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Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture
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Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture
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We consider one-dimensional quantum spin chain, which is called XX model, XX0 model or isotropic XY model in a transverse magnetic field. We study the model on the infinite lattice at zero temperature. We are interested in the entropy of a subsystem [a block of L neighboring spins]. It describes entanglement of the block with the rest of the ground state. G. Vidal, J.I. Latorre, E. Rico, and A. Kitaev showed that for large blocks the entropy scales logarithmically. We prove the logarithmic formula for the leading term and calculate the next term. We discovered that the dependence on the magnetic field interacting with spins is very simple: the magnetic field effectively reduce the size of the subsystem. We also calculate entropy of a subsystem of a small size. We also evaluated Renyi and Tsallis entropies of the subsystem. We represented the entropy in terms of a Toeplitz determinant and calculated the asymptotic analytically.
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