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arXiv:0704.1626 · Physical Review B

Magnetic exponents of two-dimensional Ising spin glasses

Frauke Liers, Olivier Martin

Published Aug 15, 2007• 5 citations
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Abstract

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.

DOI
10.1103/physrevb.76.060405

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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 \ensuremathδ\ensuremath≈1.30\ifmmode±\else\textpm\fi0.02\ensuremath{\delta}\ensuremath{\approx}1.30\ifmmode\pm\else\textpm\fi{}0.02\ensuremathδ\ensuremath≈1.30\ifmmode±\else\textpm\fi0.02 using lattice sizes up to 80\ifmmode×\else\texttimes\fi8080\ifmmode\times\else\texttimes\fi{}8080\ifmmode×\else\texttimes\fi80; this is compatible with the droplet model prediction, \ensuremathδ=1.282\ensuremath{\delta}=1.282\ensuremathδ=1.282. The reason for previous disagreements stems from the need to analyze both singular and analytic contributions in the low-field regime.
We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. 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 Tc=0{T}_{c}=0Tc​=0. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.
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 .
Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.
An approach to determine the value of the zero-temperature thermal exponent $\ensuremath{\theta}$ in spin glasses is presented. It consists in describing the energy level spectrum in spin glasses only in terms of the properties of the lowest energy droplets and the lowest droplet exponents (LDE's) ${\ensuremath{\lambda}}_{l},{\ensuremath{\theta}}_{l}$ that describe the statistics of their sizes and gaps. We show how these LDE's yield the standard thermal exponent of droplet theory $\ensuremath{\theta}$ through the relation $\ensuremath{\theta}={\ensuremath{\theta}}_{l}+d{\ensuremath{\lambda}}_{l}.$ The present approach provides a new way to measure the thermal exponent $\ensuremath{\theta}$ without any assumption about the correct procedure to generate typical low-lying excitations as is commonly done in many perturbation methods including domain wall calculations. To illustrate the usefulness of the method we present a detailed investigation of the properties of the lowest energy droplets in two-dimensional Gaussian Ising spin glasses. By independent measurements of both LDE's and an aspect-ratio analysis, we find $\ensuremath{\theta}(2d)\ensuremath{\simeq}\ensuremath{-}0.46(1)<{\ensuremath{\theta}}_{\mathrm{DW}}(2d)\ensuremath{\simeq}\ensuremath{-}0.287$ where ${\ensuremath{\theta}}_{\mathrm{DW}}$ is the thermal exponent obtained in domain-wall theory. We also discuss the origin of finite-volume corrections in the behavior of the LDE ${\ensuremath{\theta}}_{l}$ and relate them to the finite-volume corrections in the statistics of extreme values. Finally, we analyze some geometrical properties of the lowest energy droplets, finding results in agreement with those recently reported by Kawashima and Aoki [J. Phys. Soc. Jpn. 69,69,69, 169 (2000)]. All in all, we show that typical large-scale droplets are not probed by most of the present perturbation methods, since they probably do not have a compact structure as has been recently suggested. We speculate that a multifractal scenario could be at the roots of the reported discrepancies on the value of the thermal exponent $\ensuremath{\theta}$ in the two-dimensional Gaussian Ising spin glass.
With respect to the usual thermal ferromagnetic transitions, the zero-temperature finite-disorder critical point of the random field Ising model (RFIM) has the peculiarity of involving some ‘droplet’ exponent θ that enters the generalized hyperscaling relation 2 − α = ν( d − θ). In the present paper, to better understand the meaning of this droplet exponent θ beyond its role in the thermodynamics, we discuss the statistics of low-energy excitations generated by an imposed single spin-flip with respect to the ground state, as well as the statistics of equilibrium avalanches, i.e. the magnetization jumps that occur in the sequence of ground states as a function of the external magnetic field. The droplet scaling theory predicts that the distribution dl / l 1 + θ of the linear size l of low-energy excitations transforms into the distribution ds / s 1 + θ/ d f for the size s (number of spins) of excitations of fractal dimension d f ( s ∼ l d f ). In the non-mean-field region d < d c , droplets are compact d f = d , whereas in the mean-field region d > d c , droplets have a fractal dimension d f = 2θ leading to the well-known mean-field result ds / s 3/2 . Zero-field equilibrium avalanches are expected to display the same distribution ds / s 1 + θ/ d f . We also discuss the statistics of equilibrium avalanches integrated over the external field and finite-size behaviors. These expectations are checked numerically for the Dyson hierarchical version of the RFIM, where the droplet exponent θ(σ) can be varied as a function of the effective long-range interaction J ( r ) ∼ 1/ r d + σ in d = 1.
The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.