Inequalities for the Local Energy of Random Ising Models (original) (raw)

Temperature-dependent criticality in random 2D Ising models

The European Physical Journal Plus, 2021

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 Power-Law Upper Bound on the Correlations in the 2D Random Field Ising Model

Communications in Mathematical Physics, 2019

As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small. This statement is quantified here by a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems, as function of the distance to the boundary. Unlike exponential decay which is only proven for strong disorder or high temperature, the power-law upper bound is established here for all field strengths and at all temperatures, including zero, for the case of independent Gaussian random field. Our analysis proceeds through a streamlined and quantified version of the Aizenman-Wehr proof of the Imry-Ma rounding effect.

Random quantum Ising chains with competing interactions

Physical Review E, 2005

In this paper we discuss the criticality of a quantum Ising spin chain with competing random ferromagnetic and antiferromagnetic couplings. Quantum fluctuations are introduced via random local transverse fields. First we consider the chain with couplings between first and second neighbors only and then generalize the study to a quantum analog of the Viana-Bray model, defined on a small world random lattice. We use the Dasgupta-Ma decimation technique, both analytically and numerically, and focus on the scaling of the lattice topology, whose determination is necessary to define any infinite disorder transition beyond the chain. In the first case, at the transition the model renormalizes towards the chain, with the infinite disorder fixed point described by Fisher. This corresponds to the irrelevance of the competition induced by the second neighbors couplings. As opposed to this case, this infinite disorder transition is found to be unstable towards the introduction of an arbitrary small density of long range couplings in the small world models.

Self-averaging in the random two-dimensional Ising ferromagnet

Physical Review E

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size L like ∼ L ln ln(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 → ∞. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.

Organization and energy properties of metastable states for the random-field Ising model

Journal of Magnetism and Magnetic Materials, 2010

Random-field Ising model (RFIM) systems are characterized by a large number of metastable states corresponding to local minima of the system energy with respect to single spin flip. We classified the minima in a hierarchical way based on the possibility of a given state to escape from a basin of mutually reachable states. We investigate the energy properties of the metastable states in relation to the basin they belong to: states of particularly high energy, obtained by fast-quenching randomly initial spin configurations, tend to have access to a complex structure of correlated basins, opposite to what is found for low-energy states. The purpose of this paper is to investigate the connection between the properties of the basin oriented graph and the energy of the corresponding states.

One-dimensional Ising model with long-range and random short-range interactions

Journal of Magnetism and Magnetic Materials, 1999

The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a "ferrimagnetic" phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points.

Critical Behavior of the Quenched Random Mixed-Spin Ising Model

International Journal of Modern Physics C

In this work, we simulated a quenched random mixed-spin Ising model on the square lattice. The model system consists of two different particles with spins σ = 1/2 (states ±1/2) and S = 1 (states ±1, 0). These particles are randomly distributed on the lattice, and we considered only nearest-neighbor interactions. This model can represent a random magnetic binary alloy AxB1-x, obtained from the high-temperature quenching of a liquid mixture. We performed Monte Carlo simulations for several lattice sizes and temperatures, and we found its critical temperature through the reduced fourth-order cumulant. We also determined the magnetization, the susceptibility, and the specific heat as a function of temperature. We used finite-size scaling arguments to estimate the critical exponents β, γ, and ν of the model. We showed that the quenched model is in the same universality class of the two-dimensional pure Ising model. We also investigated the sample to sample fluctuations that occur in the ...