Fisher-Hartwig expansion for Toeplitz determinants and the spectrum of a single-particle reduced density matrix for one-dimensional free fermions (Q2854135)

The paper addressed the problem of the calculation of large-size Toeplitz determinants, i.e., determinants of matrices in which each element \(a_{ij}\) depends only on difference \(i-j\). These matrices play a fundamentally important role in many-body physics, as the particular one, with the \textit{sine kernel}, NEWLINE\[NEWLINE a_{ij} = \pi^{-1}(i - j)^{-1}\sin(k_F(i - j)), NEWLINE\]NEWLINE for \(i\neq j\), and \(a_{ii}=k_{F}/\pi\) at \(i=j\), describes the density matrix of a one-dimensional gas of noninteracting fermions (the \textit{symbol} of the latter matrix, i.e., its Fourier discrete transform, \(\sigma(k)=1\), at \(|k| < k_F\), and \(\sigma(k)=0\), at \(|k| > k_F\). The leading exponential asymptotic approximation for the Toeplitz determinats in the limit of its large size is well known, the problem being the calculation of corrections to it (the so-called Fisher-Hartwig expansion). The present work reports the calculation of this expansion up to the tenth order. A key role in the analysis belongs to recurrence relations borrowed from the theory of discrete Painlevé equations. The analytical results are confirmed by a numerical example.