Competing symmetry breaking and topology in quantum spin chains

Abstract

We investigate how spontaneous symmetry breaking and symmetry-protected topology compete and cooperate in one-dimensional quantum spin-1/2 systems. We introduce two families of local Hamiltonians with ${\mathbb{Z}}_2^{}\times {\mathbb{Z}}_2^{}\times {\mathbb{Z}}_2^T$ symmetry which host a zoo of broken-symmetry phases with and without gapless boundary spins, namely, six long-range-ordered gapped phases that differ by their ${\mathbb{Z}}_2^{}$-symmetry-breaking patterns (FM${}_{x,y,z}^{}$ and Neel${}_{x,y,z}^{}$), their partner symmetry-protected topological phases (FM${}_{x,y}^{\mathrm{SPT}}$ and Neel${}_{x,y}^{\mathrm{SPT}}$), and two gapless phases (gFM${}_z^{}$ and gNeel${}_z^{}$) with incommensurate ${\mathbb{Z}}_2^T$-breaking magnetic long-range order and central charge $\mathsf{c}=1$. Direct transitions between competing gapped phases can be continuous or weakly first order. We demonstrate that continuous phase transition lines have exotic critical properties. For example, a line of deconfined quantum critical points (between the Neel${}_x^{}$ and Neel${}_y^{}$ phases) described by Tomonaga-Luttinger liquid theory is terminated by the gFM${}_z^{}$ phase at a Lifshitz tricritical point which exhibits spontaneous breaking of the continuous U(1) symmetry despite the locality and one-dimensionality of the Hamiltonian, contrary to a naive expectation from the Hohenberg-Mermin-Wagner-Coleman theorem. Furthermore, critical lines between symmetry-protected topological phases with different ${\mathbb{Z}}_2^{}$-symmetry breaking (e.g., Neel${}_x^{\mathrm{SPT}}$ and Neel${}_y^{\mathrm{SPT}}$) additionally retain low-energy boundary degrees of freedom, realizing symmetry-enriched criticality (or a gapless SPT phase). Our results reveal unconventional mechanisms by which competing symmetry breaking and topology generate ordered critical phases and multicriticality.

97 pages, 37 figures