Stochastic Geometry of Classical and Quantum Ising Models
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  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  and . 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.
KeywordsIsing Model Random Graph Critical Curve Giant Component Stochastic Geometry
Unable to display preview. Download preview PDF.
- 3.Aizenman, M., Klein, A., Newman, C.M.: Percolation methods for disordered quantum Ising models, Mathematics, Physics, Biology, … R. Kotecký, ed., 1–24, World Scientific, Singapore (1993).Google Scholar
- 5.Biskup, M., Chayes, L., Crawford, N, Ioffe, D, Levit, A.: In preparation (2007).Google Scholar
- 6.Bollobás, B.: Random Graphs, London: Academic Press (1985).Google Scholar
- 9.Dorlas, T.V.: Probabilistic derivation of a noncommutative version of Varadhan's theorem. Preprint DIAS-STP-02-5 (2002).Google Scholar
- 12.Grimmett, G.: Space-time percolation, Preprint (2007).Google Scholar
- 13.Ioffe, D., Levit, A.: Long range order and giant components of quantum random graphs. submitted (2006).Google Scholar
- 16.Nachtergaele, B.: Quasi-state decompositions for quantum spin systems. Probability theory and mathematical statistics (Vilnius, 1993), 565–590, TEV, Vilnius (1994).Google Scholar
- 18.Ueltschi, D.: Geometric and probabilistic aspects of boson lattice models. In and out of equilibrium: Physics with a probability flavor, Progr. Probab. 51, 363–391, Birkhäuser (2002).Google Scholar