Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality

Abstract

In a system with chiral topological order, there is a remarkable correspondence between the edge and entanglement spectra: the low-energy spectrum of the system in the presence of a physical edge coincides with the lowest part of the entanglement spectrum (ES) across a virtual cut of the system, up to rescaling and shifting. In this paper, we explore whether the edge-ES correspondence extends to nonchiral topological phases. Specifically, we consider the Wen-plaquette model which has Z_2 topological order. The unperturbed model displays an exact correspondence: both the edge and entanglement spectra within each topological sector a (a = 1,...,4) are flat and equally degenerate. Here, we show, through a detailed microscopic calculation, that in the presence of generic local perturbations: (i) the effective degrees of freedom for both the physical edge and the entanglement cut consist of a spin-1/2 chain, with effective Hamiltonians H_edge^a and H_ent.^a, respectively, both of which have a Z_2 symmetry enforced by the bulk topological order; (ii) there is in general no match between their low energy spectra, that is, there is no edge-ES correspondence. However, if supplement the Z_2 topological order with a global symmetry (translational invariance along the edge/cut), i.e. by considering the Wen-plaquette model as a symmetry enriched topological phase (SET), then there is a finite domain in Hamiltonian space in which both H_edge^a and H_ent.^a realize the critical Ising model, whose low-energy effective theory is the c = 1/2 Ising CFT. This is achieved because the presence of the global symmetry implies that both Hamiltonians, in addition to being Z_2 symmetric, are Kramers-Wannier self-dual. Thus, the bulk topological order and the global translational symmetry of the Wen-plaquette model as a SET imply an edge-ES correspondence at least in some finite domain in Hamiltonian space.

arXiv comment23 pages, 5 figures; References added; Paper updated to match published version

References in corpus (46)

• Fault-tolerant quantum computation by anyons
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Topological orders and Edge excitations in FQH states
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Fractional Chern Insulator
• Schrieffer-Wolff transformation for quantum many-body systems
• General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
• Quantum orders in an exact soluble model
• Entanglement spectrum of topological insulators and superconductors
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• Generalised twisted partition functions
• Kramers-Wannier duality from conformal defects
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Entanglement entropy and entanglement spectrum of the Kitaev model
• Characterizing topological order by studying the ground states of an infinite cylinder
• Low-energy effective theory of the toric code model in a parallel field
• Topological multicritical point in the Toric Code and 3D gauge Higgs Models
• The entanglement gap and a new principle of adiabatic continuity
• Bulk-Edge Correspondence in the Entanglement Spectra
• Non-local order in gapless systems: Entanglement Spectrum in Spin Chains
• Entanglement spectra of quantum Heisenberg ladders
• Entanglement Spectrum, Critical Exponents and Order Parameters in Quantum Spin Chains
• Disentangling Entanglement Spectra of Fractional Quantum Hall States on Torus Geometries
• Topological Entanglement and Clustering of Jain Hierarchy States
• Real-Space Entanglement Spectrum of Quantum Hall States
• How universal is the entanglement spectrum?
• Real-space entanglement spectrum of quantum Hall systems
• Linked Cluster Series Expansions for Two-Particle Bound States
• Evaluation of ranks of real space and particle entanglement spectra for large systems
• Entanglement spectra of coupled S=1/2 spin chains in a ladder geometry
• Entanglement, combinatorics and finite-size effects in spin-chains
• Dynamics of the entanglement spectrum in spin chains
• Entanglement spectra between coupled Tomonaga-Luttinger liquids: Applications to ladder systems and topological phases
• Edge theories in Projected Entangled Pair State models
• Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State
• Entanglement Spectra of the 2D AKLT Model: VBS/CFT Correspondence
• Understanding the entanglement entropy and spectra of 2D quantum systems through arrays of coupled 1D chains
• Integrable Lattice Realizations of Conformal Twisted Boundary Conditions
• Quantum Entanglement and Thermal Reduced Density Matrices in Fermion and Spin Systems on Ladders
• Critical entanglement spectrum of one-dimensional symmetry protected topological phases
• Entanglement Spectra of the quantum hard-square model: Holographic minimal models
• RCFT with defects: Factorization and fundamental world sheets
• Majorana Edge States for Z2 Topological Orders of the Wen-plaquette Model and the Toric-code Model

Cited by in corpus (31)

• Quantum entanglement in condensed matter systems
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Entanglement Entropy From Tensor Network States for Stabilizer Codes
• Symmetry fractionalization in the topological phase of the spin-$\frac{1}{2}$ $J_1$-$J_2$ triangular Heisenberg model
• Bulk Anyons as Edge Symmetries: Boundary Phase Diagrams of Topologically Ordered States
• Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions
• Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries
• Topological Disorder Parameter
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter
• Entanglement convertibility by sweeping through the quantum phases of the alternating bonds $XXZ$ chain
• Capacity of entanglement and distribution of density matrix eigenvalues in gapless systems
• Entanglement Properties of Gauge Theories from Higher-Form Symmetries
• Momentum-Space Entanglement in Heisenberg Spin-Half Ladders
• Demonstrating the wormhole mechanism of the entanglement spectrum via a perturbed boundary
• Bulk-Edge Correspondence in Fractional Quantum Hall States
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Entanglement Hamiltonian and effective temperature of non-Hermitian quantum spin ladders
• Edge--Entanglement correspondence for gapped topological phases with symmetry
• Correspondence between bulk entanglement and boundary excitation spectra in 2d gapped topological phases
• Topologically ordered time crystals
• Duality-preserving deformation of 3+1d lattice ${\mathbb{Z}}_2$ gauge theory with exact gapped ground states
• Universal edge information from wave-function deformation
• Edge Theories for Anyon Condensation Phase Transitions
• The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology
• Anomalies of global symmetries on the lattice
• Spin chains, defects, and quantum wires for the quantum-double edge
• Localization and entanglement characterization of edge states in HgTe quantum wells in a finite strip geometry
• Temporal Entanglement Transitions in the Periodically Driven Ising Chain