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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 asks how a thermalized boundary of a fermionic MBL-SPT phase expresses the anomaly imposed by bulk order. It treats the boundary with an effective interacting SYK theory and analyzes symmetry-constrained spectra; the resulting Wigner-Dyson statistics correspond to the bulk anomaly.
Problem
The paper asks what signature the quantum anomaly of a nontrivial MBL-SPT bulk has at a thermalized boundary.
Source: pp. 1–2Method
The paper studies interacting fermionic MBL-SPT boundaries using an effective boundary theory and analyzes symmetry-constrained Hamiltonians and spectra.
Source: p. 1, PROJECTIVE REPRESENTATION ANALYSIS FOR… · p. 11Results
For a thermalized boundary, the level statistics follow a Wigner-Dyson ensemble corresponding to the global anomaly required by the bulk SPT order.
Source: p. 2Takeaways & Limitations
Boundary level statistics provide a spectral signature of the SPT anomaly, alongside protected degeneracy of every energy level.
Source: p. 2The paper calls the boundary thermalized only when its effective Hamiltonian is in the ETH phase.
Source: p. 1
Abstract
from arXiv · showhide
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.
PROJECTIVE REPRESENTATION ANALYSIS FOR SYMMETRY CLASS BDI
The BDI boundary’s projective symmetry representations provide a route to its level statistics that agrees with the Clifford-algebra analysis. Together, symmetry invariants and the level statistics resolve the Z8 anomaly pattern.
PROJECTIVE REPRESENTATION ANALYSIS FOR SYMMETRY CLASS BDI
- Fermion parity P is built from paired boundary Majoranas; odd Nχ requires an extra Majorana χ∞ to complete the physical Hilbert space.The construction also makes P an even-fermion-parity operator.Source: PROJECTIVE REPRESENTATION ANALYSIS FOR… · p. 10
- For odd Nχ, factoring χ∞ out of fermion parity defines Z, which commutes with the four-fermion interaction Hamiltonian and is an additional symmetry.Z² = 1 and Z anticommutes with P.Source: PROJECTIVE REPRESENTATION ANALYSIS FOR… · p. 10
- The Fock-basis construction of time reversal compensates for the alternating signs of Majorana operators under complex conjugation; for odd Nχ, the authors choose Tχ∞T⁻¹ = χ∞.This choice gives a unified expression for T across Nχ.Source: PROJECTIVE REPRESENTATION ANALYSIS FOR… · p. 10
- Combining (PT)² with level statistics distinguishes topological indices ν and −ν and resolves the Z8 anomaly pattern.The projective-representation invariants include T², (PT)², and, for odd Nχ, (ZT)².Source: PROJECTIVE REPRESENTATION ANALYSIS FOR… · pp. 10–11
- The analysis represents the symmetries and Hamiltonian in fermion-parity blocks, with odd Nχ requiring H+ = H− because Z is also a symmetry.The symmetry relations determine the matrix type of each Hamiltonian block.Source: PROJECTIVE REPRESENTATION ANALYSIS FOR… · p. 11
- The projective-representation analysis reproduces the Clifford-algebra conclusion that symmetry-constrained boundary Hamiltonians yield GOE, GUE, or GSE level statistics.The allowed Hamiltonian matrix types are real, complex, or quaternion Hermitian; the tabulated statistics also vary by Nχ and fermion-parity sector.Source: PROJECTIVE REPRESENTATION ANALYSIS FOR… · p. 11
SQUARE OF ANTI-UNITARY SYMMETRIES IN MANY-BODY FOCK SPACE
The paper distinguishes an anti-unitary symmetry’s square on single-particle states from its square on the many-body Fock space. It shows that the latter can carry an additional phase, with allowed values determined by the single-particle sign.
SQUARE OF ANTI-UNITARY SYMMETRIES IN MANY-BODY FOCK SPACE
- The many-body square can include an extra phase γmb that cannot be gauged away, while remaining consistent with the single-particle symmetry action.For γsp = −1, the conventional factor is fermion-number parity.Source: SQUARE OF ANTI-UNITARY SYMMETRIES IN… · p. 12
- γmb is ±1 when γsp = +1, but can be ±1 or ±i when γsp = −1.The paper demonstrates this dependence for anti-unitary symmetries acting on the many-body Fock space.Source: SQUARE OF ANTI-UNITARY SYMMETRIES IN… · p. 12
- In the examples considered, BDI time reversal and AIII chiral symmetry have γsp = +1 and γmb = ±1, while CII time reversal has γsp = −1 and γmb = (−1)^Nf.For CII, T² = γmb(−1)^F.Source: SQUARE OF ANTI-UNITARY SYMMETRIES IN… · p. 12