Interactions and the Theta Term in One-Dimensional Gapped Systems

TL;DR

This paper investigates how the heta -term behaves under interactions in one-dimensional gapped systems, revealing that it remains unaffected in some models but can be tuned to different topological phases in others.

Contribution

It demonstrates the relationship between the chiral anomaly and the heta -term in bosonized systems, and shows how interactions can induce topological phase transitions.

Findings

heta -term is insensitive to interaction strength in the Luttinger liquid.

Interaction tuning in the XXZ chain can switch between distinct topological phases.

Spontaneous breaking of charge-conjugation and parity occurs with fractional heta / \pi values.

Abstract

We study how the \theta -term is affected by interactions in certain one-dimensional gapped systems that preserve charge-conjugation, parity, and time-reversal invariance. We exploit the relation between the chiral anomaly of a fermionic system and the classical shift symmetry of its bosonized dual. The vacuum expectation value of the dual boson is identified with the value of the \theta -term for the corresponding fermionic system. Two (related) examples illustrate the identification. We first consider the massive Luttinger liquid and find the \theta -term to be insensitive to the strength of the interaction. Next, we study the continuum limit of the Heisenberg XXZ spin-1/2 chain, perturbed by a second nearest-neighbor spin interaction. For a certain range of the XXZ anisotropy, we find that we can tune between two distinct sets of topological phases by varying the second nearest-neighbor coupling. In the first, we find the standard vacua at \theta = 0, \pi, while the second contains vacua that spontaneously break charge-conjugation and parity with fractional \theta / \pi = 1/ 2, 3/2. We also study quantized pumping in both examples following recent work.