In this thesis, we investigated the evolution of low-dimensional quantum many-body systems through the lenses of their dynamics. The main tools we used are unitary dynamics and kinetic theory. The interest in this class of systems derives from the fact that they are dominated by the effects of interactions between fundamental constituents. In one dimension, the situation is even more dramatic: even the smallest interaction leads to a description that cannot be cast into a small perturbation of the initial model. Even though, a particular set of systems still allows for exact solutions; hence, they are deemed integrable and they allow for an infinite number of conserved currents. A simple way, that can also be performed experimentally, to drive many-body systems outside of the equilibrium is called quantum quench. It consists in switching instantaneously the parameters of the theory after the system is at the equilibrium and measuring an observable after the driving has occurred. Our contributions to the field amount to providing a full characterization of the evolution of the observables of an integrable system after a global quantum quench. We were also able to extend these results, valid in principle for relativistic theories, to the Lieb-Liniger model, prominent for the description of experimental setups, by the means of non-relativistic limit. For two-dimensional models, interactions are not changing the system from its foundations, although there are unexpected effects. A paradigmatic example is graphene: in a two-dimensional system where carriers are charged and massless, namely their dispersion relation does not have any gap and particles and holes can be freely excited from the thermal state, it is predicted that they are interacting via Coulomb force hundreds of times more than normal electrons in void. Hence, the system behaves in a completely different way from electrons in quantum electrodynamics: for instance, there are additional modes which are called plasmons that are also massless but neutral. They represent collective excitations of the model, as oscillations in the local density of charge. This system is not integrable but still strongly interacting; solving it as we do for one-dimensional systems is not possible. The role of conservation laws is a common theme of this thesis. In the example above exposed, total charge, momentum and energy are conserved in the pristine model, but in presence of impurities and defects they dissipate. The evolution of such quantities can though be derived exactly by the means of hydrodynamics. We were able to derive the equations that rule their evolution in time for any electronic system with two bands touching at the Dirac point. They are generalizations of the celebrated Navier-Stokes equations, and they differ from them only outside the compressible regime. We are able to show that the transport properties in the hydrodynamic regime strongly differ from the usual ballistic picture popularized by Drude and his model; this is captured for instance by violations of the Wiedemann-Franz ratio. Moreover, we prove that the role of interaction is prominent in the viscosity of these fluids.
In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator eiβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.
Recent experimental advances on ultracold atomic gases and trapped ions have made it possible to simulate exactly solvable quantum systems of interacting particles. In particular, the feasibility of making rapid changes, so-called quantum quenches, to such set-ups has allowed experimentalists to probe non-equilibrium phenomena in closed interacting quantum systems. This, in turn, has spurred a considerable theoretical interest in quantum many-body systems out of equilibrium. In this thesis, we study non-equilibrium properties of quantum many-body systems in the framework of exactly solvable quantum field theory in one spatial dimension. Specific systems include interacting fermions described by the Luttinger model and effective descriptions of spin chains using conformal field theory (CFT). Special emphasis is placed on heat and charge transport, studied from the point of view of quench dynamics, and, in particular, the effects of breaking conformal symmetries on transport properties. Examples include the Luttinger model with non-local interactions, breaking Lorentz and scale invariance, and inhomogeneous CFT, which generalizes standard CFT in that the usual propagation velocity v is replaced by a function v(x) that depends smoothly on the position x, breaking translation invariance. The quench dynamics studied here is for quantum quenches between, in general, different smooth inhomogeneous systems. An example of this is the so-called smooth-profile protocol, in which the initial state is defined by, e.g., smooth inhomogeneous profiles of inverse temperature and chemical potential, and the time evolution is governed by a homogeneous Hamiltonian. Using this protocol, we compute exact analytical results for the full time evolution of the systems mentioned above. In particular, we derive finite-time results that are universal in the sense that the same relations between the non-equilibrium dynamics and the initial profiles hold for any unitary CFT. These results also make clear that heat and charge transport in standard CFT are purely ballistic. Finally, we propose and study an inhomogeneous CFT model with v(x) given by a random function. We argue that this model naturally emerges as an effective description of one-dimensional quantum many-body systems with certain static random impurities. Using tools from wave propagation in random media, we show that such impurities lead to normal and anomalous diffusive contributions to heat transport on top of the ballistic one known from standard CFT.
Recent advances of experimental nanofabrication techniques draw increasing attention to the non-equilibrium behavior of low dimensional systems. Of particular interest are the strongly interacting one dimensional systems whose description in out of equilibrium situations remains a theoretical challenge. This thesis contributes to the understanding of important elementary processes in the non-equilibrium physics of one dimensional electron systems. Tunneling of an electron into a Luttinger liquid leads to partitioning of its charge and energy into counter-propagating modes. This thesis studies the partitioning of the energy which had previously remained unexplored. It turns out that energy partitioning is essentially independent of the charge partitioning and one can even reach conditions such that energy and charge propagate in opposite directions. Another important difference is their experimental accessibility. In contrast to the charge, energy partitioning provides a measurable characteristic of the tunneling process even in dc setups and we propose experimental geometries that allow for tuning and detecting energy partitioning. At higher excitation energies it becomes necessary to include curvature effects of the electron dispersion. Another part of this thesis discusses the consequences of curvature induced three-body collisions on the relaxation in quantum wires. This is particularly interesting due to the integrability of the Luttinger model which does not allow for thermalization within this paradigm of one dimensional systems. In this thesis we derive energy relaxation rates due to three-body processes beyond the Luttinger model within a well-defined perturbative approach. It turns out that the electron spin and the long range Coulomb interaction are important ingredients for a quantitative description of recent experiments which we provide in this thesis. Furthermore, we study the influence of three-body collisions on the energy relaxation in integer quantum Hall edge states. We specifically address different interaction induced edge reconstruction scenarios and find that edge reconstruction strongly enhances the energy relaxation. This is particularly pronounced when the reconstruction creates additional counter-propagating modes. Finally, we discuss another system which is crucially controlled by non-equilibrium effects. The so called nanoelectromechanical systems show a coupling between the electronic and mechanical degrees of freedom. The electron current can thus influence the mechanical motion which leads to a number of interesting applications. Previous theoretical studies on the basis of non-equilibrium Green's functions showed that these current induced forces can be expressed in terms of intuitive scattering matrix expressions. This thesis sheds considerable light on this observation by providing a much more satisfactory and concise derivation of the scattering theory of current induced forces.
In this thesis, we explore various universal aspects of non-equilibrium quantum many-body systems in low dimensions. The study of non-equilibrium settings is rendered possible by the existence of exact solutions in the presence of dynamical symmetries on one hand and through modeling via non-Hermitian Hamiltonians on the other. Low dimensions further offer the chance to study the interplay between the associated phenomenological curiosities, such as topological scaling or generalized statistics, and non-equilibrium dynamics. Emphasis was placed on the effects of symmetries (such as PT-symmetry, continuous
O(2)-symmetry,
SO(2,1) dynamical symmetry), quantum statistics and spatio-temporal dimension, which are shown to largely capture the universality of our results. In particular, we have successively explored the critical properties and renormalization group features of paradigmatic non-Hermitian PT-symmetric field theories within the framework of the functional renormalization group. Particular emphasis was placed on the interplay between PT-symmetry breaking and continuous symmetries as well as topological scaling. In this spirit, this led to our study of the non-Hermitian O(2)-symmetric model and non-Hermitian sine-Gordon model as well as its extensions and the corresponding fermionic duals. Our analysis not only manages to reproduce predictions made by previous studies, it extends them to new methodologies and settings and produces new results. For the O(2)-symmetric problem, our study constitutes as far as we know the first analysis based on FRG and the first exploration of critical behavior altogether. While our results for the RG flows are consistent with previous perturbative studies, it extends them to non-perturbative regimes. In addition, the fixed-point structure of the flow near
d=4 is explored for the first time in the case of Hermitian interactions and features new non-trivial non-Hermitian fixed points, which are located in the PT-broken sector. This is as far as we know the first time that the occurrence of PT-broken fixed points is predicted in a quantum many-body system. %Given their relative stability, the latter may only accessible provided significant fine-tuning of parameters. %this might not be the case in dimensions other than 4 or in the presence of non-Hermitian interactions (which might also trigger the presence of other non-trivial fixed points). Our analysis of the modified sine-Gordon models (embodying the celebrated BKT transition paradigmatic of topological phase transitions) as well as its extensions (including the Pokrovsky-Talapov and massive sine-Gordon models) is consistent with previous studies and extends the latter to non-Hermitian settings and/or to non-perturbative regimes. It predicts novel non-perturbative PT-broken phases that prove robust to the presence of mass terms and finite soliton densities. In the fermionic setting, this phase was further shown to display tachyonic and monstronic propagators and its onset was associated with the interactions between fermions (traditionally corresponding to bosonic solitons) becoming repulsive. In this case, the corresponding extensions respectively model the coupling of fermions to background and dynamical gauge fields. In both cases, the occurrence of a PT-broken phase was shown to robustly occur and gauge fields were even predicted to provide a means of controlling the imaginary mass governing the asymptotic behaviors of propagators in the PT-broken sector. Finally, we have investigated the interplay between dynamical symmetries and generalized statistics in the harmonically confined driven Calogero model. Both periodic and adiabatic drives of the trapping potential were considered, with particular focus on observables such as the ground-state echo amplitudes/fidelity and pair-momentum distribution. The former are significantly suppressed with increasing exclusion statistics (embodied by the interaction strength
λ). Given a natural trap frequency commensurate with a periodic drive, we have predicted sharp suppressions of the likelihood of intermediate echoes (between those imposed by commensurability) in the presence of interactions as well as a substantial drop in the asymptotic ground-state fidelity reached following a slow drive through
ω=0. In a similar way, these observations are also reflected in the dramatic decline in long-range spatial coherence with
λ. This is exemplified, in the case of the slow drive through the gapless point, through a reduction of the central peak of the dynamical pair-momentum distribution with heightened exclusion and interactions. These results are consistent with our interpretation in terms of defects proliferation and hindered annihilation as a direct consequence of non-trivial statistics and interactions. Universality is transversely ubiquitous in our work. Our predictions of stable PT-broken phases and fixed points in non Hermitian many-body systems embody novel universal large-scale behaviors. Moreover, the robustness of these phases was proven under various perturbations, further increasing their universal character. Our additional forecast of critical exponents further captures the nature of these novel universal behaviors. In the case of our study of driven scale-invariant fluids, universality manifests in the sole dependence of our results on universal quantities: the exponent
zν which encompasses the adiabatic limit of more complex drives through the gapless point; the effective statistics parameter
λ of the degrees of freedom; the characteristic Mathieu exponent
r associated with a periodic drive. Conversely it also manifests itself in the independence of our results on irrelevant microscopic details such as the driving rate
δ of the slow drive or the amplitude
h or base frequency
ω0 of the periodic drive. The universality of our results is also embodied by the fact that our core interpretations in terms of defect proliferation are reflected through various observables such as fidelities/echo amplitudes and pair-momentum distributions. All in all, this thesis has produced several novel results regarding universal aspects of non-equilibrium quantum many-body systems in low dimensions, which have turned into two accepted publications in high-impact journals and more in preparation. We hope that this work represents a meaningful step forward in our understanding of collective emergent phenomena in low-dimensional systems driven out of thermal equilibrium.
This thesis is concerned with non-equilibrium phenomena in interacting quantum many-body systems. Specifically, we investigate the time-evolution and relaxation dynamics of Bose gases in a highly restricted geometry, which constrains the dynamics to one spatial dimension. This leads to a description of the system in terms of a simple model Hamiltonian which permits exact many-body quantum mechanical solutions due to its integrability. In the first part of this thesis, a computational method is developed to obtain experimentally relevant correlation functions in the framework of the coordinate Bethe ansatz. We employ this method to compute exact ground-state correlation functions of the Lieb-Liniger gas for up to seven particles covering the whole regime of repulsive interactions. We also investigate the dynamics of the system after an instantaneous change of the interaction strength. This quantum quench deposits large amounts of energy that cannot be dissipated due to the system being closed, and so the dynamics far from equilibrium are probed. We prepare the system in two different initial states, and quench to the same final interaction strength. The latter is determined in such a way that the added energy due to the quench is the same for both scenarios. Conventional statistical mechanics predicts the same relaxed state, but due to the integrability of the system all considered correlation functions of the relaxed states differ from each other and also from the thermal ones. We then investigate the dynamics and relaxed state for a quench from zero to repulsive interactions in more detail, focussing on the mechanism of relaxation and the involved time-scales. We find that local correlation functions relax on time-scales determined by the interaction strength, in contrast to non-local correlation functions, whose relaxation time-scale is proportional to the system size. Next, we employ the same methodology to study the one-dimensional Bose gas with attractive interactions. In this case many-body bound states are permissible solutions of the Lieb-Liniger model. We compare exact ground-state correlation functions of up to seven particles to their corresponding mean-field solution. The latter displays a quantum phase transition at a critical interaction strength, marking the transition from a uniform-density state to a localized bright soliton. Our exact results agree remarkably well with the corresponding mean-field solution past the critical point. We also investigate the dynamics following an interaction strength quench, starting again from the non-interacting ground state. Bound states strongly influence correlation functions for all post-quench interaction strengths, and local correlation functions are largely increased compared to their initial value. In the last part of this thesis, we investigate the behavior of the one-dimensional Bose gas under periodic driving of the interaction strength. To this end, we extend the coordinate Bethe-ansatz formalism by employing Floquet theory to obtain solutions for the full time- dependent Hamiltonian. This realizes an extension to non-integrable systems in a way that allows controlled breaking of integrability. We compare the dynamics of the system following an interaction quench to a fixed final interaction strength to that of a system with periodically modulated post-quench interaction strength. For fast driving, the system evolves according to the time-averaged Hamiltonian and correlation functions are nearly identical to those of the undriven system. The response of the system qualitatively changes at resonances of the time-averaged Hamiltonian, where we observe energy absorption and a marked change in correlation functions. However, the system does not absorb energy indefinitely.
The primary area, on which we focus in the thesis, are the out-of-equilibrium properties of quantum and classical systems. The playground we use to study those properties, are interacting integrable and solitonic systems, whose structure enables us to perform analytical calculations.
In the context of the out-of-equilibrium physics we focus on two questions. The first problem is related to the set of stationary states in quantum systems. The question is especially important in the context of the homogeneous quantum quenches, since it makes possible the calculations of expectation values of local observables. The building blocks of stationary states are quasilocal conserved quantities which constitute the information that is preserved under the time evolution. In the first part of the thesis we deal with the construction of quasilocal conserved quantities in one dimensional anisotropic Heisenberg model and bosonic models. Quasilocal charges are put in the context of the standard theory of integrability, which in turn allows for the calculation of physical quantities.
The second question we deal with are the transport properties of one dimensional systems. The connection between the local integrals of motion and ideal transport was established in 90's. Here we extend the connection to the diffusive transport. This enables us to obtain explicit lower bound on diffusion constant in Heisenberg model, which proves that the Heisenberg model is not an insulator at finite temperatures.
In the last chapter we deal with the transport properties of classical celular automata. The first example we consider is the gas of charged hard core interacting particles, and the second one the gas of interacting solitons. Explicit solution of the dynamics of local observables allows for the explicit calculation of transport coeffcients, and the solution of the inhomogeneous quench problem. Despite their simplicity, the models exhibit wide range of transport phenomena ranging from insulating, through diffusive to ballistic transport.