DYNAMICAL PROPERTIES OF SPIN..'GLASSES

Abstract

• The thesis is devoted to a detailed theoretical study of the thermodynamical and dynamical properties of spin glasses. The, thermodynamics of spin glasses ha,s been studied very intensively in the past 15 years. But the focus of this work has been the infinite-ranged model, for which a variety of fascinating ideas have emerged. In this thesis our interest is mainly in short-ranged spin glasses, The most powerful method for studying short-ranged spin glasses has been the Monte-Carlo simulations. Though this method has provided very useful and reliable information about the behaviour of spin glassess, it fails to provide the physical understanding of the phenomena in simple terms. With this viowi we have investiga-ted in details another kind of approximations that seems well suited to short-ranged spin glasses. This approximation is obtained when the number of spin components n is allowed to become infinitely large. It is much like the usual Hartree-Fock approximations though there are some differences which prove quite significant from a numerical computational point of view. The basic physical idea that this scheme allows us to investigate in detail is the one enunciated by Anderson quite some time ago. Working with the linearised meanfield theory of magnetism► he pointed out a cruicia.l difference between the phase transitions in uniform and disordered systems. Since in disordered system the interaction matrix can have localised v) eigenstates (modes) ► the nature of instability leading to condensation in these modes is very different. For example, if the highest modes which is most favourable for condensation in the linearised theory happens to be localised, the corres-ponding staggered susceptibility must got renormalised as con- densation cannot occur in localised modes. The nature of this renormal is at ion which occur through mode-mode interaction was physically demonstrated by Hertz p Fleishman and Anderson. However., these authors restricted themselves to qualitative features and to temperatures above the condensation point. We have reformulated their work.. through n 4c- limit in a form which is suitable for numerical work. We have also formulated a possible form for the theory below the condensation tempera-ture. We have then carried out some numerical work for two and three dimensional systems. However due to limitations on our computational facilities t our calculation are restricted to small system sizes only. We could not go to sizes where the interplay of various physical features of this problem should become clearly visible. Further our numerical work remains restricted to high temperatures only. At low temperatures' as in other numerical schemes e.g. Monte—Carlo or mean field equations: we are plagued by difficulties associated with multi-plycity of solutions. At high temperatures,we have calculated a variety of. linear susceptibilities and spin glass susceptibi-lity. The idea is that by examining various staggered suscepti-bilit$esi one should get ap idea about the ordering tendencies in the system. Furthermore., it should allow us to understand the nature of renormalisation that occurs for susceptibilities corresponding to localised modes. Though the limited calcula- tions on small systems do not give convencing conclusions, we feel that with the extension of computing facilities., which allow procession computations on large systems' one will get further useful insights into spin glass freezing process. The dynamics of spin glasses has also been studied extensively by mean field methods, Monte Carlo methods and field theoretic techniques. Most of the analytical work has been done on the soft-spin models., as on those models the machinary of classical statistical dynamics can be utilised. In this thesis 9 we have chosen to work with fixed length planar. and vector spins. Starting from Langevin equations for such systems► we have formulated the appropriate Fokker- .Planck equations for the distribution functions. These equations are in turn used to derive equations for the correlation functions (CF) and response functions (RF) . One way to deal with these equations, which are essentially a hierarchy of equations relating lower order 'CF' S with higher order ones' is to adopt a decoupling procedure which truncates this hierarchy. Using this decoupling at the lowest levels one recovers the result of Edwards and Anderson on viscosity renormalisation. Our treatment of vector spins shows how this procedure leads to a systematic derivation of the physical results of Edwards and Anderson.