One-dimensional Ferromagnetic Ising Model Research Articles

Susceptibility Critical Exponent of a 1D Ising Ring-Type Ferromagnetic

Using the Monte Carlo method, critical behavior of the one-dimensional ferromagnetic Ising model has been investigated with allowance for the interaction of the second and third neighbors and four-particle interaction. The obtained results on the critical temperature were compared with the critical temperature of the quasi-one-dimensional Ising magnetic [(СН3)3NH] · FeCl3 · 2H2O and with the magnitude of the exchange interaction J/kB = 17.4 K. Within the scope of the finite-dimensional scaling theory, the critical susceptibility exponent has been calculated. It has been shown that values of the susceptibility exponent for the one-dimensional Ising model with periodic boundary conditions are considerably less than the known values of the exponents for three-dimensional systems. The critical susceptibility exponent strongly depends on energy parameters; namely, it decreases with an increase in them.

Критический индекс восприимчивости 1D-изинговского ферромагнетика, замкнутого в кольцо

AbstractUsing the Monte Carlo method, critical behavior of the one-dimensional ferromagnetic Ising model has been investigated with allowance for the interaction of the second and third neighbors and four-particle interaction. The obtained results on the critical temperature were compared with the critical temperature of the quasi-one-dimensional Ising magnetic [(СН_3)_3NH] · FeCl_3 · 2H_2O and with the magnitude of the exchange interaction J/k _B = 17.4 K. Within the scope of the finite-dimensional scaling theory, the critical susceptibility exponent has been calculated. It has been shown that values of the susceptibility exponent for the one-dimensional Ising model with periodic boundary conditions are considerably less than the known values of the exponents for three-dimensional systems. The critical susceptibility exponent strongly depends on energy parameters; namely, it decreases with an increase in them.

Third law of thermodynamics as a key test of generalized entropies.

The laws of thermodynamics constrain the formulation of statistical mechanics at the microscopic level. The third law of thermodynamics states that the entropy must vanish at absolute zero temperature for systems with nondegenerate ground states in equilibrium. Conversely, the entropy can vanish only at absolute zero temperature. Here we ask whether or not generalized entropies satisfy this fundamental property. We propose a direct analytical procedure to test if a generalized entropy satisfies the third law, assuming only very general assumptions for the entropy S and energy U of an arbitrary N-level classical system. Mathematically, the method relies on exact calculation of β=dS/dU in terms of the microstate probabilities p(i). To illustrate this approach, we present exact results for the two best known generalizations of statistical mechanics. Specifically, we study the Kaniadakis entropy S(κ), which is additive, and the Tsallis entropy S(q), which is nonadditive. We show that the Kaniadakis entropy correctly satisfies the third law only for -1<κ<+1, thereby shedding light on why κ is conventionally restricted to this interval. Surprisingly, however, the Tsallis entropy violates the third law for q<1. Finally, we give a concrete example of the power of our proposed method by applying it to a paradigmatic system: the one-dimensional ferromagnetic Ising model with nearest-neighbor interactions.

Diffusive-Ballistic Transition in Random Polymers with Drift and Repulsive Long-Range Interactions

This paper leads with a random polymer model in $\Z^2$ having long-range self-repulsive interactions. By comparison with a long range one-dimensional ferromagnetic Ising model we shown that the polymer models we considered here undergo a phase transition in terms of the inverse temperature $\beta$. In the second part of this work we shown, using the Lee-Yang Circle Theorem, that our random polymer model with drifts satisfies the, Wu Liming [7], $C^2$ regularity condition. As consequence we obtain a Central Limit Theorem for the model.

One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point

We consider the phase separation problem for the one-dimensional ferromagnetic Ising model with long-range two-body interaction, J(n) = n −2+α , where $${n\in {\rm {I\!N}}}$$ denotes the distance of the two spins and $${\alpha \in [0,\alpha_+[}$$ with α + = (log 3)/(log 2) −1. We prove that when α = 0 the localization of the phase separation fluctuates macroscopically with a non-uniform explicit limiting law, while when 0 < α < α + the macroscopic fluctuations disappear and mesoscopic ones appear with a gaussian behavior when conveniently scaled. The mean magnetization profile is also given.

GROUND STATES OF ONE-DIMENSIONAL LONG-RANGE FERROMAGNETIC ISING MODEL WITH EXTERNAL FIELD

A zero-temperature phase-diagram of the one-dimensional ferromagnetic Ising model is investigated. It is shown that at zero temperature spins of any compact collection of lattice points with identically oriented external field are identically oriented.

The one-dimensional long-range ferromagnetic Ising model with a periodic external field

We consider the one-dimensional ferromagnetic Ising model with very long-range interaction under a periodic, biased and weak external field and prove that at sufficiently low temperatures the model has a unique limiting Gibbs state.

ONE-DIMENSIONAL LONG-RANGE FERROMAGNETIC ISING MODEL UNDER WEAK AND SPARSE EXTERNAL FIELD

We consider the one-dimensional ferromagnetic Ising model with very long range interaction under weak and sparse biased external field and prove that at sufficiently low temperatures, the model has a unique limiting Gibbs state.

THE STEADY STATE OF THE TAPPED ISING MODEL

We consider a tapping dynamics, analogous to that in experiments on granular media, on the simple one-dimensional ferromagnetic Ising model. When unperturbed, the system undergoes a single spin flip falling dynamics where only energy lowering moves occur. With this dynamics the system has an exponentially large number of metastable states and gets stuck in blocked or jammed configurations as do granular media. When stuck, the system is tapped, in order to make it evolve, by flipping in parallel each spin with probability p (corresponding to the strength of the tapping). Under this dynamics the system reaches a steady state regime characterized by an asymptotic energy per spin E(p), which is determined analytically. Within the steady state regime we compare certain time averaged quantities with the ensemble average of Edwards based on a canonical measure over metastable states of fixed average energy. The ensemble average yields results in excellent agreement with the dynamical measurements.

COMPETING DYNAMICAL BEHAVIOR OF THE ISING MODEL IN ONE DIMENSION

We have studied the behavior of the one-dimensional ferromagnetic Ising model in contact with a heat bath and subject to an external source of energy. The contact with the heat bath is simulated by a process of Glauber type, while the continuous flux of energy into the system by a process of Kawasaki type. We show, within the dynamical pair approximation that the phase diagram exhibits a line of continuous nonequilibrium transitions between the paramagnetic and antiferromagnetic phases. However, detailed Monte Carlo simulations on the same model show clearly that the only stationary state is the paramagnetic one, whatever is the value of the competition parameter between the Glauber and Kawasaki dynamics.

Critical dynamics of the open Ising chain.

The one-dimensional ferromagnetic Ising model with open boundary conditions within the Glauber dynamics is studied. From the behavior of the order parameter, the explicit scaling form of the relaxation time is obtained. For finite systems, the dynamical critical exponent is z=1, in contrast to the value of the infinite open chain (z=2).

Yang-Lee edge singularity in the one-dimensional long-range Ising model

The finite-range scaling method has been used to study the Yang-Lee edge singularity problem for the one-dimensional ferromagnetic Ising model with long-range interactions decaying with distance r, as r-d- sigma . The phase diagram and the critical exponent nu have been calculated for different values of sigma by using finite ranges L<or=9. The critical value sigma c dividing the long-range from the short-range critical regime has been discussed.

A bromide analog of a one-dimensional Ising ferromagnet: trimethylammonium catena-bis(.mu.-bromo)diaquoiron(II) bromide

The crystal structure of the title compound, FeTAB, has been determined by X-ray diffraction and has been found to be isomorphous to that of the chloride analogue, FeTAC. The low-temperature magnetic susceptibility of polycrystalline samples can be modeled by using a one-dimensional ferromagnetic Ising model with a molecular field correction. The exchange parameters are estimated to be J/k=4 K and z'J'/k=−0.2 K, indicating that FeTAB has an intrachain exchange constant only one-fourth of that found in the chloride analogue while the interchain interactions are more than 10-time greater. Antiferromagnetic ordering is observed in the Mossbauer spectra near 1.3 K

Critical behavior of the ferromagnetic ising chain with interactionJ ij =|i?j|?2

High-temperature expansions of the susceptibility and specific heat are calculated to the eighth and ninth orders, respectively, in the reciprocal temperature for the one-dimensional ferromagnetic Ising model with interaction that decreases inversely as the square of the distance between the lattice sites. An analysis is performed of the expansion of the susceptibility that indicates that at the critical point its logarithm diverges in accordance with a power law exponent. This estimate is close to the value obtained by the renormalization-group method and is the first numerical confirmation of the predictions of the renormalization-group approach for systems with long-range interaction.

Exact solution of a one-dimensional Ising model in a random magnetic field

Analytic results for the free energy, magnetic structure factor, and Edwards-Anderson order parameter of a one-dimensional ferromagnetic Ising model in a random magnetic field are obtained. The structure factor consists of both Lorentzian and Lorentzian-squared terms at all temperatures ($T$) greater than zero; the Lorentzian-squared terms vanish at $T=0$. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.

More surprises in the general theory of lattice systems

I use Israel's methods to prove new theorems of “ubiquitous pathology” for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ϱ be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φ k } of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ϱ k , such that {ϱ k } converges to ϱ. In certain situations the perturbations Φ k −Φ can be chosen to lie in a cone of “antiferromagnetic pair interactions.” I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r 2 interaction (Thouless effect).