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arXiv:cond-mat/0504446 · Physical Review B

Scaling of entanglement entropy in the random singlet phase

Nicolas Laflorencie

Published Oct 31, 2005• 150 citations
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Abstract

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 $XX$ point up to $L=2000$ spins $\frac{1}{2}$ 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.

DOI
10.1103/physrevb.72.140408

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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 c\ifmmode~\else˜\fi=c\ifmmode×\else\texttimes\filn2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}=c\ifmmode\times\else\texttimes\fi{}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2c\ifmmode~\else˜\fi=c\ifmmode×\else\texttimes\filn2. Moreover, we provide the first visual proof of the existence of the random singlet phase with the help of quantum entanglement concept.
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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.
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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 average entanglement entropy of blocks of contiguous spins in aperiodic XXZ chains which possess an aperiodic singlet phase at least in a certain limit of the coupling ratios. In this phase, where the ground state constructed by a real space renormalization group method consists (asymptotically) of independent singlet pairs, the average entanglement entropy is found to be a piecewise linear function of the block size. The enveloping curve of this function is growing logarithmically with the block size, with an effective central charge in front of the logarithm which is characteristic for the underlying aperiodic sequence. The aperiodic sequence producing the largest effective central charge is identified, and the latter is found to exceed the central charge of the corresponding homogeneous model. For marginal aperiodic modulations, numerical investigations performed for the XX model show a logarithmic dependence, as well, with an effective central charge varying continuously with the coupling ratio.