A summary of my research activity is presented in this thesis. The first chapter starts with a short presentation of the known critical properties of the pure Potts model, mainly in two dimensions. The influence of disorder coupled to the energy density is then discussed in the general context of temperature-driven phase transitions. A physical explanation for the smoothing of the transition, due to Imry and Wortis, is presented and then applied to explain the phase diagram of the homogeneous random two and three-dimensional Potts models. It is then extended to the Potts model on irregular graphs and with layered randomness. In the third section, the critical properties at the random fixed point are examined in the light of the series expansion obtained by Renormalisation Group. The multifractal spectrum of scaling dimensions is discussed in the two-dimensional case and a new symmetry is proposed. The predictions of conformal invariance for profiles and correlation functions are compared with numerical data for the two-dimensional random Potts model in strip or square geometries. Finally, the geometrical properties of interfaces induced by symmetry-breaking boundary conditions are analysed in the context of the recent Stochastic Schramm-Loewner Evolution theory. The second chapter is devoted to the aging properties of homogeneous or frustrated Potts models submitted to a quench below or at their critical temperature. The scaling theory of two-time functions, as well as the more recent Local Scale Invariance theory , are presented in the first section. The violation of the Fluctuation-Dissipation theorem is described and the interpretation of the so-called Fluctuation-Dissipation Ratio as an out-of-equilibrium effective temperature is discussed. These ideas are applied to different spin models. First, one-dimensional systems with the Ising symmetry are considered. The equivalence of the KDH model with a gas of immobile particles subjected to pair-annihilation is exploited to obtain new analytical results. The question of the universality of dynamical exponents and ratios is then addressed in the two-dimensional case. Several lattices and several models in different universality classes are compared. A new model with the same symmetry as the three-state Potts model but with an irreversible dynamics is introduced. Its phase diagram in the steady state is determined and its aging properties on the critical line are studied. Finally, the aging of the Fully-Frustrated Ising and XY models are examined. Emphasis is put on the presence of logarithmic corrections due to the existence of topological defects. Two applications of the Jarzynski relation to the Potts model are presented at the end of the chapter. A short third chapter introduces the numerical simulations employed to analyse different set of experimental data. Monte Carlo simulations were used to reproduced some magnetic phase transitions in layered nanostructures. Spectroscopic data of surface reconstructions were compared to the electronic band structure obtained by computing the first eigenvectors of the Bloch-Schrödinger equation with a projective technique combined with a multiscale approach.
A summary of my research activity is presented in this thesis. The first chapter starts with a short presentation of the known critical properties of the pure Potts model, mainly in two dimensions. The influence of disorder coupled to the energy density is then discussed in the general context of temperature-driven phase transitions. A physical explanation for the smoothing of the transition, due to Imry and Wortis, is presented and then applied to explain the phase diagram of the homogeneous random two and three-dimensional Potts models. It is then extended to the Potts model on irregular graphs and with layered randomness. In the third section, the critical properties at the random fixed point are examined in the light of the series expansion obtained by Renormalisation Group. The multifractal spectrum of scaling dimensions is discussed in the two-dimensional case and a new symmetry is proposed. The predictions of conformal invariance for profiles and correlation functions are compared with numerical data for the two-dimensional random Potts model in strip or square geometries. Finally, the geometrical properties of interfaces induced by symmetry-breaking boundary conditions are analysed in the context of the recent Stochastic Schramm-Loewner Evolution theory. The second chapter is devoted to the aging properties of homogeneous or frustrated Potts models submitted to a quench below or at their critical temperature. The scaling theory of two-time functions, as well as the more recent Local Scale Invariance theory , are presented in the first section. The violation of the Fluctuation-Dissipation theorem is described and the interpretation of the so-called Fluctuation-Dissipation Ratio as an out-of-equilibrium effective temperature is discussed. These ideas are applied to different spin models. First, one-dimensional systems with the Ising symmetry are considered. The equivalence of the KDH model with a gas of immobile particles subjected to pair-annihilation is exploited to obtain new analytical results. The question of the universality of dynamical exponents and ratios is then addressed in the two-dimensional case. Several lattices and several models in different universality classes are compared. A new model with the same symmetry as the three-state Potts model but with an irreversible dynamics is introduced. Its phase diagram in the steady state is determined and its aging properties on the critical line are studied. Finally, the aging of the Fully-Frustrated Ising and XY models are examined. Emphasis is put on the presence of logarithmic corrections due to the existence of topological defects. Two applications of the Jarzynski relation to the Potts model are presented at the end of the chapter. A short third chapter introduces the numerical simulations employed to analyse different set of experimental data. Monte Carlo simulations were used to reproduced some magnetic phase transitions in layered nanostructures. Spectroscopic data of surface reconstructions were compared to the electronic band structure obtained by computing the first eigenvectors of the Bloch-Schrödinger equation with a projective technique combined with a multiscale approach.
This thesis divides very naturally into three chapters, reflecting the three separate areas I have worked in throughout my PhD studies. During my PhD, I published two papers, one relating to the work in the first chapter, and one to that of the second. At the time of submission, the project which the third chapter relates to was still ongoing, with plans for a future publication. The first chapter discusses work done on the real space renormalisation group (RG) for Ising and Potts models in 2 dimensions. Taking inspiration from Hasenbusch's work, a new framework for carrying out the RG is developed, its computational implementation discussed in some detail, as well as the results for a variety of systems and the implications of these results. The numerical scaling of the procedure and the consequences of this for future work are also covered. The results documented in chapter two are purely theoretical and presented in closed form. The research conducted is to do with properties of free interfaces. The second chapter is by and large critical of previous assumptions which have been made about the so-called wave vector dependent, in order to attempt to experimentally measure it and use these measurements to make further predictions. Using some simple toy models, many of these assumptions are shown to be false. In a sense, the goal of the research presented in the chapter two, is not to motivate further research, but to dissuade research in a direction we consider to be misguided, due to the faulty assumptions it is based on. The third chapter covers a small subset of a project concerned with studying correlations within the so-called Abelian Manna Model. The majority of the project involves (computational) Monte Carlo simulations of the dynamics of such systems, but as these results are not ready to present at the time of writing, the chapter is mainly concerned with some analytical results which were derived in order to validate our models for small systems, explain certain quirky phenomena arising from our simulations, and help quantify errors. Finally, there is an appendix which expands upon various topics from the first two chapters.
Among the many non-equilibrium processes encountered in nature we deal with two different but related aspects. One is the non-equilibrium relaxation process that is at the origin of \'aging phenomena••, and the other one is a non-equilibrium phase transition, called ••dynamic phase transition••. One of the main purposes of our research is to explore more realistic situations than studied previously. Indeed, in the study of aging phenomena certain kinds of disorder effects are considered, and we introduce the ••surface•• as a spatial boundary to the system undergoing the dynamic phase transition. In order to observe these processes as clearly as possible, we study in both cases simple spin systems. Using Monte Carlo simulations we first investigate aging in three-dimensional Ising spin glasses as well as in two-dimensional Ising models with disorder quenched to low temperatures. The time-dependent dynamical correlation length L(t) is determined numerically and the scaling behavior of various two-time quantities as a function of L(t)/L(s) is discussed where t and s are two different times. For disordered Ising models deviations of L(t) from algebraic growth law show up. The generalized scaling forms as a function of L(t)/L(s) reveal a generic simple aging scenario for Ising spin glasses as well as for disordered Ising ferromagnets. We also study the local critical phenomena at a dynamic phase transition by means of numerical simulations of kinetic Ising models with surfaces subjected to a periodic oscillating field. We examine layer-dependent quantities, such as the period-averaged magnetization per layer Q(z) and the layer susceptibility ¥ö(z), and determine local critical exponents through finite size scaling. Both for two and three dimensions, we find that the values of the surface exponents differ from those of the equilibrium critical surface. It is revealed that the surface phase diagram of the non-equilibrium system is not identical to that of the equilibrium system in three dimensions.
PART I: DISORDER The effect of quenched impurities on systems which undergo first-order phase transitions is studied within the framework of the q-state Potts model. For large q a mapping to the random field Ising model is introduced which provides a simple physical explanation of the absence of any latent heat in two dimensions, and suggests that in higher dimensions such systems should exhibit a tricritical point with a correlation length exponent related to the exponents of the random field model by \nu = \nu_RF/(2-\alpha_RF-\beta_RF). A phase diagram unifying pure, percolative and non-trivial random behaviour is proposed. In two dimensions we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a magnetic exponent \beta/\nu which varies continuously with q, and a weakly varying correlation length exponent \nu ~ 1. For q>4 the first-order transitions of the pure model are softened due to the impurities, and the resulting universality class is different from that of the pure Ising model. We find strong evidence for multiscaling of the correlation functions, as expected for such random systems. PART II: FRUSTRATION Exact results for conformational statistics of compact polymers are derived from the two-flavour fully packed loop model on the square lattice. This loop model exhibits a two-dimensional manifold of critical fixed points each one characterised by an infinite set of geometrical scaling dimensions. We calculate these dimensions exactly by mapping the loop model to an interface model whose scaling limit is described by a Liouville field theory. The formulae for the central charge and the first few scaling dimensions are compared to numerical transfer matrix results and excellent agreement is found. Compact polymers are identified with a particular point in the phase diagram of the loop model, and the non-mean field value of the conformational exponent \gamma = 117/112 is calculated for the first time. We also present the very precise numerical estimate \kappa = 1.472801(10) for the connective constant. Interacting compact polymers are described by a line of fixed points along which \gamma and \kappa vary continuously. Our identification of compact polymers with a critical model has bearings on the cooperativity of protein folding thermodynamics. Namely, in the large chain limit, the absence of an energy gap separating the denatured states from the native (compact) ones implies that homopolymer collapse is a one-state process. We comment on the prospects of modeling specific sequence (amino acid) information through the imposition of quenched disorder.
The theory of statistical mechanics provides a powerful conceptual framework within which the relevant (macroscopic) features of systems at equilibrium can be described. As there is currently no equivalent capable of encompassing the much richer class of non-equilibrium phenomena, research in this direction proceeds mainly on an instance-by-instance basis. The aim of this Thesis is to describe in some detail three such attempts, which involve different dynamical aspects of classical and quantum systems. As summarised below, each of the last three Chapters of this document delves into one of these different topics, while Chapter 2 provides a brief introduction on the study of non-equilibrium dynamics. In Chapter 3 we investigate the purely relaxational dynamics of classical critical ferromagnetic systems in the proximity of surfaces, paying particular attention to the effects that the latter induce on the early stages of the evolution following an abrupt change in the temperature of the sample. When the latter ends close enough to the critical value which separates the paramagnetic from the ferromagnetic phase, it effectively introduces a temporal boundary which can be treated as if it were a surface. Within this picture, we highlight the emergence of novel effects near the effective edge formed by the intersection of the two spatial and temporal boundaries. Our findings are apparently in disagreement with previous predictions which were based on the assumption that the presence of such an edge would not affect the scaling behaviour of observables; in order to explain this discrepancy, we propose an alternative for the original power-counting argument which, at least, correctly predicts the emergence of novel field-theoretical divergences in our one-loop calculations. We show that said singularities are associated with the scaling at the edge. Moreover, by encoding our findings in a boundary renormalisation group framework, we argue that the new predicted behaviour represents a universal feature associated to the short-distance expansion of the order parameter of the transition near the edge; we also calculate explicitly its anomalous dimension at the first-order in a dimensional expansion. As a qualitative feature, this anomalous dimension depends on the type of phase transition occurring at the surface. We exploit this fact in order to provide numerical support to our predictions via Monte Carlo simulations of the dynamical behaviour of a three-dimensional Ising model. The main results reported in Chap. 3 have appeared in Ref. [1]. In Chapter 4 we revisit the Euclidean mapping to imaginary times which has been recently proposed [2, 3] as an alternative for approaching the problem of quantum dynamics following a quench. This is expected to allow one to reformulate the original problem as a static one confined in a film geometry. We show that this interpretation actually holds only if the initial state of the dynamics is pure. Statistical mixtures, instead, intertwine the effects due to the two boundaries, which therefore cannot be regarded as being independent. We emphasize that, although the aforementioned reinterpretation as a confined static problem fails, one is still able, in principle, to write down and solve the corresponding equations. We also discuss in some detail the relation between this approach and the real-time field-theoretical one which makes use of the two-time Keldysh contour. For this purpose, we study the analytical structure of relevant observables — such as correlation functions — in the complex plane of times, identifying a subdivision of this domain into several sectors which depend on the ordering of the imaginary parts of the involved time coordinates. Within each of these subdomains, the analytic continuation to the real axis provides in principle a different result. This feature allows one to reconstruct from the Euclidean formalism all possible non-time-ordered functions, which in particular include all those which can be calculated via the Keldysh two-time formalism. Moreover, we give a prescription on how to retrieve response functions, discussing some simple examples and rationalising some recent numerical data obtained for one of these observables in a one-dimensional quantum Ising chain [4]. We also highlight the emergence of a light-cone effect fairly similar to the one previously found for correlation functions [2], which therefore provides further confirmation to the fact that information travels across the system in the form of the entanglement of quasi-particles produced by the quenching procedure. We have reported part of this analysis in Ref. [5]. Chapter 5 presents part of our recent work on effective relaxation in quantum systems following a quench and on the observed prethermalisation. We analyse the effects caused by the introduction of a long-range integrability-breaking interaction in the early stages of the dynamics of an otherwise integrable quantum spin chain following a quench in the magnetic field. By employing a suitable transformation, we redefine the theory in terms of a fully-connected model of hard-core bosons, which allows us to exploit the (generically) low density of excitations for rendering our model exactly solvable (in a numerical sense, i.e., by numerically diagonalising an exact matrix). We verify that, indeed, as long as the parameters of the quench are not too close to the critical point, the low-density approximation captures the dynamical features of the elementary operators, highlighting the appearance of marked plateaux in their dynamics, which we reinterpret as the emergence of a prethermal regime in the original model. As expected, the latter behaviour is reflected also on extensive observables which can be constructed as appropriate combinations of the mode populations. For these quantities, the typical approach to the quasi-stationary value is algebraic with exponent a ≈ 3, independently of the size of the system, the strength of the interaction and the amplitude of the magnetic field (as long as it is kept far from the critical point). The plateaux mentioned above last until a recurrence time — which can be approximately identified with tR ≈ N/2 for single modes and t′R ≈ N/4 for extensive quantities — after which quantum oscillations due to the finite size of the chain reappear. Our procedure allows us to shed some light over prethermal features without having to considerably limit the size of the system, which we can choose to be quite large, as we discuss in Ref. [6].
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.
The present thesis adresses three different topics that are all in direct connection to the statistical physics of disordered systems. The first chapter, which is the main part of this work, is devoted to driven interfaces in random media. The main focus is on the case of an ac driving. Our investigations start with the associated mean field theory. At the beginning we consider the case of a constant driving force and discuss the critical behaviour close to the depinning transition. Subsequently, we examine the velocity hysteresis curve for small frequencies by use of analytical and numerical methods. It turns out that all observables under consideration exhibit scaling behaviour on approaching the adiabatic case. Next, we analyse perturbation theory for ac-driven interfaces. Tt turns out, that perturbation theory can only work for sufficiently high interface dimensionalities D>4. The failure of perturbation theory in low dimensions is discussed. The second part of this thesis is concerned with the influence of electroelastic coupling on the behaviour of domain walls in ferroelectrics. For the case of electrostrictive coupling in the high temperature phase we find, that merely the interface tension coefficients are changed in an anisotropic fashion. On the contrary, a piezo effect in the paraphase entails a long range term. After deriving the interface hamiltonian in the harmonic approximation, we determine the roughness of domain walls due to random field disorder using an Imry-Ma type argument. On large length scales, domain walls turn out to be flat. In the last chapter, we consider one-dimensional electronic systems, so-called Luttinger liquids. The subject of the first part is the competition of two phases of one-dimensional disordered fermionic systems, namely the Mott and the Anderson insulator. By relating both, the compressibility as well as a finite optical conductivity to a vanishing mass of charged excitations, the existence of an intermediate Mott glass phase can be excluded. Combining informations about the renormalisation group flow and the energy of topological excitations, as well as using simple scaling arguments, we construct the phase diagram. In the second part we study the replica trick to describe the quantum creep and the linear conductivity at finite temperatures for Luttinger liquids with relevant disorder. We provide approximate solutions to the Euler-Lagrange equations for the replica action. The replica limit turns out to be problematic. At the example of a toy model we illustrate a method for the replica limit and demonstrate the emergent difficulties.