We study the real-time dynamics of the Kondo effect after a quantum quench in which a magnetic impurity is coupled to two metallic Hubbard chains. Using an effective field theory approach, we find that for noninteracting electrons the charge current across the impurity is given by a scaling function that involves the Kondo time. In the interacting case, we show that the Kondo time decreases with the strength of the repulsive interaction and the time dependence of the current reveals signatures of the Kondo effect in a Luttinger liquid. In addition, we verify that the relaxation of the impurity magnetization does not exhibit universal scaling behavior in the perturbative regime below the Kondo time. Our results highlight the role of nonequilibrium dynamics as a valuable tool in the study of quantum impurities in interacting systems.
We study the real-time dynamics of the Kondo effect after a quantum quench in which a magnetic impurity is coupled to two metallic Hubbard chains. Using an effective field theory approach, we find that for noninteracting electrons the charge current across the impurity is given by a scaling function that involves the Kondo time. In the interacting case, we show that the Kondo time decreases with the strength of the repulsive interaction and the time dependence of the current reveals signatures of the Kondo effect in a Luttinger liquid. In addition, we verify that the relaxation of the impurity magnetization does not exhibit universal scaling behavior in the perturbative regime below the Kondo time. Our results highlight the role of nonequilibrium dynamics as a valuable tool in the study of quantum impurities in interacting systems.
We study a quantum quench in which a magnetic impurity is suddenly coupled to Hubbard chains, whose low-energy physics is described by Tomonaga-Luttinger liquid theory. Using the time-dependent density-matrix renormalization-group (tDMRG) technique, we analyze the propagation of charge, spin, and entanglement in the chains after the quench and relate the light-cone velocities to the dispersion of holons and spinons. We find that the local magnetization at the impurity site decays faster if we increase the interaction in the chains, even though the spin velocity decreases. We derive an analytical expression for the relaxation of the impurity magnetization which is in good agreement with the tDMRG results at intermediate timescales, providing valuable insight into the time evolution of the Kondo screening cloud in interacting systems.
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.
In this thesis we investigate the physics of two strongly correlated electron systems. The part I of this document is dedicated to the study of the real-time dynamics of the Kondo effect produced by a quench protocol. For it, we consider one or two metallic chains of interacting electrons (Hubbard chains) coupled to a magnetic impurity of spin 1/2. The quench protocol consists of assuming that, for times τ < 0, the chains are decoupled from the impurity, at τ = 0 the coupling is suddenly turned on and, for τ > 0 , the system evolves unitarily in time. We determine an analytical expression for the relaxation of the impurity magnetization in the time regime τ ≪ τK , where τK is the Kondo time. This result was confirmed by numerical calculations performed in our group. Based on these numerical results, we observe that the increase in the interaction U between the electrons in the chains leads to a faster decay of the impurity magnetization, showing that the electron-electron interaction favors the formation of the Kondo effect. Further we studied the time-dependent transport through the impurity, from which we determine, in the interacting case, an analytical expression for the time scale equivalent to the Kondo time. According to the expression we obtained, we conclude that the Kondo scale is reduced when U increases, in agreement with the numerical results mentioned above. In part II of this thesis we investigate the physics of superconducting cuprates, motivated by electronic Raman spectroscopy measurements of the charge density wave (CDW) phase observed in these materials. Through a phenomenological approach, we describe a charge order in the CuO2 plane. Analyzing the spectral properties of the system, we establish that the CDW phase should be more easily detectable in the ω > 0 (above the Fermi level) region of the local density of states. Our results for the B2g Raman response show a general dip-hump behavior, as actually seen in the experiments, which is independent of the system band structure. In contrast, the B1g Raman response is strongly band structure dependent. In the second part of the work, a more general model was proposed, based on the Yang-Rice-Zhang ansatz, which also takes into account the pseudogap (PG) phase. We conclude that the PG phase drives the doping dependence of the CDW energy scale, in agreement with the experiments. This result is suggestive of a PG phase governing the different energy scales of the exotic phases that appear in the underdoped region of the cuprate phase diagram.
The Anderson impurity model is a paradigmatic example in the study of strongly correlated quantum systems and describes an interacting quantum dot coupled to electronic leads. Here we investigate its dynamics following a quantum quench based on matrix product state simulations. We examine the behavior of its impurity magnetization. Its relaxation allows us to extract the predicted scaling of the Kondo temperature as a function of the impurity-lead hybridization and quantum dot repulsion. Additionally, our simulations provide estimates of the currents in the nonequilibrium quasisteady state appearing after the quench. Through their values, we examine the dependence of the conductance on the voltage bias Vb and on the impurity chemical potential Vg, which displays a zero-bias Kondo peak. Our results are relevant for transport measurements in Coulomb blockaded devices, and, in particular, in quantum dots induced in nanowires.
A detailed derivation of the recently proposed time-dependent numerical renormalization-group (TD-NRG) approach to nonequilibrium dynamics in quantum-impurity systems is presented. We demonstrate that the method is suitable for fermionic as well as bosonic baths. Comparisons with exact analytical results for the charge relaxation in the resonant-level model and for dephasing in the spin-boson model establish the accuracy of the method. The real-time dynamics of a single spin coupled to each type of bath is investigated. We use the TD-NRG to calculate the spin relaxation and spin precession of a single Kondo impurity. The short- and long-time dynamics are studied as a function of temperature in the ferromagnetic and antiferromagnetic regimes. The short-time dynamics agrees very well with analytical results obtained at second order in the exchange coupling J. In the ferromagnetic regime, the transient spin decay is described by the scaling variable $x=2{\ensuremath{\rho}}_{F}\ensuremath{\mid}J(T)\ensuremath{\mid}Tt$. In the antiferromagnetic regime, the long-time decay is governed for T<TK by the Kondo time scale 1∕TK. Here ${\ensuremath{\rho}}_{F}$ is the conduction-electron density of states, TK is the Kondo temperature, and J(T) is the effective exchange coupling at temperature T. Results for spin precession are obtained by rotating the external magnetic field from the x axis to the z axis.
We study non-equilibrium electron transport through a quantum impurity coupled to metallic leads using the equation of motion technique at finite temperature T . Assuming that the interactions are taking place solely in the impurity and focusing on the infinite Hubbard limit, we compute the out of equilibrium density of states and the differential conductance G2(T, V ) in order to test several scaling laws. We find that G2(T, V )/G2(T, 0) is a universal function of both eV/TK and T/TK, TK being the Kondo temperature. The effect of an in-plane magnetic field on the splitting of the zero bias anomaly in the differential conductance is also analyzed. For a Zeeman splitting , the computed differential conductance peak splitting depends only on /TK, and for large fields approaches the value of 2. Besides studying the traditional two leads setup, we also consider other configurations that mimic recent experiments, namely, an impurity embedded in a mesoscopic wire and the presence of a third weakly coupled lead. In these cases, a double peak structure of the Kondo resonance is clearly obtained in the differential conductance while the amplitude of the highest peak is shown to decrease as ln(eV/TK). Several features of these results are in qualitative agreement with recent experimental observations reported on quantum dots.