We determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix‐valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.

中文翻译：

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具有分段连续符号的块Toeplitz行列式的渐近


我们确定了在温和附加条件下具有有限多次跳跃的矩阵值分段连续函数的块托普利茨行列式的渐近性。特别是，我们证明其中 、 和 是依赖于矩阵符号的常数，并在我们的主要结果中进行了描述。我们的方法基于托普利茨行列式的新定位定理、使用分段连续矩阵值符号计算托普利茨算子 Fredholm 指数的新方法以及其他算子理论方法。作为我们结果的应用，我们考虑在量子自旋链模型中的纠缠熵研究中出现的分段连续符号。