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TY - THES AU - Kischel, Florian TI - Phase transitions in classical systems : anisotropic models, computational methods, and universality predictions PB - RWTH Aachen University VL - Dissertation CY - Aachen M1 - RWTH-2024-05870 SP - 1 Online-Ressource : Illustrationen PY - 2024 N1 - Veröffentlicht auf dem Publikationsserver der RWTH Aachen University N1 - Dissertation, RWTH Aachen University, 2024 AB - This thesis is mostly concerned with phase transitions in classical systems, with a focus on the anisotropic Ising model in two dimensions and on parallelogram lattices. After first discussing the history of phase transitions and specifically how anisotropies were treated in the renormalization group approach, an introduction to multi-parameter universality is given. This is followed by a derivation, using anti-commuting Grassmann variables, of the exact solution of the fully anisotropic 2d Ising model on a finite parallelogram lattice for all temperatures and couplings; from there, the scaling function near the critical point in the ferromagnetic regime is recovered. Additionally, some predictions made by multi-parameter universality regarding non-universal prefactors, modular invariance and behavior at criticality are confirmed. Finally, the strip limit of the model is discussed and connections to previous results of more restricted cases are made. In the next chapter, the investigation of anisotropic systems in 2d is continued, now by discussing the q-state Potts model and attempting to measure its angle dependent correlation lengths, a characterizing quantity according to multi-parameter universality, via an tensor network approach. More specifically, the Corner Transfer Matrix Renormalization Group (CTMRG) algorithm is used to numerically extract the quantities of interest. A range of checks and comparisons to the few exactly known results are made to ensure a continued high accuracy of the simulation method. Finally, the discrete to continuous crossover behavior in a modified 3d clock model, a relative of the Potts model, is investigated. This model exhibits a first order phase transition between an ordered and disordered phase and, based on prior work, predictions can be made for how much the phases contribute at the transition point when the clock has either three different states or, on the other extreme, infinitely many. This behavior is simulated at and between these extremes using the Wang-Landau Monte Carlo algorithm, which is very well suited for systems that exhibit complicated energy distributions, as present near and at first order phase transitions. A wide range of system sizes are simulated and care is taken to carefully determine the bulk transition temperature on which the accuracy of the final results depends very crucially. LB - PUB:(DE-HGF)11 DO - DOI:10.18154/RWTH-2024-05870 UR - https://publications.rwth-aachen.de/record/987839 ER -