Emergent universality in critical quantum spin chains: entanglement Virasoro algebra

TL;DR

This paper reveals that the eigenvectors of the reduced density matrix in critical quantum spin chains exhibit a universal structure related to the Virasoro algebra of boundary conformal field theory, confirmed through numerical analysis.

Contribution

It demonstrates the emergent Virasoro algebra structure in the Schmidt vectors of critical quantum spin chains, linking lattice models to boundary CFT.

Findings

Schmidt vectors show universal structure matching boundary CFT Virasoro algebra.

Matrix elements of weighted sums of Hamiltonian density are universal up to finite-size effects.

Numerical confirmation using critical Ising and free-fermion models supports the theoretical predictions.

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 $Schmidt~ values$, or eigenvalues $\lambda_{\alpha}$ of the reduced density matrix $\rho_A$ for region $A$. In this paper we draw attention instead to the $Schmidt~ vectors$, or eigenvectors $|v_{\alpha}\rangle$ 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 $|v_{\alpha}\rangle$ 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 $\langle v_{\alpha}H_n |v_{\alpha'}\rangle$ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for $H_n^{\tiny \text{CFT}} = \frac 1 2 (L_n + L_{-n})$ 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.