Crystal-to-Fracton Tensor Gauge Theory Dualities

TL;DR

This work establishes explicit dualities between 2D crystal elasticity and U(1) fracton tensor gauge theories, mapping disclinations to fracton charges and dislocations to dipoles, with phonons corresponding to gauge modes. Extending to bosonic quantum melting yields a hybrid vector–tensor gauge theory describing a supersolid, where axion-like couplings attach quantum numbers across sectors and enforce symmetry-enriched mobility constraints. The framework reproduces classical 2D melting results, predicts finite-temperature fracton phase behavior, and outlines a quantum phase diagram for bosons with intertwined crystalline and superfluid orders, including potential deconfined quantum critical points. The duality also suggests connections to TCIs and provides a versatile lens to explore both fracton physics and elasticity theory, offering new avenues for experimental probes via pinch-point signatures and disclination dynamics.

Abstract

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 analogue 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.

Figures (9)

• 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 Reference seung.)
• 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 Reference seung.)
• 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 Reference seung.)
• 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 non-collinear quadrupole moment.