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arXiv:1307.3987 · Physical Review B

Exact ground states of one-dimensional long-range random-field Ising magnets

Timo Dewenter, Alexander K. Hartmann

Published Jul 31, 2014• 10 citations
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Abstract

We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where $r$ is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to $L={2}^{19}$ via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength $h$ of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength ${h}_{c}>0$.

DOI
10.1103/physrevb.90.014207

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We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability p\ensuremath∼r\ensuremath−1\ensuremath−\ensuremathσp\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}p\ensuremath∼r\ensuremath−1\ensuremath−\ensuremathσ, where rrr is the distance between these spins and \ensuremathσ\ensuremath{\sigma}\ensuremathσ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to L=219L={2}^{19}L=219 via graph theoretical algorithms for four different values of \ensuremathσ\ensuremath∈0.25,0.4,0.5,1.0\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}\ensuremathσ\ensuremath∈0.25,0.4,0.5,1.0 while varying the strength hhh of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values hc(\ensuremathσ){h}_{c}(\ensuremath{\sigma})hc​(\ensuremathσ) as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for \ensuremathσ=1/2\ensuremath{\sigma}=1/2\ensuremathσ=1/2 the data is compatible with a critical random-field strength hc>0{h}_{c}>0hc​>0.
The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to 324{32}^{4}324 are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness hc=4.18(1){h}_{c}=4.18(1)hc​=4.18(1) and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$
Exact ground states of three-dimensional random field Ising magnets with Gaussian distribution of the disorder are calculated using graph-theoretical algorithms. Systems for different strengths h of the random fields and sizes up to N=963{N=96}^{3}N=963 are considered. By numerically differentiating the bond-energy with respect to h a specific-heat-like quantity is obtained, which does not appear to diverge at the critical point but rather exhibits a cusp. We also consider the effect of a small uniform magnetic field, which allows us to calculate the T=0T=0T=0 susceptibility. From a finite-size scaling analysis, we obtain the critical exponents $\ensuremath{\nu}=1.32(7),$ $\ensuremath{\alpha}=\ensuremath{-}0.63(7),$ $\ensuremath{\eta}=0.50(3)$ and find that the critical strength of the random field is hc=2.28(1).{h}_{c}=2.28(1).hc​=2.28(1). We discuss the significance of the result that $\ensuremath{\alpha}$ appears to be strongly negative.
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 the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ra{r}^{a}ra, where rrr is the distance between two lattice sites and a<0a<0a<0. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.
The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to 603.{60}^{3}.603. For each realization of the normalized random fields, the strength of the random field, $\ensuremath{\Delta}$ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the $H\ensuremath{-}\ensuremath{\Delta}$ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.
We investigate the critical properties of the d=3d=3d=3 random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength hhh and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength hc{h}_{\mathrm{c}}hc​ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.