scholarly journals Rigorous solution of a mean field spin glass model

2000 ◽  
Vol 13 (2) ◽  
pp. 147-160
Author(s):  
T. C. Dorlas ◽  
J. R. Wedagedera
Keyword(s):  
Spin Glass ◽  
Large Deviation ◽  
Mean Field ◽  
Almost Sure Convergence ◽  
Thermodynamic Quantities ◽  
Convergence Criteria ◽  
Spin Glass Model ◽  
Rigorous Solution ◽  
Glass Model ◽  
Null Set

A separable spin glass model whose exchange integral takes the form Jij=J(ξi1ξj2+ξi2ξj1) which was solved by van Hemmen et al. [12] using large deviation theory [14] is rigorously treated. The almost sure convergence criteria associated with the cumulant generating function C(t) with respect to the quenched random variables ξ is carefully investigated, and it is proved that the related excluded null set 𝒩 is independent of t. The free energy and hence the other thermodynamic quantities are rederived using Varadhan's Large Deviation Theorem. A simulation is also presented for the entropy when ξ assumes a Gaussian distribution.

scholarly journals An investigation of the hidden structure of states in a mean-field spin-glass model

1997 ◽  
Vol 30 (20) ◽  
pp. 7021-7038 ◽  
Author(s):  
Andrea Cavagna ◽  
Irene Giardina ◽  
Giorgio Parisi
Keyword(s):  
Spin Glass ◽  
Mean Field ◽  
Spin Glass Model ◽  
Glass Model

scholarly journals The Free Energy of a Quantum Sherrington–Kirkpatrick Spin-Glass Model for Weak Disorder

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Hajo Leschke ◽  
Sebastian Rothlauf ◽  
Rainer Ruder ◽  
Wolfgang Spitzer
Keyword(s):  
Free Energy ◽  
Spin Glass ◽  
Large Deviation ◽  
Integral Formula ◽  
Main Tool ◽  
Spin Operator ◽  
Spin Glass Model ◽  
Rigorous Results ◽  
Static Approximation ◽  
Glass Model

AbstractWe extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington–Kirkpatrick spin-glass model without external magnetic field to the quantum case with a “transverse field” of strength $$\mathsf {b}$$ b . More precisely, if the Gaussian disorder is weak in the sense that its standard deviation $$\mathsf {v}>0$$ v > 0 is smaller than the temperature $$1/\beta $$ 1 / β , then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any $$\mathsf {b}/\mathsf {v}\ge 0$$ b / v ≥ 0 . The macroscopic annealed free energy turns out to be non-trivial and given, for any $$\beta \mathsf {v}>0$$ β v > 0 , by the global minimum of a certain functional of square-integrable functions on the unit square according to a Varadhan large-deviation principle. For $$\beta \mathsf {v}<1$$ β v < 1 we determine this minimum up to the order $$(\beta \mathsf {v})^{4}$$ ( β v ) 4 with the Taylor coefficients explicitly given as functions of $$\beta \mathsf {b}$$ β b and with a remainder not exceeding $$(\beta \mathsf {v})^{6}/16$$ ( β v ) 6 / 16 . As a by-product we prove that the so-called static approximation to the minimization problem yields the wrong $$\beta \mathsf {b}$$ β b -dependence even to lowest order. Our main tool for dealing with the non-commutativity of the spin-operator components is a probabilistic representation of the Boltzmann–Gibbs operator by a Feynman–Kac (path-integral) formula based on an independent collection of Poisson processes in the positive half-line with common rate $$\beta \mathsf {b}$$ β b . Its essence dates back to Kac in 1956, but the formula was published only in 1989 by Gaveau and Schulman.

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