In this master’s dissertation, we explore non-equilibrium phenomena in strongly interacting electron systems. Our main focus is on investigating the emergence of charge and spin orderings in one-dimensional systems under non-equilibrium conditions. Starting from an initially disordered non-interacting state, we track its time evolution under an interacting Hamiltonian—a protocol known as a quantum quench. Specifically, we concentrate on the development of charge and spin orderings within the half-filled one-dimensional extended Hubbard model, characterized by repulsive interactions and exhibiting the charge density wave (CDW) and spin density wave (SDW) phases. We simulate the progression of the initial state under the Hamiltonian of the mentioned model, considering that the electronic interactions progressively scale up over a finite time interval, eventually reaching values corresponding to the CDW or SDW phase. The non-equilibrium dynamics are examined using the time-dependent extension of the density matrix renormalization group (t-DMRG) method, implemented through the ITensor software. Inspired by tensor diagrams, ITensor facilitates efficient and accurate numerical simulations, enabling comprehensive analyses of the system time evolution under different degrees of electronic interactions. After reproducing results obtained in our research group for the evolution during the quench, we proceed to investigate the post-quench free evolution. Analyzing the order parameters of the CDW and SDW phases, we identify three temporal regimes for the quenches: instantaneous, intermediate, and adiabatic. The system that achieved the adiabatic regime during the quench remained stable in the post-quench analysis. The system subject to the instantaneous regime during the quench did not converge to the ground state in the post-quench, exhibiting a significant increase in entropy. In the case of the intermediate regime, after reaching a peak during the quench, the parameters exhibited oscillatory behavior, which was also observed in larger chains.
In this master’s dissertation, we explore non-equilibrium phenomena in strongly interacting electron systems. Our main focus is on investigating the emergence of charge and spin orderings in one-dimensional systems under non-equilibrium conditions. Starting from an initially disordered non-interacting state, we track its time evolution under an interacting Hamiltonian—a protocol known as a quantum quench. Specifically, we concentrate on the development of charge and spin orderings within the half-filled one-dimensional extended Hubbard model, characterized by repulsive interactions and exhibiting the charge density wave (CDW) and spin density wave (SDW) phases. We simulate the progression of the initial state under the Hamiltonian of the mentioned model, considering that the electronic interactions progressively scale up over a finite time interval, eventually reaching values corresponding to the CDW or SDW phase. The non-equilibrium dynamics are examined using the time-dependent extension of the density matrix renormalization group (t-DMRG) method, implemented through the ITensor software. Inspired by tensor diagrams, ITensor facilitates efficient and accurate numerical simulations, enabling comprehensive analyses of the system time evolution under different degrees of electronic interactions. After reproducing results obtained in our research group for the evolution during the quench, we proceed to investigate the post-quench free evolution. Analyzing the order parameters of the CDW and SDW phases, we identify three temporal regimes for the quenches: instantaneous, intermediate, and adiabatic. The system that achieved the adiabatic regime during the quench remained stable in the post-quench analysis. The system subject to the instantaneous regime during the quench did not converge to the ground state in the post-quench, exhibiting a significant increase in entropy. In the case of the intermediate regime, after reaching a peak during the quench, the parameters exhibited oscillatory behavior, which was also observed in larger chains.
In this thesis, we study non-equilibrium phenomena in strongly interacting systems. We focus on the out-of-equilibrium formation of charge and spin ordering in one dimension, starting from a disordered non-interacting state and that we let evolve in time according to an interacting Hamiltonian. We analyze the formation of charge and spin ordering within the half-filled one-dimensional extended Hubbard model with repulsive interactions, which presents the charge density wave (CDW) and spin density wave (SDW) phases. We perform finite time quenches to simulate the evolution of the initial state under a Hamiltonian in which electronic interactions increase linearly in time up to final values corresponding to the CDW or SDW phase. The non-equilibrium dynamics is described by the time extension of the density matrix renormalization group (DMRG) method. By linearly turning on the nearest-neighbor or the onsite interaction, we describe the formation of the CDW and SDW ordering, respectively. Our analysis is based on the behavior, along the quench, of the CDW and SDW order parameters, the entanglement entropy, and the fidelity between the evolved state and the corresponding equilibrium ground state. We classify the system behaviour according to the timescale in which the quench is performed: we find the existence of impulse, intermediate, and adiabatic regimes. During the quench, in the impulse regime, we observe that the evolved state remains in the same configuration as the initial state, while in the adiabatic regime it follows the predicted behavior for the ground state of the instantaneous Hamiltonian. For the quenches we analyze, our results indicate that the timescale associated with adiabatic behavior depends on the final state ordering. We observe that the adiabatic regime is reached more slowly when the final state has CDW ordering, compared to the case in which the final state has spin ordering. In the intermediate regime, preceding the adiabatic one in the first case we see an increase in entanglement entropy beyond its initial value which is not observed in the quench towards SDW. We conclude that the adiabatic regime for the case of CDW ordering requires a longer time to prevent entangled excited states from being accessed during the quench. In our work, we also compare cases in which only one of the interactions (onsite or nearest-neighbor) is turned on during the quench with those in which we turn on both interactions simultaneously. Our findings show that the breaking of the system integrability, by turning on the nearest-neighbor interactions, does not give rise to significant changes in the nonequilibrium behavior within the adiabatic approximation.
We investigate the formation of charge and spin ordering by starting from a noninteracting state and studying how it evolves in time under a Hamiltonian with finite electronic interactions. We consider the one-dimensional, half-filled extended Hubbard model, which we solve within time-dependent density matrix renormalization group. By employing linear finite-time quenches in on-site and nearest-neighbor interactions, we find the existence of impulse, intermediate, and adiabatic regimes of time evolution. For the quenches we analyze, we observe that the adiabatic regime is reached with distinct ramping timescales depending on whether the charge density wave (CDW) or the spin density wave (SDW) is formed. The former needs to be slower than the latter to prevent entangled excited states from being accessed during the quench. More interestingly, in the intermediate regime, we observe an enhancement of the entanglement entropy with respect to its initial value, which precedes the formation of the CDW ordering; a similar enhancement is not seen in the quench toward SDW. Our findings also show that the breaking of the system integrability, by turning on the nearest-neighbor interactions, does not give rise to significant changes in the nonequilibrium behavior within the adiabatic approximation.
In the first part of this thesis we study the Aubry-André model for interacting fermions. We numerically describe its phase diagram at half filling, performing both DMRG and QMC simulations. We show the existence of a localized phase and other three regimes: luttinger liquid, charge density wave and productstate. We study the properties of the excited states of the Hamiltonian, looking for a many-body mobility edge in the spectrum, i.e. an energy threshold that separates localized from ergodic states. Analyzing many indicators we prove its existence. Finally we propose a quench-spectroscopy method for detecting the mobility edge dynamically. In the second part we study the expansion dynamics of two bosons in a one-dimensional lattice as ruled by the Bose-Hubbard model Hamiltonian, both in the attractive and repulsive regime. Using the Bethe Ansatz we identify the bound states effects and how the two-particles state evolves in time. We show that, independently from the initial state, there exists a strong relation between the expansion velocity and the presence of bound states in the spectrum. Moreover, we discuss the role of the lattice in the system expansion. In the third part we study the time evolution of the entanglement entropy in the Ising model, when it is dynamically driven across a quantum phase transition with different velocities. We computed the time-evolution of the half chain entanglement entropy and we found that, depending on the velocity at which the critical point is reached, it displays different regimes: an adiabatic one when the system evolves according to the instantaneous ground state; a a sudden quench regime when the system remains frozen to its initial state; and an intermediate one, where the entropy starts growing linearly but then displays oscillations. Moreover, we discuss the Kibble-Zurek mechanism for the transition between the paramagnetic and the ordered phase.
In this thesis, strongly correlated quantum many-body systems in equilibrium and in out-of-equilibrium situations are investigated. This is done by applying and developing well established numerical methods. The focus of the thesis lies in the development and application of the density matrix renormalization group method (DMRG) to quantum many-body systems out of equilbrium. In this thesis, in addition to the DMRG, methods for the exact diagonalization of the Hamiltonian of the system, like the Lanczos- or the Jacobi-Davidson method, are treated. An extension of the Lanczos method makes it possible to treat the time evolution of strongly correlated quantum systems with an accuracy comparable to machine precision. This method is the basis for a possible extension of the DMRG for the treatment of systems out of equilibrium, the so-called adaptive time-dependent DMRG ("adaptive t-DMRG"). A second variant of the adaptive t-DMRG uses the Suzuki-Trotter decomposition of the time-evolution operator. An error analysis demonstrates that both methods, for a suitable choice of control parameters, have errors < 1% at the end of the time evolution in complicated quantities like, e.g., the momentum distribution function. We apply these numerical methods to investigate the quantum critical behavior of a variant of the Hubbard model and to treat two non-equilibrium situations of current interest. Extensive use of the DMRG makes it possible to clarify the quantum-critical behavior of the ionic Hubbard model; in particular, the numerical results demonstrate quite clearly that the correct scenario has two critical points; at the first critical point, only the charge degrees of freedom are critical, while at the second one only the spin degrees of freedom are critical. Quite surprisingly, the Mott-insulator phase in the strong-coupling regime shows a divergent electrical susceptibility. Next, the dynamics of a system of so-called soft-core bosons is treated. The particles initially are trapped in a deep box-potential and the dynamics is investigated after releasing them from this trapping potential. Similar to exact results obtained for so-called hard-core bosons, (quasi-)coherent matter waves emerge, demonstrating the possibility of realizing an atom laser in experiments on optical lattices. We find that the wave vector of the emerging matter wave can be tuned by changing the strength of the interaction between the particles. The second non-equilibrium situation is a so-called "quantum-quench" of a system of strongly correlated fermions, i.e., the evolution of the system after suddenly changing an intrinsic parameter like, e.g., the interaction strength between the particles, is investigated. In this case, we are mainly interested in the long-time behavior; in particular, the question arises whether, relying on Boltzmann's ergodic hypothesis, relaxation to a thermal state is obtained. We find that, in general, non-thermal final states are reached, which can be described by generalized Gibbs-Boltzmann ensembles.
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.
Abstract We investigate the non-equilibrium dynamics of the one-dimensional extended Hubbard model after interaction quenches. In strong-coupling regime with large on-site interaction, the ground states of this model with small and large nearest-neighbor interactions are in spin-density-wave and charge-density-wave phases, respectively. Combining twisted boundary conditions with the time-dependent Lanczos method, we obtain snapshots of the time-dependent single-particle spectrum after quenches. We find that for quench within the same phase, the single-particle spectrum becomes close to that of the quenched Hamiltonian immediately after the quench. While for quench across the critical point, the afterward evolution process depends mainly on the distribution of the initial state among the eigenstates of the quenched Hamiltonian. Our finding may serve as a way to detect the phase transition in ultracold atom systems with interactions.