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 $\mathbb{Z}^{\,}_{2}\times\mathbb{Z}^{\,}_{2}\times\mathbb{Z}^{T}_{2}$ 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$^{\mathrm{SPT}}_{x,y}$ and Neel$^{\mathrm{SPT}}_{x,y}$), and two gapless phases (gFM$^{\,}_{z}$ and gNeel$^{\,}_{z}$) with incommensurate $\mathbb{Z}^{T}_{2}$-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$^{\mathrm{SPT}}_{x}$ and Neel$^{\mathrm{SPT}}_{y}$) 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.