Thermodynamics and Statistical mechanics of loops: A Monte Carlo study

Abstract

Thermodynamics and Statistical Mechanics of Multi-colored Loop models in three dimensions: A Monte-Carlo study.

Soumya Kanti Ganguly

Department of Physics.

ABSTRACT

A perfectly crystalline solid is a regular arrangement of atoms with a given periodicity. However in reality, perfect solids do not exist but contain topological defects known as a dislocation which cannot be removed by any smooth deformation of the crystalline order parameter field. In three dimensions, a dislocation is characterized by the Burgers vector and the dislocation line vector. One can construct dyadic products of these two quantities to construct second rank tensors known as dislocation line density. Dislocations break translational symmetry, have long-range interactions and assume closed loop-like structures due to a continuity condition. They play a key role in the properties of crystalline solids. Hence, studies of their properties have been a subject of great interest for both physicists and materials scientists. Owing to long-range interactions among dislocations, analytic and numerical studies of the statistical mechanics of a collection of dislocation lines can be difficult. An alternative is to construct a dual model with short-range interactions. Duality is an important concept that is often used to bridge connections between two distinct statistical mechanical models. Since duality transformations are exact, if the original model exhibits a phase transition, so does its dual. Objects dual to dislocations in a three dimensional lattice also have loop structures, but with short-range interactions. The dual objects are symmetric tensor loops which give rise to a first-order melting transition. This is consistent with the observed transition in three dimensional crystalline solids. From the point of view of the nature (symmetric or non-symmetric) of the loops, one can now ask the following question, "Is the nature of the transition intimately related to the nature of the loops ?" In an attempt to answer this question, one can construct a tensor loop model which is non-symmetric in its indices and has short-range interactions. This model is similar to the dual model for vortices in the three dimensional XY model, but has three "colors' associated with the components of the Burgers vector. Contrary to the symmetric loop model, the dual model for vortices in the XY model is known to undergo a continuous phase transition. This suggests that the multi-color non-symmetric loop model also exhibits a continuous transition. Circumstances under which the non-symmetric loop model undergoes a first-order transition are explored and its equilibrium properties are studied using Monte Carlo simulations and finite-size scaling. In addition, the geometrical properties of the loops near the phase transition, leading to a percolation transition are studied in detail. The non-equilibrium process of the development of order in this model after a quench from a high temperature is also studied.