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arXiv:1012.3444 · Physical Review Letters

Zero- and Low-Temperature Behavior of the Two-Dimensional ±J Ising Spin Glass

Creighton K. Thomas, David A. Huse, A. Alan Middleton

Published Jul 20, 2011• 54 citations
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Abstract

Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.

DOI
10.1103/physrevlett.107.047203

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Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.
The standard short-range two-dimensional Ising spin glass is numerically well accessible, in particular, because there are polynomial-time ground-state algorithms. On the other hand, in contrast to higher dimensional spin glasses, it does not exhibit a rich behavior, i.e., no ordered phase at finite temperature. Here, we investigate whether long-range correlated bonds change this behavior. This would still keep the model numerically well accessible while exhibiting a more interesting behavior. The bonds are drawn from a Gaussian distribution with a two-point correlation for bonds at distance r that decays as (1+r^{2})^{-a/2}, a≥0. We study numerically with exact algorithms the ground-state and domain-wall excitations. Our results indicate that the inclusion of bond correlations still does not lead to a spin-glass order at any finite temperature. A further analysis reveals that bond correlations have a strong effect at local length scales, inducing ferro- and antiferromagnetic domains into the system. The length scale of ferro- and antiferromagnetic order diverges exponentially as the correlation exponent approaches a critical value, a→a_{crit}=0. Thus, our results suggest that the system becomes a ferro- or antiferromagnet only in the limit a→0.
A phenomenological theory of the ordered phase of short-range Ising spin glasses is developed in terms of droplet excitations and presented in detail. These excitations have free energies with a broad distribution whose characteristic magnitude grows with length scale L as ${L}^{\ensuremath{\theta}}$. A small fraction of droplets of all scales are thermally active; these dominate much of the physics. The mean-square correlation functions are found to decay with distance as 1/${r}^{\ensuremath{\theta}}$ for all TTc{T}_{c}Tc​ and the autocorrelations decay logarithmically with time because of large activation barriers for creation and annihilation of the droplet excitations. A renormalization procedure is sketched in order to define excitations at positive temperature. It is found that the long-distance equilibrium correlation functions are extremely sensitive to small temperature changes, yielding breakdown of certain relations between fluctuations and thermodynamic derivatives. The behavior near to the critical temperature is discussed and some of the ideas are extended to systems with power-law interactions and to spin glasses with X-Y or Heisenberg symmetry. The inequality \ensuremath{\theta}\ensuremath{\le}(d-1)/2 is also derived.
We investigate the zero-temperature glassy transitions in the square-lattice ±J Ising model, with bond distribution P(J{xy})=pδ(J{xy}-J)+(1-p)δ(J{xy}+J) ; p=1 and p=1/2 correspond to the pure Ising model and to the Ising spin glass with symmetric bimodal distribution, respectively. We present finite-temperature Monte Carlo simulations at p=4/5 , which is close to the low-temperature paramagnetic-ferromagnetic transition line located at p≈0.89 , and at p=1/2 . Their comparison provides a strong evidence that the glassy critical behavior that occurs for 1-p{0}<p<p{0} , p{0}≈0.897 , is universal, i.e., independent of p . Moreover, we show that glassy and magnetic modes are not coupled at the multicritical zero-temperature point where the paramagnetic-ferromagnetic transition line and the T=0 glassy transition line meet. On the theoretical side we discuss the validity of finite-size scaling in glassy systems with a zero-temperature transition and a discrete Hamiltonian spectrum. Because of a freezing phenomenon which occurs in a finite volume at sufficiently low temperatures, the standard finite-size scaling limit in terms of TL(1/ν) does not exist; the renormalization-group invariant quantity ξ/L should be used instead as basic variable.
For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J ( r ) ∼ r − σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 2 (including the usual Gaussian case μ = 2), we consider the region σ > 1/ μ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings J L ∝ L θ μ ( σ ) is found to be whenever the long-ranged couplings are relevant θ μ ( σ ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θ μ ( σ ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(− u ) 1+ μ for u → − ∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.
The stability of the spin-glass phase against a magnetic field is studied in the three- and four-dimensional Edwards-Anderson Ising spin glasses. Effective couplings J(eff) and effective fields H(eff) associated with length scale L are measured by a numerical domain-wall renormalization-group method. The results obtained by scaling analysis of the data strongly indicate the existence of a crossover length beyond which the spin-glass order is destroyed by field H. The crossover length well obeys a power law of H which diverges as H --> 0 but remains finite for any nonzero H, implying that the spin-glass phase is absent even in an infinitesimal field. These results are well consistent with the droplet theory for short-range spin glasses.
Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations.