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arXiv:cond-mat/0105310 · Physical review. B, Condensed matter

Specific-heat exponent of random-field systems via ground-state calculations

Alexander K. Hartmann, A. P. Young

Published Nov 13, 2001• 81 citations
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Abstract

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 ${N=96}^{3}$ 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 $T=0$ 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 ${h}_{c}=2.28(1).$ We discuss the significance of the result that $\ensuremath{\alpha}$ appears to be strongly negative.

DOI
10.1103/physrevb.64.214419

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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 $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$
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 rrr 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=219L={2}^{19}L=219 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 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 ${h}_{c}(\ensuremath{\sigma})$ 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 hc>0{h}_{c}>0hc​>0.
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We investigate the low-temperature critical behavior of the three-dimensional random-field Ising ferromagnet. By a scaling analysis we find that in the limit of temperature T→0 the usual scaling relations have to be modified as far as the exponent α of the specific heat is concerned. At zero temperature, the Rushbrooke equation is modified to α+2β+γ=1, an equation which we expect to be valid also for other systems with similar critical behavior. We test the scaling theory numerically for the three-dimensional random-field Ising system with Gaussian probability distribution of the random fields by a combination of calculations of exact ground states with an integer optimization algorithm and Monte Carlo methods. By a finite-size scaling analysis we calculate the critical exponents ν≈1.0, β≈0.05, γ̄≈2.9, γ≈1.5 and α≈−0.55.