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Interactions and the Theta Term in One-Dimensional Gapped Systems

We study how the θ-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 θ-term for the corresponding fermionic system. Two (related) examples illustrate the identification. We first consider the massive Luttinger liquid and find the θ-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 θ= 0, π, while the second contains vacua that spontaneously break charge-conjugation and parity with fractional θ/ π= 1/ 2, 3/2. We also study quantized pumping in both examples following recent work.