This thesis deals with different aspects of statistical physics of correlated systems. The first part is related to the fluctuations of global quantities in correlated systems. Various studies clam that such fluctuations are well described by the BHP distribution. We use the 2D Ising model to test and quantify this proposition. Using observations from Monte Carlo simulations, we build a theoretical analysis, showing that the apparent universality of the BHP distribution is related to the Gaussian model obtained from perturbation expansion. Deviations from BHP, due to a non-linear term are expected. In the second part we consider a new model for a 1/f classical intermittent noise and study its effects on the dephasing of a two-level system.Within this model, the evolution of the relative phase between the two states is described as a continuous time random walk. Using renewal theory, we find exact expressions for the dephasing factor and identify the physically relevant various regimes in terms of the coupling to the noise. In particular, we point out the consequences of the non-stationarity and pronounced non-Gaussian features of this noise, including some new anomalous and aging dephasing scenarios. In the last part we present an alternative method to obtain some exact results for the 2D Ising model with a boundary magnetic field, for a finite size system. This method is a generalisation of ideas from Plechko presented for the 2D Ising model in zero field, based on the representation of the Ising model using a Grassmann algebra. A Gaussian 1D action is obtained for a general configuration of the boundary magnetic field. When the magnetic field is homogeneous, our results are in agreement with McCoy and Wu's previous work. This 1D action is used to compute in an efficient way the free energy in a special case of inhomogeneous boundary magnetic field. This allows us to compute the phase diagram of a wetting transition induced by a boundary defect.
This thesis deals with different aspects of statistical physics of correlated systems. The first part is related to the fluctuations of global quantities in correlated systems. Various studies clam that such fluctuations are well described by the BHP distribution. We use the 2D Ising model to test and quantify this proposition. Using observations from Monte Carlo simulations, we build a theoretical analysis, showing that the apparent universality of the BHP distribution is related to the Gaussian model obtained from perturbation expansion. Deviations from BHP, due to a non-linear term are expected. In the second part we consider a new model for a 1/f classical intermittent noise and study its effects on the dephasing of a two-level system.Within this model, the evolution of the relative phase between the two states is described as a continuous time random walk. Using renewal theory, we find exact expressions for the dephasing factor and identify the physically relevant various regimes in terms of the coupling to the noise. In particular, we point out the consequences of the non-stationarity and pronounced non-Gaussian features of this noise, including some new anomalous and aging dephasing scenarios. In the last part we present an alternative method to obtain some exact results for the 2D Ising model with a boundary magnetic field, for a finite size system. This method is a generalisation of ideas from Plechko presented for the 2D Ising model in zero field, based on the representation of the Ising model using a Grassmann algebra. A Gaussian 1D action is obtained for a general configuration of the boundary magnetic field. When the magnetic field is homogeneous, our results are in agreement with McCoy and Wu's previous work. This 1D action is used to compute in an efficient way the free energy in a special case of inhomogeneous boundary magnetic field. This allows us to compute the phase diagram of a wetting transition induced by a boundary defect.
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].
This thesis addresses the nature of global (many body) fluctuations in highly correlated systems. We begin with the question of temperature dependence in finite two dimensional XY (2dXY) model magnets. Such systems have a fully critical low temperature phase. It is shown analytically, backed up by extensive Monte Carlo simulations, that the non-Gaussian distribution of order parameter fluctuations is not strictly universal but has an explicit temperature dependence - contrary to previous findings. The temperature dependence is used to explain why past studies derived the same distribution for fluctuations of the full order parameter and an approximate linearized form. The appearance of spin vortices in the related Harmonic model is discussed and an argument is presented for why these defects must always appear as bound pairs. The linearized order parameter of the 2dXY model leads to a family of dimensionally dependent models defined in reciprocal space. An argument is presented for the interpretation of these systems as being critical and a direct space Hamiltonian is derived for the one dimensional case. This model has order parameter fluctuations distributed according to the Fisher-Tippett-Gumbel distribution from extreme value statistics (EVS). The link between criticality and EVS is investigated, as are the origins of the non-Gaussianity. The ability to distinguish between critical distributions is discussed. It is seen that for one, two and three dimensions the critical models presented lead to functionally similar fluctuation distributions. A previously reported link between EVS and l/f noise is investigated. Our one dimensional critical model is mapped onto the action used to generate l/f signals and we propose an alternative interpretation of the link in the context of a 1/q dispersion of spatial normal modes. The experimental observability of the FTG distribution in l/f signals is considered with emphasis on imperfections in the noise. A physically relevant method of generating l/f noise from the superposition of random telegraph signals is also examined.
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 is concerned with several aspects of the Ising and tri-critical Ising models in two-dimensions. These are much-studied models relevant in both condensed matter physics as descriptions of the critical phenomena of two- dimensional systems and in String theory as building blocks of the string world sheet theory. The first part of the thesis is concerned with the derivation of differential equations for the critical four-point function in the Ising model. We present a method which provides the well-known standard solutions by a new and efficient route. The second part of the thesis is concerned with off-critical behaviour, and in particular the numerical study of perturbations of conformal field theory through the truncated conformal space approach. We show that the coupling constant undergoes significant renormalization in this scheme, and in particular the Ising model can be found as a fixed point for a finite value of the bare coupling constant. The renormalization group equations we find are of general use in the TCSA approach. The final part of the thesis considers off-critical boundary conditions in the tri-critical Ising model. We study them using a variant of the mean-field method and find a qualitative description of the space of boundary conditions that is in accord with the exact conformal field theory description. This is both a test of the method and its applicability in new domains, and also shows that previously published results are in error.
This thesis is a collection of three exact results on correlation and response functions in integrable systems. In the first part we study transport (two-point functions) in the 1D Hubbard model. First, we analyze the limit of large on-site repulsion and characterize spin transport as a function of temperature. Then, we consider the case where the model displays non-abelian symmetries, e.g., spin SU(2), and argue that in this case transport is anomalous with dynamical scaling exponent z=3/2 and follows \acrshort{kpz} scaling. In the next two parts we focus instead on (perturbative) nonlinear response functions, which can be expressed as n-point correlators with n>2. Given the lack of a systematic understanding of the information which can be extracted from these, we compute them in integrable 1D systems. We therefore focus on two distinct scenarios. In the first scenario, we consider a finite temperature system perturbed by some inhomogeneous field coupling to local charges --- thus the perturbations can only accelerate quasiparticles, but not create/annihilate them. In this context, we develop a diagrammatic framework based on generalized hydrodynamics that allows a systematic calculation of nonlinear response functions. We show that, in the hydrodynamic limit, nonlinear response functions qualitatively distinguish between non-interacting and interacting systems. In the second scenario, we instead consider a zero temperature system whose ground state coincides with the quasiparticle vacuum. In this simpler setting we consider more general perturbations that can also create and annihilate quasiparticles. Focusing on the Ising chain as a paradigmatic model we show that the long-time limit of four-point functions (third-order response) grows linearly in time. We interpret these divergences in terms of semiclassical processes where quasiparticles propagate ballistically and scatter when their trajectories intersect.
The study of non-equilibrium behaviors of quantum matter is one of the research directions at the frontiers of condensed matter physics and, in general, of theoretical and experimental physics. Understanding the mechanisms which prevent many quantum particles from reaching thermal equilibrium can ultimately shed light on the foundations of statistical physics and provide insight into the anomalous long-time collective behaviors observed in experiments. In this thesis, we present new theoretical developments in this direction. In the first part of the thesis we discuss the non-equilibrium dynamics of the two-dimensional quantum Ising model, the prototypical model of magnetism. In particular, we investigate the dynamics of interfaces separating different spin domains. In some regimes, we can provide an exact description of their unitary time evolution, but we also discuss the robustness of our predictions away from the exactly-solvable limit. Moreover, we address the same problem in the presence of disorder and provide a hydrodynamic interpretation of our findings. In the second part of the thesis, we investigate the localization properties of disordered quantum systems. In particular, we develop a renormalization-group picture of the Anderson localization transition in generic geometries. This provides a new framework to understand the phase diagram of such systems and their critical behavior beyond the usual finite-size scaling analysis. We also provide further insights on this topic both by studying resonances and their connection with observables and by employing unsupervised non-parametric learning techniques to probe disordered systems in an experimentally accessible form. The results presented in this thesis constitute a step forward in understanding the non-equilibrium phases and regimes of quantum matter and they pave the way to interesting future developments.