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Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States

Yi-Zhuang You, Andreas W. W. Ludwig, Cenke Xu

TL;DR

The paper links the thermalized boundaries of 1D fermionic MBL-SPT states to quantum-chaotic SYK dynamics, showing that boundary anomalies imprint topology-dependent Wigner-Dyson statistics that cycle with the bulk's interaction-reduced classification. By analyzing boundary Majorana modes in BDI and complex boundary modes in AIII and CII, it demonstrates eight-fold WD statistics periodicity in BDI and corresponding patterns in the other classes, grounded in Clifford-algebra representations and projective symmetry actions. A central result is the general many-body squaring rule for anti-unitary symmetries, Theta^2 = gamma_mb (gamma_sp)^F, which restricts gamma_mb to a finite set and accounts for the observed spectral statistics across classes. These findings provide a diagnostic link between bulk topological order in disordered systems and the chaotic spectra of their boundaries, with implications for SYK-like boundary theories of interacting SPTs.

Abstract

We consider the Sachdev-Ye-Kitaev (SYK) model as an effective theory arising at the zero-dimensional boundary of a many-body localized, Fermionic symmetry protected topological (SPT) phase in one spatial dimension. The Fermions at the boundary are always fully interacting. We find that the boundary is thermalized and investigate how its boundary anomaly, dictated by the bulk SPT order, is encoded in the quantum chaotic eigenspectrum of the SYK model. We show that depending on the SPT symmetry class, the boundary many-body level statistics cycle in a systematic manner through those of the three different Wigner-Dyson random matrix ensembles with a periodicity in the topological index that matches the interaction-reduced classification of the bulk SPT states. We consider all three symmetry classes BDI, AIII, and CII, whose SPT phases are classified in one spatial dimension by $\mathbb{Z}$ in the absence of interactions. For symmetry class BDI, we derive the eight-fold periodicity of the Wigner-Dyson statistics by using Clifford algebras.

Sachdev-Ye-Kitaev Model and Thermalization on the Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States

TL;DR

The paper links the thermalized boundaries of 1D fermionic MBL-SPT states to quantum-chaotic SYK dynamics, showing that boundary anomalies imprint topology-dependent Wigner-Dyson statistics that cycle with the bulk's interaction-reduced classification. By analyzing boundary Majorana modes in BDI and complex boundary modes in AIII and CII, it demonstrates eight-fold WD statistics periodicity in BDI and corresponding patterns in the other classes, grounded in Clifford-algebra representations and projective symmetry actions. A central result is the general many-body squaring rule for anti-unitary symmetries, Theta^2 = gamma_mb (gamma_sp)^F, which restricts gamma_mb to a finite set and accounts for the observed spectral statistics across classes. These findings provide a diagnostic link between bulk topological order in disordered systems and the chaotic spectra of their boundaries, with implications for SYK-like boundary theories of interacting SPTs.

Abstract

We consider the Sachdev-Ye-Kitaev (SYK) model as an effective theory arising at the zero-dimensional boundary of a many-body localized, Fermionic symmetry protected topological (SPT) phase in one spatial dimension. The Fermions at the boundary are always fully interacting. We find that the boundary is thermalized and investigate how its boundary anomaly, dictated by the bulk SPT order, is encoded in the quantum chaotic eigenspectrum of the SYK model. We show that depending on the SPT symmetry class, the boundary many-body level statistics cycle in a systematic manner through those of the three different Wigner-Dyson random matrix ensembles with a periodicity in the topological index that matches the interaction-reduced classification of the bulk SPT states. We consider all three symmetry classes BDI, AIII, and CII, whose SPT phases are classified in one spatial dimension by in the absence of interactions. For symmetry class BDI, we derive the eight-fold periodicity of the Wigner-Dyson statistics by using Clifford algebras.

Paper Structure

This paper contains 2 sections, 40 equations, 1 figure, 4 tables.

Table of Contents

  1. Projective Representation Analysis for Symmetry Class BDI
  2. Square of Anti-Unitary Symmetries in Many-Body Fock Space

Figures (1)

  • Figure 1: Many-body level statistics (in term of the $\ln r$ distribution) of the random interaction model in Eq. \ref{['eq: Kitaev']}, for $N_\chi=8,\cdots,15$ (a full $\mathbb{Z}_8$ period) by exact diagonalization. The background gray curves describe the "Wigner-surmise" given by Eq. \ref{['eq: r-stat MBL']} and Eq. \ref{['eq: r-stat ETH']}: from wide to narrow, they correspond to Poisson, GOE, GUE, and GSE statistics respectively. The level statistics in the even (odd) Fermion parity sector is shown in green (yellow).