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arXiv:1803.09076 · Physical review. B./Physical review. B

Ground-state magnetization of the Ising spin glass: A recursive numerical method and Chen-Ma scaling

Reza Sepehrinia, Fartash Chalangari

Published Mar 5, 2018• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.97.104201

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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.
We obtain the exact ground-state energy of the quasi-one-dimensional restricted ±J Ising spin glass in a uniform magnetic field using the transfer matrix method. Magnetic field dependence allows us to derive the magnetization as a function of concentration and magnetic field. It turns out that, in the limit of zero field, the magnetization tends to a nonzero value with a singular dependence on the magnetic field. We derive the explicit form of the singularity in thermodynamic quantities such as energy E≃E_{0}+m_{0}h+E_{1}e^{-h_{0}/h}, which is an essential singularity known as Griffiths singularity. We confirm our analytical results using the numerical approach based on iterative equations for energy.
Griffiths singularity in quasi-one dimensional restricted ±J\pm J±J Ising spin glass
We obtain the exact ground state energy of the quasi-one dimensional restricted ±J\pm J±J Ising spin glass in a uniform magnetic field using the transfer matrix method. Magnetic field dependence allows us to derive the magnetization as a function of concentration and magnetic field. It turns out that, in the limit of zero field, the magnetization tends to a nonzero value with a singular dependence on the magnetic field. We derive the explicit form of the singularity in thermodynamic quantities such as energy E≃E0+m0h+E1e−h0/hE\simeq E_0+m_0 h + E_1e^{-h_0/h}E≃E0​+m0​h+E1​e−h0​/h, which is an essential singularity known as Griffiths singularity. We confirm our analytical results using the numerical approach based on iterative equations for energy.
For the spin-glass chain in an external field hhh, a nonzero weight at the origin of the bond distribution $\ensuremath{\rho}(J)$ is known to induce a nonanalytical magnetization at zero temperature; for $\ensuremath{\rho}(J)\ensuremath{\sim}A{\ensuremath{\mid}J\ensuremath{\mid}}^{\ensuremath{\mu}\ensuremath{-}1}$ near $J\ensuremath{\rightarrow}0$, the magnetization follows the Chen-Ma scaling $M\ensuremath{\sim}{h}^{\ensuremath{\mu}∕(2+\ensuremath{\mu})}$. In this paper, we numerically revisit this model to obtain detailed statistical information on the ground-state configuration and on the low-energy two-level excitations that govern the low-temperature properties. The ground state consists of long unfrustrated intervals separated by weak frustrated bonds. We accordingly compute the strength distribution of these frustrated bonds, as well as the length and magnetization distributions of the unfrustrated intervals. We find that the low-energy excitations are of two types: (i) one frustrated bond of the ground state may have two positions that are nearly degenerate in energy and (ii) two neighboring frustrated bonds of the ground state may be annihilated or created with nearly zero energy cost. For each excitation type, we compute its probability density as a function of its length. Moreover, we show that the contributions of these excitations to various observables (specific heat, Edwards-Anderson order parameter, susceptibility) are in full agreement with the direct transfer matrix evaluations at low temperature. Finally, following the special bimodal case $\ifmmode\pm\else\textpm\fi{}J$, where a Ma-Dasgupta, RG procedure has been previously used to compute explicitly the above observables, we discuss the possibility of an extended RG procedure. We find that the ground state can be seen as the result of a hierarchical ``fragmentation'' procedure that we describe.
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.
We have studied numerically the remanent magnetization in the six- and eight-dimensional Ising spin glass and we have compared it with the behaviour observed in the SK model, that we have also computed analytically. We also report the value of the dynamical critical exponent z in six dimensions measured in three different ways: from the behaviour of the energy and the susceptibility as a function of the Monte Carlo time and by studying the overlap - overlap correlation function as a function of space and time. These three results are in very good agreement with the mean field prediction z = 4. Finally we have checked numerically the analytical prediction, obtained by assuming spontaneously broken replica symmetry, for the most singular part of the propagator in the spin-glass phase. This last result supports the existence of spontaneously broken replica symmetry in finite-dimensional spin glasses.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.