Emergent universality in critical quantum spin chains: entanglement Virasoro algebra

Abstract

Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. Given a pure state of the system and a division into regions $A$ and $B$, they can be obtained in terms of the $Schmidtvalues$, or eigenvalues ${\lambda }_{\alpha }$ of the reduced density matrix ${\rho }_A$ for region $A$. In this paper we draw attention instead to the $Schmidtvectors$, or eigenvectors $|v_{\alpha }⟩$ of ${\rho }_A$. We consider the ground state of critical quantum spin chains whose low energy/long distance physics is described by an emergent conformal field theory (CFT). We show that the Schmidt vectors $|v_{\alpha }⟩$ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT (a chiral version of the original CFT). Indeed, we build weighted sums $H_n$ of the lattice Hamiltonian density $h_{j,j+1}$ over region $A$ and show that the matrix elements $⟨v_{\alpha }{H}_n|v_{{\alpha }^{\prime }}⟩$ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for $H_n^{\text{CFT}}=\frac{1}{2}(L_n+L_{-n})$ in the boundary CFT, where $L_n$'s are (one copy of) the Virasoro generators. We numerically confirm our results using the critical Ising quantum spin chain and other (free-fermion equivalent) models.

arXiv comment6+7 pages, 12 figures

References in corpus (28)

• Entanglement Entropy and Quantum Field Theory
• Concentrating Partial Entanglement by Local Operations
• Entanglement in quantum critical phenomena
• Geometric and Renormalized Entropy in Conformal Field Theory
• 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
• General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
• Entanglement spectrum in one-dimensional systems
• Entanglement spectrum of topological insulators and superconductors
• Entanglement Spectrum of a Disordered Topological Chern Insulator
• Entanglement Spectrum, Critical Exponents and Order Parameters in Quantum Spin Chains
• Spin-resolved entanglement spectroscopy of critical spin chains and Luttinger liquids
• Physics at the entangling surface
• How universal is the entanglement spectrum?
• Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems
• Scaling of the entanglement spectrum near quantum phase transitions
• Holographic entanglement contour, bit threads, and the entanglement tsunami
• Entanglement Hamiltonians of lattice models via the Bisognano-Wichmann theorem
• Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories
• Area law and its violation: A microscopic inspection into the structure of entanglement and fluctuations
• Fine structure in holographic entanglement and entanglement contour
• Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
• Operator content of real-space entanglement spectra at conformal critical points
• Universal Entanglement Spectra of Gapped One-dimensional Field Theories
• Universal aspects in the behavior of the entanglement spectrum in one dimension: scaling transition at the factorization point and ordered entangled structures
• Operator content of entanglement spectra after global quenches in the transverse field Ising chain
• Spectrum of the Wilson-Fisher conformal field theory on the torus
• Universal entanglement spectra in critical spin chains

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