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

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

2008-07-16

arxiv.org/abs/0807.2553v2

Phys. Rev. B 78, 134428 (2008)

Physics

Condensed Matter

Statistical Mechanics

25 pages, 15 figures, typos corrected

Scientific paper

10.1103/PhysRevB.78.134428

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

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