We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
These notes give an introduction to the physics of the infinite range version of the Edwards--Anderson model, the so-called Sherrington--Kirkpatrick model. In a first part, I motivate and introduce the Edwards--Anderson and Sherrington--Kirkpatrick models. In the second part, I explain the analytical solution of the Sherrington--Kirkpatrick model, following Giorgio Parisi. I next give the physical interpretation of this solution. This is a vast subject, and I concentrate on the major points and give references for more details. The third part presents the simulation approach and compare its results to theoretical expectations: thermodynamics, finite size scaling, determination of the critical temperature using "parameters" like the "Binder parameter", and fluctuations. The last part gives a summary of our current understanding of finite size effects for the free energy and internal energy of the Sherrington--Kirkpatrick model
Critical point scaling in a field
H applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function
f(R) defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically
f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension
d=3, we have determined the exponents and the critical scaling function
f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for
f(R) in
d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of
R for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function
f(R) at some negative value of
R, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.
The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.
The nature of static chaos in Ising spin glasses is studied. For the problem of chaos with magnetic field, scaling relations in the case of the SK model and short-range models are presented. Our results also suggest that if there is de Almeida-Thouless line then it is similar to that of mean-field theory for d=4 and close to the h=0 axis for d=3. We estimate the lower critical dimension to be in the range 2.7-2.9. Numerical studies at d=4 show that there is chaos against temperature changes dT and the correlation length diverges like 1/dT.