Abstract
Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space.
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Caputo, P., Martinelli, F., Simenhaus, F., Toninelli, F.L.: “Zero” temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion. Commun. Pure Appl. Math. 64, 778–831 (2011)
Cipriani, A., Dai Pra, P.: Decay of correlations for quantum spin systems with a transverse field: a dynamic approach. arXiv:1005.3547 (2010)
Ioffe, D.: Stochastic geometry of classical and quantum Ising models. In: Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Math., vol. 1970, pp. 87–127. Springer, Berlin (2009)
Kreĭn, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Transl. Am. Math. Soc. 1950(26), 128 (1950)
Krzakala, F., Rosso, A., Semerjian, G., Zamponi, F.: Path-integral representation for quantum spin models: Application to the quantum cavity method and Monte Carlo simulations. Phys. Rev. B 78(13), 134428 (2008)
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. Am. Math. Soc., Providence (2009). With a chapter by J.G. Propp and D.B. Wilson
Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F.L.: Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality. J. Eur. Math. Soc. (to appear). arXiv:1012.1271 (2011)
Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics, Saint-Flour, 1997. Lecture Notes in Math., vol. 1717, pp. 93–191. Springer, Berlin (1999)
Martinelli, F., Sinclair, A.: Mixing time for the solid-on-solid model. Ann. Appl. Probab. (to appear). arXiv:1008.0125 (2010). A preliminary version appeared in Proc. 41st Annual ACM Symposium on Theory of Computing, pp. 571–580 (2009)
Martinelli, F., Toninelli, F.L.: On the mixing time of the 2D stochastic Ising model with “plus” boundary conditions at low temperature. Commun. Math. Phys. 296, 175–213 (2010)
Martinelli, F., Sinclair, A., Weitz, D.: Glauber dynamics on trees: boundary conditions and mixing time. Commun. Math. Phys. 250(2), 301–334 (2004)
Peres, Y., Winkler, P.: Can extra updates delay mixing? arXiv:1112.0603v1 (2011)
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This work was supported by the European Research Council throughout the “Advanced Grant” PTRELSS 228032.
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Martinelli, F., Wouts, M. Glauber Dynamics for the Quantum Ising Model in a Transverse Field on a Regular Tree. J Stat Phys 146, 1059–1088 (2012). https://doi.org/10.1007/s10955-012-0436-7
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DOI: https://doi.org/10.1007/s10955-012-0436-7