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arXiv:1401.6342 · Physical Review B

Zero-temperature spin-glass–ferromagnetic transition: Scaling analysis of the domain-wall energy

Cécile Monthus, Thomas Garel

Published May 13, 2014• 13 citations
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Abstract

For the Ising model with Gaussian random coupling of average ${J}_{0}$ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter ${J}_{0}$ can be studied via the size-$L$-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension $d=2,3,4,5,6$. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.

DOI
10.1103/physrevb.89.184408

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Critical point scaling in a field HHH applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function f(R)f(R)f(R) defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically f(R)f(R)f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d=3d=3d=3, we have determined the exponents and the critical scaling function f(R)f(R)f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for f(R)f(R)f(R) in d=3d=3d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of RRR for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function f(R)f(R)f(R) at some negative value of RRR, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length LLL, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows tuning of the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic ground state. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the average domain-wall energy $(\ensuremath{\sim}{L}^{\ensuremath{\theta}})$ and the average domain-wall length $(\ensuremath{\sim}{L}^{{d}_{f}})$. Performing a finite-size scaling analysis for systems with a side length up to L=512L=512L=512, we find that both exponents remain constant in the spin-glass phase, i.e., $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.28$ and ${d}_{f}\ensuremath{\approx}1.275$. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of stochastic Loewner evolutions via ${d}_{f}\ensuremath{-}1=3/[4(3+\ensuremath{\theta})]$. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.
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We study domain-wall excitations in two-dimensional random-bond Ising spin systems on a square lattice with side length L, subject to two different continuous disorder distributions. In both cases an adjustable parameter allows to tune the disorder so as to yield a transition from a spin-glass ordered ground state to a ferromagnetic groundstate. We formulate an auxiliary graph-theoretical problem in which domain walls are given by undirected shortest paths with possibly negative distances. Due to the details of the mapping, standard shortest-path algorithms cannot be applied. To solve such shortest-path problems we have to apply minimum-weight perfect-matching algorithms. We first locate the critical values of the disorder parameters, where the ferromagnet to spin-glass transition occurs for the two types of the disorder. For certain values of the disorder parameters close to the respective critical point, we investigate the system size dependence of the width of the the average domain-wall energy (~L^\theta) and the average domain-wall length (~L^df). Performing a finite-size scaling analysis for systems with a side length up to L=512, we find that both exponents remain constant in the spin-glass phase, i.e. \theta ~- 0.28 and df~1.275. This is consistent with conformal field theory, where it seems to be possible to relate the exponents from the analysis of Stochastic Loewner evolutions (SLEs) via df-1=3/[4(3+\theta)]. Finally, we characterize the transition in terms of ferromagnetic clusters of spins that form, as one proceeds from spin-glass ordered to ferromagnetic ground states.
For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco (1979 Phys. Rev. D 19 3173 ) is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent ν = 1. Recently, Miyazaki and Nishimori (2013 Phys. Rev. E 87 032154 ) have proposed to study the disordered quantum Ising model in dimensions d > 1 by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the d directions are treated on the same footing. For the pure model, this leads to the correlation length exponents ν ≃ 0.625 in d = 2 (to be compared with the 3D classical Ising model exponent ν ≃ 0.63) and ν ≃ 0.5018 (to be compared with the 4D classical Ising model mean-field exponent ν = 1/2). For the disordered model in dimension d = 2, either ferromagnetic or spin-glass, the numerical application of the renormalization rules to samples of linear size L = 4096 yields that the transition is governed by an Infinite Disorder Fixed Point, with the activated exponent ψ ≃ 0.65, the typical correlation exponent ν typ ≃ 0.44 and the finite-size correlation exponent ν FS ≃ 1.25. We discuss the similarities and differences with the Strong Disorder Renormalization results.