具有自守边界条件的泛函积分与XY和XX海森堡链自旋Z分量的相关
Functional Integration with "Automorphic" Boundary Conditions and Correlators of Z-Components of Spins in the XY and XX Heisenberg Chains
摘要：
给出了XY和$XX$海森堡自旋链中$z$ -自旋分量的静态关联函数的生成泛函的费米子泛函积分和的表示.所讨论的泛函积分的特殊性是因为积分变量“自同构”地依赖于虚时间。换句话说，当虚时间移动一个周期时，积分变量乘以某个复数。因此，在虚时间段两端的相应边界条件不是对应于非零温度下松原表示中的费米子或玻色子变量的形式。实际上，模型的一部分站点对应于受到异常边界条件的积分变量，而其他站点上的变量通常依赖于虚时间，即，费米子（或玻色子）。因此，一种情况下，当一个“自同构”的边界条件是相同的所有网站的链自旋模型，是广义的。泛函积分的结果以矩阵算子的行列式的形式给出，这些矩阵算子是通过广义zeta函数方法正则化的。明确地得到了非零温度下模型的配分函数和某些关联函数，从而证明了所提出的泛函积分表示的正确性。
英文摘要：
Representations for the generating functionals of static correlators of $z$-components of spins in the XY and $XX$ Heisenberg spin chains are obtained in the form of sums of the fermionic functional integrals. The peculiarity of the functional integrals in question is because of the fact that the integration variables depend on the imaginary time ``automorphically''. In other words, the integration variables are multiplied with a certain complex number when the imaginary time is shifted by a period. Therefore, the corresponding boundary conditions at the ends of the imaginary time segment are not of the form corresponding to fermionic, or bosonic, variables taken in the Matsubara representation at nonzero temperature. In fact, one part of sites of the models corresponds to the integration variables which are subjected to the unusual boundary conditions, while the variables on the other sites depend on the imaginary time conventionally, i.e., as fermions (or bosons). Thus a situation, when an ``automorphic'' boundary condition is the same for all sites of a chain spin model, is generalized. The results of the functional integration are obtained in the form of determinants of the matrix operators which are regularized by means of the generalized zeta-function approach. The partition functions of the models and certain correlation functions at nonzero temperature are obtained explicitly thus demonstrating correctness of the functional integral representations proposed.
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