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.
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.
This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.
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.
The Luttinger liquid model, which describes interacting electrons in a single-channel quantum wire, is completely integrable in the absence of disorder and as such does not exhibit any relaxation to equilibrium. We consider relaxation processes induced by inelastic electron-electron interactions in a disordered Luttinger liquid, focusing on the equilibration rate and its essential differences from the electron-electron scattering rate as well as the rate of phase relaxation. In the first part of the paper, we review the basic concepts in a disordered Luttinger liquid at equilibrium. These include the elastic renormalization, dephasing, and interference-induced localization. In the second part, we formulate a conceptually important framework for systematically studying the nonequilibrium properties of a strongly correlated (non-Fermi) Luttinger liquid. We derive a coupled set of kinetic equations for the fermionic and bosonic distribution functions that describe the evolution of a nonequilibrium Luttinger liquid. Remarkably, the energy equilibration rate in the conducting disordered quantum wire (at sufficiently high temperature, when the localization effects are suppressed by dephasing) is shown to be on the order of the rate of elastic scattering off disorder, independent of the interaction constant and temperature.
This thesis investigates quantum-many body systems out of equilibrium in a variety of physically relevant and intriguing settings. It provides an overview of the recent developments in this field and based on the author’s recent review illuminates many interesting connections and summarises open questions. In this general framework, the various results of the author are embedded. While they often rely on advanced mathematics, great care is taken to present them in an intuitive way and most technical material is discussed separately in appendices. In the context of equilibration, work created as part of this thesis is able to capture the Gaussification of correlated initial states for a large class of free models. This allows to significantly extend equilibration results of free models, which are of crucial importance, as they provide reasonable relaxation time scales and immediately connect to our physical intuition in terms of ballistic spreading. Building on two surprising connections between static and dynamic features of Hamiltonians, which in general are hard to obtain, the results of this thesis further greatly contribute to the recent debate on the true nature of quantum many- body localisation. For those interacting models, we derive the localised structure of eigenstates from a dynamical suppression of information propagation on the low-energy sector. Further, we show how a non-degenerate spectrum, indicating the presence of interactions, and the existence of an approximately local constant of motion are sufficient to show information propagation, if arbitrary energies are allowed. In the context of quantum phase transitions, we continue such connections between static and dynamic features. We specifically investigate the Mott-superfluid transition of the Bose-Hubbard model and ask to what extent it has a universal dynamical signature. In a joint experimental, numerical and analytical effort, complex behaviour of these dynamics is uncovered, thus challenging the common believe that the Kibble-Zurek mechanism is sufficient to fully capture such transitions. We embed such experimental investigation in the recent debate on quantum simulators, which give the exciting perspective of solving notoriously and even provably hard problems efficiently in the laboratory. For such devices, we argue that the final read-out of the results is an important out- standing problem to be solved. We approach this issue in the case of a continuous quantum field, in which the notion of reconstructing the quantum state is even conceptually unclear. Based on tensor network methods, we demonstrate that in an experiment of ultra-cold atoms in a continuous setup, states can nevertheless efficiently be obtained. Thus, this thesis constitutes not only an important review of the field of quantum many-body systems out of equilibrium, but, using advanced mathematical and numerical tools, has significantly contributed to our understanding of various important out- standing questions in interacting many-body systems.
This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional.
With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades later, this eventually led to the theoretical prediction and the experimental realization of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we will study the interplay between topology and another key topic in condensed matter physics, namely the study of inter-particle interactions in many-body systems. This interplay is analyzed from two different perspectives. Firstly, we studied how the presence of electron-electron interactions affects single-electron injection into a couple of counter-propagating one-dimensional edge channels. The latter appear at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel interactions, the injected electron splits up into a couple of counter-propagating fractional excitations. Here, we carefully study and discuss their properties by means of an analytical approach based on the Luttinger liquid theory and the bosonization method. Our results are quite relevant in the context of the so-called electron quantum optics, a fast developing field which deeply exploits the topological protection of one-dimensional edge states to study the coherent propagation of electrons in solid-state devices. As an aside, we also showed that similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger liquid subject to a sudden change of the interaction strength, a protocol known as quantum quench which is gaining more and more attention, especially within the cold-atoms community. Secondly, we study how inter-particle interactions can enhance the topological properties of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at its ends. The latter are extremely relevant in the context of topological quantum computation because of their non-Abelian anyonic exchange statistics. Here we show that, by properly adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models which feature zero-energy parafermionic modes, an even more intriguing generalization of Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an exactly solvable fermionic model hosting zero-energy parafermions. We study their properties and numerically investigate their signatures and robustness even when parameters are tuned away from the exactly solvable point.