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Stochastic Geometry of Classical and Quantum Ising Models

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Book cover Methods of Contemporary Mathematical Statistical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1970))

Abstract

These lecture notes are based on a mini-course which I taught at Prague school in September 2006. The idea was to try to develop and explain to probabilistically minded students a unified approach to the Fortuin-Kasteleyn (FK) and to the random current (RC) representation of classical and quantum Ising models via path integrals. No background in quantum statistical mechanics was assumed.

In Section 1 familiar classical Ising models are rewritten in the quantum language. In this way usual FK and RC representations emerge as different instances of Lie-Trotter product formula. Then I am following [4] and set up a general notation for the Poisson limits.

In Section 2 both FK and the RC representations are generalized to quantum Ising models in transverse field. The FK representation was originally derived in [8] and [3]. The observation regarding the RC representation seems to be new. Both representations are used to derive formulas for one and two point functions and for the matrix and reduced density matrix elements.

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Authors and Affiliations

  1. Faculty of Industrial Engineering and Management, Technion, Israel

    Dmitry Ioffe

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  1. Dmitry Ioffe
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Correspondence to Dmitry Ioffe .

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Editors and Affiliations

  1. Ctr. for Theoretical Study, Charles University, Jilska 1, Praha 1, 110 00, Czech Republic

    Roman Kotecký

  2. Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK

    Roman Kotecký

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Ioffe, D. (2009). Stochastic Geometry of Classical and Quantum Ising Models. In: Kotecký, R. (eds) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics(), vol 1970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92796-9_2

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