TY - JOUR

T1 - Universal entanglement spectra of gapped one-dimensional field theories

AU - Cho, Gil Young

AU - Ludwig, Andreas W.W.

AU - Ryu, Shinsei

N1 - Funding Information:
We are grateful to the KITP Program Entanglement in Strongly-Correlated Quantum Matter (Apr. 6 - Jul. 2, 2015), where some part of the work was performed. This work is supported by the NSF under Grants No. NSF PHY11-25915, No. DMR-1064319 (G.Y.C.), and No. DMR-1309667 (A.W.W.L.), the Brain Korea 21 PLUS Project of Korea Government (G.Y.C.), Grant No. 2016R1A5A1008184 under NRF of Korea (G.Y.C.), and by the Alfred P. Sloan foundation (S.R.).
Publisher Copyright:
© 2017 American Physical Society.

PY - 2017/3/13

Y1 - 2017/3/13

N2 - We discuss the entanglement spectrum of the ground state of a (1+1)-dimensional system in a gapped 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 ξ 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 Lξ=ln(ξ/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. While the entanglement spectrum of the ground state of a gapped theory on a finite interval of length 2R with suitable boundary conditions, bipartitioned into two equal pieces, turns out to exhibit a crossover between the finite-size spectra of the same CFT with in general different boundary conditions as the system size R crosses the correlation length from the "critical regime'' Rξ to the "gapped regime'' Rξ, the physical spectrum on a finite interval of length R with the same boundary conditions, on the other hand, is known to undergo a dramatic reorganization during the same crossover from being discrete to being continuous.

AB - We discuss the entanglement spectrum of the ground state of a (1+1)-dimensional system in a gapped 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 ξ 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 Lξ=ln(ξ/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. While the entanglement spectrum of the ground state of a gapped theory on a finite interval of length 2R with suitable boundary conditions, bipartitioned into two equal pieces, turns out to exhibit a crossover between the finite-size spectra of the same CFT with in general different boundary conditions as the system size R crosses the correlation length from the "critical regime'' Rξ to the "gapped regime'' Rξ, the physical spectrum on a finite interval of length R with the same boundary conditions, on the other hand, is known to undergo a dramatic reorganization during the same crossover from being discrete to being continuous.

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U2 - 10.1103/PhysRevB.95.115122

DO - 10.1103/PhysRevB.95.115122

M3 - Article

AN - SCOPUS:85015991810

SN - 2469-9950

VL - 95

JO - Physical Review B

JF - Physical Review B

IS - 11

M1 - 115122

ER -