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.
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.
The Voronoi construction is ubiquitous across the natural sciences and engineering. In statistical mechanics, however, only its dual, the Delaunay triangulation, has been considered in the investigation of critical phenomena. In this paper we set to fill this gap by studying three prominent systems of classical statistical mechanics, the equilibrium spin-1/2 Ising model, the nonequilibrium contact process, and the conserved stochastic sandpile model on two-dimensional random Voronoi graphs. Particular motivation comes from the fact that these graphs have vertices of constant coordination number, making it possible to isolate topological effects of quenched disorder from node-intrinsic coordination number disorder. Using large-scale numerical simulations and finite-size scaling techniques, we are able to demonstrate that all three systems belong to their respective clean universality classes. Therefore, quenched disorder introduced by the randomness of the lattice is irrelevant and does not influence the character of the phase transitions. We report the critical points to considerable precision and, for the Ising model, also the first correction-to-scaling exponent.
The concept of non-trivial electronic topology in condensed matter physics is crucial for understanding the electronic spectrum of solids and for tailoring various thermodynamic and transport properties, useful for real-world applications. Research on topological insulators and superconductors primarily utilizes the electronic band theory and symmetry classes of gapped quantum systems, with the existence of spectral or mobility gaps ensured even in the presence of impurities and defects. However, structural disorder, naturally manifested in materials such as glasses and amorphous solids, can significantly impact the critical properties of phase transitions in topological quantum matters. This dissertation systematically studies topological phase transitions in non-crystalline free-fermion systems via tight-binding quantum Hamiltonians on 2D random geometries. Leveraging various theoretical and experimentally accessible measures, we demonstrate the resilience of topological phases in non-crystalline systems against structural randomness, described by classical percolation theory, and characterize their unique critical behavior through finite-size scaling. We also extend the notion of amorphous topological states to non-integer dimensional random systems. We reveal that these 2D-embedded random fractals can admit well-defined thermodynamic topological phases with unconventional critical properties. We further study quantum dynamics in these random topological systems. By employing a family of discrete-time topological quantum walks on random percolation lattices, we address wavefunction propagation, superdiffusive quantum speedup, and localization transition by identifying the critical probabilities and timescales. Lastly, we introduce topological lattices with adjustable spatial correlation to investigate Landau levels and quantum oscillations in a low magnetic field regime.
Disordered systems arise in many physical contexts. Not all matter is uni- form, and impurities or heterogeneities can be modeled by fixed random disor- der. Numerous complex networks also possess fixed disorder, leading to appli- cations in transportation systems [1], telecommunications [2], social networks [3, 4], and epidemic modeling [5], to name a few. Due to their random nature and power law critical behavior, disordered systems are difficult to study analytically. Numerical simulation can help overcome this hurdle by allowing for the rapid computation of system states. In order to get precise statistics and extrapolate to the thermodynamic limit, large systems must be studied over many realizations. Thus, innovative al- gorithm development is essential in order reduce memory or running time requirements of simulations. This thesis presents a review of disordered systems, as well as a thorough study of two particular systems through numerical simulation, algorithm de- velopment and optimization, and careful statistical analysis of scaling proper- ties. Chapter 1 provides a thorough overview of disordered systems, the his- tory of their study in the physics community, and the development of tech- niques used to study them. Topics of quenched disorder, phase transitions, the renormalization group, criticality, and scale invariance are discussed. Several prominent models of disordered systems are also explained. Lastly, analysis techniques used in studying disordered systems are covered. In Chapter 2, minimal spanning trees on critical percolation clusters are studied, motivated in part by an analytic perturbation expansion by Jackson and Read [6] that I check against numerical calculations. This system has a direct mapping to the ground state of the strongly disordered spin glass [7]. We compute the path length fractal dimension of these trees in dimensions d = {2, 3, 4, 5} and find our results to be compatible with the analytic results suggested by Jackson and Read. In Chapter 3, the random bond Ising ferromagnet is studied, which is es- pecially useful since it serves as a prototype for more complicated disordered systems such as the random field Ising model and spin glasses. We investigate the effect that changing boundary spins has on the locations of domain walls in the interior of the random ferromagnet system. We provide an analytic proof that ground state domain walls in the two dimensional system are de- composable, and we map these domain walls to a shortest paths problem. By implementing a multiple-source shortest paths algorithm developed by Philip Klein [8], we are able to efficiently probe domain wall locations for all possible configurations of boundary spins. We consider lattices with uncorrelated dis- order, as well as disorder that is spatially correlated according to a power law. We present numerical results for the scaling exponent governing the probabil- ity that a domain wall can be induced that passes through a particular location in the system’s interior, and we compare these results to previous results on the directed polymer problem.
Disorder is an inevitable part of any condensed matter system and therefore its study has always been of great importance. The effect of quenched randomness on a system that exhibits a continuous phase transition in the absence of any impurity has been studied in the past and the results are relatively well understood. However, the effect of quenched randomness on \\emph{first-order} phase transitions is still not well understood. In this dissertation, we study the effect of quenched bond-randomness on the classical and quantum first-order phase transitions.In Chapter 2, we study the effect of the disordered three-color Ashkin-Teller model, whose pure version undergoes a first-order phase transition. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as a critical point. We conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model. Furthermore, we find that the exponents beta and nu vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to 128times128 spins, each of which is represented by three colors taking values pm1, we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents, nu and beta, change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound 2/Dlenu, where D is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in (1+1) dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.
Emergence in systems of many electrons can lead to macroscopic demonstrations of quantum mechanics which are intrinsically many-body. A primary focus of this dissertation is the phenomena of fractionalization, where the effective quasiparticles which emerge at long distances exhibit fractional quantum numbers of the microscopic degrees of freedom. The interplay between these emergent degrees of freedom and the microscopic symmetries can lead to a number of exotic properties including competing orders and unconventional phase transitions. I explore this phenomena in a number of platforms. This includes quantum antiferromagnets which can give rise to \textit{quantum spin liquids}, where we predict critical properties of phase transitions between spin liquids and conventionally-ordered phases - due to the fractionalized excitations, such critical theories deviate from traditional Landau-Ginzburg predictions. I also study the interplay between fractionalized spin and charge excitations, as well as fractionalization in non-equilibrium contexts. The second goal of this dissertation is to investigate the properties of disordered, strongly interacting, zero dimensional systems which exemplify many notable properties of higher-dimensional systems, such as doping-induced quantum criticality and non-Fermi liquid transport. The effects of disorder play an essential role in these models, leading to phenomena such as spin glass phases and conductance fluctuations which we investigate through both numerical and analytical methods.
Frustrated spin systems exhibit a wide range of phenomena such as novel phases, residual entropy and glassy dynamics. In contrast to the usual assumption of universality, microscopic details such as the underlying lattice structure often play a crucial role in these systems. In this dissertation the frustrated J1-J2 Ising model on the honeycomb lattice is considered. Earlier work shows that the coupling ratio R = J2 /|J1 | = −1/4 separates regions of two different ground states. Above this threshold the ground state is (anti-)ferromagnetically ordered for (anti-)ferromagnetic J1 , whereas below it infinitely many states of minimal energy exist. A recent Monte Carlo (MC) study by Žukovič finds a continuous transition with Ising critical exponents for R ≥ −0.2, leaving open the question of criticality for −1/4 < R < −0.2, for which the study was obfuscated by the presence of strong metastability. The regime with the highly degenerate ground states (R < −1/4), on the other hand, is characterized by feature-rich specific heat curves, and slow low-temperature dynamics reminiscent of spin glasses, although there is no disorder in this system. Prior to this work, no order parameter has been identified, and there is no clear understanding of its potentially critical behavior. Here, the question of criticality for −1/4 < R < −0.2 is addressed, and Ising universality is found for lower values of R, namely, at least for R ≥ −0.23. This result is achieved by carrying out finite-size scaling analyses of response functions like the magnetic susceptibility, and of the complex partition function zeros. In both cases, the underlying numerical data was generated using population annealing (PA) MC, a framework devised for spin glasses but generally applicable to systems with metastability. Furthermore, the region of degenerate ground states is studied using different boundary conditions, which are found to strongly affect the thermodynamic observations. A broken symmetry at zero temperature is identified, which can be quantified using a complex nematic order parameter. The MC data are consistent with a continuous symmetry-breaking transition. However, due to the limited system sizes that could be studied, and the presence of strong finite-size effects, one cannot rule out a higher-order transition or a cross-over without any singularities. In the process of studying this system, also the understanding of the PA framework was further developed. Specifically, the role of the resampling step in the algorithm was investigated. While it is crucial for efficient equilibration, no benefit is found in noisier resampling schemes such as the multinomial resampling method. Instead, two resampling schemes, which have the lowest sampling variance, are identified as preferential.:1 Introduction I Background and Methodology 2 Background 2.1 Fundamentals of Statistical Mechanics 2.1.1 Equilibrium ensembles 2.1.2 Phase transitions and critical behavior 2.1.3 Common spin models 2.2 Frustrated Spin Models 2.2.1 Frustrated J1-J2 Ising model on the square lattice 2.2.2 Gonihedric model in two dimensions 2.3 Monte Carlo Simulations 2.3.1 Fundamentals 2.3.2 Autocorrelation and error bars in MCMC time series 2.3.3 Possible implementations 2.3.4 Generalized ensemble methods 3 Methodology 3.1 General Algorithmic Design 3.2 Population Annealing 3.2.1 Algorithm 3.2.2 Assessing population annealing performance 3.2.3 Adaptive temperature steps 3.2.4 Weighted averages 3.2.5 Population annealing and first-order transitions 3.3 Estimating Critical Exponents from Finite Systems 3.3.1 Standard finite size scaling 3.3.2 Complex partition function zeros II Results 4 Algorithmic Improvements and Extensions to the Population Annealing Framework 4.1 Resampling Step in Population Annealing 4.1.1 Conceptual overview 4.1.2 Possible resampling methods 4.1.3 Benchmarking different resampling methods 4.1.4 Simulation results 4.1.5 PA simulations with finite populations in the limit of perfect equilibration 4.1.6 PA simulations in the limit of infinite populations and perfect equilibration 4.1.7 Conclusion 4.2 Using Population Annealing on Many Computing Nodes 4.2.1 Literature overview and different rebalancing schemes 4.2.2 Simulations of simulations 4.2.3 Conclusion and outlook 4.3 Adaptive Number of Spin Updates in Population Annealing 4.4 Microcanonical Population Annealing 5 The Frustrated J1-J2 Ising Model on the Honeycomb Lattice 5.1 Overview 5.2 Regime of Ising Universality Class 5.2.1 Finite-size scaling analysis 5.2.2 Energy barriers 5.2.3 Exploring the complex partition function zeros 5.3 Finite Ground State Entropy per Spin for J2/|J1| = −1/4 5.4 Highly Degenerate Striped Phase for J2/|J1| < −1/4 5.4.1 Boundary conditions 5.4.2 Energy minimization and ground states 5.4.3 Topological defects 5.4.4 Observables 5.4.5 Simulation details 5.4.6 Simulation results for R = −0.5 5.4.7 Simulation results for R = −1 5.4.8 Discussion 5.4.9 Conclusion and outlook 6 Parallel tempering and its infinite-swap limit 6.1 Formalism 6.2 Details on Implementation 6.3 Numerical Results from INS-PT Simulations 6.3.1 One-dimensional Ising model 6.3.2 Two-dimensional Ising model 6.3.3 q-state Potts model 6.3.4 Lennard-Jones clusters 6.4 Discussion 7 Conclusions and outlook Bibliography Appendices A Algorithmic Improvements and Extensions to the Population Annealing framework: Further details A.1 Generating correlated energy time series B J1-J2 honeycomb Ising model: Further plots and details B.1 GPU domain decomposition B.2 Finite-size scaling using Metropolis population annealing B.3 The decoupled antiferromagnet limit B.4 Zero temperature value of the nematic order parameter B.5 Percolation properties for J2 = −0.5 B.6 Binder parameter of the one-dimensional Ising model C Application for Ethical Approval