dorsal/arxiv
View SchemaRandom Matrix Theory and Entanglement in Quantum Spin Chains
| Authors | J. P. Keating, F. Mezzadri |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0407047 |
| URL | https://arxiv.org/abs/quant-ph/0407047 |
| DOI | 10.1007/s00220-004-1188-2 |
| Journal | Commun. Math. Phys., Vol. 252 (2004), 543-579 |
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 --> infinity . 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 Painlev\'e type. In some cases these solutions can be evaluated to all orders using recurrence relations.
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"abstract": "We compute the entropy of entanglement in the ground states of a general\nclass of quantum spin-chain Hamiltonians - those that are related to quadratic\nforms of Fermi operators - between the first N spins and the rest of the system\nin the limit of infinite total chain length. We show that the entropy can be\nexpressed in terms of averages over the classical compact groups and establish\nan explicit correspondence between the symmetries of a given Hamiltonian and\nthose characterizing the Haar measure of the associated group. These averages\nare either Toeplitz determinants or determinants of combinations of Toeplitz\nand Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture\nare used to compute the leading order asymptotics of the entropy as N --\u003e\ninfinity . This is shown to grow logarithmically with N. The constant of\nproportionality is determined explicitly, as is the next (constant) term in the\nasymptotic expansion. The logarithmic growth of the entropy was previously\npredicted on the basis of numerical computations and conformal-field-theoretic\ncalculations. In these calculations the constant of proportionality was\ndetermined in terms of the central charge of the Virasoro algebra. Our results\ntherefore lead to an explicit formula for this charge. We also show that the\nentropy is related to solutions of ordinary differential equations of\nPainlev\\\u0027e type. In some cases these solutions can be evaluated to all orders\nusing recurrence relations.",
"arxiv_id": "quant-ph/0407047",
"authors": [
"J. P. Keating",
"F. Mezzadri"
],
"categories": [
"quant-ph",
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"math.MP"
],
"doi": "10.1007/s00220-004-1188-2",
"journal_ref": "Commun. Math. Phys., Vol. 252 (2004), 543-579",
"title": "Random Matrix Theory and Entanglement in Quantum Spin Chains",
"url": "https://arxiv.org/abs/quant-ph/0407047"
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