Fractons from vector gauge theory

TL;DR

This paper introduces a vector gauge theory framework to describe fracton phases, revealing their origin from an unusual Gauss law and connecting to known tensor gauge theories, with lattice models illustrating the condensation mechanisms.

Contribution

It develops a vector gauge theory approach to fractons, linking elasticity duality to fracton order and constructing lattice models demonstrating string condensation.

Findings

Fracton order arises from a novel Gauss law in vector gauge theories.

Low-energy limit reduces to known symmetric tensor gauge theories.

Lattice models exhibit fracton phases via string-like excitation condensation.

Abstract

Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fracton phases via a condensation of string-like excitations built out of charged particles, analogous to the p-string condensation mechanism of the gapped X-cube fracton phase.

Equations

$$\tilde{\mathcal{H}}$$

$$\nabla \cdot {\mathbf{E}}_k$$

$$\nabla \cdot \mathbf{e}$$

$$\mathcal{L}$$

$$L$$

$$\tilde{\mathcal{L}}$$

$${\partial }_i{\partial }_k{E}_{ik}$$

$${\mathbf{A}}_k$$

$$a_k$$

$$u_k$$

$$\theta$$

$$H$$

$${\partial }_i{e}_{ik}$$

$${\partial }_i{E}_{ik}$$

$$\mathcal{L}$$

$${\mathcal{A}}_{0k}$$

$${\mathcal{A}}_{jk}$$

$${\mathcal{B}}_0$$

$${\mathcal{B}}_j$$

$$a_{ij}$$

$$A_{ij}$$

$$a_{ij}=\frac{1}{2}{a}_{ij}^s+{ϵ}_{ijk}{\Omega }_k\text{,}$$

$${\partial }_t{a}_{ij}+{\partial }_i{a}_{0j}+{ϵ}_{kij}{A}_{0k}=\frac{1}{2}{\partial }_t{a}_{ij}^s+{\partial }_i{a}_{0j}+{ϵ}_{kij}(A_{0k}+{\partial }_t{\Omega }_k)\text{,}$$

$$b_{ij}=\frac{1}{2}{ϵ}_{ikl}{\partial }_k{a}_{lj}^s+(A_{ji}-{\partial }_j{\Omega }_i)-{\delta }_{ij}(A_{kk}-{\partial }_k{\Omega }_k)\text{.}$$

$$\mathcal{L}$$

$$c_{ij}=\frac{1}{2}{ϵ}_{ikl}{\partial }_k{a}_{lj}^s\text{,}$$

$${\beta }_{ij}=2{ϵ}_{jkl}{\partial }_k{c}_{il}={ϵ}_{imn}{ϵ}_{jkl}{\partial }_m{\partial }_k{a}_{nl}^s\text{,}$$

$$(\nabla \times {\mathbf{A}}_k{)}_i\to (\nabla \times {\mathbf{A}}_k{)}_i-\frac{1}{2}{\beta }_{ki}\text{.}$$

$$\mathcal{L}$$

$$\mathcal{L}=\frac{1}{8K}\sum _{ij}({\partial }_t{\alpha }_{ij}+{\partial }_i{a}_{0j}+{\partial }_j{a}_{0i}{)}^2-\frac{1}{8}\sum _{ij}{\beta }_{ij}^2\text{,}$$

$${\partial }_i{e}_{ik}$$

Full text

Introduction. Motivated by continued interest in topological quantum matter and by a search of fault-tolerant quantum memory, recent studies have led to fascinating developments in an exotic class of quantum spin-liquid modelsChamon (2005); Bravyi et al. (2011); Haah (2011); Vijay et al. (2015, 2016); Slagle and Kim (2017); Nandkishore and Hermele (2019). These are characterized by many nontrivial properties, the most unusual of which are system-size-dependent ground state degeneracy and the existence of quasi-particles, dubbed “fractons”, that exhibit restricted mobility. Namely, there are quasi-particles confined to zero-, one- and/or two-dimensional subspaces of the full three-dimensional space of the model. While such fracton phases were originally discovered in fully gapped phases of commuting projector lattice spin Hamiltonians, it was more recently pointed out Pretko (2017), that fractonic charges are also realized in gapless phases of U(1) symmetric tensor gauge theories Rasmussen et al. (2016).

In a parallel development, it was observed by one of us (L.R.) Radzihovsky that such restricted quasi-particle mobility is strongly reminiscent of the immobile disclinations and glide-only dislocations in an ordinary two-dimensional (2D) crystal, described by a symmetric strain tensor field. This conjecture of fracton-elasticity duality was reported and moreover explicitly demonstrated in Pretko and Radzihovsky (2018a), utilizing a generalization of boson-vortex duality Dasgupta and Halperin (1981); Fisher and Lee (1989). It was shown that a 2+1D quantum crystal is dual to a symmetric tensor gauge theory, with disclinations and dislocations mapping onto fractonic charges and their dipolar bound states, and with stress tensor $\sigma_{ik}$ and momentum vector $\pi_{k}$ fields respectively corresponding to the electric tensor $E_{ik}$ and magnetic vector $B_{k}$ fields.

Motivation and results. An important source of insight into fracton physics has been to relate apparently exotic fracton states to more familiar quantum phases of matter. Indeed, the fracton-elasticity duality is an example of such a relationship. Related progress has also been made for certain gapped fracton phases, via a construction of these phases in terms of coupled layers of ordinary 2D topologically ordered states Ma et al. (2017); Vijay (2017). So far there is a relative paucity of relationships between gapless fracton phases and better understood phases or theories. Most U(1) tensor gauge theories are not dual to elasticity, and even for those that are, developing alternative viewpoints is highly desirable.

Remarkably, the fracton-elasticity duality itself contains the seed of another such point of view. There is a sense in which elasticity, formulated in terms of a symmetrized strain tensor $u_{ik}={\frac{1}{2}}(\partial_{i}u_{k}+\partial_{k}u_{i})$ is a system of two “spin” flavors of XY models, joined together via “spin-space” coupling, as in systems with spin-orbit interaction. However, in the absence of spin-space coupling, such a system dualizes to two independent flavors of U(1) vector gauge theory, and lacks fractonic charges. It is thus natural to ask whether fractonic tensor gauge theories can be formulated in terms of coupled vector gauge theories and if so, what minimal ingredients are required for such coupling. Some progress has been made along these lines from a different point of view, starting with a vector U(1) gauge theory and gauging certain global symmetries to obtain a fractonic tensor gauge theory Williamson et al. (2018). We make contact with this result below in greater detail.

In this Letter, we first utilize a reformulated fracton-elasticity duality to derive a 2+1D U(1) vector gauge theory that hosts fractonic charges and is equivalent at low energy to the rank-2 symmetric tensor U(1) gauge theory with scalar charge, dubbed the “scalar-charge theory”. We then discuss a lattice version of the same theory, which allows the scalar-charge theory to be understood starting from two decoupled vector U(1) gauge theories, and condensing certain charged loops. Finally, we discuss generalizations of our lattice construction that provide constructions of a new class of fractonic tensor gauge theories as coupled vector gauge theories.

The continuum Hamiltonian density we obtain is given by

[TABLE]

which involves three U(1) vector gauge fields with electric fields ${\bf E}_{k}$ (with flavors $k=x,y$) and ${\bf e}$ and corresponding canonically conjugate vector potentials ${\bf A}_{k}$ and ${\bf a}$. We denote the corresponding charge densities by $p_{k}$ and $\rho$, and currents by ${\bf J}_{k}$ and ${\bf j}$, with $p_{k}$ and ${\bf J}_{k}$ referred to as the dipole charge and dipole current, respectively. Moreover, $A_{a}=\epsilon_{ik}A_{ik}$, and we use a short-hand notation where the curl of a 2D vector field is implicitly its scalar z-component, i.e., ${\bm{\nabla}}\times{\bf a}\equiv{\hat{\bf z}}\cdot{\bm{\nabla}}\times{\bf a}$.

The Hamiltonian is supplemented by the Gauss’ law constraints:

[TABLE]

Crucially, the components of the electric field $e_{k}$ appear as additional dipole charge in the Gauss’s law (2).

The generalization to $d$ dimensions is straightforward and consists of $d+1$ U(1) gauge fields obeying the same Gauss’ laws but with $k=1,\dots,d$. The main difference in the Hamiltonian is that the $({\bm{\nabla}}\times{\bf a}+A_{a})^{2}$ term is replaced by a sum of the $d(d-1)/2$ terms of the form $(\partial_{i}a_{j}-\partial_{j}a_{i}+A_{ij}-A_{ji})^{2}$. The resulting theory is equivalent at low energy to the $d$-dimensional scalar-charge theory, as we will detail below.

The fractonic nature of the $\rho$ charges can be seen by observing that moving such a charge requires creating or destroying field lines of the ${\bf e}$ electric field, but since these field lines themselves carry gauge charge, a single “piece” of field line cannot be locally created or destroyed. The immobility of these charges is also manifest in that gauge invariance requires the current ${\bf j}$ to vanish identically, as we elaborate below. We next turn to the derivation of this fractonic coupled vector gauge theory and its connection to the previously studied tensor scalar-charge theory.

Derivation. To this end, we pass to Lagrangian formalism and begin with an elastic theory of a 2+1D quantum crystal formulated in terms of the phonon field $u_{k}$ and its canonically conjugate momentum $\pi_{k}$. For simplicity we take the elastic tensor $C_{ij,kl}$ to be $C_{ij,kl}=C\delta_{ik}\delta_{jl}$. The generalization to an arbitrary $C_{ij,kl}$ is straightforward.

Ref.Pretko and Radzihovsky, 2018a started with such a theory written in terms of the symmetrized strain $u_{ik}=\partial_{i}u_{k}+\partial_{k}u_{i}$, and showed it is dual to a symmetric tensor gauge theory, where fractonic charges correspond to disclinations and dipoles to dislocations. To get to an equivalent flavored vector gauge theory description, we reformulate the elastic theory in terms of “minimally”-coupled quantum XY models, introducing the orientational bond-angle field, $\theta$ and its canonically conjugate angular momentum density $L$. The Lagrangian density is given by

[TABLE]

Due to the coupling to $\theta$, the anti-symmetric part of the unsymmetrized strain ${\partial}_{i}u_{k}$ is massive, below a scale set by $C$, similar to the Higgs mechanism for gauge fields. Integrating out $\theta$ results in the standard elasticity theory formulated in terms of the symmetrized strain $u_{ik}$, which is the starting point of Ref.Pretko and Radzihovsky, 2018a. To proceed, it is convenient to decouple the elastic and orientational terms in (4) via Hubbard-Stratonovich vector fields, stress ${\bm{\sigma}}_{k}$ and torque ${\bm{\tau}}$, resulting in the Lagrangian density

[TABLE]

where $\sigma_{a}\equiv\epsilon_{ik}\sigma_{ik}$.

For a complete description, in addition to the single-valued (smooth) Goldstone mode degrees of freedom, $\theta^{e}$ and $u^{e}_{k}$, we must also include topological defects. A disclination defect is defined by a nonsingle-valued bond angle with winding $\oint d\theta^{s}=2\pi s/n$ around the disclination position, or equivalently in a differential form, ${\bm{\nabla}}\times{\bm{\nabla}}\theta^{s}=\frac{2\pi s}{n}\delta^{2}({\bf r})\equiv\rho({\bf r})$. The integer disclination charge $s$ corresponds to an integer-multiple of $2\pi/n$ missing (added) wedge of atoms for $s>0$ ($s<0$) in a $C_{n}$ symmetric crystal, with most common case of a hexagonal lattice, $n=6$.

A dislocation is a point vector defect, around which the displacement $u_{k}$ is not single-valued, with winding $\oint du_{k}=b_{k}$, or equivalently in a differential form, ${\bm{\nabla}}\times{\bm{\nabla}}u^{s}_{k}=b_{k}\delta^{2}({\bf r})\equiv b_{k}({\bf r})$. An elementary dislocation is a dipole of $\pm 2\pi/n$ disclinations and is characterized by a 2D Burgers vector charges, $b_{k}$, that takes values in the lattice. An edge dislocation corresponds to a ray of missing or extra lattice sites, with a Burgers vector lying in the 2d plane of the crystal. A nontrivial configuration of dislocations, ${\bf b}({\bf r})$ can also contribute to a disclination density, given by $s_{b}({\bf r})={\hat{\bf z}}\cdot{\bm{\nabla}}\times{\bf b}({\bf r})$, with a single disclination corresponding to an end point of a ray of dislocations.

Expressing the phonon and bond-angle fields in terms of corresponding singular (s) and elastic (e) parts, $u_{k}=u_{k}^{e}+u_{k}^{s}$, $\theta=\theta^{e}+\theta^{s}$, and integrating over the elastic parts, gives the conservation of linear and angular momentum, $\partial_{t}\pi_{k}-{\bm{\nabla}}\cdot{\bm{\sigma}}_{k}=0$ and $\partial_{t}L-{\bm{\nabla}}\cdot{\bm{\tau}}=\sigma_{a}$.

Expressing linear momentum conservation constraint in terms of dual magnetic and electric fields, $\pi_{k}=\epsilon_{kj}B_{j}$, $\sigma_{ik}=-\epsilon_{ij}\epsilon_{k\ell}E_{j\ell}$, leads to the $k$-flavored Faraday equations, $\partial_{t}B_{k}+{\bm{\nabla}}\times{\bf E}_{k}=0$. As in standard electrodynamics, the Faraday law is solved by $k$-flavored vector ${\bf A}_{k}$ and scalar $A_{0k}$ gauge potentials, $B_{k}={\bm{\nabla}}\times{\bf A}_{k}$, ${\bf E}_{k}=-\partial_{t}{\bf A}_{k}-{\bm{\nabla}}A_{0k}$. We emphasize that, in contrast to the symmetric tensor approachPretko and Radzihovsky (2018a, b), here, the $k=(x,y)$-flavored vector gauge field ${\bf A}_{k}$ has components $A_{ik}$ that form an unsymmetrized tensor field.

Using these definitions reduces the conservation of angular momentum to $\partial_{t}(L-A_{a})-{\bm{\nabla}}\cdot({\bm{\tau}}-{\hat{\bf z}}\times{\bf A}_{0})=0$, which is then solved by introducing another set of vector ${\bf a}$ and scalar $a_{0}$ gauge fields, giving

[TABLE]

Using these gauge fields to eliminate ${\bm{\sigma}}_{k}$, $\pi_{k}$, ${\bm{\tau}}$ and $L$, gives an effective Lagrangian density

[TABLE]

Here, the dipole charge $p_{k}$ is given by the dislocation density $p_{k}=\epsilon_{lk}b_{l}=({\hat{\bf z}}\times{\bf b})_{k}$, the fracton charge $\rho$ is the disclination density, and the corresponding currents are given by ${\bf J}_{k}=\epsilon_{lk}{\hat{\bf z}}\times(\partial_{t}{\bm{\nabla}}u_{l}-{\bm{\nabla}}\partial_{t}u_{l})$ and ${\bf j}={\hat{\bf z}}\times(\partial_{t}{\bm{\nabla}}\theta-{\bm{\nabla}}\partial_{t}\theta)$. Finally, using Hubbard-Stratonovich transformations to introduce electric fields canonically conjugate to each vector potential, we obtain (1) in Lagrangian form.

The unusual Gauss’s law (2) couples three vector gauge theories, with $k$-th component of the $\bf e$ field acting as an additional source of charge in the Gauss’s law for ${\bf E}_{k}$. We also note that taking the divergence on the second index $k$ of the Gauss’s law for ${\bf E}_{k}$, (2) and using the second law for ${\bf e}$ to eliminate ${\bm{\nabla}}\cdot{\bf e}$ from the resulting right hand side gives,

[TABLE]

This thereby recovers the generalized Gauss’s law of scalar-charge tensor gauge theory, with $\tilde{\rho}\equiv-\rho+{\bm{\nabla}}\cdot{\bf p}$ the total charge contribution,Pretko and Radzihovsky (2018a) that encodes the additional dipole conservation responsible for immobility of fractonic charges.Pretko (2017) We note that, in contrast to the scalar-charge theory, $E_{ik}$ is not a symmetric tensor, but effectively becomes symmetric at low energy as we demonstrate below.

We note that the Lagrangian (Fractons from vector gauge theory) is invariant under a deformed gauge transformation,

[TABLE]

with (10) ensuring that ${\bm{\nabla}}\times{\bf a}+A_{a}$ is gauge invariant. Under the $\chi_{k}$ gauge transformation, the current source terms in (Fractons from vector gauge theory) shift by $-\chi_{k}j_{k}+{\partial}_{t}\chi_{k}\tilde{p}_{k}+{\bm{\nabla}}\chi_{k}\cdot{\bf J}_{k}$, where $\tilde{p}_{k}=p_{k}-e_{k}$ is the effective dipole density, that is a combination of microscopic dipoles and electric field generated by pairs of fracton charges. Requiring gauge invariance then leads to the dipole continuity equation ${\partial}_{t}\tilde{p}_{k}+{\bm{\nabla}}\cdot{\bf J}_{k}=-j_{k}$, where dipole conservation is violated by a nonzero fracton current ${\bf j}$. It follows that in the absence of gapped dipoles, ${\bf j}=0$ for on-shell processes, i.e. isolated disclinations are immobile fractonic charges.

The harmonic 11endnote: 1In a fully nonlinear elasticity, the Hamiltonian is an expansion in the nonlinear strain tensor, $u_{ij}={\frac{1}{2}}({\partial}_{i}{\bf R}\cdot{\partial}_{j}{\bf R}-\delta_{ij})={\frac{1}{2}}({\partial}_{i}u_{j}+{\partial}_{j}u_{i}+{\partial}_{i}{\bf u}\cdot{\partial}_{j}{\bf u})$, that, by construction is a target-space ${\bf R}(r_{i})$ scalar, i.e., invariant under rotation of ${\bf R}\rightarrow{\bf R^{\prime}}=O\cdot{\bf R}$. This corresponds to $u^{\prime}_{k}=(O_{kj}-\delta_{kj})x_{j}+O_{kj}u_{j}$, that, in a linear approximation (small rotation angle $\beta$) reduces to (11). elasticity theory (4) enjoys the symmetries

[TABLE]

The constant shift of $u_{k}$ by $\alpha_{k}$ can be interpreted as continuous translational symmetry. The terms proportional to $\beta$ are a small-angle rotation, where the displacements $u_{k}$ in the initial configuration (before symmetry transformation) are also small. By introducing background gauge fields for these symmetries (see Appendix) and carrying out the duality in the presence of the background fields, we identify corresponding conserved currents on the gauge theory side. Associated with the $\alpha_{k}$ translational symmetry, we have the linear momentum current $J^{m}_{\mu k}$, where the conserved density $J^{m}_{0k}=\epsilon_{kj}B_{j}$ is the magnetic flux, and $J^{m}_{ik}=\epsilon_{kj}\epsilon_{i\ell}E_{\ell j}$. Operators transforming under $\alpha_{k}$ are thus monopole operators of the ${\bf A}_{k}$ gauge fields. Associated with the $\beta$ rotation symmetry and conservation of angular momentum, we have the magnetic flux current of the ${\bf a}$ gauge field, $j^{m}_{\mu}$. Due to the coupling between the vector gauge fields, the naive magnetic flux $\nabla\times{\bf a}$ is not gauge-invariant, requiring modified expressions $j^{m}_{0}=\epsilon_{ij}(\partial_{i}a_{j}+A_{ij})-r_{i}\epsilon_{ij}J^{m}_{0j}$ and $j^{m}_{j}=\epsilon_{ij}(\partial_{i}a_{0}+\partial_{t}a_{i}-A_{0i})-r_{i}\epsilon_{ik}J^{m}_{jk}$. We note that the current $j^{m}_{\mu}$ is explicitly position-dependent, similar to the symmetries discussed in Seiberg (2019). If the position-dependent terms are dropped, $j^{m}_{\mu}$ remains gauge-invariant but is no longer conserved. This identification of magnetic flux currents in the coupled vector gauge theory with symmetries of the elasticity theory is useful in our discussion of lattice models below.

We conclude by demonstrating that in fact this dual coupled vector U(1) gauge theory, at low energies is indeed equivalent to the symmetric tensor gauge theory. To this end, we observe that the enlarged gauge redundancy allows us to completely eliminate $a_{k}$ from the Lagrangian (Fractons from vector gauge theory), by choosing $\chi_{k}=a_{k}$. The term ${\frac{1}{2}}({\bm{\nabla}}\times{\bf a}+A_{a})^{2}$ reduces to ${\frac{1}{2}}A_{a}^{2}$, thereby gapping out the antisymmetric component $A_{a}=\epsilon_{ij}A_{ik}$. Thus, at energies well below this gap, $\epsilon_{ik}A_{ik}\approx 0$, and only the symmetric components of $A_{ik}$ remain as active degrees of freedom. Furthermore, the electric field term reduces to ${\frac{1}{2}}K^{-1}({\partial}_{t}{\bf a}+{\bm{\nabla}}a_{0}-A_{0k})^{2}\rightarrow{\frac{1}{2}}K^{-1}({\bm{\nabla}}a_{0}-A_{0k})^{2}$, enforcing $A_{0k}=\partial_{k}\alpha_{0}$. Thus at low energies, this reduces the Lagrangian exactly to that of the symmetric tensor gauge theory, with the Gauss’ law (8).

Lattice model and charged loop condensation. We now consider a lattice version of the Hamiltonian (1). For simplicity of presentation we first work in 2D, then discuss the generalization to arbitrary dimension. The 3D version of the model appeared previously in Williamson et al. (2018). We use the resulting model to obtain a physical picture of the scalar-charge theory in terms of condensation of certain charged loops. This differs in perspective from the results of Williamson et al. (2018), where the fracton phase arises upon gauging certain global symmetries. There are subtleties particular to the 2D case that we discuss, having to do with the role of compact gauge fields and associated symmetries.

The lattice geometry consists of three interpenetrating 2D square lattices as shown in Fig. 1. One of these we refer to as the “underlying lattice,” and the $k$-lattice ($k=x,y$) is a square lattice with vertices the $k$-directed links of the underlying lattice. We place the electric field ${\bf e}$ and vector potential ${\bf a}$ on the links of the underlying lattice, with ${\bf E}_{k},{\bf A}_{k}$ placed on the links of the $k$-lattice. The ${\bf e},{\bf a}$ gauge field is taken to be compact, thereby allowing a loop-condensation phase transition into the fracton phase, with ${\bf e}$ taking integer eigenvalues and ${\bf a}$ a $2\pi$-periodic phase. For simplicity (apart from 2D), ${\bf E}_{k}$ and ${\bf A}_{k}$ are taken non-compact, with real eigenvalues. The Gauss’s laws are $\nabla\cdot\bm{E}_{k}=e_{k}$ and $\nabla\cdot\bm{e}=0$, where the derivatives denote lattice finite differences. For simplicity, we do not include any additional charged matter; including it does not affect the following discussion.

We consider the following lattice Hamiltonian counterpart of (1):

[TABLE]

Here, $k=x,y$ is summed over in those terms where it appears, and $L$ and $L_{k}$ are the sets of links in the underlying lattice and $k$-lattice, respectively. Similarly $\square$ and $\square_{k}$ denote plaquettes of the underlying lattice and the $k$-lattice. The expression $(\nabla\times{\bf a})_{\square}$ is the lattice line integral of ${\bf a}$ taken counterclockwise around the perimeter of the plaquette $\square$. To understand the last term, note that a single $x$-lattice link $\ell_{x}$ and single $y$-lattice link $\ell_{y}$ pass through the center of each plaquette $\square$, as shown on the left of Fig. 1. This term is thus a gauge-invariant operator that creates electric field configurations like that shown on the right of Fig. 1.

For the generalization to $d$ dimensions, we start with an underlying hypercubic lattice, and $d$ hypercubic $k$-lattices, with $k=1,\dots,d$, whose vertices are centered on $k$-directed links of the underlying lattice. The $K_{e}$ operator is replaced with a term proportional to {\cos\big{[}(\nabla\times{\bf a})_{\square}+A_{ij}-A_{ji}\big{]}}, where $\square$ is one of the ${d(d-1)/2}$ types of plaquettes in the underlying lattice, bisected by the two perpendicular links $\ell_{i}$ and $\ell_{j}$.

Considering the 3D case, Ref. Williamson et al. (2018) showed that one obtains the Hilbert space of the scalar-charge theory upon taking the limit $K_{e}\to\infty$. We take a different point of view, considering the phases of the Hamiltonian (12). When $U_{e}\gg K_{e}$, we put the ${\bf e}$ gauge field into its confining phase, with ${\bf e}\approx 0$, and electric field loops costing an energy proportional to their length. In this limit ${\bf e}$ can be integrated out and we wind up with $d$ decoupled ${\bf E}_{k},{\bf A}_{k}$ non-compact U(1) vector gauge theories.

Starting from this phase, we increase $K_{e}$, thus increasing the fluctuations of the ${\bf e}$ loops. After $K_{e}$ is raised above a critical value, the ${\bf e}$ loops proliferate and condense. This is analogous to the $p$-string condensation in the coupled layer construction of the X-cube fracton model Ma et al. (2017); Vijay (2017), because each ${\bf e}$ loop is built from point-like charged particles of the ${\bf E}_{k}$ gauge fields. We can access the condensed phase by taking $K_{e}$ large and expanding the cosine in the last term. The resulting Gaussian model is simply a lattice regularization of the continuum Hamiltonian (1), and is identical at low energy to the scalar-charge tensor gauge theory.

In $d=2$ this picture breaks down, and – as for a single compact ${\rm U}(1)$ vector gauge theory Polyakov (1977) – there is only one phase. This occurs because when ${\bf e}$ is compact, we only have the two U(1) symmetries associated with conservation of $\nabla\times{\bf A}_{k}$ magnetic flux, with currents $J^{m}_{\mu k}$ discussed above. When $U_{e}\gg K_{e}$, we have two decoupled non-compact U(1) gauge theories in their deconfined phase, which is of course dual to the ordered phase of two decoupled XY models, where the two XY currents are $J^{m}_{\mu k}$. When $U_{e}\ll K_{e}$, naively one may expect to be able to expand the cosine to obtain the Hamiltonian (1). But, as we can see from the dual elasticity description, the resulting theory is only distinct from two decoupled XY models in the presence of the third $j^{m}_{\mu}$ U(1) symmetry, which is absent for compact ${\bf a}$ for any finite $K_{e}/U_{e}$. The absence of a phase transition is obvious on the elasticity side, corresponding to an incommensurate substrate that breaks rotational symmetry and thereby gaps out $\theta$ in (4), reducing to two XY models for $u_{x}$ and $u_{y}$. Therefore there is only one phase in the $d=2$ case. Alternatively we could have taken ${\bf e}$ to be non-compact, in which case we do have all three U(1) symmetries, but there is still only one phase.

These considerations have an interesting consequence for the $d=2$ scalar-charge theory. Namely, if only the two $J^{m}_{\mu k}$ symmetries are present, there is no difference between this theory and two non-compact vector U(1) gauge theories. We note that these conclusions can be obtained working entirely on the lattice; that is, we can follow standard techniques to obtain a dual lattice theory of (12), which at the Gaussian level and in the continuum limit is identical to elasticity.

Generalization to vector-charge theory. We briefly describe a generalization of the above construction that reduces to a different fractonic symmetric-tensor gauge theory at low energy, the vector-charge theory Pretko (2017). In the vector-charge theory, as with the scalar-charge theory, the electric field $\varepsilon_{ij}$ and gauge potential $\alpha_{ij}$ are symmetric tensors, but the Gauss’ law constraint is different, given by $\partial_{i}\varepsilon_{ij}=\rho_{j}$, with the gauge charge $\rho_{j}$ now carrying a vector index.

Focusing on 3D and working in the continuum for simplicity, we introduce a theory of coupled vector gauge theories that reduces to the vector-charge tensor gauge theory at low energy. We introduce six ${\rm U}(1)$ gauge fields, with electric fields ${\bf e}_{k}$ and ${\bf E}_{k}$ ($k=x,y,z$), and corresponding vector potentials ${\bf a}_{k}$ and ${\bf A}_{k}$. We take the Gauss’ law constraints to be

[TABLE]

which express that the anti-symmetric components of $e_{ij}$ act as sources of gauge charge for the ${\bf E}_{k}$ electric fields. These constraints are encoded in the Lagrangian density

[TABLE]

where $b_{ij}=\epsilon_{ikl}\partial_{k}a_{lj}+A_{ji}-\delta_{ij}A_{kk}$. In the Appendix, we show that this theory reduces at low energy to the vector-charge theory, and discuss how to carry out the construction on the lattice. We also briefly remark on a generalization to the 2D vector charge theory. The Appendix also discusses some further generalizations of the lattice coupled vector gauge theory construction of the scalar-charge theory, that reduce at low energy to the $(m,n)$ theories of Bulmash and Barkeshli (2018), and the version of the scalar-charge theory where the electric field is a traceless, symmetric tensor Pretko (2017).

In summary, motivated by the fracton-elasticity dualityRadzihovsky ; Pretko and Radzihovsky (2018a, b), we utilized its reformulation to derive a fractonic coupled U(1) vector gauge theory representation in terms of $d+1$-coupled gauge fields, where components of one type of electric field act as charges for the remaining $d$ gauge theories. At low energies this vector description is identical to fractonic scalar-charge tensor gauge theory. We used a lattice version of this model to discuss fracton order in terms of proliferation of electric field loops. We also proposed a number of generalizations of this construction, making contact with fractonic tensor gauge theories that are not dual to elasticity.

Acknowledgments. We thank Michael Pretko for discussions. MH also acknowledges discussions with Nathan Seiberg. The work of LR was supported by the Simons Investigator Award from the Simons Foundation, and by the Soft Materials Research Center under NSF MRSEC Grants DMR-1420736. The work of MH is supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES) under Award number DE-SC0014415, and also partly supported by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440).

Appendix A Appendix

Background gauge fields for translation and rotation symmetries. To identify the $J^{m}_{\mu k}$ linear momentum and the $j^{m}_{\mu}$ angular momentum currents on the gauge theory side, we introduce background gauge fields ${\cal A}_{\mu k}$ and ${\cal B}_{\mu}$ for the $\alpha_{k}$ and $\beta$ symmetries of Eq. (11) in the main text, respectively. The elasticity Lagrangian with background fields included is

[TABLE]

This is invariant under Eq. (11) of the main text with $\alpha_{k}$ and $\beta$ now space and time dependent, combined with the transformations of the background gauge fields,

[TABLE]

From this starting point, it is straightforward to carry through the duality as in the main text to identify the conserved currents on the gauge theory side. We note that for this purpose, the singular parts of the elastic and bond angle fields can be ignored.

Coupled vector gauge theory construction of the vector-charge theory. Here we show that the coupled vector gauge theory Lagrangian in Eq. (15) of the main text reduces to the 3D vector-charge theory at low energy, and briefly describe the corresponding lattice model. We also briefly remark on the generalization of this construction to the 2D vector charge theory.

The Lagrangian is invariant under the gauge transformations

[TABLE]

We would like to eliminate the antisymmetric part of $a_{ij}$ from the Lagrangian. To do this, we write

[TABLE]

where $a^{s}_{ij}=a_{ij}+a_{ji}$, and using the fact that an antisymmetric rank-2 tensor can be expressed in the form of the second term. We have

[TABLE]

and

[TABLE]

Then clearly if we make a partial gauge transformation, where we only transform $A_{ij}$ using $\chi_{i}=\Omega_{i}$, and do not transform $a_{ij}$, we eliminate the antisymmetric part of $a_{ij}$. Integrating out the electric fields ${\bf e}_{k}$ and ${\bf E}_{k}$, the resulting Lagrangian is

[TABLE]

It is convenient to define

[TABLE]

and observe $c_{ii}=0$. Under a gauge transformation, $c_{ij}\to c_{ij}+\frac{1}{2}\epsilon_{ikl}\partial_{k}\partial_{j}\phi_{l}$, and we define the gauge invariant quantity

[TABLE]

which will play the role of the magnetic field tensor in the low-energy vector-charge theory.

We proceed by making the change of variables $A_{ij}\to A_{ij}-c_{ji}$, where the position of the indices should be carefully noted. We observe that

[TABLE]

The Lagrangian becomes

[TABLE]

We want to integrate out the massive $A_{0k}$ and $A_{ij}$ fields, and neglect all their subdominant higher-derivative terms, that are unimportant for the long-distance, low-energy behavior. The generated leading terms in $a^{s}_{ij}$ have either two time derivatives, or four spatial derivatives, resulting in the dynamical critical exponent $z=2$ scaling expected for the vector-charge theory. All of the $c_{ji}$ terms proportional to $1/C$ lead to subdominant contributions, as does the second term on the last line, and we drop these to obtain

[TABLE]

Now integrating out the $A_{ij}$, $A_{0k}$ gauge fields gives

[TABLE]

where we have identifed $\alpha_{ij}=a^{s}_{ij}$. This is precisely the Lagrangian for the vector-charge theory.

To carry out this construction on the lattice, we consider an underlying cubic lattice, and three cubic $k$-lattices ($k=x,y,z$), whose vertices are the $k$-directed links of the underlying lattice. We place the ${\bf e}_{k}$ electric field on the links of the $k$-lattice. To understand where to place the ${\bf E}_{k}$ electric fields, we refer to the Gauss law, Eq. (14) of the main text, which asserts that $\epsilon_{kij}e_{ij}$ is the density of ${\bf E}_{k}$ gauge charge. On the lattice, $\epsilon_{kij}e_{ij}$ naturally resides on the plaquettes of the underlying lattice normal to the $k$-direction. For instance, $\epsilon_{zij}e_{ij}=e_{xy}-e_{yx}$, and this variable naturally resides on $xy$-plane underlying lattice plaquettes, because a $y$-link of the $x$-lattice crosses an $x$-link of the $y$-lattice at the center of each such plaquette. Therefore we introduce $xy$, $yz$ and $xz$ cubic lattices, whose vertices lie at plaquette centers of the corresponding underlying lattice plaquettes, and we place ${\bf E}_{x}$ on the links of the $yz$-lattice, ${\bf E}_{y}$ on the links of the $xz$-lattice, and ${\bf E}_{z}$ on the links of the $xy$-lattice. This results in the usual lattice vector-charge theory where diagonal components of $\varepsilon_{ij}$ reside on sites of the underlying lattice, with off-diagonal components residing on underlying lattice plaquettes.

Finally, we briefly comment on the generalization of this construction to the 2D vector charge theory. In this case we introduce two ${\bf e}_{k}$ electric fields ($k=x,y$) and one ${\bf E}$ electric field, for a total of three ${\rm U}(1)$ vector gauge fields. The Gauss’ law constraints are

[TABLE]

The construction then proceeds as in the 3D case, and one obtains the 2D vector charge theory at low energy. We note that the gauge-invariant magnetic field operators in the 2D construction are $B=\epsilon_{ij}\partial_{i}A_{j}$ and $b_{k}=\epsilon_{ij}\partial_{i}a_{jk}+A_{k}$, where ${\bf a}_{k}$ and ${\bf A}$ are the vector potentials conjugate to ${\bf e}_{k}$ and ${\bf E}$, respectively.

Further lattice-model generalizations. We briefly state some generalizations of the lattice construction of the scalar-charge theory, a detailed analysis of which will be presented elsewhere. We focus on three dimensions, and introduce an underlying cubic lattice and three cubic $k$-lattices ($k=x,y,z$), whose vertices are centered on $k$-directed links of the underlying lattice. We introduce a compact U(1) gauge field on each lattice with electric fields ${\bf e}$ and ${\bf E}_{k}$, writing $E_{ik}\equiv({\bf E}_{k})_{i}$. We consider Gauss’s laws of the form

[TABLE]

where $\Delta_{i}$ is the lattice derivative operator and $M$ is a $3\times 3$ matrix with $m$ in the diagonal entries and $n$ in the off-diagonal entries, with $m$ and $n$ positive, relatively prime integers. $m=n=1$ corresponds to an isotropic Gauss’s law for the ${\bf E}_{k}$ electric fields, with $e_{k}$ carrying ${\bf E}_{k}$ electric charge, while $m\neq n$ makes the Gauss’s laws anisotropic. Starting from a phase where ${\bf e}$ loops are confined and the three ${\bf E}_{k}$ gauge fields are deconfined, condensing ${\bf e}$ loops produces the deconfined phase of the $(m,n)$ scalar-charge tensor gauge theories introduced in [20]. These theories have a symmetric-tensor electric field $E^{s}_{ij}$ obeying the Gauss’s law $\sum_{i\leq j}M_{ij}\Delta_{i}\Delta_{j}E^{s}_{ij}=0$.

This construction can be further generalized by adding another vector gauge field on the underlying lattice, with electric field $\bm{{\cal E}}$, and supplementing the above Gauss’s laws with

[TABLE]

Constructing a Hamiltonian for these gauge fields and taking the coefficients of all the Maxwell terms large, so that all cosines can be expanded, one obtains the traceless version of the $(m,n)$ scalar-charge theory, which obeys the additional constraint $\sum_{i}E^{s}_{ii}=0$.