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We study the spectrum of the Toeplitz matrix with a sine kernel, whichcorresponds to the single-particle reduced density matrix for free fermions onthe one-dimensional lattice. For the spectral determinant of this matrix, aFisher--Hartwig expansion in the inverse matrix size has been recentlyconjectured. This expansion can be verified order by order, away from the lineof accumulation of zeros, using the recurrence relation known from the theoryof discrete Painleve equations. We perform such a verification to the tenthorder and calculate the corresponding coefficients in the Fisher-Hartwigexpansion. Under the assumption of the validity of the Fisher-Hartwig expansionin the whole range of the spectral parameter, we further derive expansions foran equation on the eigenvalues of this matrix and for the von Neumannentanglement entropy in the corresponding fermion problem. These analyticalresults are supported by a numerical example.

Fisher-Hartwig expansion for Toeplitz determinants and the spectrum of a single-particle reduced density matrix for one-dimensional free fermions