Many important concepts used in the study of complex systems have their origin in the lattice statistical mechanics of cooperative phase transitions. The classic order-disorder transitions of Potts/Ising models illustrate emergent phenomena such as spontaneous symmetry breaking, scale-free critical behaviour and power-law singularities. However, a class of lattice statistics models show that rather more complex behaviour can be obtained, under conditions which are suffciently general that they might also occur in less idealised models. A 5-state lattice statistics model is analysed to explore behaviour in and around a so-called "massless phase". The massless phase, characterised by power law decay of correlations, appears as an intermediate regime between high temperature disorder and low-temperature ordering. The critical exponents (and fractal spatial distributions) occur over a finite range of temperatures. The state of the massless phase is characterised by a topological ordering with analogues of spin-waves being the dominant perturbations. The transition from the ordered to massless phases is analysed using new exact series expansions obtained by the finite lattice method. The results are also compared to Monte Carlo simulations. The results are compared to other studies in the small number of cases where such series expansions exist for comparable models, including the 6-state "clock model". One of the properties that makes the massless phase dificult to study is the weak nature of the associated Kosterlitz-Thouless transition with exponentially-weak behaviour rather than power-law behaviour. As a further diffculty, the limiting behaviour is confined to a narrow regime, outside which one sees an apparent "cross-over" to power law behaviour. This suggests that behaviour analogous to the "massless phases" will be diffcult to characterise as one moves beyond idealised lattice systems.
Many important concepts used in the study of complex systems have their origin in the lattice statistical mechanics of cooperative phase transitions. The classic order-disorder transitions of Potts/Ising models illustrate emergent phenomena such as spontaneous symmetry breaking, scale-free critical behaviour and power-law singularities. However, a class of lattice statistics models show that rather more complex behaviour can be obtained, under conditions which are suffciently general that they might also occur in less idealised models. A 5-state lattice statistics model is analysed to explore behaviour in and around a so-called "massless phase". The massless phase, characterised by power law decay of correlations, appears as an intermediate regime between high temperature disorder and low-temperature ordering. The critical exponents (and fractal spatial distributions) occur over a finite range of temperatures. The state of the massless phase is characterised by a topological ordering with analogues of spin-waves being the dominant perturbations. The transition from the ordered to massless phases is analysed using new exact series expansions obtained by the finite lattice method. The results are also compared to Monte Carlo simulations. The results are compared to other studies in the small number of cases where such series expansions exist for comparable models, including the 6-state "clock model". One of the properties that makes the massless phase dificult to study is the weak nature of the associated Kosterlitz-Thouless transition with exponentially-weak behaviour rather than power-law behaviour. As a further diffculty, the limiting behaviour is confined to a narrow regime, outside which one sees an apparent "cross-over" to power law behaviour. This suggests that behaviour analogous to the "massless phases" will be diffcult to characterise as one moves beyond idealised lattice systems.
Many important concepts used in the study of complex systems have their origin in the lattice statistical mechanics
of cooperative phase transitions. The classic order-disorder transitions of Potts/Ising models illustrate
emergent phenomena such as spontaneous symmetry breaking, scale-free critical behaviour and power-law singularities.
However, a class of lattice statistics models show that rather more complex behaviour can be obtained,
under conditions which are suffciently general that they might also occur in less idealised models.
A 5-state lattice statistics model is analysed to explore behaviour in and around a so-called phase.
The massless phase, characterised by power law decay of correlations, appears as an intermediate regime between
high temperature disorder and low-temperature ordering. The critical exponents (and fractal spatial distributions)
occur over a finite range of temperatures. The state of the massless phase is characterised by a topological
ordering with analogues of spin-waves being the dominant perturbations.
The transition from the ordered to massless is analysed using new exact series expansions obtained by
the finite lattice method. The results are also compared to Monte Carlo simulations. The results are compared
to other studies in the small number of cases where such series expansions exist for comparable models, including
the 6-state clock model.
One of the properties that makes the massless phase dificult to study is the weak nature of the associated
Kosterlitz-Thouless transition with exponentially-weak behaviour rather than power-law behaviour. As a further
diffculty, the limiting behaviour is confined to a narrow regime, outside which one sees an apparent cross-over
to power law behaviour. This suggests that behaviour analogous to the phases will be diffcult to
characterise as one moves beyond idealised lattice systems.
The Ising and Q-state Potts models are statistical mechanical models of spins interaction on crystal lattices. \nWe study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena − how higher level structure is affected by the detailed activity of a very large number of lower level structures. \nWe investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. \nOur study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. \nHere we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. \nThe partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. \nWe extend the study of Q-state Potts models to a more general case called the ZQ-symmetric model. \nWe evidence the existence of multiple phase transitions for this model in case Q=5,6, and discuss the possible connection to different stages of disordered state. \nGiven sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. \nOur analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. \nWe evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent alpha.
The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.
We have examined the critical properties of the q=6 clock (vector Potts) model in two dimensions through Monte Carlo simulations. The model was investigated on L\ifmmode\times\else\texttimes\fi{}L square lattices of size L=4 to L=72 with periodic boundary conditions. We found an intermediate XY-like phase between a low-temperature ordered phase and a high-temperature disordered phase. The phase transitions occur at kBT1/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and kBT2/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at T1 and 0.275 at T2.
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.
Clearly, in nature, but also in technological applications, complex systems built in an entirely ordered and regular fashion are the exception rather than the rule. In this thesis we explore how critical phenomena are influenced by quenched spatial randomness. Specifically, we consider physical systems undergoing a continuous phase transition in the presence of topological disorder, where the underlying structure, on which the system evolves, is given by a non-regular, discrete lattice. We therefore endeavour to achieve a thorough understanding of the interplay between collective dynamics and quenched randomness. According to the intriguing concept of universality, certain laws emerge from collectively behaving many-body systems at criticality, almost regardless of the precise microscopic realization of interactions in those systems. As a consequence, vastly different phenomena show striking similarities at their respective phase transitions. In this dissertation we pursue the question of whether the universal properties of critical phenomena are preserved when the system is subjected to topological perturbations. For this purpose, we perform numerical simulations of several prototypical systems of statistical physics which show a continuous phase transition. In particular, the equilibrium spin-1/2 Ising model and its generalizations represent -- among other applications -- fairly natural approaches to model magnetism in solids, whereas the non-equilibrium contact process serves as a toy model for percolation in porous media and epidemic spreading. Finally, the Manna sandpile model is strongly related to the concept of self-organized criticality, where a complex dynamic system reaches a critical state without fine-tuning of external variables. Our results reveal that the prevailing understanding of the influence of topological randomness on critical phenomena is insufficient. In particular, by considering very specific and newly developed lattice structures, we are able to show that -- contrary to the popular opinion -- spatial correlations in the number of interacting neighbours are not a key measure for predicting whether disorder ultimately alters the behaviour of a given critical system.