Random Ising Model Research Articles

The diluted random field mixed spin Ising model: thermodynamical properties

The diluted mixed spin Ising system consisting of spin-1/2and spin-1 with a random field is studied by the use of the finite clusterapproximation within the framework of a single-site cluster theory. The stateequations are derived using a probability distribution method based on the use ofVan der Waerden identities. In this approach, the complete phase diagramsare investigated for the simple cubic lattice, when the random field isbimodally and trimodally distributed. The internal energy, specific heat andsusceptibility are also calculated for both kinds of probability distributions.

Short-time dynamics of a random Ising model with long-range interaction.

Short-time critical dynamics of a random Ising model (model A) with long-range interaction decaying as r(-(d+sigma)) (where sigma is the parameter controlling the range of the interaction), is studied by the theoretic renormalization-group approach. In dimensions d<2sigma, the initial slip exponents theta(') describing the initial increase of the order parameter, and theta for the growth of the response function, which govern the short-time scaling behaviors, are calculated to the second order in sqrt[epsilon] with epsilon=2sigma-d. The crossover between the long-range interaction and the short-range interaction, which occurs at some sigma>2, is also discussed.

Quantum annealing by the path-integral Monte Carlo method: The two-dimensional random Ising model

Quantum annealing was recently found experimentally in a disordered spin-$\frac{1}{2}$ magnet to be more effective than its classical, thermal counterpart. We use the random two-dimensional Ising model as a test example and perform on it both classical and quantum (path-integral) Monte Carlo annealing. A systematic study of the dependence of the final residual energy on the annealing Monte Carlo time quantitatively demonstrates the superiority of quantum relative to classical annealing in this system. In order to determine the parameter regime for optimal efficiency of the quantum annealing procedure we explore a range of values of Trotter slice number P and temperature T. This identifies two different regimes of freezing with respect to efficiency of the algorithm, and leads to useful guidelines for the optimal choice of quantum annealing parameters.

Random exchange-type electro-optic behavior of Pb0.865La0.09(Zr0.65Ti0.35)O3 relaxor ferroelectrics

Random exchange nature of Pb0.865La0.09(Zr0.65Ti0.35)O3 relaxor ferroelectrics was revealed by electro-optic study. Time decay behavior of birefringence was measured after switching off the external bias field, and showed random exchange-type behavior. Specific heat behavior was also obtained from the temperature derivatives of birefringence, and its critical exponent was α=−0.09 as predicted by the random exchange Ising model. We think the random exchange interaction is induced by nonpolar La ions and vacancies.

Nonlinear susceptibilities of a weakly disordered uniaxial ferromagnet in the critical region

For the three-dimensional random Ising model, the effective sextic coupling constant v_6 and the nonlinear susceptibilities of the fourth (chi_4) and sixth (chi_6) orders are calculated at criticality. These quantities are shown to differ markedly from their counterparts for pure uniaxial magnets. In particular, the ratio v_6/v_{4}^2 entering the equation of state of the random Ising model turns out to be equal to 0.87, while in pure magnets v_6/v_{4}^2 = 1.65. The universal susceptibility ratios m^3 chi_4/chi^2 and m^6 chi_6/chi^3 (m - the inverse correlation length) are found to differ by factors 1.6 and 2.7, respectively, for random and uniform Ising models. These big differences of the universal quantities can be measured both in physical and computer experiments, and such measurements may be considered as a tool for an identification of the random critical behavior.

Monte Carlo study of the magnetic properties of Fe0.9−qMn0.1Alq-disordered alloys

Within the framework of a random site-diluted Ising model with nearest-neighbor interactions, and using the Metropolis algorithm for equilibration and energy minimization, we have computed the ensemble and configurational averages for magnetization per site, magnetic susceptibility and specific heat of Fe 0.9− q Mn 0.1 Al q -disordered alloys with 0.1⩽ q ⩽0.55. In the model, atoms have been randomly distributed on a body-centered cubic lattice in order to simulate the disorder and structure as that obtained in arc-melted Fe 0.9− q Mn 0.1 Al q alloys treated at high temperatures during long periods of time and followed by fast quenching. Competitive interactions coming from Fe–Fe ferromagnetic bonds and Fe–Mn and Mn–Mn antiferromagnetic couplings, as well as the Al dilutor effect, have been taken into account in our study. Results allow us to conclude that, in agreement with previous Mössbauer data of the average hyperfine field, for which a comparison is also carried out, the Fe 0.9− q Mn 0.1 Al q -disordered alloys are well characterized by a critical concentration at room temperature at around 40 at% Al, for which the system undergoes a transition from a ferromagnetic state to a paramagnetic one. The finite size scaling analysis to obtain the critical Al concentration in the thermodynamic limit, as well as the critical exponents, is also presented and discussed.

Hysteretic behavior of Fe0.9−qMn0.1Alq-disordered alloys: a Monte Carlo study

In this work we have simulated the Fe 0.9− q Mn 0.1 Al q alloy series with Al contents ranging from 10 up to 50 at%, and for several system sizes. In the simulation, the atoms are randomly distributed on a body-centered cubic according to the atomic disorder achieved through quenching techniques for the experimental samples. In computing the thermodynamic quantities such as the magnetization per site as a function of an external applied magnetic field, we have employed a Monte Carlo algorithm based on a Metropolis dynamics implemented with a random site-diluted Ising model. In this model, we have taken into account nearest-neighbor interactions for which both the ferromagnetic (Fe–Fe) and antiferromagnetic (Fe–Mn, Mn–Mn) interactions are present. From the simulation of the hysteresis loops at room temperature, the remanence and the coercive force as a function of the Al concentration have been obtained. Finally, a comparison with the previous experimental data on coercivity obtained by means of vibrating sample magnetometry is also carried out and discussed.

Theory of Quantum Annealing of an Ising Spin Glass

Probing the lowest energy configuration of a complex system by quantum annealing was recently found to be more effective than its classical, thermal counterpart. By comparing classical and quantum Monte Carlo annealing protocols on the two-dimensional random Ising model (a prototype spin glass), we confirm the superiority of quantum annealing relative to classical annealing. We also propose a theory of quantum annealing based on a cascade of Landau-Zener tunneling events. For both classical and quantum annealing, the residual energy after annealing is inversely proportional to a power of the logarithm of the annealing time, but the quantum case has a larger power that makes it faster.

Fat and thin Fisher zeroes

We show that it is possible to determine the locus of Fisher zeroes in the thermodynamic limit for the Ising model on planar (“fat”) φ 4 random graphs and their dual quadrangulations by matching up the real part of the high- and low-temperature branches of the expression for the free energy. Similar methods work for the mean-field model on generic, “thin” graphs. Series expansions are very easy to obtain for such random graph Ising models.

Thermodynamical properties of the random field mixed spin Ising model

A theoretical study of a mixed spin Ising system consisting of spin-1/2 and spin-1 with a random field is studied by the use of the finite cluster approximation within the framework of a single-site cluster theory. In this approach, the phase diagrams and magnetizations are investigated for the simple cubic lattice when the random field is bimodally and trimodally distributed. The internal energy, specific heat and susceptibility are also calculated for both kinds of the probability distributions.

Dynamics of the Random Ising Model with Long-Range Interaction**The project supported by National Natural Science Foundation of China under Grant No. 19772074 and Advance Research Center of Zhongshan University

Critical dynamics of the random Ising model with long-range interaction decaying as (where is the dimensionality) is studied by the theoretic renormalization-group approach. The system is released to an evolution within a model dynamics. Asymptotic scaling laws are studied in a frame of the expansion in . In dimensions , the dynamic exponent is calculated to the second order in at the random fixed point.

Magnetism of Fe–Al disordered alloys: An Ising–Monte Carlo approach

Using a model of atoms randomly distributed on a cubic lattice, we have simulated the atomic disorder induced in quenched binary alloys. Our study has been developed within the framework of a random site-diluted Ising model with nearest-neighbor interactions and used a Monte Carlo algorithm implemented with Metropolis kinetics for sampling states. After equilibration, ensemble and configurational averages for magnetization, magnetic susceptibility, and heat capacity were computed. We conclude that, in agreement with previous experimental Mössbauer data for which a comparison is carried out, Fe–Al disordered alloys exhibit room-temperature ferromagnetic behavior up to around 43 at. % Al, beyond which the system becomes paramagnetic. This result contrasts with that for alloys with atomic order, which exhibit ferromagnetism only up to 30 at. % Al.

Dynamical response of quantum spin-glass models at t = 0.

We study the behavior of two archetypal quantum spin glasses at T = 0 by exact diagonalization techniques: the random Ising model in a transverse field and the random Heisenberg model. The behavior of the dynamical spin response is obtained in the spin-glass ordered phase. In both models it is gapless and has the general form chi(")(omega) = qdelta(omega)+chi(")(reg)(omega), with chi(")(reg)(omega) approximately omega for the Ising and chi(")(reg)(omega) approximately const for the Heisenberg, at low frequencies. The method provides new insight to the physical nature of the low-lying excitations.

Hysteresis and avalanches in the random anisotropy Ising model

The behavior of the random anisotropy Ising model at $T=0$ under local relaxation dynamics is studied. The model includes a dominant short-range ferromagnetic interaction and assumes an infinite anisotropy at each site along local anisotropy axes which are randomly aligned. As a consequence, some of the effective interactions become antiferromagneticlike and frustration appears. Two different random distributions of anisotropy axes have been studied. Both are characterized by a parameter that allows control of the degree of disorder in the system. By using numerical simulations we analyze the hysteresis loop properties and characterize the statistical distribution of avalanches occurring during the metastable evolution of the system driven by an external field. A disorder-induced critical point is found in which the hysteresis loop changes from displaying a typical ferromagnetic magnetization jump (large avalanche spanning a macroscopic fraction of the system) to a rather smooth loop exhibiting only tiny avalanches. The critical point is characterized by a set of critical exponents, which are consistent with the universal values proposed from the study of other simpler models.

Dynamics and transport in random quantum systems governed by strong-randomness fixed points

We present results on the low-frequency dynamical and transport properties of random quantum systems whose low temperature $(T),$ low-energy behavior is controlled by strong-disorder fixed points. We obtain the momentum- and frequency-dependent dynamic structure factor in the random singlet (RS) phases of both spin-1/2 and spin-1 random antiferromagnetic chains, as well as in the random dimer and Ising antiferromagnetic phases of spin-1/2 random antiferromagnetic chains. We show that the RS phases are unusual ``spin metals'' with divergent low-frequency spin conductivity at $T=0,$ and we also follow the conductivity through ``metal-insulator'' transitions tuned by the strength of dimerization or Ising anisotropy in the spin-1/2 case, and by the strength of disorder in the spin-1 case. We work out the average spin and energy autocorrelations in the one-dimensional random transverse-field Ising model in the vicinity of its quantum critical point. All of the above calculations are valid in the frequency-dominated regime $\ensuremath{\omega}\ensuremath{\gtrsim}T,$ and rely on previously available renormalization group schemes that describe these systems in terms of the properties of certain strong-disorder fixed-point theories. In addition, we obtain some information about the behavior of the dynamic structure factor and dynamical conductivity in the opposite ``hydrodynamic'' regime $\ensuremath{\omega}&lt;T$ for the special case of spin-1/2 chains close to the planar limit (the quantum $x\ensuremath{-}y$ model) by analyzing the corresponding quantities in an equivalent model of spinless fermions with weak repulsive interactions and particle-hole symmetric disorder.

A three-dimensional random Ising model: Resummation of five-loop series

A three-dimensional random Ising model: Resummation of five-loop series

Hysteresis and avalanches in disordered systems

Rate-independent hysteresis is studied in magnetic systems driven by an external field for which the influence of thermal fluctuations is negligible. In such systems, the hysteresis cycles are not continuous, but rather are composed of a sequence of magnetisation jumps or avalanches between metastable states; the so-called Barkhausen noise. The study of the statistical distribution of such avalanches provides an alternative description to the more common procedure of measuring properties of the loop shape. We focus on four different zero-temperature 3d lattice models: the random field Ising model, the random bond Ising model, the site-diluted Ising model and the random anisotropy Ising model. By defining appropriate local dynamics, we have studied the metastable evolution by numerical simulations. We analyse the avalanche size distribution as a function of the degree of quenched disorder in these systems. For specific amounts of disorder, the distributions exhibit critical behaviour that can be characterised by universal exponents.

Tricritical Points and Reentry in the Quantum Hopfield Neural-Network Model**The project supported by the Fifth Fok Ying-Tung Foundation and the Trans-Century Training Programme Foundation for the Talents by the State Education Commission of China

We examine a quantum Hopfield neural-network model in the presence of trimodal random transverse fields and random neuronal thresholds within the method of statistical physics. We use the Trotter decomposition to map the problem into an equivalent classical random Hopfield-type Ising model and obtain phase transitions between the ferromagnetic retrieval and the paramagnetic phases. The influence of competition between the diluted random transverse fields and the diluted random thresholds on the system is discussed, and some interesting results such as tricritical points and reentrance are analyzed.

Randomly dilute spin models: A six-loop field-theoretic study

We consider the Ginzburg-Landau MN-model that describes M N-vector cubic models with O(M)-symmetric couplings. We compute the renormalization-group functions to six-loop order in d=3. We focus on the limit N -> 0 which describes the critical behaviour of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model (M=1). We obtain for the critical exponents: gamma = 1.330(17), nu = 0.678(10), eta = 0.030(3), alpha=-0.034(30), beta = 0.349(5), omega = 0.25(10). For M > 1 we show that the O(M) fixed point is stable, in agreement with general non-perturbative arguments, and that no random fixed point exists.

Stability of a cubic fixed point in three dimensions: Critical exponents for genericN

The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Pad\'e-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to ${N}_{c}=2.89\ifmmode\pm\else\textpm\fi{}0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ensuremath{\varepsilon}$ expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for $N&gt;~3,$ and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when $N=0$ is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.