Using the renormalization-group technique in the expansion of \ensuremath{\epsilon} up to first order, where \ensuremath{\epsilon}=4-d and d is the dimension, we calculated the critical exponents of a mixture of disordered spins (quenched) composed of two different species of magnetic atoms A and B (${A}_{x}$${B}_{y}$${C}_{1\mathrm{\ensuremath{-}}x\mathrm{\ensuremath{-}}y}$). We observed that the two different magnetic momentums of this mixture couple with each other so that the field of spins which have intense coupling orders itself while the field of spins which have weak interaction remains disordered. The results show that the disordered systems are universal in the sense that they have the same critical exponents although there are additional fixed points.
Using the renormalization-group technique in the expansion of \ensuremath{\epsilon} up to first order, where \ensuremath{\epsilon}=4-d and d is the dimension, we calculated the critical exponents of a mixture of disordered spins (quenched) composed of two different species of magnetic atoms A and B ({A}_{x}$${B}_{y}$${C}_{1\mathrm{\ensuremath{-}}x\mathrm{\ensuremath{-}}y}). We observed that the two different magnetic momentums of this mixture couple with each other so that the field of spins which have intense coupling orders itself while the field of spins which have weak interaction remains disordered. The results show that the disordered systems are universal in the sense that they have the same critical exponents although there are additional fixed points.
Changes in magnetic critical behaviour of quenched structurally-disordered magnets are usually exemplified in experiments and in MC simulations by diluted systems consisting of magnetic and non-magnetic components. In our study we aim to show that similar effects can be observed not only for diluted magnets with non-magnetic impurities but may be implemented, e.g., by the presence of two (and more) chemically different magnetic components as well. Therefore we consider a model of the structurally-disordered quenched magnet where all lattice sites are occupied by Ising-like spins of different lengths L. In such a random spin length Ising model, the length L of each spin is a random variable governed by the distribution function p(L). We demonstrate that this model belongs to the universality class of the site-diluted Ising model. This proves that both models are described by the same values of asymptotic critical exponents. However, their effective critical behaviour differs. As a case study, we consider a quenched mixture of two different magnets with values of elementary magnetic moments L1=1 and L2=s, and of concentration c and 1−c, correspondingly. We apply field-theoretical renormalization group approach to analyse the renormalization group flow for different initial conditions, triggered by s and c, and to calculate effective critical exponents further away from the fixed points of the renormalization group transformation. We show how the effective exponents are governed by difference in properties of the magnetic components.
The large scale behavior of the simplest non-mean-field spin-glass system is analyzed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the ε-expansion technique with two independent methods. The techniques presented show how the underlying ideas of the renormalization group apply also in this disordered model, in such a way that an ε-expansion can be consistently set up. By pushing such calculation to high orders in ε, a consistent non-mean-field theory for such disordered system could be established, giving a substantial contribution the development of a predictive theory for real spin glasses.
"Renorma1ization-group methods have been applied in the study of quenched random magnetic systems in recent years. We begin with a brief review of second-order phase transitions in pure, homogeneous systems and also of the .. renorma1ization group framework. Then we provide an introduction to quenched random magnetic systems. Next, momentum-space methods and position-space techniques as applied to quenched random magnets are outlined and compared. Grinstein and Luther applied the Wilson-Fisher E-expansion to random n-vector models; Khme1'nitsky discovered that the random Ising model (n = 1) possessed a ""random"" fixed point of 0(squareroot(e)). This fixed point was found to have one marginal and one irrelevant operator. We have investigated the stability of this fixed point using Ca11an-Symanzik equations and renorma1ized perturbation theory. We find the fixed point stable in the next order; we have also obtained critical exponents to one higher order. Next, position-space techniques are used to study some simple model systems. In addition to critical exponents, global thermodynamic properties are determined. These calculations are based on the Migda1-Kadanoff approximate recursion relations suitably generalized to the inhomogeneous case. Firstly we study the randomly bond-dilute two-dimensional nearest- neighbor Ising model on a square lattice. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold Pc. This provides a possible explanation for the apparent specific-heat rounding in certain experiments. We then study the Edwards-Anderson model of a spin glass. The current theoretical situation, which is far from satisfactory at present, is briefly reviewed. We treat the spin-l/2 Ising model with independently random nearestneighbor interactions in dimensionalities d = 2, 3, and 4. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spinglass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific heat anomaly. Global specific-heat curves are also displayed for d = 2."
In a recent work [M. Castellana and G. Parisi, Phys. Rev. E 82, 040105(R) (2010)], the large-scale behavior of the simplest non-mean-field spin-glass system has been analyzed, and the critical exponent related to the divergence of the correlation length has been computed at two loops within the ε-expansion technique by two independent methods. By performing the explicit calculation of the critical exponents at two loops, one obtains that the two methods yield the same result. This shows that the underlying renormalization group ideas apply consistently in this disordered model, in such a way that an ε-expansion can be set up. The question of the extension to high orders of this ε-expansion is particularly interesting from the physical point of view. Indeed, once high orders of the series in ε for the critical exponents are known, one could check the convergence properties of the series, and find out if the ordinary series resummation techniques, yielding very accurate predictions for the Ising model, work also for this model. If this is the case, a consistent and predictive non-mean-field theory for such a disordered system could be established. In that regard, in this work we expose the underlying techniques of such a two-loop computation. We show with an explicit example that such a computation could be quite easily automatized, i.e., performed by a computer program, in order to compute high orders of the ε-expansion, and so eventually make this theory physically predictive. Moreover, all the underlying renormalization group ideas implemented in such a computation are widely discussed and exposed.
This report will study the theory of renormalization group used on a ferromagnet to calculate magnetic phase transitions. As the ferromagnet's temperature changes, so does interatomic interaction, causing either a lingering magnetization at low temperatures, or a random state at high temperatures. At the phase transition, where the magnetization first appears/disappears, depending from which side the critical temperature is approached, the ferromagnet will behave in a certain way that causes some quantities to receive power law behaviours, so called critical exponents. The theory of renormalization group is used to find the critical point and these power units on two different two-dimensional lattices, the square lattice and the triangular lattice. The first step is to study how atoms couple to each other in one-dimension. It is possible to group atoms together into blocks. These blocks will interact with each other through new couplings. It is possible to calculate the critical point and critical exponents by comparing the block couplings with the former atomic couplings and then find the exponents. The theory of renormalization does seem to work in general, as the result will be able to explain experimentally observed relation between exponents.
We review and compare different approaches for studying the influence of quenched, random disorder in three-dimensional Ising and Potts models for ferromagnets subject to impurities.From a theoretical view point, field theoretic renormalization group studies provide quite accurate results.Experiments carried out on crystalline mixtures of compounds lead to measurements of criticial exponents as accurate as three digits.Numerically, extensive Monte Carlo simulations are shown to be of comparable accuracy.Finally, we also discuss recently generated high-temperature series expansions for the free energy and susceptibility.Within this approach, using the star-graph expansion technique, quenched disorder averages can be calculated exactly while keeping the disorder strength p as well as the dimension d as symbolic parameters.