arXiv:quant-ph/0504179 · Physical Review Letters

Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory

Jonathan P. Keating, Francesco Mezzadri

Published Feb 7, 2005• 123 citations

Abstract

We compute the entropy of entanglement between the first $N$ spins and the rest of the system in the ground states of a general class of quantum spin chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like $\ensuremath{\kappa}{log}_{2}N+\stackrel{\texttildelow{}}{\ensuremath{\kappa}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, where $\ensuremath{\kappa}$ and $\stackrel{\texttildelow{}}{\ensuremath{\kappa}}$ are determined explicitly. In an important class of systems, $\ensuremath{\kappa}$ is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for $\ensuremath{\kappa}$ therefore provides an explicit formula for the central charge.

DOI: 10.1103/physrevlett.94.050501

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arXiv:quant-ph/0504179 · Physical Review Letters

Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory

Jonathan P. Keating, Francesco Mezzadri

Published Feb 7, 2005• 123 citations

arXiv HTML PDF

Abstract

We compute the entropy of entanglement between the first

N

spins and the rest of the system in the ground states of a general class of quantum spin chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like

\ensuremath{\kappa}{log﻿}_{2}N+\stackrel{\texttildelow{}}{\ensuremath{\kappa}}

N\ensuremath{\rightarrow}\ensuremath{\infty}

, where

\ensuremath{\kappa}

and

\stackrel{\texttildelow{}}{\ensuremath{\kappa}}

are determined explicitly. In an important class of systems,

\ensuremath{\kappa}

is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for

\ensuremath{\kappa}

therefore provides an explicit formula for the central charge.

DOI: 10.1103/physrevlett.94.050501

Related papers

Entanglement Entropy of Random Quantum Critical Points in One Dimension

Gil Refael, Joel E. Moore

For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization-group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization-group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.

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Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to "nonunitary quantum mechanics," which has seen growing interest from areas as diverse as open quantum systems, noninteracting electronic disordered systems, or nonunitary conformal field theory (CFT). We propose and investigate such an extension here, by focusing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of Rényi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well-understood cases. The example of the sl(2|1) alternating spin chain for percolation is discussed in detail.Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to "nonunitary quantum mechanics," which has seen growing interest from areas as diverse as open quantum systems, noninteracting electronic disordered systems, or nonunitary conformal field theory (CFT). We propose and investigate such an extension here, by focusing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of Rényi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well-understood cases. The example of the sl(2|1) alternating spin chain for percolation is discussed in detail.

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Entanglement entropy scaling of the XXZ chain

Pochung Chen, Zhi-Long Xue, Ian P. McCulloch +3 more

We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge $c$ in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of $c$ . In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge $c$ in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of $c$. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.

arXiv:1306.5828Journal of Statistical Mechanics Theory and ExperimentOct 11, 201312 citations

We compute the entropy of entanglement between the first

N

\ensuremath{\kappa}{log﻿}_{2}N+\stackrel{\texttildelow{}}{\ensuremath{\kappa}}

N\ensuremath{\rightarrow}\ensuremath{\infty}

, where

\ensuremath{\kappa}

and

\stackrel{\texttildelow{}}{\ensuremath{\kappa}}

are determined explicitly. In an important class of systems,

\ensuremath{\kappa}

is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for

\ensuremath{\kappa}

therefore provides an explicit formula for the central charge.

Random spin chains at quantum critical points exhibit an entanglement entropy between a segment of length

L

and the rest of the chain that scales as

{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}L

with a universal coefficient. Since for pure quantum critical spin chains this coefficient is fixed by the central charge of the associated conformal field theory, the universal coefficient in the random case can be understood as an effective central charge. In this paper we calculate the entanglement entropy and effective central charge of the spin-1 random Heisenberg model in its random-singlet phase and also at the critical point at which the Haldane phase breaks down. The latter is the first entanglement calculation for an infinite-randomness fixed point that is not in the random-singlet universality class. Our results are consistent with a

c

-theorem for flow between infinite-randomness fixed points. The formalism we use can be generally applied to calculation of quantities that depend on the RG history in $s\ensuremath{\geqslant}1$ random Heisenberg chains.

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

λ_α

of the reduced density matrix

ρ_A

for region

A

. In this paper we draw attention instead to the

Schmidt~ vectors

, or eigenvectors

|v_α\rangle

ρ_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_α\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_αH_n |v_{α'}\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.

c

in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of

c

. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.