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The cavity method is a well established technique for solving classical spinmodels on sparse random graphs (mean-field models with finite connectivity).Laumann et al. [arXiv:0706.4391] proposed recently an extension of this methodto quantum spin-1/2 models in a transverse field, using a discretizedSuzuki-Trotter imaginary time formalism. Here we show how to take analyticallythe continuous imaginary time limit. Our main technical contribution is anexplicit procedure to generate the spin trajectories in a path integralrepresentation of the imaginary time dynamics. As a side result we also showhow this procedure can be used in simple heat-bath like Monte Carlo simulationsof generic quantum spin models. The replica symmetric continuous time quantumcavity method is formulated for a wide class of models, and applied as a simpleexample on the Bethe lattice ferromagnet in a transverse field. The results ofthe methods are confronted with various approximation schemes in thisparticular case. On this system we performed quantum Monte Carlo simulationsthat confirm the exactness of the cavity method in the thermodynamic limit.

On the path integral representation for quantum spin models and its application to the quantum cavity method and to Monte Carlo simulations