Crystal-to-fracton tensor gauge theory dualities

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 particlevortex 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 timereversal 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.

Fracton phases of matter

“…While the models we have considered so far have taken the form of complicated spin models, without an immediate connection to material realization, it is important to note that fractons have a much more down-to-earth physical realization as the topological lattice defects of ordinary two-dimensional solids. This connection between fractons and lattice defects can be seen by studying the conventional elasticity theory of two-dimensional crystals, which turns out to have an exact duality mapping with the scalar charge fracton tensor gauge theory (enriched by an extra global symmetry) [9,46]. Within this duality, disclination defects play the role of immobile fractons, while dislocation defects act as dipoles exhibiting one-dimensional motion.…”

“…(This can also be seen more explicitly in an alternative formulation of elasticity theory [46,54].) This action describes the behavior of two gapless modes, corresponding to transverse and longitudinal phonons.…”

“…For example, the duality has been used to provide a simplified derivation of the Halperin-Nelson-Young theory of two-dimensional melting [44]. It has also been proposed that the duality may aid in the classification of interacting topological crystalline insulators [9,46].…”

“…This connection is made precise via a duality transformation, often referred to as "fracton-elasticity duality," which maps the elasticity theory of crystals onto a symmetric tensor gauge theory [9]. We discuss this duality in detail in Section V, along with its various generalizations [41][42][43][44][45][46][47]. For example, the duality can be extended to three-dimensional elasticity theory, giving rise to the concept of fractonic lines, i.e line-like excitations without the ability to move [41].…”

Quantum robustness of fracton phases

“…The physical origin of the thermal fragility in two-dimensional stabilizer codes is the finite energy barrier between different topological ground states, e.g., the toric code becomes thermally stable only for dimensions larger three [14]. The recently discovered fracton phases reach beyond such topological codes and have interesting crosslinks to other domains in physics like elasticity [38][39][40][41][42][43], localization [15,44,45], gravity [46][47][48], Majorana fermions [49,50], and deconfined quantum criticality [51]. * matthias.muehlhauser@fau.de † matthias.walther@fau.de ‡ david.reiss@fu-berlin.de § kai.phillip.schmidt@fau.de…”