In the theory of stochastic many particle systems one models phenomena of non-equilibrium statistical mechanics by means of a Markov process. The master equation, describing the time evolution of such a process, can formally be interpreted as a Schrodinger equation in imaginary time. In this approach, the (non-Hermitian) generator of the process plays the role of the Hamiltonian. This mathematical equivalence permits the successful application to stochastic systems of a number of exact, approximative or numerical methods from quantum mechanics.A particularly interesting type of stochastic systems is formed by the processes having a phase transition between an absorbing state and an active state. Examples of these systems can be found in physics, chemistry and even biology. The best known process of this kind is the contact process. In this thesis real-space renormalisation group techniques, which were originally developed for quantum mechanical systems, are applied to study stochastic models on a one-dimensional lattice, more precisely the critical behaviour and the universality of absorbing state phase transitions. First, the standard renormalisation group (SRG) is adapted to a stochastic context resulting in one of the few analytical techniques capable of producing approximations for systems that can not be solved exactly. The SRG is successfully applied to the contact process for which it characterises the stationary critical behaviour with a competitive accuracy. Next a generalisation of the contact process to a model with several absorbing states is studied by means of the density matrix renormalisation group (DMRG) algorithm. The results for the cases with one and two absorbing states are consistent with what was known from simulations done by an other author. For the models with more than two absorbing states new results were found, which are supported by analytical considerations in a particular limit. Finally the experience of the previous two projects is used in the investigation of the quenched random contact process. In this model the local interactions on a lattice site are time-invariant stochastic variables, independently drawn from a given distribution with a certain variance, which gives a measure for the disorder. Very little was known about the stationary behaviour of this model since its extremely slow dynamics makes simulations inefficient. The renormalisation techniques in this thesis, suffer less from this problem. First, for the case of small disorder the SRG indicates that the critical behaviour is changed by the presence of disorder and the cross-over exponent is consistent with the Harris criterion. For intermediate disorder a DMRG study reveals that the critical exponents are continuously varying with the amount of disorder. Finally, an SRG calculation, suitably adapted to strong disorder, shows that in this case the phase transition is characterised by a fixed point at infinite randomness. In this limit the renormalisation is expected to be exact. It is the first time exact critical exponents are found for a phase transition out of an absorbing state. Moreover, the exponents are the same as those previously found in a class of disordered quantum chains, suggesting a new kind of universality for strongly disordered systems.
In the theory of stochastic many particle systems one models phenomena of non-equilibrium statistical mechanics by means of a Markov process. The master equation, describing the time evolution of such a process, can formally be interpreted as a Schrodinger equation in imaginary time. In this approach, the (non-Hermitian) generator of the process plays the role of the Hamiltonian. This mathematical equivalence permits the successful application to stochastic systems of a number of exact, approximative or numerical methods from quantum mechanics.A particularly interesting type of stochastic systems is formed by the processes having a phase transition between an absorbing state and an active state. Examples of these systems can be found in physics, chemistry and even biology. The best known process of this kind is the contact process. In this thesis real-space renormalisation group techniques, which were originally developed for quantum mechanical systems, are applied to study stochastic models on a one-dimensional lattice, more precisely the critical behaviour and the universality of absorbing state phase transitions. First, the standard renormalisation group (SRG) is adapted to a stochastic context resulting in one of the few analytical techniques capable of producing approximations for systems that can not be solved exactly. The SRG is successfully applied to the contact process for which it characterises the stationary critical behaviour with a competitive accuracy. Next a generalisation of the contact process to a model with several absorbing states is studied by means of the density matrix renormalisation group (DMRG) algorithm. The results for the cases with one and two absorbing states are consistent with what was known from simulations done by an other author. For the models with more than two absorbing states new results were found, which are supported by analytical considerations in a particular limit. Finally the experience of the previous two projects is used in the investigation of the quenched random contact process. In this model the local interactions on a lattice site are time-invariant stochastic variables, independently drawn from a given distribution with a certain variance, which gives a measure for the disorder. Very little was known about the stationary behaviour of this model since its extremely slow dynamics makes simulations inefficient. The renormalisation techniques in this thesis, suffer less from this problem. First, for the case of small disorder the SRG indicates that the critical behaviour is changed by the presence of disorder and the cross-over exponent is consistent with the Harris criterion. For intermediate disorder a DMRG study reveals that the critical exponents are continuously varying with the amount of disorder. Finally, an SRG calculation, suitably adapted to strong disorder, shows that in this case the phase transition is characterised by a fixed point at infinite randomness. In this limit the renormalisation is expected to be exact. It is the first time exact critical exponents are found for a phase transition out of an absorbing state. Moreover, the exponents are the same as those previously found in a class of disordered quantum chains, suggesting a new kind of universality for strongly disordered systems.
Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).
It is explained how field-theoretic methods and the dynamic renormalisation group (RG) can be applied to study the universal scaling properties of systems that either undergo a continuous phase transition or display generic scale invariance, both near and far from thermal equilibrium. Part 1 introduces the response functional field theory representation of (nonlinear) Langevin equations. The RG is employed to compute the scaling exponents for several universality classes governing the critical dynamics near second-order phase transitions in equilibrium. The effects of reversible mode-coupling terms, quenching from random initial conditions to the critical point, and violating the detailed balance constraints are briefly discussed. It is shown how the same formalism can be applied to nonequilibrium systems such as driven diffusive lattice gases. Part 2 describes how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then used to analyse simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which
The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real space renormalisation group approach, originally developped for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d=1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalisation.
The contact process is a paradigmatic classical stochastic system displaying critical behavior even in one dimension. It features a nonequilibrium phase transition into an absorbing state that has been widely investigated and shown to belong to the directed percolation universality class. When the same process is considered in a quantum setting, much less is known. So far, mainly semiclassical studies have been conducted and the nature of the transition in low dimensions is still a matter of debate. Also, from a numerical point of view, from which the system may look fairly simple-especially in one dimension-results are lacking. In particular, the presence of the absorbing state poses a substantial challenge, which appears to affect the reliability of algorithms targeting directly the steady state. Here we perform real-time numerical simulations of the open dynamics of the quantum contact process and shed light on the existence and on the nature of an absorbing state phase transition in one dimension. We find evidence for the transition being continuous and provide first estimates for the critical exponents. Beyond the conceptual interest, the simplicity of the quantum contact process makes it an ideal benchmark problem for scrutinizing numerical methods for open quantum nonequilibrium systems.
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].
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.