Free energy of quantum spin systems: Functional integral representation

In this work, we propose a method for calculating the free energy of anisotropic quantum spin systems.We use the Hubbard-Stratonovich transformation to express the partition function of a generic bilinear superexchange Hamiltonian in terms of a functional integral over classical time-dependent fields. In the general case the result is presented as a product of traces over single spins subject to a time-dependent field. The traces may be evaluated in closed form in the case of systems with large Ising anisotropy. In the general case we derive a compact expression for the contribution of Gaussian spin fluctuations to the free energy.