussed in the main text, the crystal also exhibits scalar non-topological point defects, corresponding to deficiency and excess in atom density, which permits the climb process of the dislocations (See Fig.  7). Combined with the bosonic statistics of the underlying particles, the quantum crystal can first develop into a super-solid phase (incommensurate crystal), featuring both the crystalline order and the superfluid order. The condensation of vacancies or interstitials in the super-solid phase, therefore, frees these symmetryforbidden climb events. Therefore, for a complete description, we also need to add the superfluid part of the underlying bosonic particles, L sf , and the minimal gaugeinvariant coupling between the dislocation climb operators and superfluid order parameter ,L sf-dis , into the full Lagrangian 47 , where, Ψ sf is the superfluid order parameter, O climb,p = ψ † p (r + p)e ipiAij pj ψ p (r) is the p-dipole climb operator, and γ is the coupling constant 47 . Fig. 8 shows an example of terms in L sf-dis .
where, in the second line, we have assumed that ϕ s (r) varies slowly in space and dropped the term, 1 4msρ 0 s ∇ 2 , and in the last line, we have written it as that of a sound mode, in a Lorentz-invariant form for simplicity, with ∂ µ ≡ 1 c ph ∂ t , ∇ and c ph = ρ 0 s U ms . Combined, L dis , with, L sf and L sf-dis , the resulting phase is a supersolid with the spontaneous breaking of both partice number conservation U(1) symmetry and translational symmetry. Writing ψ k = √ ρ k e iϕ k , and integrating out the massive magnitude fluctuations, the effective Lagrangian density for the super-solid phase is given bỹ Freezing the superfluid phase by fixing ϕ s = 0, and rescaling the longitudinal and transverse gradients, we get (correct to quadratic order in the argument of cosine terms), L cr = ρ k 2 (∂ t ϕ k + pA 0k ) 2 − ρ k 2m p cos (∇ϕ k + pA k ) + L cr where in the second form, we have written it as an equivalent "soft-spin" description in terms of ψ k , with the landau U(1)-invariant potential, V ({ψ k }). Note that the fugacity e −Ẽ b k and density ρ k are related by the relation: mp d 2 e −Ẽ b k = ρ k to the lowest order, such that ρ k mp = 2g b k . Therefore, Eq.(A7) is in the same form as Eq. (61). Similar analysis applies for the quantum smectic phase with just one species of dipoles, i.e.,x−dipoles (ŷ−dislocations), which leads to the full mobility of dipoles (dislocations) in the smectic and the effective Lagrangian density of the super-smectic phase given by, which is in the same form as Eq.(63).
where, P L ij = q i q j /q 2 is the longitudinal projector, P T ij,z = δ ij − q ⊥ i q ⊥ j /q 2 (1 − δ iz δ jz ) and P T ij,⊥ = δ ij − q 2 ⊥ /q 2 (δ ix δ jx + δ iy δ jy ) are the transverse-to-ẑ and transverse-to-layer projector, respectively, and in the second line, we have transformed into the momentum space. For long-wavelength limit, with the wave number q χ/K 2 , χ/K 3 , we can integrate out the higherenergy terms, which sets δn = ∇ ⊥ u and reduces H 3d sm into the form, matching exactly with the standard form of the elastic energy of a 3D smectic. In Section IV, for a simple anal-ysis with losing much qualitative physics, we have set K = χ, and made the isotropic elasticity approximation with K 1 = K 2 = K 3 , replacing them by K for simplicity. Below, we give a briefly discussion of dislocations and their energies in a 3D smectic, based on H 3d sm given by Eq.(C3). A more detailed discussion based on H 3d sm given by Eq.(C1) can be found standard textbooks 2 . For a single positive screw dislocation with its line core, located in the origin of xy−plane, in theẑ direction, as shown in Fig. 11(a) , from the Eq. (9), that determines the positions of the layer planes, we get the layer displacement given by, taking place in the xy plane only, and then, ∂ z u screw = 0, shuch that, and therefore, the energy of a single screw dislocation in a smectic is 0. For a single positive edge dislocation with its line core perpendicular to theẑ direction, say in thex direction, as shown in Fig. 11(b), the layer displacement is then given by, and then, ∂ z u edge = y y 2 + z 2 , such that, and, the energy of a single edge dislocation in a smectic can be shown to be divergent as the length scale of the system, after integrating (C3) over space. * Electronic address: radzihov@colorado.edu, zhzh3530@colorado.edu