Gauging on the lattice and gapped/gapless topological phases
Takamasa Ando
TL;DR
This work develops a unified lattice framework for effective gauging and fermionization, where Gauss-law constraints are enforced energetically rather than strictly, to realize both gapped and gapless topological phases with a nontrivial action of an enlarged symmetry $\Gamma$ that extends a finite abelian group $A$ by $G$ via $1\rightarrow A\rightarrow \Gamma\rightarrow G\rightarrow 1$. By coupling to background fields $\hat{A}$, $G$, and $A$ and using explicit lattice constructions, the authors derive precise topological response actions and classify emergent anomalies, including mixed $G\times\hat{A}$ anomalies and Gu-Wen fermionic anomalies, in terms of group cohomology data $[e]\in H^2(G,A)$ and cocycles in $H^3(G\times\hat{A},U(1))$. They demonstrate how effective gauging yields both gapped and intrinsically gapless SPTs (igSPTs), with concrete 1+1D examples (TFI, XXZ, clock models) and higher-dimensional remarks, and show how gapless phases can host emergent anomalies by incorporating appropriate symmetry extensions and condensations. The field-theory perspective ties these lattice constructions to bulk-boundary inflow, describing how $-\int g^*e\cup \hat{A}$ cancels boundary anomalies and how the surviving global symmetry acts faithfully on the full Hilbert space. Overall, the paper provides a versatile toolkit for constructing and diagnosing both gapped and gapless topological phases via effective gauging/fermionization, with clear routes to higher dimensions and connections to established igSPT frameworks and Gu-Wen anomaly data.
Abstract
In this work, we explore topological phases of matter obtained by effectively gauging or fermionizing a system, where the Gauss law constraint is only enforced energetically. In contrast to conventional gauging or fermionization, the symmetry that is effectively gauged at low energies still generates a global symmetry that acts on the whole Hilbert space faithfully. This symmetry turns out to protect a nontrivial topological phase together with other symmetries, or it can carry a nontrivial emergent 't Hooft anomaly. We provide a precise formula for the topological response action involving these symmetries in a general setup, as well as a formula for 't Hooft anomalies. As an application, we apply the general treatment of the procedure to gapless systems and find various new gapless SPT phases, such as the one carrying the Gu-Wen fermionic anomalies at low energy.