Topological Mott Insulators and Discontinuous $U(1)$ $\theta$-Terms

TL;DR

This paper introduces a lattice field theory describing the transition between superfluid and topological Mott insulator phases, revealing a level-shift symmetry and predicting quantized Hall conductance in the topological phase.

Contribution

It presents a novel lattice model with a topological term for bosonic topological Mott insulators, establishing a connection to the well-known 2+1d XY transition and uncovering a level-shift symmetry.

Findings

Critical exponents match those of the 2+1d XY transition.

Topological phase exhibits quantized Hall conductance.

Level-shift symmetry relates models differing by the topological term.

Abstract

We introduce a lattice field theory that describes the transition between a superfluid (SF) and a bosonic topological Mott Insulator (tMI) -- a $U(1)$ symmetry protected topological phase labeled by an integer level $k$ and possessing an even integer $2k\frac{e^2}{h}$ quantized Hall conductance. Our model differs from the usual $2+1$d XY model by a topological term that vanishes on closed manifolds and in the absence of an applied gauge field, which implies that the critical exponents of the SF-tMI transition are identical to those of the well-studied $2+1$d XY transition. Our formalism predicts a "level-shift" symmetry: in the absence of an applied gauge field, the bulk correlation functions of all local operators are identical for models differing by the topological term. %, for example, near the SF-MI and SF-tMI transitions, and hence the extremely well-studied critical exponents of the $2+1$d XY model apply to the SF-tMI transition. In the presence of a background gauge field, the topological term leads to a quantized Hall response in the tMI phase, and we argue that this quantized Hall effect persists in the vicinity of the phase transition into the SF phase. Our formalism paves the way for other exact lattice descriptions of symmetry-protected-topological (SPT) phases, and mappings of critical exponents between transitions from symmetry-breaking to trivial states and transitions from symmetry-breaking to SPT states. A similar "level shift" symmetry should appear between all group cohomology SPT states protected by the same symmetry.