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Fisher-Hartwig expansion for Toeplitz determinants and the spectrum of a single-particle reduced density matrix for one-dimensional free fermions | | Dmitri A. Ivanov
; Alexander G. Abanov
; | | Date: |
21 Jun 2013 | | Abstract: | We study the spectrum of the Toeplitz matrix with a sine kernel, which
corresponds to the single-particle reduced density matrix for free fermions on
the one-dimensional lattice. For the spectral determinant of this matrix, a
Fisher--Hartwig expansion in the inverse matrix size has been recently
conjectured. This expansion can be verified order by order, away from the line
of accumulation of zeros, using the recurrence relation known from the theory
of discrete Painleve equations. We perform such a verification to the tenth
order and calculate the corresponding coefficients in the Fisher-Hartwig
expansion. Under the assumption of the validity of the Fisher-Hartwig expansion
in the whole range of the spectral parameter, we further derive expansions for
an equation on the eigenvalues of this matrix and for the von Neumann
entanglement entropy in the corresponding fermion problem. These analytical
results are supported by a numerical example. | | Source: | arXiv, 1306.5017 | | Services: | Forum | Review | PDF | Favorites | |
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