scholarly journals Some rigorous results on the Sherrington-Kirkpatrick spin glass model

1988 ◽  
Vol 116 (3) ◽  
pp. 527-527 ◽  
Author(s):  
M. Aizenman ◽  
J. L. Lebowitz ◽  
D. Ruelle
Keyword(s):  
Spin Glass ◽  
Spin Glass Model ◽  
Rigorous Results ◽  
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.

scholarly journals Some rigorous results on the Sherrington-Kirkpatrick spin glass model

1987 ◽  
Vol 112 (1) ◽  
pp. 3-20 ◽  
Author(s):  
M. Aizenman ◽  
J. L. Lebowitz ◽  
D. Ruelle
Keyword(s):  
Spin Glass ◽  
Spin Glass Model ◽  
Rigorous Results ◽  
Glass Model

scholarly journals ABOUT THE OVERLAP DISTRIBUTION IN MEAN FIELD SPIN GLASS MODELS

1996 ◽  
Vol 10 (13n14) ◽  
pp. 1675-1684 ◽  
Author(s):  
FRANCESCO GUERRA
Keyword(s):  
Spin Glass ◽  
Order Parameter ◽  
Short Range ◽  
Mean Field ◽  
Spin Glass Model ◽  
Replica Symmetry ◽  
Replica Symmetry Breaking ◽  
Full Agreement ◽  
Rigorous Results ◽  
Glass Model

We continue our presentation of mathematically rigorous results about the Sherrington-Kirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica symmetry breaking. As a byproduct, we show that the self-averaging of the Edwards-Anderson fluctuating order parameter, with respect to the external quenched noise, implies that the overlap distribution is given by the Sherrington-Kirkpatrick replica symmetric Ansatz. This extends previous results of Pastur and Scherbina. Finally, we show how to generalize our results to realistic short range spin glass models.

scholarly journals Dynamical solutions of a quantum Heisenberg spin glass model

2004 ◽  
Vol 41 (4) ◽  
pp. 525-533 ◽  
Author(s):  
M. Bechmann ◽  
R. Oppermann
Keyword(s):  
Spin Glass ◽  
Spin Glass Model ◽  
Heisenberg Spin ◽  
Glass Model

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
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