We study the effects of topological (connectivity) disorder on phase transitions. We identify a broad class of random lattices whose disorder fluctuations decay much faster with increasing length scale than those of generic random systems, yielding a wandering exponent of ω=(d-1)/(2d) in d dimensions. The stability of clean critical points is thus governed by the criterion (d+1)ν>2 rather than the usual Harris criterion dν>2, making topological disorder less relevant than generic randomness. The Imry-Ma criterion is also modified, allowing first-order transitions to survive in all dimensions d>1. These results explain a host of puzzling violations of the original criteria for equilibrium and nonequilibrium phase transitions on random lattices. We discuss applications, and we illustrate our theory by computer simulations of random Voronoi and other lattices.
We study the effects of topological (connectivity) disorder on phase transitions. We identify a broad class of random lattices whose disorder fluctuations decay much faster with increasing length scale than those of generic random systems, yielding a wandering exponent of ω=(d-1)/(2d) in d dimensions. The stability of clean critical points is thus governed by the criterion (d+1)ν>2 rather than the usual Harris criterion dν>2, making topological disorder less relevant than generic randomness. The Imry-Ma criterion is also modified, allowing first-order transitions to survive in all dimensions d>1. These results explain a host of puzzling violations of the original criteria for equilibrium and nonequilibrium phase transitions on random lattices. We discuss applications, and we illustrate our theory by computer simulations of random Voronoi and other lattices.
Using the two-dimensional Ashkin-Teller (AT) model, we compare criticality on three non-periodic lattices: the Smith-hat aperiodic tiling, Voronoi-Delaunay (VD) random triangulations, and uncorrelated diluted square lattices. The decay of the block-averaged coordination fluctuation σQ with exponent α is used to describe the connectivity disorder. The first two lattices share the same exponent α, which differs from that of the third. We consider the regime where the correlation-length exponent ν<1, where randomness is relevant according to the Harris criterion dν≤2, but should be irrelevant in cases of the Smith-hat tiling and VD triangulations, where αν>1, according to the Harris--Barghathi--Vojta (HBV) criterion. For the diluted lattice, we indeed find that the clean universality behavior breaks down along the entire critical line, indicating a crossover to a fixed line dominated by disorder, in line with both the Harris criterion and the HBV criterion. In contrast, both VD and Smith-hat lattices display critical exponents consistent with the clean AT universality class, as validated by a Coulomb-gas self-consistency check, violating the Harris criterion while conforming to the HBV criterion.
We employ scaling arguments and optimal fluctuation theory to establish a general relation between quantum Griffiths singularities and the Harris criterion for quantum phase transitions in disordered systems. If a clean critical point violates the Harris criterion, it is destabilized by weak disorder. At the same time, the Griffiths dynamical exponent z' diverges upon approaching the transition, suggesting unconventional critical behavior. In contrast, if the Harris criterion is fulfilled, power-law Griffiths singularities can coexist with clean critical behavior, but z' saturates at a finite value. We present applications of our theory to a variety of systems including quantum spin chains, classical reaction-diffusion systems and metallic magnets, and we discuss modifications for transitions above the upper critical dimension. Based on these results we propose a unified classification of phase transitions in disordered systems.
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.
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.
Dynamical quantum phase transitions (DQPTs) represent a counterpart in nonequilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multiorbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions and construct basic counterexamples to the occurrence of DQPTs in multiband topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multiband to disordered systems, we approach random disorder by increasing the (super)unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.
In clean and weakly disordered systems, topological and trivial phases having a finite bulk energy gap can transit to each other via a quantum critical point. In presence of strong disorder, both the nature of the phases and the associated criticality can fundamentally change. Here we investigate topological properties of a strongly disordered fermionic chain where the bond couplings are drawn from normal probability distributions which are defined by characteristic standard deviations. Using numerical strong disorder renormalization group methods along with analytical techniques, we show that the competition between fluctuation scales renders both the trivial and topological phases gapless with Griffiths like rare regions. Moreover, the transition between these phases is solely governed by the fluctuation scales, rather than the means, rendering the critical behavior to be determined by an infinite randomness fixed point with an irrational central charge. Our work points to a host of novel topological phases and atypical topological phase transitions which can be realized in systems under strong disorder.