The focus of this thesis is to investigate non equilibrium properties of a spin 1/2 chain considered as a closed or an open quantum system. In the first case, the non equilibrium situation is generated by means of a quench (sudden change of the hamiltonian); in the second case, we connect unbalanced magnetic reservoirs at the chain edges. In this thesis, we present two works in the context of closed systems and another one in the case of open systems. To describe the dynamics we use the time-dependent extension of Density MatrixRenormalization Group (DMRG) technique. In the first work, we investigate the non equilibrium dynamics generated by a local quenchin a spin system described by the XXZ Heisenberg model. More specifically, the quench involves connecting two chains, one that is prepared in the ferromagnetic phase and another that is in the paramagnetic phase. After the quench, we observe effective light cones inthe maps of the on-site magnetization and the entanglement entropy as a function of site and time, when these quantities are calculated in the chain that had no magnetic order.We show that these light cones are associated with two kinds of excitation characteristic of the XXZ model, more specifically spinons and bound states. Furthermore, we see that, for long times, the system approaches the equilibrium state of the connected chains. systemwill go to the equilibrium state at infinity time.In the second work, we analyze shielding properties that appear in the transverse Ising model. If the site i=L of the chain has null field and the global state is a Gibbs state, the reduced state of the chain in the right side of the i=L site is shown to have no dependence on the parameters of the left chain. We refer to this effect as a shielding property. Wealso study the chain dynamics after a local quench that changes the magnetic field in the first site. We observe a light cone in the on-site magnetization, however the magnetization pulse is reflected by the site with null field.Finally, in the third work, we connect two magnetic reservoirs at the ends of the XXZ Heisenberg spin chain and calculate the spin and energy currents in the non equilibrium steady state (NESS). We investigate if it is possible to control and manipulate such currents and their respective rectifications by the application of an external magnetic field. Forthat, we introduce a graded asymmetry (gradual change of the parameters) in the field and in the anisotropy parameter of the model. For appropriate choice of the field, we observe that it is possible to force the energy and the spin currents to propagate in opposite directions. This means that we invert not only the currents but also their rectificationsthrough external mechanisms.
The focus of this thesis is to investigate non equilibrium properties of a spin 1/2 chain considered as a closed or an open quantum system. In the first case, the non equilibrium situation is generated by means of a quench (sudden change of the hamiltonian); in the second case, we connect unbalanced magnetic reservoirs at the chain edges. In this thesis, we present two works in the context of closed systems and another one in the case of open systems. To describe the dynamics we use the time-dependent extension of Density MatrixRenormalization Group (DMRG) technique. In the first work, we investigate the non equilibrium dynamics generated by a local quenchin a spin system described by the XXZ Heisenberg model. More specifically, the quench involves connecting two chains, one that is prepared in the ferromagnetic phase and another that is in the paramagnetic phase. After the quench, we observe effective light cones inthe maps of the on-site magnetization and the entanglement entropy as a function of site and time, when these quantities are calculated in the chain that had no magnetic order.We show that these light cones are associated with two kinds of excitation characteristic of the XXZ model, more specifically spinons and bound states. Furthermore, we see that, for long times, the system approaches the equilibrium state of the connected chains. systemwill go to the equilibrium state at infinity time.In the second work, we analyze shielding properties that appear in the transverse Ising model. If the site i=L of the chain has null field and the global state is a Gibbs state, the reduced state of the chain in the right side of the i=L site is shown to have no dependence on the parameters of the left chain. We refer to this effect as a shielding property. Wealso study the chain dynamics after a local quench that changes the magnetic field in the first site. We observe a light cone in the on-site magnetization, however the magnetization pulse is reflected by the site with null field.Finally, in the third work, we connect two magnetic reservoirs at the ends of the XXZ Heisenberg spin chain and calculate the spin and energy currents in the non equilibrium steady state (NESS). We investigate if it is possible to control and manipulate such currents and their respective rectifications by the application of an external magnetic field. Forthat, we introduce a graded asymmetry (gradual change of the parameters) in the field and in the anisotropy parameter of the model. For appropriate choice of the field, we observe that it is possible to force the energy and the spin currents to propagate in opposite directions. This means that we invert not only the currents but also their rectificationsthrough external mechanisms.
We investigate the out-of-equilibrium dynamics after a local quench that connects two spin-1/2 XXZ chains prepared in the ground state of the Hamiltonian in different phases, one in the ferromagnetic phase and the other in the critical phase. We analyze the time evolution of the on-site magnetization and bipartite entanglement entropy via adaptive time-dependent density matrix renormalization group. In systems with short-range interactions, such as the one we consider, the velocity of information transfer is expected to be bounded, giving rise to a light-cone effect. Interestingly, our results show that, when the anisotropy parameter of the critical chain is sufficiently close to that of the isotropic ferromagnet, the light cone is determined by the velocity of spin-wave bound states that propagate faster than single-particle (``spinon'') excitations. Furthermore, we investigate how the system approaches equilibrium in the inhomogeneous ground state of the connected system, in which the ferromagnetic chain induces a nonzero magnetization in the critical chain in the vicinity of the interface.
We addressed the issue of energy and magnetization rectification in the quantum one-dimensional open XXZ spin chain driven out of equilibrium by magnetization pumping applied at its boundaries and in the presence of an external magnetic field. We analyzed the effects of the introduction of an inner graded structure in the chain, as well as the effects of a graded magnetic field in the system. We present theoretical results for the fluxes and also exact solutions, obtained by means of numerical computation, for a small chain. We verified that the energy flux exhibits rectification in both the inner graded structure case and the graded magnetic field one, although the magnetization flux exhibits rectification only in the latter situation. With a view to showing that the rectification is not a phenomenon limited to the studied model, we additionally investigated, by means of computational analysis, distinct configurations of the spin chain regarding the chain size, the internal interactions, and the boundary conditions. We found that the same conditions apply to the existence of rectification, except for one situation: with regard to the twisted XY boundary gradients model, the magnetization flux exhibits rectification for a uniform magnetic field as well.
In this thesis, we study the properties of the transverse field Ising (TFI) chain in several non-equilibrium setups. One such setup is finite-time quantum quench—quench in which a Hamiltonian parameter is continuously changed during a finite-time, as opposed to the usual instantaneous change in the sudden quenches. A different type of quantum quench, which does not involve changing a Hamiltonian parameter, can be realised by the so-called partitioning protocol. In this protocol, two independent chains are prepared at different temperatures and then coupled into a single system and evolved as such. Another possibility to take the system out of equilibrium is to introduce a driving—for example, by suddenly and continuously changing a Hamiltonian parameter over long times. We discuss the general treatment of each of these situations, and apply them to several quench and driving protocols in the TFI chain. We analyse the behaviour of several observables, such as the total energy, spin correlation functions and Loschmidt echo, during and after quenches and during driving. While we choose the specific protocols to display the dependence of observables on the protocol features, we also investigate the universal behaviours of the observables in the late-time limit. Isolated quenched TFI chains do not relax at late times, however, local observables in these systems do—they relax to a special stationary state described by the generalised Gibbs ensemble (GGE). A similar situation happens in the case of periodic driving of the TFI chain—the late time state of the driven TFI chain, at least as far as the local observables are concerned, it is a periodic steady state described by the periodic Gibbs ensemble (PGE). In this thesis, we observe the relaxation behaviour for various quench and driving protocols and observables. In addition to studying statistical ensembles in the context of relaxation of out-of-equilibrium systems, we describe a machine learning algorithm based on the generalised Gibbs ensemble. We first describe a traditional machine learning algorithm, the Boltzmann machine, and its quantum version. Based on our knowledge of the relaxation of the out-of-equilibrium TFI chain, we modify the existing Boltzmann machine algorithm to include the many GGE charges and test it on the commonly used MNIST dataset. Rather than learning a complicated energy function and using the Boltzmann distribution, we then use the relatively simple TFI Hamiltonian and learn the effective temperatures of the GGE and find reasonably low error rates, while learning a comparatively low number of parameters.
We consider the non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that stationary values of local correlation functions can be described by the generalised Gibbs ensemble (GGE). Then we study the full time evo-lution of the transverse magnetisation by means of stationary phase methods. The equal time two-point longitudinal correlation function is analytically derived for a par-ticular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches. ar
We consider the non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that stationary values of local correlation functions can be described by the generalized Gibbs ensemble. Then we study the full time evolution of the transverse magnetization by means of stationary phase methods. The equal-time two-point longitudinal correlation function is analytically derived for a particular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches.
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.