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arXiv:cond-mat/0406737 · Physical Review Letters

Entanglement Entropy of Random Quantum Critical Points in One Dimension

Gil Refael, Joel E. Moore

Published Dec 21, 2004• 361 citations
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Abstract

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.

DOI
10.1103/physrevlett.93.260602

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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.
Random spin chains at quantum critical points exhibit an entanglement entropy between a segment of length LLL and the rest of the chain that scales as log2L{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}Llog2​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 ccc-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.
It is known that the entropy of a block of spins of size L embedded in an infinite pure critical spin chain diverges as the logarithm of L with a prefactor fixed by the central charge of the correspondent conformal field theory. For a class of strongly random spin chains, it has been shown that the correspondent block entropy still remains universal and diverges logarithmically with an ‘effective’ central charge. By computing the entanglement entropy for a family of models which includes the N -states random Potts chain and the Z N clock model, we give some definitive answer to recent conjectures about the behaviour of this effective central charge. In particular, we show that the ratio between the entanglement entropy in the pure and in the disordered system is model dependent and we provide a series of critical models where the entanglement entropy grows from the pure to the random case.
We study the scaling properties of entanglement entropy (EE) near the quantum critical points in interacting random antiferromagnetic (AFM) spin chains. Using density-matrix renormalization group, we compute the half-chain EE near the topological phase transition between the Haldane and random singlet phases in a disordered spin-1 chain. It is found to diverge logarithmically in system size with an effective central charge ceff=1.17(4){c}_{\mathrm{eff}}=1.17(4)ceff​=1.17(4) at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $\ensuremath{\nu}=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on the strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on the Imry-Ma argument. We provide a possible explanation.
We discuss the scaling of entanglement entropy in the random singlet phase (RSP) of disordered quantum magnetic chains of general spin SSS. Through an analysis of the general structure of the RSP, we show that the entanglement entropy scales logarithmically with the size of a block, and we provide a closed expression for this scaling. This result is applicable for arbitrary quantum spin chains in the RSP, being dependent only on the magnitude SSS of the spin. Remarkably, the logarithmic scaling holds for the disordered chain even if the pure chain with no disorder does not exhibit conformal invariance, as is the case for Heisenberg integer-spin chains. Our conclusions are supported by explicit evaluations of the entanglement entropy for random spin-1 and spin-3∕23∕23∕2 chains using an asymptotically exact real-space renormalization group approach.
We present numerical evidences for the logarithmic scaling of the entanglement entropy in critical random spin chains. Very large scale exact diagonalizations performed at the critical XXXXXX point up to L=2000L=2000L=2000 spins 12\frac{1}{2}21​ lead to a perfect agreement with recent real-space renormalization-group predictions of [Refael and Moore Phys. Rev. Lett. 93, 260602 (2004)] for the logarithmic scaling of the entanglement entropy in the random singlet phase with an effective central charge $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}=c\ifmmode\times\else\texttimes\fi{}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2$. Moreover, we provide the first visual proof of the existence of the random singlet phase with the help of quantum entanglement concept.
We study the entanglement entropy of blocks of contiguous spins in non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg, XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and relevant aperiodic modulations, the entanglement entropy is found to be a logarithmic function of the block size with log-periodic oscillations. The effective central charge, c eff , defined through the constant in front of the logarithm may depend on the ratio of couplings and can even exceed the corresponding value in the homogeneous system. In the strong modulation limit, the ground state is constructed by a renormalization group method and the limiting value of c eff is exactly calculated. Keeping the ratio of the block size and the system size constant, the entanglement entropy exhibits a scaling property, however, the corresponding scaling function may be nonanalytic.