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. We show how anisotropic spin interactions lead to anisotropies in the free energy, giving rise to pinning of the spontaneous magnetization along preferred directions
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We thank Ioannis Rousochatzakis for useful discussions. P.W. thanks the Department of Physics at the University of Wisconsin-Madison for hospitality during several stays as a visiting professor. P.W. also acknowledges partial support by an ICAM senior fellowship. Part of this work was performed at the Aspen Center for Physics, which is supported by NSF Grant No. PHY-1066293. N.P. and Y.S. acknowledge the support from NSF DMR-1511768 Grant. N.P. acknowledges the hospitality of KITP and partial support by the National Science Foundation under Grant No. NSF PHY11-25915.
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