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.
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.
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.
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.
We discuss a phase transition in spin glass models that have been rarely considered in the past, namely, the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e., at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that, even in a paramagnetic phase, the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite dimensional spin glasses. This is a notoriously difficult computational problem, because of considerable finite size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, h > 0 and T c ( h ) < T < T c ( h = 0 ) , of the d = 1 spin glass model with long range interactions decaying fast enough to be outside the regime of validity of the mean field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.
We review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the existence of a low-temperature (high-transverse-field) ergodic region. We also find in high-temperature (low-transverse-field) spin glass phase the order parameter distribution has nonzero value for all values of the order parameter even in infinite system size limit, which essentially indicates the nonergodic behavior of the system. We study the annealing dynamics by the paths which pass through both ergodic and nonergodic spin glass regions. We find the average annealing time becomes system size independent for the paths which pass through the quantum-fluctuation-dominated ergodic spin glass region. In contrast to that, the annealing time becomes strongly system size dependent for annealing down through the classical-fluctuation-dominated nonergodic spin glass region. We investigate the behavior of the spin autocorrelation in the spin glass phase. We observe that the decay rate of autocorrelation towards its equilibrium value is much faster in the ergodic region with respect to the nonergodic region of the spin glass phase.
We present a constructive approach to obtain information about the compactness and shape of large-scale lowest excitations in disordered systems by studying order parameter fluctuations (OPF) at low temperatures. We show that the parameter G that measures OPF is 1/3 at T = 0 provided the ground state is unique and the probability distribution for the lowest excitations is gapless and with finite weight at zero excitation energy. We then apply zero temperature scaling to describe the energy and volume spectra of the lowest large-scale excitations that scale with the system size and have a weight at zero energy v (0)~ l -θ' with v = l d . A low-temperature expansion reveals that, OPF vanish like L -θ , if θ>0 and remain finite for space-filling lowest excitations with θ = 0. The method can be extended to extract information about the shape and fractal surface of the large-scale lowest excitations.