Numerical data on scaling of the normalized Binder cumulant and the normalized correlation length are shown for the Thermodynamic limit regime, first for canonical Ising ferromagnet models and then for a range of Ising spin glass models. A fundamental Renormalization Group Theory rule linking the critical exponents for the two observables is well obeyed in the Ising models, but not for the Ising spin glasses in dimensions three and four. We conclude that there is a violation of a standard Josephson hyperscaling rule in Ising spin glasses.
Among the Renormalization Group Theory scaling rules relating critical exponents, there are hyperscaling rules involving the dimension of the system. It is well known that in Ising models hyperscaling breaks down above the upper critical dimension. It was shown by M. Schwartz [Europhys. Lett. {\bf 15}, 777 (1991)] that the standard Josephson hyperscaling rule can also break down in Ising systems with quenched random interactions. A related Renormalization Group Theory hyperscaling rule links the critical exponents for the normalized Binder cumulant and the correlation length in the thermodynam…
Large numbers of ground states of the three-dimensional $\pm J$ random-bond Ising model are calculated for sizes up to $14^3$ using a combination of a genetic algorithm and Cluster-Exact Approximation. Several quantities are calculated as function of the concentration $p$ of the antiferromagnetic bonds. The critical concentration where the ferromagnetic order disappears is determined using the Binder cumulant of the magnetization. A value of $p_c=0.222\pm 0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents $ν=1.1 \pm 0.3$ and $β=0.2…
Localization Transitions in a Half-Filled Helical Aubry-André Model
Bilkent University（比尔肯特大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；MTA-BME Lendület “Momentum” Open Quantum Systems Research Group, Institute of Physics, Budapest University of Technology and Economics（布达佩斯技术与经济大学物理研究所 MTA-BME Lendület “Momentum”开放量子系统研究组）；Institute for Solid State Physics and Optics, HUN-REN Wigner Research Centre for Physics（匈牙利科学院Wigner物理研究所凝聚态物理与光学研究所）
The critical behavior of the Binder cumulant for Ising spin glasses in dimension four are studied through simulation measurements. Data for the bimodal interaction model are compared with those for the Laplacian interaction model. Special attention is paid to scaling corrections. The limiting infinite size value at criticality for this dimensionless variable is a parameter characteristic of a universality class. This critical limit is estimated to be equal to $0.523(3)$ in the bimodal model and to $0.473(3)$ in the Laplacian model.
We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $β=1/T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state and the specific heat. We confirm that $T_c=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $ν$. We find $-1/ν=-0.295(30)$, whi…
I discuss results from numerical simulations of finite dimensional spin glass models, and show that they show all signatures of a mean field like behavior, basically coinciding with the one of the Parisi solution. I discuss the Binder cumulant, the probability distribution of the order parameter, the non self-averaging behavior. The determination of correlation function and of spatially blocked observables quantities helps in qualifying the behavior of the system.
In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models which are in this class have been intensively studied. Here numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlat…
We compare the probability distributions and Binder cumulants of the overlap in the 3D Ising spin glass with those of the magnetization in the ferromagnetic 2D XY model. We analyze similarities and differences. Evidence for the existence of a phase transition in the spin glass model is obtained thanks to the crossing of the Binder cumulant. We show that the behavior of the XY model is fully compatible with the Kosterlitz-Thouless scenario. Finite size effects have to be dealt with by using great care in order to discern among two very different physical pictures that can look very similar if a…
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 re…
We study the critical behavior of two-dimensional short-range quantum spin glasses by numerical simulations. Using a parallel tempering algorithm, we calculate the Binder cumulant for the Ising spin glass in a transverse magnetic field with two different short-range bond distributions, the bimodal and the Gaussian ones. Through an exhaustive finite-size scaling analysis, we show that the universality class does not depend on the exact form of the bond distribution but, most important, that the quantum critical behavior is governed by an infinite randomness fixed point.
The use of parameters measuring order-parameter fluctuations (OPF) has been encouraged by the recent results reported in \cite{RS} which show that two of these parameters, $G$ and $G_c$, take universal values in the $\lim_{T\to 0}$. In this paper we present a detailed study of parameters measuring OPF for two mean-field models with and without time-reversal symmetry which exhibit different patterns of replica symmetry breaking below the transition: the Sherrington-Kirkpatrick model with and without a field and the Ising p-spin glass (p=3). We give numerical results and analyze the consequences…
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；School of Science, Constructor University（康斯坦茨大学理学院）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）
Direct measurements of the spin glass correlation function $G(R)$ for Gaussian and bimodal Ising spin glasses in dimension two have been carried out in the temperature region $T \sim 1$. In the Gaussian case the data are consistent with the known anomalous dimension value $η\equiv 0$. For the bimodal spin glass in this temperature region $T > T^{*}(L)$, well above the crossover $T^{*}(L)$ to the ground state dominated regime, the effective exponent $η$ is clearly non-zero and the data are consistent with the estimate $η\sim 0.28(4)$ given by McMillan in 1983 from similar measurements. Measurem…
We study order-parameter fluctuations (OPF) in disordered systems by considering the behavior of some recently introduced paramaters $G,G_c$ which have proven very useful to locate 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 $T\to 0$ limit for any {\em finite} volume provided the ground state is {\em unique} and there is no gap in the ground state local-field distributions, conditions which are met in generic spin-glass models with continuous…
We study the distribution of equilibrium avalanches (shocks) in Ising spin glasses which occur at zero temperature upon small changes in the magnetic field. For the infinite-range Sherrington-Kirkpatrick model we present a detailed derivation of the density rho(Delta M) of the magnetization jumps Delta M. It is obtained by introducing a multi-component generalization of the Parisi-Duplantier equation, which allows us to compute all cumulants of the magnetization. We find that rho(Delta M) ~ (Delta M)^(-tau) with an avalanche exponent tau=1 for the SK model, originating from the marginal stabil…
We compare numerical estimates from different sources for the ordering temperature $T_g$ and the critical exponents of the Ising spin glass in dimension three with binomial ($\pm J$) interactions. Corrections to finite size scaling turn out to be important especially for parameters such as the Binder cumulant. For non-equilibrium parameters it is easier to approach the large size limit and to allow for corrections to scaling. Relying principally on such data, a crossing point defines the freezing temperature $T_g$; the possibility that the ordering temperature is zero can definitively be exclu…
It is shown by Monte Carlo method that the finite size scaling (FSS) holds in the two dimensional random-coupled Ising ferromagnet. It is also demonstrated that the form of universal FSS function constructed via novel FSS scheme depends on the strength of the random coupling for strongly disordered cases. Monte Carlo measurements of thermodynamic (infinite volume limit) data of the correlation length ($ξ$) up to $ξ\simeq 200$ along with measurements of the fourth order cumulant ratio (Binder's ratio) at criticality are reported and analyzed in view of two competing scenarios. It is demonstrate…
