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Phase transitions in magnetic clusters and other finite systems

Heinrich Stamerjohanns

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

This cumulative dissertation consists of seven articles dealing with phase transitions of magnetic clusters or other finite systems. We show that phase transitions in finite systems, which can be experimentally seen only in floppy humps of the specific heat or the magnetic susceptibility, can be quantitatively described by an analysis of an optimized estimation of the probability distribution of the system states combined with Landau free energy, or by analyzing the partition function in the complex temperature plane, or other complex-time response functions such as the complex specific heat. We show that new efficient methods, such as the Ferrenberg analysis, not applicable to infinite systems, can be applied to finite systems, even if the systems have more than one external parameter. With an Optimized Multidimensional Data Analysis of histograms produced by canonical Monte Carlo simulations, we are able to compute the phase behaviour of finite magnetic systems over a wide temperature and magnetic field range just by determining observables at a few given temperatures and magnetic fields.

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This cumulative dissertation consists of seven articles dealing with phase transitions of magnetic clusters or other finite systems. We show that phase transitions in finite systems, which can be experimentally seen only in floppy humps of the specific heat or the magnetic susceptibility, can be quantitatively described by an analysis of an optimized estimation of the probability distribution of the system states combined with Landau free energy, or by analyzing the partition function in the complex temperature plane, or other complex-time response functions such as the complex specific heat. We show that new efficient methods, such as the Ferrenberg analysis, not applicable to infinite systems, can be applied to finite systems, even if the systems have more than one external parameter. With an Optimized Multidimensional Data Analysis of histograms produced by canonical Monte Carlo simulations, we are able to compute the phase behaviour of finite magnetic systems over a wide temperature and magnetic field range just by determining observables at a few given temperatures and magnetic fields.
The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.
In this project, the location of the phase transition in the two dimensional Ising model will be determined using Monte Carlo simulation with importance sampling. the magnetization per site [m ], energy per site [J ], magnetic susceptibility, specific heat of a Ferromagnetic materials are Calculated as a function of temperature T for 10´10, 20´ 20, 40´ 40 , 50´50 spin lattice interaction by using Monte Carlo Simulation of the 2D Ising Model for some experimental values of ferromagnetic materials such as Gadolinium Chloride ( ) 3 GdCl at Curie temperature c T B = 2.2 J k , and ferromagnetic thin film from Nickel ( ) i N growth on cooper (Cu)at Curie temperature c B T = 2.772 J k , in zero and nonzero magnetic field. It was noticed that above a certain temperature ( ) c T the material will be in a paramagnetic state, this will lead to that the average magnetization will be decrease and the average energy increase, while below that temperature ,it will be in a ferromagnetic state, and the average magnetization will increase and the average energy decrease. Moreover, above a certain temperature spontaneous magnetization will be zero
By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.
The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.
The aim of this contribution is to draw attention to some recent developments in the theory of critical phenomena and phase transitions that may be of interest to the magnetism community in that magnetic systems have provided a fertile testing ground for modern theoretical ideas concerning ordering and phase transitions while, on the other hand, many aspects of observed magnetic phase behavior have required sophisticated theory for their proper elucidation. The new works in question have been published or will appear shortly: accordingly, only an outline is presented here together with references to the main literature sources which, in turn, include fuller references to earlier work not explicitly mentioned. Four linked topics are addressed: (i) finite-size effects at first-order transitions; (ii) asymptotic degeneracy in transfer matrices, which is intimately related to long-range order; (iii) some consequences of the conformal covariance of fluctuations at a critical point in bulk and in finite systems, especially in two spatial dimensions; (iv) the discovery of ‘‘exactly soluble’’ two-dimensional models exhibiting two infinite classes of distinct critical and multicritical points.
We present extensive Monte Carlo simulations on a two-dimensional XY model with a modified form of interaction potential. Thermodynamic quantities other than energy, specific heat, etc. (such as magnetization, susceptibility, and fourth-order cumulant of magnetization) are obtained using multiple histogram reweighting of the data obtained from the simulations. We employ an approach which eliminates the need to construct two-dimensional histograms. This approach makes judicious use of computer memory as well as CPU time. Lee-Kosterlitz's method of finite size scaling for a first-order transition and analysis using Binder's cumulant method allow us to make an accurate determination of the transition temperature.