Fracton-Elasticity Duality (1711.11044v2)

Abstract: Motivated by recent studies of fractons, we demonstrate that elasticity theory of a two-dimensional quantum crystal is dual to a fracton tensor gauge theory, providing a concrete manifestation of the fracton phenomenon in an ordinary solid. The topological defects of elasticity theory map onto charges of the tensor gauge theory, with disclinations and dislocations corresponding to fractons and dipoles, respectively. The transverse and longitudinal phonons of crystals map onto the two gapless gauge modes of the gauge theory. The restricted dynamics of fractons matches with constraints on the mobility of lattice defects. The duality leads to numerous predictions for phases and phase transitions of the fracton system, such as the existence of gauge theory counterparts to the (commensurate) crystal, supersolid, hexatic, and isotropic fluid phases of elasticity theory. Extensions of this duality to generalized elasticity theories provide a route to the discovery of new fracton models. As a further consequence, the duality implies that fracton phases are relevant to the study of interacting topological crystalline insulators.

Fracton-Elasticity Duality: A Study on Quantum Crystals and Tensor Gauge Theories

The paper by Michael Pretko and Leo Radzihovsky explores the intriguing duality between elasticity theory in two-dimensional quantum crystals and fracton phenomena via tensor gauge theories. It illustrates how ordinary solid-state systems can provide concrete realizations of fracton models, enriching our understanding of quantum phases with restricted mobility.

Overview of Fracton Elasticity Duality

Fractons are characterized by their unusual dynamics, particularly their limited mobility, which contrasts sharply with more conventional quasiparticle behaviors. Many fracton models show that the excitations are immobile when isolated but can traverse certain paths through interactions. The paper establishes a duality where a quantum crystal's elasticity is mapped to fracton tensor gauge theory. This mapping offers new insights, linking the topological defects in elasticity theory to fractons and dipoles in gauge theory.

Key Correspondences

1. Phonons and Gauge Modes: The duality maps longitudinal and transverse phonons in crystals to the gapless gauge modes present in fracton gauge theory. This provides a natural pair of correspondences between the systems.
2. Topological Defects and Charge: Disclinations and dislocations in elasticity theory correspond to fractons and dipoles in the gauge framework. The restricted dynamics inherent in fractons find analogy in the constraints on motion for lattice defects within crystals.
3. Structural and Dynamical Implications: The paper predicts phase transitions within fracton systems by correlating them to known transitions in elasticity. Notably, the fracton gauge system exhibits analogs to crystal, hexatic, and fluid phases.

Phase and Transition Insights

Drawing from the well-characterized physics of quantum crystals, the duality suggests a roadmap for understanding fracton behaviors. The analogy points to finite-temperature phase transitions akin to those found in two-dimensional crystals undergoing BKT-like melting transitions. Dislocations and disclinations play key roles in these transitions, analogous to dipole and fracton proliferation studies within gauge theories.

Practical and Theoretical Implications

Practically, this insight from elasticity theory into fracton models suggests it can serve as a framework for discovering novel fracton phases. On a theoretical level, understanding fractons through this duality might provide tools for classifying phases in topological crystalline insulators (TCIs) and symmetry-protected topological phases. Gauging spatial symmetries in TCIs and enzymes topological order might open new pathways in the synthesis and classification of interacting crystalline topological phases.

Future Directions

The exploration of fracton-elasticity duality contributes significantly to the field by establishing connections with established physics frameworks, aiding the deciphering of fracton phases. Future research could refine these predictions and extend their applications to real-world materials and condensed matter phenomena. It sets the stage for a deeper understanding of non-conventional phases and interactions in solid-state systems. As the research progresses, it may elucidate the fundamental aspects of intertwined quantum systems, potentially leading to novel technological applications and advancements in material science.