Phase transitions in low dimensional systems via spontaneous symmetry breaking and topology

Kazuo Ueda, 上田 和夫

Salvatore D. Pace

We explore the rich landscape of higher-form and non-invertible symmetries that emerge at low energies in generic ordered phases. Using that their charge is carried by homotopy defects (i.e., domain walls, vortices, hedgehogs, etc.), in the absence of domain walls we find that their symmetry defects in D D -dimensional spacetime are described by (D-1) ( D − 1 ) -representations of a (D-1) ( D − 1 ) -group that depends only on the spontaneous symmetry-breaking (SSB) pattern of the ordered phase. These emergent symmetries are not spontaneously broken in the ordered phase. We show that spontaneously breaking them induces a phase transition into a nontrivial disordered phase that can have symmetry-enriched (non-)Abelian topological orders, photons, and even more emergent symmetries. This SSB transition is between two distinct SSB phases–an ordinary and a generalized one–making it a possible generalized deconfined quantum critical point. We also investigate the ’t Hooft anomalies of these emergent symmetries and conjecture that there is always a mixed anomaly between them and the microscopic symmetry spontaneously broken in the ordered phase. One way this anomaly can manifest is through the fractionalization of the microscopic symmetry’s quantum numbers. Our results demonstrate that even the most exotic generalized symmetries emerge in ordinary phases and provide a valuable framework for characterizing them and their transitions.

A. P. Young

Anthony Rey, Ömer M. Aksoy, Claudio Chamon +2 more

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.

Yasuhiro Hatsugai

Linhao Li, Rui-Zhen Huang, Weiguang Cao

Noninvertible symmetry generalizes traditional group symmetries, advancing our understanding of quantum matter, especially one-dimensional gapped quantum systems. In critical lattice models, it is usually realized as emergent symmetries in the corresponding low-energy conformal field theories. In this work, we study critical lattice models with the noninvertible Rep($D_8$) symmetry in one dimension. This leads us to a new class of quantum critical points (QCP), noninvertible symmetry-enriched QCPs, as a generalization of known group symmetry-enriched QCPs. They are realized as phase transitions between one noninvertible symmetry-protected topological (SPT) phase and another different one or spontaneous symmetry breaking (SSB) phase. We identify their low-energy properties and topological features through the Kennedy-Tasaki (KT) duality transformation. We argue that distinct noninvertible symmetry-enriched QCPs can not be smoothly connected without a phase transition or a multi-critical point.