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arXiv:1407.4891 · Journal of Statistical Mechanics Theory and Experiment

Critical behaviour of the random-field Ising model with long-range interactions in one dimension

Ivan Balog, Gilles Tarjus, Matthieu Tissier

Published Oct 9, 2014• 16 citations
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Abstract

We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.

DOI
10.1088/1742-5468/2014/10/p10017

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We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.
Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.
We show that, contrary to previous suggestions based on computer simulations or erroneous theoretical treatments, the critical points of the random-field Ising model out of equilibrium, when quasistatically changing the applied source at zero temperature, and in equilibrium are not in the same universality class below some critical dimension ${d}_{DR}\ensuremath{\approx}5.1$. We demonstrate this by implementing a nonperturbative functional renormalization group for the associated dynamical field theory. Above dDR{d}_{DR}dDR​, the avalanches, which characterize the evolution of the system at zero temperature, become irrelevant at large distance, and hysteresis and equilibrium critical points are then controlled by the same fixed point. We explain how to use computer simulation and finite-size scaling to check the correspondence between in and out of equilibrium criticality in a far less ambiguous way than done so far.
The critical properties of a random Ising model with long-range isotropic interactions decaying as R -(d+ sigma ) are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon 1 2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behaviour of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1/rd + σ, with σ = 0.75. The critical temperature, as well as the critical exponents, are evaluated from the power-law behaviour of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different truncated interaction ranges. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν) are in good agreement with renormalization group (RG) results and previously reported numerical data obtained under equilibrium conditions. On the other hand, the dynamic exponent of the initial increase of the magnetization (θ) was close to RG predictions. However, the dynamic exponent z of the time correlation length is slightly different to the RG results probably due to the fact that it may depend on the specific dynamics used or because the two-loop expansion used in the RG analysis may be insufficient.
We investigate the critical behavior of the one-dimensional Ising model with long-range interactions using the functional renormalization group in the local potential approximation (LPA), and compare our findings with Dyson's hierarchical model (DHM). While the DHM lacks translational invariance, it admits a field-theoretical description closely resembling the LPA, up to minor but nontrivial differences. After reviewing the real-space renormalization group approach to the DHM, we demonstrate a remarkable agreement in the critical exponent ννν between the two methods across the entire range of power-law decays 1/2<σ<11/2 < σ< 11/2<σ<1. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime, σ≲1σ\lesssim 1σ≲1.