An overview is presented of non-equilibrium systems in the presence of disorder, which typically strongly affects their behaviour. The examples considered here are those for which most progress is currently being made, namely low-dimensional collective models of the lattice-based exclusion type with substitutional disorder. Discussions are given of such effects as localization, and of disorder-induced crossovers to new critical or singular behaviour at steady state non-equilibrium transitions, and to new asymptotic or critical dynamics. Mappings to disordered quantum spin systems are used to provide some exact results (exploiting Goldstone symmetries or free fermion equivalences) and to introduce non-equilibrium/disorder effects analogous to Griffiths phases and McCoy-Wu singularities. Among other topics also discussed here are criteria for crossovers, and non-equilibrium transitions and dynamics on diluted and possibly scale-invariant backgrounds. As well as mappings, direct approaches of various sorts are used, such as mean field methods, and scaling procedures typically scaling a disorder distribution. Scaling for the non-equilibrium dynamics is relatively simple and very effective when the equations of motion are linear or linearizable by certain transformations presented here. The current situation regarding scaling approaches to the more resistant non-linear cases is briefly discussed.
An overview is presented of non-equilibrium systems in the presence of disorder, which typically strongly affects their behaviour. The examples considered here are those for which most progress is currently being made, namely low-dimensional collective models of the lattice-based exclusion type with substitutional disorder. Discussions are given of such effects as localization, and of disorder-induced crossovers to new critical or singular behaviour at steady state non-equilibrium transitions, and to new asymptotic or critical dynamics. Mappings to disordered quantum spin systems are used to provide some exact results (exploiting Goldstone symmetries or free fermion equivalences) and to introduce non-equilibrium/disorder effects analogous to Griffiths phases and McCoy-Wu singularities. Among other topics also discussed here are criteria for crossovers, and non-equilibrium transitions and dynamics on diluted and possibly scale-invariant backgrounds. As well as mappings, direct approaches of various sorts are used, such as mean field methods, and scaling procedures typically scaling a disorder distribution. Scaling for the non-equilibrium dynamics is relatively simple and very effective when the equations of motion are linear or linearizable by certain transformations presented here. The current situation regarding scaling approaches to the more resistant non-linear cases is briefly discussed.
Recent advances in the theory of non-equilibrium systems are reviewed. The emphasis is on collective behaviour, particularly non-equilibrium stochastic dynamics and steady state phase transitions. The models discussed are largely lattice-based 'exclusion' models (with hard core interactions) which can include a wide variety of collective stochastic processes and phenomena. The techniques employed range from intuitive reasoning and approximate mean field approaches to scaling and renormalization group methods, and to exact descriptions in terms of operator algebras, where many new results are included. Mappings to quantum systems, particularly lattice spin models, are also exploited, which provide new viewpoints and allow the use of standard techniques including Bethe ansatz and free fermion solutions.
The study of collective behaviour in many-body systems often explores fundamentally new ideas absent from the mere constituents of such a system. A paradigmatic model for these studies is the spin-1/2 XXZ chain and its fermionic equivalent. This thesis can be broadly divided into the study of two fundamental aspects of this model. Firstly, we discuss localisation phenomena in one dimensional lattices as often experimentally realised in cold atom systems. Secondly, we investigate how disorder and symmetry influence heat transport in spin chains. More specifically, in the first part we consider a system of non-interacting fermions in one dimension subject to a single-particle potential consisting of a strong optical lattice, a harmonic trap, and uncorrelated on-site disorder. We investigate a global inhomogeneous quantum quench and present numerical and analytical results for static and dynamical properties. We show that the approach to the non-thermal equilibrium state is extremely slow and that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localisation. We also consider the above system in a strong non-uniform electric field. In the non-interacting case, due to Wannier-Stark localisation, the single-particle wave functions are exponentially localised without quenched disorder. We show that this system remains localised in the presence of nearest-neighbour interactions and exhibits physics analogous to models of conventional many-body localisation. The second part explores the hydrodynamics of the disordered XYZ spin chain. Using time-evolving block decimation on open chains of up to 400 spins attached to thermal baths, we probe the energy transport of this system. Our principal findings are as follows. For weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Then, for disorder strengths above this critical value energy transport becomes increasingly subdiffusive.
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].
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.
A central goal in the study of modern condensed matter physics is the characterization of the dynamical properties of quantum systems. Many decades of effort towards this goal, studying a diverse range of (near-equilibrium) quantum matter, from Fermi liquids, to quantum two-level systems, to interacting spin models, and more, has revealed a remarkable pervasiveness of the simple dynamical description of these complex systems in terms of quasi-particles that carry spin, charge, and heat, and that are generally able to equilibrate systems. \n\nThis thesis is an examination of some exceptions to this rule. Specifically, we study a number of instances of quantum matter where equilibration phenomena happens at rather long time scales, or does not occur at all. Particular emphasis is laid on the role of dimensionality, disorder, and dissipation in engendering such novel dynamical behavior.\n\nFirst, we consider non-equilibrium dynamics in one-dimensional quasi-condensates. Low dimensionality inhibits scattering in these systems, and low-energy excitations are long-lived phase fluctuations that exhibit an enriched conformal symmetry. Utilizing this symmetry, we generalize sudden quenches typically used to study non-equilibrium dynamics to quenches along general relativistic and conformal trajectories. Gases never truly equilibrate after such a quench; instead, they evolve into a `prethermal' state with thermal-looking correlations and a chiral asymmetry. \n\nWe then study the problem of the dynamical transition driven by disorder, from an ergodic to a non-ergodic phase, in one-dimensional quantum spin chains. In particular, in XXZ chains with on-site disorder, we find a unique intermediate phase straddling the boundary of the dynamical phase transition, wherein rare-region effects lead to long-time tails in equilibration and vanishing DC conduction before the onset of non-ergodicity. We propose generalizations of such `Griffiths' behavior to arbitrary dimensions. We also study the dynamics of random-bond Heisenberg chains by developing a strong-disorder renormalization group protocol for these systems. We discuss how magnetic noise from such disordered systems contains signatures of their anomalous dynamical properties. \n\nNext, we re-examine the phenomenological theory of two-level systems in amorphous materials in the light of new experimental evidence that these states have large electric/magnetic dipole moments. We propose and justify an interpretation of the model as one of tunneling electrons slowed down by a large phonon drag and discuss the dynamical consequences of such polaronic effects. \n\nFinally, we discuss how magnetic noise measurements can be used to non-invasively access the anomalous properties of systems such as those discussed above. In particular, we examine how scattering properties of isolated magnetic impurities and non-local transport in a variety of two-dimensional materials can be probed experimentally using NV centers as noise magnetometers.
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.