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arXiv:2411.17219

Topological domain-wall pump with $\mathbb{Z}_2$ spontaneous symmetry breaking

Yoshihito Kuno, Yasuhiro Hatsugai

Published Nov 26, 2024• 0 citations
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Abstract

A domain-wall pump by an extended cluster model of $S=1/2$ spins is proposed with local $U(1)$ gauge invariance. Its snapshot ground state is gapped and doubly degenerated due to $\mathbb{Z}_2$ invariance, which is broken by an infinitesimal boundary magnetic field. The ground state associated with the spontaneous symmetry breaking (SSB) is still symmetry-protected with additional spatial inversion that is characterized by the $\mathbb{Z}_2$ Berry phase. We investigate the topological domain-wall pump with/without boundaries. The topological pump associated with the inversion symmetry-breaking path induces a non-trivial Chern number of bulk and a singular behavior of edge states of the domain-wall. Generalization to the multi-spin interaction is also explicitly given.

DOI
10.48550/arxiv.2411.17219

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Topological domain-wall pump with Z2\mathbb{Z}_2Z2​ spontaneous symmetry breaking
A domain-wall pump by an extended cluster model of S=1/2S=1/2S=1/2 spins is proposed with local U(1)U(1)U(1) gauge invariance. Its snapshot ground state is gapped and doubly degenerated due to Z2\mathbb{Z}_2Z2​ invariance, which is broken by an infinitesimal boundary magnetic field. The ground state associated with the spontaneous symmetry breaking (SSB) is still symmetry-protected with additional spatial inversion that is characterized by the Z2\mathbb{Z}_2Z2​ Berry phase. We investigate the topological domain-wall pump with/without boundaries. The topological pump associated with the inversion symmetry-breaking path induces a non-trivial Chern number of bulk and a singular behavior of edge states of the domain-wall. Generalization to the multi-spin interaction is also explicitly given.
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Topological pump and its plateau transitions of NNN-leg spin ladder
A topological pump on an N-N\textrm{-}N-leg spin ladder is discussed by introducing spatial clusterization whose adiabatic limit is a set of 2N-2N\textrm{-}2N-site staircase clusters. We set a pump path in the parameter space that connects two different symmetry protected topological phases. By introducing a symmetry breaking staggered magnetic field, the system is always gapped during the pump. In the topological pump {thus obtained}, the bulk Chern number is given by the number of the critical points enclosed by the pump path. Plateau transitions characterized by the Chern number are demonstrated associated with deformation of the pump path. We find that there are NNN critical points enclosed by the pump path for the N-N\textrm{-}N-leg ladder. The ground state phase diagram without symmetry breaking terms is numerically investigated by using the quantized Berry phase. We also discuss the physical picture of edge states in the diagonal boundary, and numerically demonstrate the bulk-edge correspondence for N=2,3N=2,3N=2,3 cases.
Emergent Symmetry and Phase Transitions on the Domain Wall of Z2\mathbb{Z}_{2}Z2​ Topological Orders
The one-dimensional (1D) domain wall of 2D Z2\mathbb{Z}_{2}Z2​ topological orders is studied theoretically. The Ising domain wall model is shown to have an emergent SU(2)1_{1}1​ conformal symmetry because of a hidden nonsymmorphic octahedral symmetry. While a weak magnetic field is an irrelevant perturbation to the bulk topological orders, it induces a domain wall transition from the Tomonaga-Luttinger liquid to a ferromagnetic order, which spontaneously breaks the anomalous Z2\mathbb{Z}_{2}Z2​ symmetry and the time-reversal symmetry on the domain wall. Moreover, the gapless domain wall state also realizes a 1D topological quantum critical point between a Z2T\mathbb{Z}_{2}^{T}Z2T​-symmetry-protected topological phase and a trivial phase, thus demonstrating the holographic construction of topological transitions.
We propose a general construction scheme of topological pump for correlated systems with local gauge symmetry. The bulk-edge correspondence is one of its natural consequences. As a concrete example, an SU(Q)SU(Q)SU(Q) quantum chain associated with a local ${[U(1)]}^{\ensuremath{\bigotimes}Q}$ gauge invariance of colored fermions is discussed in detail. The SU(Q)SU(Q)SU(Q) invariant dimer phases are characterized by the ZQ{Z}_{Q}ZQ​ Berry phases as a topological order parameter with a ddd-dimensional twist space ($d=Q\ensuremath{-}1$) as a synthetic Brillouin zone. By inclusion of the symmetry breaking perturbation specified by a rational parameter $\mathrm{\ensuremath{\Phi}}=P/Q$, the pump that encloses around the phase boundary, is characterized by the QQQ Chern numbers associated with the currents due to uniform infinitesimal twists. The analysis of the systems under the open/periodic/twisted boundary conditions clarifies the bulk-edge correspondence of the pump where the large gauge transformation generated by the center-of-mass (CoM) plays a central role. An explicit formula for the Chern number is given by using the Diophantine equation. Numerical demonstrations by the exact diagonalization and the DMRG for finite systems (Q=3,4,Q=3,4,Q=3,4, and 5) have been presented to confirm the general discussions for low-energy spectra, edge states, CoM's, Chern numbers, and the bulk-edge correspondence. A modified Lieb-Schultz-Mattis type argument for the general SU(Q)SU(Q)SU(Q) quantum chain is also mentioned.
We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry-breaking transitions or transitions between symmetry-protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single- and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.
We explore the states of matter arising from the spontaneous symmetry breaking (SSB) of Z2\mathbb{Z}_2Z2​ non-onsite symmetries. In one spatial dimension, we construct a frustration-free lattice model exhibiting SSB of a non-onsite symmetry, which features the coexistence of two ground states with distinct symmetry-protected topological (SPT) orders. We analytically prove the two-fold ground-state degeneracy and the existence of a finite energy gap. Fixing the symmetry sector yields a long-range entangled ground state that features long-range correlations among non-invertible charged operators. We also present a constant-depth measurement-feedback protocol to prepare such a state with a constant success probability in the thermodynamic limit, which may be of independent interest. Under a symmetric deformation, the SSB persists up to a critical point, beyond which a gapless phase characterized by a conformal field theory emerges. In two spatial dimensions, the SSB of 1-form non-onsite symmetries leads to a long-range entangled state (SPT soup) - a condensate of 1d SPT along any closed loops. On a torus, there are four such locally indistinguishable states that exhibit algebraic correlations between local operators, which we derived via a mapping to the critical O(2)O(2)O(2) loop model. This constitutes an intriguing example of `topological quantum criticality'.