This thesis consists of two parts. In part one we discuss the Kondo model. First we point out the differences in physical properties between the one- and multi-channel Kondo model using a renormalization group analysis. We also point out the challenges in obtaining exact solutions. Then we give an overview of two approaches, bosonization and spin chain, for finding exact solution of the one-, two- and four-channel Kondo model. Using bosonization we review a mapping from the one- and two-channel model to the exactly solvable resonant level model for a specific coupling strength. Following previous work, we derive the Kondo effect for a spin chain construction with XX and Ising coupling, showing the low energy behavior to be equivalent to the two- or four-channel Kondo model. We also discuss why one, two and four channels are special and we point out the difficulties in extending these approaches to other multi-channel Kondo models.\nThen in part two we investigate survival of the topological edge effects of the Kitaev chain (Majorana edge modes) after a quantum quench. We determine a stability region for the edge modes after chemical potential and superconductivity quenches, by analyzing the wave functions using two measure: inverse participation ratio and overlap. We see that this region does not cover the full topological phase in parameter space. The Majoranas already disappear before the quenched system has reached the phase transition.
This thesis consists of two parts. In part one we discuss the Kondo model. First we point out the differences in physical properties between the one- and multi-channel Kondo model using a renormalization group analysis. We also point out the challenges in obtaining exact solutions. Then we give an overview of two approaches, bosonization and spin chain, for finding exact solution of the one-, two- and four-channel Kondo model. Using bosonization we review a mapping from the one- and two-channel model to the exactly solvable resonant level model for a specific coupling strength. Following previous work, we derive the Kondo effect for a spin chain construction with XX and Ising coupling, showing the low energy behavior to be equivalent to the two- or four-channel Kondo model. We also discuss why one, two and four channels are special and we point out the difficulties in extending these approaches to other multi-channel Kondo models.\nThen in part two we investigate survival of the topological edge effects of the Kitaev chain (Majorana edge modes) after a quantum quench. We determine a stability region for the edge modes after chemical potential and superconductivity quenches, by analyzing the wave functions using two measure: inverse participation ratio and overlap. We see that this region does not cover the full topological phase in parameter space. The Majoranas already disappear before the quenched system has reached the phase transition.
We consider the Kondo effect in Y-junctions of anisotropic XY models in an applied magnetic field along the critical lines characterized by a gapless excitation spectrum. We find that, while the boundary interaction Hamiltonian describing the junction can be recasted in the form of a four-channel, spin-1/2 antiferromagnetic Kondo Hamiltonian, the number of channels effectively participating in the Kondo effect depends on the chain parameters, as well as on the boundary couplings at the junction. The system evolves from an effective four-channel topological Kondo effect for a junction of XX-chains with symmetric boundary couplings into a two-channel one at a junction of three quantum critical Ising chains. The effective number of Kondo channels depends on the properties of the boundary and of the bulk. The XX-line is a “critical” line, where a four-channel topological Kondo effect can be recovered by fine-tuning the boundary parameter, while along the line in parameter space connecting the XX-line and the critical Ising point the junction is effectively equivalent to a two-channel topological Kondo Hamiltonian. Using a renormalization group approach, we determine the flow of the boundary couplings, which allows us to define and estimate the critical couplings and Kondo temperatures of the different Kondo (pair) channels. Finally, we study the local transverse magnetization in the center of the Y-junction, eventually arguing that it provides an effective tool to monitor the onset of the two-channel Kondo effect.
The two-impurity Kondo model with particle-hole symmetry is known to have a critical point at comparable RKKY interaction and Kondo temperature. Using bosonization and controlled low energy projection, we identify the critical point and map it to a solvable resonant level model. The physics of the critical point is shown to be the same as the two-channel Kondo model. An effective Hamiltonian is derived from which the solutions for the critical point and surrounding Fermi-liquid phase follow immediately. The universality of the critical point is strongly supported by the overall excellent agreement with the numerical renormalization group and conformal field theory results.
This thesis investigates the properties of entanglement in one-dimensional many-body systems. In the first part, the non-equilibrium dynamics following a sudden global quench are exploited for the purpose of generating long-range entanglement. A number of initial states are considered. It is shown that the dynamics following the considered quench can be mapped to the problem of a state transfer. The quench can then be optimised by exploiting the literature about quantum state transfer to generate maximal long-range entanglement and maximal block entropy. \nIn the second part of the thesis, a spin chain emulation of the two-channel, Kondo (2CK) model is proposed. Studying the local magnetisation and susceptibility we show that the spin-only emulation truly represent the two-channel Kondo model and extract the Kondo temperature. A detailed entanglement analysis is presented. Using density matrix renormalisation group (DMRG), which allow for real space analysis, Kondo temperature and Kondo length are evaluated. An entanglement measure, namely the negativity, as well as the Schmidt gap are used as possible order parameters predicting the critical point. An extensive analysis of the block entropy of the system is presented for different limiting values of Kondo coupling. A universal scaling of the impurity contribution to the entropy is found and the 2CK residual entropy is extracted. \nThe last part explores quench dynamics in Kondo systems using time-dependent DMRG. For a quench in the Kondo coupling a travelling and breathing clouds are ob-served. A measurement-induced dynamics lead to an oscillation between an effective singlet and triplet states of the impurity and the Kondo cloud. Kondo temperature can be extracted from the frequency of the oscillation.
It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, since only the spin degrees of freedom are involved in the Kondo interaction, and due to spin–charge separation in 1D, the universal low energy long distance physics of the Kondo model also arises when a magnetic impurity is coupled to the end of a gapless antiferromagnetic J 1 –J 2 spin- chain, where J 1 ( J 2 ) is the (next) nearest neighbor coupling. Experimental realizations of such spin chain models are possible and, using various analytical and numerical techniques, we present a detailed and quantitative comparison between the usual free electron Kondo model and such spin chain versions of the Kondo problem. For the gapless J 1 –J 2 spin chain two cases are studied, with zero next nearest neighbor coupling, J 2 = 0, and with a critical second-neighbor coupling, J 2 = J 2 c . We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange where a bulk marginal coupling present in the spin chain model for J 2 < J 2 c vanishes . There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J 2 c the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. We then analyze the nearest neighbor model ( J 2 = 0) where a new kind of Kondo effect occurs due to the presence of the bulk marginal coupling. This marginal coupling slightly alters the β function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T K , with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model ( J 2 = 0), Frahm and Zvyagin noted this relation as well as the connection to the Kondo problem. Here, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.
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.
Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here, we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it binds an emergent gauge flux, while otherwise it becomes free at low temperatures. We show how Majorana degrees of freedom determine the fixed-point properties, make contact with Kondo screening in pseudogap Fermi systems, and discuss effects away from the dilute limit.