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arXiv:2610.10694

Competing symmetry breaking and topology in quantum spin chains

Anthony Rey, Ömer M. Aksoy, Claudio Chamon, Akira Furusaki, Christopher Mudry

Published Oct 7, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2610.10694

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We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, Z3×Z2T\mathbb Z_3\times\mathbb Z_2^TZ3​×Z2T​, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy and a nontrivially degenerate entanglement spectrum based on the density-matrix renormalization group (DMRG) method.Next, we clarify that chirality degrees of freedom form the Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane state based on a degenerate perturbation theory. Last but not least, we discuss the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local Z3\mathbb{Z}_3Z3​ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the Z3\mathbb{Z}_3Z3​ on-site symmetry, the Z2T\mathbb{Z}_2^TZ2T​ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.