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arXiv:2411.19620 · Physical Review Letters

Bulk-Boundary Correspondence of Fractonic Field Theories

Bhandaru Phani Parasar, Yuval Gefen, Vijay B. Shenoy

Published May 19, 2025• 2 citations
arXivHTMLPDF

Abstract

We develop a theory of edge excitations of fractonic systems in two dimensions, and elucidate their connections to bulk transport properties and quantum statistics of bulk excitations. The system we consider has immobile point charges, dipoles constrained to move only along lines perpendicular to their moment, and freely mobile quadrupoles and higher multipoles, realizing a bulk fractonic analog of fractional quantum Hall phases. We demonstrate that a quantized braiding phase between two bulk excitations is obtained only in two cases: when a point quadrupole braids around an immobile point charge, or when two non-orthogonal point dipoles braid with one another. The presence of a boundary edge in the system entails two types of gapless edge excitation modes, one that is fractonic with immobile charges and longitudinal dipoles, and a second non-fractonic mode consisting of transverse dipoles. We derive a novel current algebra of the fractonic edge modes. Further, investigating the effect of local edge-to-edge tunneling on these modes, we find that such a process is a relevant perturbation suggesting the possibility of edge deformation.

DOI
10.1103/ggls-zhl8

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We develop a theory of edge excitations of fractonic systems in two dimensions, and elucidate their connections to bulk transport properties and quantum statistics of bulk excitations. The system we consider has immobile point charges, dipoles constrained to move only along lines perpendicular to their moment, and freely mobile quadrupoles and higher multipoles, realizing a bulk fractonic analog of fractional quantum Hall phases. We demonstrate that a quantized braiding phase between two bulk excitations is obtained only in two cases: when a point quadrupole braids around an immobile point charge, or when two non-orthogonal point dipoles braid with one another. The presence of a boundary edge in the system entails two types of gapless edge excitation modes, one that is fractonic with immobile charges and longitudinal dipoles, and a second non-fractonic mode consisting of transverse dipoles. We derive a novel current algebra of the fractonic edge modes. Further, investigating the effect of local edge-to-edge tunneling on these modes, we find that such a process is a relevant perturbation suggesting the possibility of edge deformation.
In three dimensions, gapped phases can support "fractonic" quasiparticle excitations, which are either completely immobile or can only move within a low-dimensional submanifold, a peculiar topological phenomenon going beyond the conventional framework of topological quantum field theory.In this work we explore fractonic topological phases using three-dimensional coupled wire constructions, which have proven to be a successful tool to realize and characterize topological phases in two dimensions.We find that both gapped and gapless phases with fractonic excitations can emerge from the models.In the gapped case, we argue that fractonic excitations are mobile along the wire direction, but their mobility in the transverse plane is generally reduced.We show that the excitations in general have infinite-order fusion structure, distinct from previously known gapped fracton models.Like the two-dimensional coupled-wire constructions, many models exhibit gapless (or even chiral) surface states, which can be described by infinite-component Luttinger liquids.However, the universality class of the surface theory strongly depends on the surface orientation, thus revealing a different type of bulk-boundary correspondence unique to fracton phases.
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More profound than bulk topological order of quantum materials is only its unwinding via gapless excitations along boundaries of the sample. We recast this bulk-edge correspondence -- for the experimentally relevant case of fractional quantum Hall (FQH) systems -- in terms of effective relative higher gauge theory, controlled by choices of classifying fibrations. For FQH systems, we identify the complex Hopf fibration as classifying the bulk/boundary topological effects, and find that it yields a non-Lagrangian reconstruction of Floreanini-Jackiw/Wess-Zumino-Witten chiral edge currents. Remarkably, the resulting effective FQH higher gauge theory turns out to be "geometrically engineered" on M2/M5-branes probing A-type orbi-singularities in 11D supergravity, globally completed by flux-quantization in twisted equivariant differential (TED) Cohomotopy: Here the M-string ends of M2-branes on M5-branes engineer the FQH liquid's boundary. This geometric engineering on M-branes might naturally elucidate the curious combination of W∞W_\inftyW∞​-symmetry and of super-symmetry that is known to govern the collective excitations of FQH liquids at long wavelengths.
Fractons are emergent particles which are immobile in isolation, but which can move together in dipolar pairs or other small clusters. These exotic excitations naturally occur in certain quantum phases of matter described by tensor gauge theories. Previous research has focused on the properties of small numbers of fractons and their interactions, effectively mapping out the ``standard model'' of fractons. In the present work, however, we consider systems with a finite density of either fractons or their dipolar bound states, with a focus on the U(1)U(1)U(1) fracton models. We study some of the phases in which emergent fractonic matter can exist, thereby initiating the study of the ``condensed matter'' of fractons. We begin by considering a system with a finite density of fractons, which we show can exhibit microemulsion physics, in which fractons form small-scale clusters emulsed in a phase dominated by long-range repulsion. We then move on to study systems with a finite density of mobile dipoles, which have phases analogous to many conventional condensed matter phases. We focus on two major examples: Fermi liquids and quantum Hall phases. A finite density of fermionic dipoles will form a Fermi surface and enter a Fermi liquid phase. Interestingly, this dipolar Fermi liquid exhibits a finite-temperature phase transition, corresponding to an unbinding transition of fractons. Finally, we study chiral two-dimensional phases corresponding to dipoles in ``quantum Hall'' states of their emergent magnetic field. We study numerous aspects of these generalized quantum Hall systems, such as their edge theories and ground state degeneracies.
We demonstrate the existence of a fundamentally new type of excitation, fractonic lines, which are linelike excitations with the restricted mobility properties of fractons. These excitations, described using an amalgamation of higher-form gauge theories with symmetric tensor gauge theories, see direct physical realization as the topological lattice defects of ordinary three-dimensional quantum crystals. Starting with the more familiar elasticity theory, we show how theory maps onto a rank-4 tensor gauge theory, with phonons corresponding to gapless gauge modes and disclination defects corresponding to linelike charges. We derive flux conservation laws which lock these linelike excitations in place, analogous to the higher moment charge conservation laws of fracton theories. This way of encoding mobility restrictions of lattice defects could shed light on melting transitions in three dimensions. This new type of extended object may also be a useful tool in the search for improved quantum error-correcting codes in three dimensions.
We present a coupled-wire construction of a model with chiral fracton topological order. The model combines the known construction of ν=1/m\nu=1/mν=1/m Laughlin fractional quantum Hall states with a planar p-string condensation mechanism. The bulk of the model supports gapped immobile fracton excitations that generate a hierarchy of mobile composite excitations. Open boundaries of the model are chiral and gapless, and can be used to demonstrate a fractional quantized Hall conductance where fracton composites act as charge carriers in the bulk. The planar p-string mechanism used to construct and analyze the model generalizes to a wide class of models including those based on layers supporting non-Abelian topological order.