Crystal-to-fracton tensor gauge theory dualities

We demonstrate several explicit duality mappings between elasticity of two-dimensional crystals and fracton tensor gauge theories, expanding on recent works by two of the present authors. We begin by dualizing the quantum elasticity theory of an ordinary commensurate crystal, which maps directly onto a fracton tensor gauge theory, in a natural tensor analog of the conventional particle-vortex duality transformation of a superfluid. The transverse and longitudinal phonons of a crystal map onto the two gapless gauge modes of the tensor gauge theory, while the topological lattice defects map onto the gauge charges, with disclinations corresponding to isolated fractons and dislocations corresponding to dipoles of fractons. We use the classical limit of this duality to make new predictions for the finite-temperature phase diagram of fracton models, and provide a simpler derivation of the Halperin-Nelson-Young theory of thermal melting of two-dimensional solids. We extend this duality to incorporate bosonic statistics, which is necessary for a description of the quantum melting transitions. We thereby derive a hybrid vector-tensor gauge theory which describes a supersolid phase, hosting both crystalline and superfluid orders. The structure of this gauge theory puts constraints on the quantum phase diagram of bosons, and also leads to the concept of symmetry enriched fracton order. We formulate the extension of these dualities to systems breaking time-reversal symmetry. We also discuss the broader implications of these dualities, such as a possible connection between fracton phases and the study of interacting topological crystalline insulators.

Figure 1

The excitations and operators of the scalar charge theory are in one-to-one correspondence with those of elasticity theory. (Pictures of lattice defects adapted from Ref. [56].)

Figure 2

Disclinations are orientational defects of the crystal, as depicted above on the triangular lattice. Notice that the central site touches only five other sites, indicating a missing bond angle of $\pi /3$. (Figure adapted from Ref. [56].)

Figure 3

Dislocations correspond to bound states of two equal and opposite disclinations, representing translational defects of the crystal. Note that the Burgers vector $\vec{b}$ is perpendicular to the vector between the two disclinations. (Figure adapted from Ref. [56].)

Figure 4

A bound state of two dislocations of opposite charge, separated by a single lattice constant, carries a unit of vacancy number, as can be seen by the depleted density of atoms.

Figure 5

(a) A fracton is immobile since motion of a fracton requires creation of a conserved dipole moment. (b) A dipole is immobile in the longitudinal direction in a phase without superfluid order, since such motion corresponds to creation of a collinear quadrupole, carrying conserved boson number, protected by global U(1) symmetry. (c) A dipole is always fully mobile in the transverse direction, since it corresponds to creation of a U(1)-neutral noncollinear quadrupole moment.

Figure 6

A supersolid features both superfluid order, with its associated confined vortex defects, and crystalline order, with its associated topological lattice defects, namely, disclinations and dislocations. Other typical quantum phases of bosons, such as normal solids, superfluids, and superhexatics, can be obtained from the supersolid upon condensation of lattice defects. Note, however, that normal (i.e., nonsuper) liquids and hexatics are ruled out as quantum phases on general principles from the LSM theorem [64, 65, 66].

Figure 7

A schematic quantum (i.e., zero-temperature) phase diagram of bosons in two dimensions.

Figure 8

As the temperature is raised, a two-dimensional fracton tensor gauge theory exhibits a dipole unbinding transition, analogous to the solid-hexatic transition of elasticity theory. At a higher temperature, the system will then undergo a fracton unbinding transition, analogous to the hexatic-liquid transition.

Figure 9

At left is a schematic plot of $\langle E^x\left(q\right)E^y(-q)\rangle$ in the $q_x\text{−}{q}_y$ plane, displaying the characteristic twofold pinch-point singularity of conventional gauge theories. At right is an analogous plot of $\langle E^{xx}\left(q\right)E^{yy}(-q)\rangle$, displaying the fourfold pinch-point singularity of a rank-2 tensor gauge theory. Figure adapted from Ref. [125].