The Berezinskii-Kosterlitz-Thouless transition of mobile Hamiltonian polar particles

Abstract

The phase of matter is the state of a system in a region on a macroscopic scale. At this macroscopic scale, in equilibrium, physical properties of the system are homogeneous. Statistical physics tries to explain the homogeneity and the associated properties from microscopic interactions and by laws of nature. On a macroscopic scale, the systems to investigate are many-body systems like gases, liquids or solids. They are characterized by thermodynamic functions like the free energy. Order parameters, like for example the magnetization in magnetic systems, play a special role here. Most of the time they are associated with a breaking of symmetry when transitioning from one phase to another. This however is not a universal observation. One example for which this is not the case is the Berezinskii-Kosterlitz-Thouless phase transition (KT-transition). This particular transition can be studied in two-dimensional XY-models, where there is a phase transition without the breaking of continuous symmetries. First theoretical descriptions trace back to the works of Berezinskii [Ber71], as well as Kosterlitz and Thouless [Kos73, Kos74, Kos16]. They initially investigated a system of classical spins fixed on the nodes of a square lattice. Consequently, only the rotational degree of freedom remains. In this work we start with the model of [LB16] who introduces a coupling of spins to velocities of mobile particles. After a numerical analysis of different integration methods for the equations of motion and investigations of this system, we remove the coupling. By that, we mimic an XY- model with additional translational degrees of freedom. For this system we examine the system with respect to the occurrence of the KT-transition by computer simulations. We apply several general tools for the analysis of magnetic systems. Subsequently we look into topological defects in the system, namely vortices and anti-vortices. For temperatures lower than the transition temperature T < T KT , defects are closely bound together, while for temperatures T > T KT they can proliferate in the system and can be treated like a gas of unbound defects. Moreover, we investigate spatial and temporal spin-correlations to estimate the critical exponent in the system, as well as the critical temperature. For a reference we first simulate an XY-model on a trigonal lattice as a comparison for our results in the mobile model. In both systems we do find the KT-transition by the occurrence of unbound topological defects and from the critical exponents we calculate from the spin-correlations functions.