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arXiv:cond-mat/0201441 · Physical review. B, Condensed matter

Critical exponents of four-dimensional random-field Ising systems

Alexander K. Hartmann

Published May 7, 2002• 30 citations
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Abstract

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 ${32}^{4}$ 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 ${h}_{c}=4.18(1)$ 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 $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$

DOI
10.1103/physrevb.65.174427

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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.
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