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Random Matrix Theory and Entanglement in Quantum Spin Chains

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Abstract

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N→∞. This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    J.P. Keating & F. Mezzadri

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  1. J.P. Keating
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  2. F. Mezzadri
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Correspondence to J.P. Keating.

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Communicated by P. Sarnak

Acknowledgement We gratefully acknowledge stimulating discussions with Estelle Basor, Peter Forrester and Noah Linden. We are also grateful for the kind hospitality of the Isaac Newton Institute for the Mathematical Sciences, Cambridge, while this research was completed. Francesco Mezzadri was supported by a Royal Society Dorothy Hodgkin Research Fellowship.

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Keating, J., Mezzadri, F. Random Matrix Theory and Entanglement in Quantum Spin Chains. Commun. Math. Phys. 252, 543–579 (2004). https://doi.org/10.1007/s00220-004-1188-2

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