Topological Disorder Parameter

Abstract

We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in (2+1)d. TDP is defined as the constant correction that appears in the ground state expectation value of a partial symmetry transformation applied to a connected spatial region $M$, the absolute value of which scales generically as $\exp (-\alpha l+\gamma )$ where $l$ is the perimeter of $M$ and $\gamma$ is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that $e^{\gamma }$ can be related to the quantum dimension of the symmetry defect, and provide a general formula for $\gamma$ when the entanglement Hamiltonian of the topological phase can be described by a (1+1)d conformal field theory (CFT). A special case of TDP is equivalent to the topological Rényi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish a new tool to detect quantum topological order.

21 pages, 6 figures; v2 published version

References in corpus (23)

• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Towards a derivation of holographic entanglement entropy
• Identifying Topological Order by Entanglement Entropy
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum
• Genons, twist defects, and projective non-Abelian braiding statistics
• Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
• Universal terms for the entanglement entropy in 2+1 dimensions
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models
• Scaling of entanglement entropy at deconfined quantum criticality
• Measuring Rényi entanglement entropy with high efficiency and precision in quantum Monte Carlo simulations
• Scaling of disorder operator at deconfined quantum criticality
• Universal Features of Higher-Form Symmetries at Phase Transitions
• Higher-form symmetry breaking at Ising transitions
• Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality
• Scaling of disorder operator at $(2+1)d$ U(1) quantum criticality
• Flux crystals, Majorana metals, and flat bands in exactly solvable spin-orbital liquids
• Cornering the universal shape of fluctuations
• Categorical Symmetries at Criticality
• A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

Cited by in corpus (19)

• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Fermion disorder operator at Gross-Neveu and deconfined quantum criticalities
• Unlocking the general relationship between energy and entanglement spectra via the wormhole effect
• Stable computation of entanglement entropy for 2D interacting fermion systems
• Disorder Operator and Rényi Entanglement Entropy of Symmetric Mass Generation
• Extracting subleading corrections in entanglement entropy at quantum phase transitions
• An integral algorithm of exponential observables for interacting fermions in quantum Monte Carlo simulation
• Detecting Subsystem Symmetry Protected Topological Order Through Strange Correlators
• Different temperature-dependence for the edge and bulk of entanglement Hamiltonian
• Disentangling critical quantum spin chains with Clifford circuits
• Distinguishing Quantum Phases through Cusps in Full Counting Statistics
• Probing the Topology of Fermionic Gaussian Mixed States with {U(1)} symmetry by Full Counting Statistics
• Corner Charge Fluctuations and Many-Body Quantum Geometry
• Kramers-Wannier self-duality and non-invertible translation symmetry in quantum chains: a wave-function perspective
• Universal contributions to charge fluctuations in spin chains at finite temperature
• Crosscap states and duality of Ising field theory in two dimensions
• Disorder operators in two-dimensional Fermi and non-Fermi liquids through multidimensional bosonization
• Finite Correlation Length Scaling of Disorder Parameter at Quantum Criticality
• Symmetrizing the Constraints -- Density Matrix Renormalization Group for Constrained Lattice Models