Non-self-averaging in the Critical Point of a Random Ising Ferromagnet
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ontribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary
Non-self-averaging in the Critical Point of a Random Ising Ferromagnet V. Dotsenkoa,b aSorbonne
Université, LPTMC, Paris, F-75005 France Landau Institute for Theoretical Physics, Moscow, 119334 Russia e-mail: [email protected]
b
Received January 31, 2019; revised February 22, 2019; accepted February 22, 2019
Abstract—In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In particular, in terms of the renormalized replica Ginzburg–Landau Hamiltonian in dimensions D < 4 an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations is derived. Next, using known fixed-point values for the renormalized coupling parameters the universal curve for such PDF in the dimension D = 3 is obtained. For the specific case of the two-dimensional Ising model, using replica calculations in the renormalization group framework, we derive explicit expressions for the PDF of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energy is Gaussian, and its typical sample-to-sample fluctuations as well as its average value scale with the system size L like ~ Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ-peak in the thermodynamic limit L → ∞. DOI: 10.1134/S1063776119100194
1. INTRODUCTION After intensive studies during last several decades it is well established now that the presence of weak quenched disorder in a ferromagnetic system can essentially modify its critical behavior in the vicinity of the phase transition point such that new universal critical exponents may set in [1–7]. According to the so called Harris criteria [1] weak disorder is relevant for the critical behavior only if the specific heat of the pure system is divergent, i.e., the corresponding critical exponent α > 0. The critical behavior is then governed by a new, random renormalization-group fixed point, and the pure fixed point becomes unstable. On the other hand in recent years it is argued that due to the presence of disorder the statistical properties of some thermodynamical quantities at the critical point can become non-self-averaging [8–11], i.e., the behavior of a large sample with a specific realization of impurities will not be well described by the ensemble average normally calculated in an analytical or numerical approach. This clearly has profound consequences for the physical interpretation of the outcomes and the possibilities for comparing theoretical and experimental results. Recently, an explicit expression for the probability distribution function of the critical free-energy fluctuations for a weakly disordered Ising ferromagnet was
derived for D < 4 and its universal shape was obtained at D = 3 [12]. Away from the critical point at scales much bigger than the correlation l
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