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arXiv:2210.11646 · Physical review. B./Physical review. B

Topological pump of SU(Q) quantum chain and Diophantine equation

Yasuhiro Hatsugai, Yoshihito Kuno

Published Jun 2, 2023• 5 citations
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Abstract

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 $SU(Q)$ quantum chain associated with a local ${[U(1)]}^{\ensuremath{\bigotimes}Q}$ gauge invariance of colored fermions is discussed in detail. The $SU(Q)$ invariant dimer phases are characterized by the ${Z}_{Q}$ Berry phases as a topological order parameter with a $d$-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 $Q$ 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 ($Q=3,4,$ 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 $SU(Q)$ quantum chain is also mentioned.

DOI
10.1103/physrevb.107.235106

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A domain-wall pump by an extended cluster model of S=1/2 spins is proposed with local U(1) gauge invariance. In contrast to the conventional pump, the current and the gauge field are defined on sites instead of links. Its snapshot ground state is gapped and doubly degenerated due to Z_{2} invariance, which is broken by an infinitesimal boundary magnetic field. The ground state associated with the spontaneous symmetry breaking is, at the same time, symmetry-protected with additional spatial inversion that is characterized by the Z_{2} Berry phase defined by the twist on the boundary site. We investigate the topological domain-wall pump with and without boundaries. The topological pump associated with the inversion symmetry-breaking path induces a nontrivial Chern number of bulk and a singular behavior of edge states of the domain wall. Generalization to the multispin interaction is also explicitly given. These extended quantum degrees of freedom are prominent qubits that are topologically protected.
Sequential plateau transitions of quantum spin chains (S=1S=1S=1, 3/2, 2, and 3) are demonstrated by a spin pump using dimerization and a staggered magnetic field as synthetic dimensions. The bulk is characterized by the Chern number associated with the boundary twist and the pump protocol as the time. It counts the number of critical points in the loop that is specified by the Z2{Z}_{2}Z2​ Berry phases. With an open boundary condition, the discontinuity of the spin weighted center of mass due to emergent effective edge spins also characterizes the pump as the bulk-edge correspondence. It requires extra level crossings in the pump as a superselection rule that is consistent with the valence bond solid picture.
An extended Bose-Hubbard model (EBHM) with three- and four-body constraints can be feasible in cold atoms in an optical lattice. A rich phase structure including various symmetry-protected topological (SPT) phases is obtained numerically with suitable parameter settings and particle filling. The SPT phase is characterized by the Berry phase as a local topological order parameter and the structure of the entanglement spectrum (ES). Based on the presence of various topological phases, separated by gapless phase boundaries, the EBHM exhibits various bosonic topological pumps, which are constructed by connecting the different SPT phases without gap closing. The bulk topological pumps exhibit the plateau transitions characterized by many-body Chern numbers. For the system with boundary, the center of mass (CoM) under grand canonical ensemble elucidates the contributions of multiple edge states and reveals the topology of the system. We demonstrate that the interacting bosonic pumps obey the bulk-edge correspondence.
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The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems, quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here, we show that quantized Thouless pumps connecting C_{4}-symmetric HOSPTs can be described by a tuple of four Chern numbers that measure quantized bulk charge transport in a direction-dependent fashion. Moreover, this tuple of Chern numbers allows to predict the sign and value of fractional corner charges in the HOSPTs. We show that the topologically nontrivial phase can be characterized by both quadrupole and dipole configurations, shedding new light on current debates about the multipole nature of the HOSPT bulk. By employing corner-periodic boundary conditions, we generalize Restas's theory to HOSPTs. Our approach provides a simple framework for understanding topological invariants of general HOSPTs and paves the way for an in-depth description of future dynamical experiments.
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.