The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.
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.
Abstract. We study order-parameter fluctuations (OPF) in disordered systems by considering the behaviour of some recently introduced parameters G, Gc which have proven very useful in locating phase transitions. We prove that both parameters G (for disconnected overlap disorder averages) and Gc (for connected disorder averages) take the respective universal values 1 13 3 and 31 in the T → 0 limit for any finite volume provided the ground state is unique and there is no gap in the ground-state local-field distributions, conditions which are met in generic spin-glass models with continuous couplings and no gap at zero coupling. This makes G, Gc ideal parameters to locate phase transitions in disordered systems much like the Binder cumulant for ordered systems. We check our results by exactly computing OPF in a simple example of uncoupled spins in the presence of random fields and the one-dimensional Ising spin glass. At finite temperatures, we discuss under which conditions the value 1 3 for G may be recovered by conjecturing different scenarios depending on whether OPF are finite or vanish in the infinite-volume limit. In particular, we discuss replica equivalence and its natural consequence limV → ∞ G(V, T) = 1 3 when OPF are finite. As an example of a model where OPF vanish and replica equivalence does not give information about G we study the Sherrington–Kirkpatrick spherical spin-glass model by performing numerical simulations for small sizes. Again we find results compatible with G = 1 3 in the spin-glass phase. 1.
We study order-parameter fluctuations (OPF) in disordered systems by considering the behaviour of some recently introduced parameters G , G c which have proven very useful in locating phase transitions. We prove that both parameters G (for disconnected overlap disorder averages) and G c (for connected disorder averages) take the respective universal values (1/3) and (13/31) in the T →0 limit for any finite volume provided the ground state is unique and there is no gap in the ground-state local-field distributions, conditions which are met in generic spin-glass models with continuous couplings and no gap at zero coupling. This makes G , G c ideal parameters to locate phase transitions in disordered systems much like the Binder cumulant for ordered systems. We check our results by exactly computing OPF in a simple example of uncoupled spins in the presence of random fields and the one-dimensional Ising spin glass. At finite temperatures, we discuss under which conditions the value (1/3) for G may be recovered by conjecturing different scenarios depending on whether OPF are finite or vanish in the infinite-volume limit. In particular, we discuss replica equivalence and its natural consequence lim V → ∞ G ( V , T ) = (1/3) when OPF are finite. As an example of a model where OPF vanish and replica equivalence does not give information about G we study the Sherrington-Kirkpatrick spherical spin-glass model by performing numerical simulations for small sizes. Again we find results compatible with G = (1/3) in the spin-glass phase.
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
We consider the statistical properties over disordered samples
(J) of the overlap distribution
PJ(q) which plays the role of an order parameter in spin glasses. We show that near zero temperature (i) the typical overlap distribution is exponentially small in the central region of $\ensuremath{-}1<q<1$: ${P}^{\text{typ}}(q)={e}^{\overline{\mathrm{ln}{P}_{\mathcal{J}}(q)}}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}{N}^{\ensuremath{\theta}}\ensuremath{\phi}(q)}$, where $\ensuremath{\theta}$ is the droplet exponent defined here with respect to the total number
N of spins (in order to consider also fully connected models in which the notion of length does not exist); (ii) the rescaled variable $v=\ensuremath{-}[\mathrm{ln}{P}_{\mathcal{J}}(q)]/{N}^{\ensuremath{\theta}}$ remains an
O(1) random positive variable describing sample-to-sample fluctuations; (iii) the averaged distribution
PJ(q) is nontypical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution ${I}_{\mathcal{J}}({q}_{0})\ensuremath{\equiv}{\ensuremath{\int}}_{0}^{{q}_{0}}dq\phantom{\rule{0.16em}{0ex}}{P}_{\mathcal{J}}(q)$. These results are derived explicitly for the spherical mean-field model with $\ensuremath{\theta}=1/3$, $\ensuremath{\phi}(q)=1\ensuremath{-}{q}^{2}$, and the random variable
v corresponds to the rescaled difference between the two largest eigenvalues of Gaussian orthogonal ensemble random matrices. Then we compare numerically the typical and averaged overlap distributions for the long-ranged one-dimensional Ising spin glass with random couplings decaying as $J(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}$ for various values of the exponent $\ensuremath{\sigma}$, corresponding to various droplet exponents $\ensuremath{\theta}(\ensuremath{\sigma})$, and for the mean-field Sherrington-Kirkpatrick model (corresponding formally to the $\ensuremath{\sigma}=0$ limit of the previous model). Our conclusion is that future studies on spin glasses should measure the typical values of the overlap distribution
Ptyp(q) or of the cumulative overlap distribution
Ityp(q0)=elnIJ(q0) to obtain clearer conclusions on the nature of the spin-glass phase.