Functional integration and correlators of $z$-components of local spins in the XY and XX Heisenberg magnets

The generating function of static correlators of $z$-components of local spins in the XY and XX Heisenberg magnets is calculated as a combination of functional integrals over anticommuting variables. The peculiarity of the Gaussian integrals in question consists in the fact that the integration variables are subjected to an “automorphic” boundary condition in the imaginary time, i.e., they are multiplied by a certain complex number when the imaginary time is shifted by a period. Zeta-regularization is used for the calculation of the integrals. The answers are obtained for several correlation functions at nonzero temperature, and some reductions are checked. The generating function of the XX model is obtained in the frequency-coordinate representation in the form of a rational expression build up out of the Chebyshev polynomials. Some estimations are provided for the correlators of the XX model in strong, i.e., exceeding a critical value, magnetic field. Asymptotics of the modified Bessel function $I_\nu(z)$ both at moderate and large values of the index are used in the estimations.