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Physical review. B, Condensed matter

Ground-state magnetization of Ising spin glasses

Francisco Barahona

Published May 1, 1994• 20 citations
Publisher

Abstract

We study the magnetization of the ground states of Ising spin glasses as a function of the magnetic field. The grids are two-dimensional with Gaussian bond distribution. Finding a ground state in this case is a difficult combinatorial problem. We present a method that guarantees finding a true ground state. Although it is unlikely that one can have an algorithm with a polynomial bound for the running time, in all cases we tried, our method found a solution in a reasonable amount of time. We used this to obtain estimations of the critical exponents.

DOI
10.1103/physrevb.49.12864

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We study the magnetization of the ground states of Ising spin glasses as a function of the magnetic field. The grids are two-dimensional with Gaussian bond distribution. Finding a ground state in this case is a difficult combinatorial problem. We present a method that guarantees finding a true ground state. Although it is unlikely that one can have an algorithm with a polynomial bound for the running time, in all cases we tried, our method found a solution in a reasonable amount of time. We used this to obtain estimations of the critical exponents.
A new method to find ground states is proposed for random spin systems. It is applicable to systems with any boundary conditions, any bond distribution and any magnetic field. The efficiency of this method is confirmed numerically in the case of the two-dimensional Ising spin glass with Gaussian bond distribution in a uniform field. The introduction of more than two replicas improves the efficiency of the method considerably. It is also found that the renormalization process is effective. The increase in computational time with respect to system size is moderate and well fitted by a power law up to L=32. Magnetizations are calculated for various magnetic field using the new method. The size dependence of the susceptibility is found to be chi (L) varies as L x with x=0.476(5). This is somewhat larger than predictions using domain-wall renormalization group arguments.
The ground-state properties of quasi-one-dimensional (Q1D) Ising spin glass are investigated using an exact numerical approach and analytical arguments. A set of coupled recursive equations for the ground-state energy are introduced and solved numerically. For various types of coupling distribution, we obtain accurate results for magnetization, particularly in the presence of a weak external magnetic field. We show that in the weak magnetic field limit, similar to the 1D model, magnetization exhibits a singular power-law behavior with divergent susceptibility. Remarkably, the spectrum of magnetic exponents is markedly different from that of the 1D system even in the case of two coupled chains. The magnetic exponent makes a crossover from being dependent on a distribution function to a constant value independent of distribution. We provide an analytic theory for these observations by extending the Chen-Ma argument to the Q1D case. We derive an analytical formula for the exponent which is in perfect agreement with the numerical results.
The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using an exact ground-state determination, we extract the properties of clusters flipped when a uniform field is increased continuously. We show that these clusters have many holes but otherwise have statistical properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization exponent $\ensuremath{\delta}\ensuremath{\approx}1.30\ifmmode\pm\else\textpm\fi{}0.02$ using lattice sizes up to $80\ifmmode\times\else\texttimes\fi{}80$; this is compatible with the droplet model prediction, $\ensuremath{\delta}=1.282$. The reason for previous disagreements stems from the need to analyze both singular and analytic contributions in the low-field regime.
A lower bound result about the approximability of the energy function of Ising spin glasses in the quantum context is determined. The ground-state problem for the three-dimensional Ising spin glasses represented on two-level grids such that the vertical interactions are at most (where n is the number of spins set on the grid), is considered. We prove that, unless , there is a constant such that every quantum approximate polynomial-time algorithm finds a solution with absolute error greater than , with high probability, infinitely often.
The critical exponents for of the two-dimensional Ising spin-glass model with Gaussian couplings are determined with the help of exact ground states for system sizes up to L = 50 and by a Monte Carlo study of a pseudo-ferromagnetic order parameter. We obtain: for the stiffness exponent , for the magnetic exponent and for the chaos exponent . From Monte Carlo simulations we get the thermal exponent . The scaling prediction is fulfilled within the error bars, whereas there is a disagreement with the relation .
Large numbers of ground states of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model are calculated for sizes up to 143{14}^{3}143 using a combination of a genetic algorithm and cluster-exact approximation. Several quantities are calculated as functions 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 ${p}_{c}=0.222\ifmmode\pm\else\textpm\fi{}0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\beta}=0.2\ifmmode\pm\else\textpm\fi{}0.1 are calculated. The behavior of the distribution of overlaps P(q)P(q)P(q) is used to investigate how the spin-glass phase evolves with increasing concentration p.p.p. The spin-glass order is characterized by a broad distribution of overlaps which extends down to q=0.q=0.q=0.