The condition of quasi-periodicity in imaginary time as a constraint at the functional integration and the time-dependent ZZ-correlator of the XX Heisenberg magnet

A functional integration approach for calculation of the longitudinal correlation functions of the XY Heisenberg magnet in the constant homogeneous magnetic field is considered in the present paper. The generating functionals of the correlators are defined in the form of the functional integrals over anti-commuting variables. The peculiarity of the functional integrals consists in the fact that the integration variables depend on the imaginary time quasi-periodically. The corresponding boundary conditions are accounted for as constraints which reduce the integration domain. For the XX Heisenberg magnet an application of the approach is given in the case of two-point correlator of third components of spins with an explicit dependence on time.