Higher-form Gauge Symmetries in Multipole Topological Phases

TL;DR

The paper develops a 1-form symmetry framework for dipole-conserving multipole topological phases by introducing a background 2-form field $\mathcal{B}$ that couples to electric lines, yielding a generalized Peierls substitution $A_{xy}=\frac{e}{\hbar}\iint_{p_{xy}}\mathcal{B}$. It constructs a topological quadrupole response action, $S_{Q}=-\frac{e\theta}{2\pi}\int d\mathcal{B}$, whose Dixmier-Douady invariant quantizes the quadrupole moment $q_{xy}$ and accounts for corner and edge polarizations consistent with crystal symmetries. The framework also recasts the rank-2 Berry phase as a large gauge transformation of $\mathcal{B}$ and proves a generic higher-form LSM constraint: a unique ground state in dipole-conserving systems requires the total polarization to vanish modulo a polarization quantum, with an equivalently stringent flux-threading argument. Together, these results unify lattice dipole/quadrupole models with higher-form gauge theories, connect fracton-like physics to multipole topological responses, and suggest extensions to higher-form and subsystem symmetries in multipole topological phases.

Abstract

In this article we study field-theoretical aspects of multipolar topological insulators. Previous research has shown that such systems naturally couple to higher-rank tensor gauge fields that arise as a result of gauging dipole or subsystem $U(1)$ symmetries. Here we propose a complementary framework using electric higher-form symmetries. We utilize the fact that gauging 1-form electric symmetries results in a 2-form gauge field which couples naturally to extended line-like objects: Wilson lines. In our context the Wilson lines are electric flux lines associated to the electric polarization of the system. This allows us to define a generalized 2-form Peierls' substitution for dipoles that shows that the off-diagonal components of a rank-2 tensor gauge field $A_{ij}$ can arise as a lattice Peierls factor generated by the background antisymmetric 2-form gauge field. This framework has immediate applications: (i) it allows us to construct a manifestly topological quadrupolar response action given by a Dixmier-Douady invariant -- a generalization of a Chern number for 2-form gauge fields -- which makes plain the quantization of the quadrupole moment in the presence of certain crystal symmetries; (ii) it allows for a clearer interpretation of the rank-2 Berry phase calculation of the quadrupole moment; (iii) it allows for a proof of a generic Lieb-Schultz-Mattis theorem for dipole-conserving systems.

Figures (8)

• Figure 1: A hierarchical refinement of the notion of 'insulator.'
• Figure 2: Lattice structure of the quadrupole ring-excahnge model.
• Figure 3: A uniformly electrically-neutral chain carrying a polarization and an associated electric line which can be revealed by breaking the periodic boundary conditions.
• Figure 4: Two opposite Wilson lines carrying electric flux $\Phi_E$ wrapping around $x$-direction may be modified by arbitrary phases under the action of $U_g(\mathcal{M}^1)$ if the parameter $g$ depends on $y$. The manifold $\mathcal{M}^1$ is represented by the blue line.
• Figure 5: A pair of opposite running electric lines created by a successive application of the dipole hopping operator, that transports $y$-dipole along the $\hat{x}$-direction. In other words, dragging a dipole all around the manifold results in an operator that is a product of two parallel and oppositely running operators.