Interactions and the θ term in one-dimensional gapped systems

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. 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.