Universal Entanglement Spectra of Gapped One-dimensional Field Theories

TL;DR

This paper demonstrates that the low-lying entanglement spectrum of gapped (1+1)D Lorentz invariant field theories near a quantum phase transition is universally equivalent to the finite-size spectrum of a boundary conformal field theory, independent of integrability.

Contribution

It establishes a universal correspondence between the entanglement spectrum of gapped Lorentz invariant theories and boundary CFT spectra, extending known results beyond integrable models.

Findings

Low-lying entanglement spectrum equals the finite-size boundary CFT spectrum.

The spectrum is discrete and universal, determined by boundary conditions.

Different behaviors observed in entanglement and physical spectra across critical and gapped regimes.

Abstract

We discuss the entanglement spectrum of the ground state of a gapped (1+1)-dimensional system in a phase near a quantum phase transition. In particular, in proximity to a quantum phase transition described by a conformal field theory (CFT), the system is represented by a gapped Lorentz invariant field theory in the "scaling limit" (correlation length $\xi$ much larger than microscopic 'lattice' scale $a$), and can be thought of as a CFT perturbed by a relevant perturbation. We show that for such (1+1) gapped Lorentz invariant field theories in infinite space, the low-lying entanglement spectrum obtained by tracing out, say, left half-infinite space, is precisely equal to the physical spectrum of the unperturbed gapless, i.e. conformal field theory defined on a finite interval of length $L_\xi=$ $\log(\xi/a)$ with certain boundary conditions. In particular, the low-lying entanglement spectrum of the gapped theory is the finite-size spectrum of a boundary conformal field theory, and is always discrete and universal. Each relevant perturbation, and thus each gapped phase in proximity to the quantum phase transition, maps into a particular boundary condition. A similar property has been known to hold for Baxter's Corner Transfer Matrices in a very special class of fine-tuned, namely integrable off-critical lattice models, for the entire entanglement spectrum and independent of the scaling limit. In contrast, our result applies to completely general gapped Lorentz invariant theories in the scaling limit, without the requirement of integrability, for the low-lying entanglement spectrum. - The finite-size spectra of the entanglement Hamiltonian and of the physical Hamiltonian on an interval of length R exhibit entirely different behaviors upon crossover from the critical regime $R \ll \xi$ to the gapped regime $R \gg \xi$.