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Journal of Physics Condensed Matter

The renormalisation of probability distributions in the random field problem

B. Boechat, M. A. Contínentino

Published Jun 11, 1990• 9 citations
Publisher

Abstract

The authors study the Ising ferromagnet in a random field following the renormalisation of the distribution of random fields. The disorder in the local field induces fluctuations in the exchange couplings which are taken into account. The method allows a straightforward determination of the exponents associated with the zero-temperature fixed point which governs the critical behaviour along the critical line and in particular the critical slowing down. They find the correlation length exponent v=1.0 which is in very good agreement with numerical calculations. They also obtain the crossover exponent for small random fields.

DOI
10.1088/0953-8984/2/23/020

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The authors study the Ising ferromagnet in a random field following the renormalisation of the distribution of random fields. The disorder in the local field induces fluctuations in the exchange couplings which are taken into account. The method allows a straightforward determination of the exponents associated with the zero-temperature fixed point which governs the critical behaviour along the critical line and in particular the critical slowing down. They find the correlation length exponent v=1.0 which is in very good agreement with numerical calculations. They also obtain the crossover exponent for small random fields.
The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.
We compute the crossover exponent $\ensuremath{\varphi}$ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as ${\mathrm{Fe}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2}$ and ${\mathrm{Mn}}_{x}{\mathrm{Zn}}_{1\ensuremath{-}x}{\mathrm{F}}_{2},$ when applying an external magnetic field. Our result $\ensuremath{\varphi}=1.42(2)$ for the crossover exponent is in good agreement with the available experimental estimates.
Abstract We consider 2D random Ising ferromagnetic models, where quenched disorder is represented either by random local magnetic fields (random-field Ising model) or by a random distribution of interaction couplings (random-bond Ising model). In both cases, we first perform zero- and finite-temperature Monte Carlo simulations to determine how the critical temperature depends on the disorder parameter. We then focus on the reversal transition triggered by an external field and study the associated Barkhausen noise. Our main result is that the critical exponents characterizing the power law associated with the Barkhausen noise exhibit a temperature dependence in line with existing experimental observations.
A review is given on some recent developments in the theory of the Ising model in a random field. This model is a good representation of a large number of impure materials. After a short repetition of earlier arguments, which prove the absence of ferromagnetic order in d≤2d\le 2d≤2 space dimensions for uncorrelated random fields, we consider different random field correlations and in particular the generation of uncorrelated from anti-correlated random fields by thermal fluctuations. In discussing the phase transition, we consider the transition to be characterized by a divergent correlation length and compare the critical exponents obtained from various methods (real space RNG, Monte Carlo calculations, weighted mean field theory etc.). The ferromagnetic transition is believed to be preceded by a spin glass transition which manifests itself by replica symmetry breaking. In the discussion of dynamical properties, we concentrate mainly on the zero temperature depinning transition of a domain wall, which represents a critical point far from equilibrium with new scaling relations and critical exponents.
Phase transition in d=3d=3d=3 ferromagnetic Ising models with random fields is analyzed directly at the zero-temperature critical point. Critical behavior is extracted from correlation functions averaged over an ensemble of exact ground states obtained with a new integer optimization algorithm. For Gaussian distribution of random fields finite-size scaling demonstrates a continuous phase transition with effective disconnected susceptibility exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremath{\approx}\ensuremath{-}0.9$, correlation-length exponent $\ensuremath{\nu}\ensuremath{\approx}1.0$, and magnetization exponent $\ensuremath{\beta}\ensuremath{\approx}0.05$.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.