Multipolar Topological Field Theories: Bridging Higher Order Topological Insulators and Fractons

TL;DR

The paper develops a unified, field-theoretic framework of topological multipole responses to describe both fracton-like subsystem-symmetric phases and higher-order topological insulators (HOTIs) in 2D and 3D. It introduces 2D quadrupole and 3D dipolar Chern-Simons terms, explains their quantization and boundary anomalies, and connects these to corner/hinge modes via dimensional reduction and parton constructions. The work shows how subsystem symmetry can map to global symmetry cases, yielding robust, quantized quadrupole moments $q_{xy}$ and hinge currents, and extends the construction to higher dimensions with octupole and generalized multipole CS/axion theories. These theories provide concrete, experimentally testable predictions for multipole-induced transport and edge/corner phenomena, while outlining a hierarchy that links HOTIs, fractons, and their interacting descendants. Overall, the framework offers a versatile toolkit for exploring topological responses beyond conventional polarization and Hall conductance, with implications for strongly interacting systems and engineered metamaterials.

Abstract

Two new recently proposed classes of topological phases, namely fractons and higher order topological insulators (HOTIs), share at least superficial similarities. The wide variety of proposals for these phases calls for a universal field theory description that captures their key characteristic physical phenomena. In this work, we construct topological multipolar response theories that capture the essential features of some classes of fractons and higher order topological insulators. Remarkably, we find that despite their distinct symmetry structure, some classes of fractons and HOTIs can be connected through their essentially identical topological response theories. More precisely, we propose a topological quadrupole response theory that describes both a 2D symmetry enriched fracton phase and a related bosonic quadrupolar HOTI with strong interactions. Such a topological quadrupole term encapsulates the protected corner charge modes and, for the HOTI, predicts an anomalous edge with fractional dipole moment. In 3D we propose a dipolar Chern-Simons theory with a quantized coefficient as a description of the response of both second order HOTIs harboring chiral hinge currents, and of a related fracton phase. This theory correctly predicts chiral currents on the hinges and anomalous dipole currents on the surfaces. We generalize these results to higher dimensions to reveal a family of multipolar Chern-Simons terms and related $θ$-term actions that can be reached via dimensional reduction or extension from the Chern-Simons theories.

Figures (8)

• Figure 1: There are four bosons/spins (red dots) in each unit cell (shaded island), and each of them participates in one plaquette ring-exchange term. Each ring-exchange term involves four bosons living at the four corners of the plaquette interacting at the quartic level. Bosons at the edges can also be gapped using quadratic terms, but only within the unit cell. A single boson zero mode (alternatively free spin-1/2) survives on each of the corners and is protected by $\mathcal{T}$ and U(1) subsystem symmetry.
• Figure 2: Parton state for $h$(blue/left) and $v$(green/right) bosons. The full system has these two layers superimposed. Each parton resides in a 1D SPT chain. On the boundary rows, the dangling spins are paired within the site so there is only one non-trivial SPT chain per edge. Both parton types contribute a single protected corner mode on each corner that eventually projects to a physical boson corner mode $a=hv$.
• Figure 3: 2D Topological quadrupole insulator with four spinless fermion orbitals per-unit cell. Each intra- and inter-cell plaquette contains a $\pi$ flux. The fermion only hops along the blue bonds within each plaquette. The dotted lines represent hopping terms with a relative minus sign compared to the solid lines. The edges resemble gapped SSH chains and the corners carry a fermion zero mode.
• Figure 4: Summary of 2D Chern-Simons response and related effects in 1D and 3D.
• Figure 5: Summary of 3D dipole Chern-Simons response and related effects in 2D and 4D.