Toeplitz行列式的Fisher-Hartwig展开和一维自由费米子单粒子约化密度矩阵的谱
Fisher-Hartwig expansion for Toeplitz determinants and the spectrum of a single-particle reduced density matrix for one-dimensional free fermions
摘要：
本文研究了一维格点上自由费米子的单粒子约化密度矩阵所对应的具有正弦核的Toeplitz矩阵的谱。对于这个矩阵的谱行列式，最近已经证明了逆矩阵大小的Fisher-Hartwig展开。这种扩展可以通过使用离散Painleve方程理论中已知的递归关系，远离零点积累线，逐阶验证。我们执行这样的验证到第十阶，并计算相应的系数在Fisher-Hartwig展开。在假定Fisher-Hartwig展开式在整个谱参数范围内都是有效的情况下，我们进一步导出了一个方程关于该矩阵的本征值的展开式以及相应费米子问题中的von Neumann纠缠熵的展开式.这些分析结果得到了数值例子的支持。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
