Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name

Computer Science


Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Rao, B. V. (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

In this introductory chapter, we begin with a brief description of spin glasses in section 1. We are not physicists. The purpose of this section is to trace the history of the models. Section 2 gives a brief summary of the thesis and section 3 recalls certain known facts which will be used later in the thesis.Origin of the problem The models considered in this thesis have their origin in spin glass theory. Roughly, spin glass is a glassy state in a spin system or a disordered material exhibiting high magnetic frustration. The origin of this behavior can be either a disordered structure (such as that of a conventional, chemical glass) or a disordered magnetic doping in an otherwise regular structure. But what is a glass? Loosely speaking, it is a state of spins with local ordering (in solid state physics, this is called local ‘freezing’ - locally, the system looks more like an ordered solid rather than a disordered liquid) but no global ordering. Spin glass can not remain in a single lowest energy state (the ground state). Rather it has many ground states which are never explored on experimental time scales. The freezing of the spins, in spin glasses, is not a deterministic one like ferromagnetic materials. Rather they freeze in random with some memory effect.Experiments show that the susceptibility obtained by cooling the spin glass system in the presence of a magnetic field yielded a higher value than that obtained by first cooling in zero field and then applying the magnetic field. If the spin glass is cooled below Tc (a certain critical temperature) in the absence of an external field, and then a magnetic field is applied, there is a rapid increase towards a value, called the zerofield-cooled magnetization. This value is less than the field-cooled magnetization. The following phenomenon has also been observed in the measurement of remanent magnetization (the permanent magnetization that remains after the external field is removed). We can cool in the presence of external field, remove the external field and then measure the remanent magnetization. Alternatively, first cool with out the external field, then apply the external field and measure the remanent magnetization after removing the external field. The first value is larger than the second one.The other peculiarity of the spin glasses is its time dependence, which will be explained now, that makes it different from other magnetic systems. Above the spin glass transition temperature, Tc, the spin glass exhibits typical magnetic behavior. In other words, at temperature above Tc, if an external magnetic field is applied and the magnetization is plotted versus temperature, it follows the typical Curie law (in which magnetization is inversely proportional to temperature). This happens until Tc is reached, at which point the magnetization becomes virtually constant. This is the onset of the spin glass phase. When the external field is removed, the spin glass has a rapid decrease of magnetization to a value called the remnant magnetization, and then a slow decay as the magnetization approaches zero (or some small fraction of the original value).


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