We study the phenomenon of the locking of the order parameter (or synchronization) in spin glasses at low temperatures. When two systems with independent disorders are coupled, their overlaps become similar. A crucial question is how this effect depends on the strength of the coupling between the two systems. Non-perturbative phenomena are present when $1 \ll ΔH \ll N$, being $ΔH$ the coupling Hamiltonian and $N$ the size of the system. In this intermediate-coupling region, the effect is related to finite-size free-energy corrections and to the correlations in the Dyson hierarchical spin glass, a model that mimics the physics of finite-dimensional systems. We study this phenomenon in the mean-field approach, both analytically and numerically, and we finally compute the critical exponents for finite-volume corrections in mean-field theory and for the decay of correlations in the Dyson hierarchical model.
We study the phenomenon of the locking of the order parameter (or synchronization) in spin glasses at low temperatures. When two systems with independent disorders are coupled, their overlaps become similar. A crucial question is how this effect depends on the strength of the coupling between the two systems. Non-perturbative phenomena are present when 1≪ΔH≪N, being ΔH the coupling Hamiltonian and N the size of the system. In this intermediate-coupling region, the effect is related to finite-size free-energy corrections and to the correlations in the Dyson hierarchical spin glass, a model that mimics the physics of finite-dimensional systems. We study this phenomenon in the mean-field approach, both analytically and numerically, and we finally compute the critical exponents for finite-volume corrections in mean-field theory and for the decay of correlations in the Dyson hierarchical model.
The large scale behavior of the simplest non-mean-field spin-glass system is analyzed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the ε-expansion technique with two independent methods. The techniques presented show how the underlying ideas of the renormalization group apply also in this disordered model, in such a way that an ε-expansion can be consistently set up. By pushing such calculation to high orders in ε, a consistent non-mean-field theory for such disordered system could be established, giving a substantial contribution the development of a predictive theory for real spin glasses.
We present a numerical study of an Ising spin glass with hierarchical interactions--the hierarchical Edwards-Anderson model with an external magnetic field (HEA). We study the model with Monte Carlo (MC) simulations in the mean-field (MF) and non-mean-field (NMF) regions corresponding to d ≥ 4 and d < 4 for the d-dimensional ferromagnetic Ising model respectively. We compare the MC results with those of a renormalization-group (RG) study where the critical fixed point is treated as a perturbation of the MF one, along the same lines as in the -expansion for the Ising model. The MC and the RG method agree in the MF region, predicting the existence of a transition and compatible values of the critical exponents. Conversely, the two approaches markedly disagree in the NMF case, where the MC data indicates a transition, while the RG analysis predicts that no perturbative critical fixed point exists. Also, the MC estimate of the critical exponent ν in the NMF region is about twice as large as its classical value, even if the analog of the system dimension is within only ~2% from its upper-critical-dimension value. Taken together, these results indicate that the transition in the NMF region is governed by strong non-perturbative effects.
In a recent work [M. Castellana and G. Parisi, Phys. Rev. E 82, 040105(R) (2010)], the large-scale behavior of the simplest non-mean-field spin-glass system has been analyzed, and the critical exponent related to the divergence of the correlation length has been computed at two loops within the ε-expansion technique by two independent methods. By performing the explicit calculation of the critical exponents at two loops, one obtains that the two methods yield the same result. This shows that the underlying renormalization group ideas apply consistently in this disordered model, in such a way that an ε-expansion can be set up. The question of the extension to high orders of this ε-expansion is particularly interesting from the physical point of view. Indeed, once high orders of the series in ε for the critical exponents are known, one could check the convergence properties of the series, and find out if the ordinary series resummation techniques, yielding very accurate predictions for the Ising model, work also for this model. If this is the case, a consistent and predictive non-mean-field theory for such a disordered system could be established. In that regard, in this work we expose the underlying techniques of such a two-loop computation. We show with an explicit example that such a computation could be quite easily automatized, i.e., performed by a computer program, in order to compute high orders of the ε-expansion, and so eventually make this theory physically predictive. Moreover, all the underlying renormalization group ideas implemented in such a computation are widely discussed and exposed.
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.
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 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.