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, 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.
The main topic of this thesis is the study of many-body effects in strongly correlated one- or quasi one-dimensional condensed matter systems. These systems are characterized by strong quantum and thermal fluctuations, which make mean-field methods fail and demand for a fully numerical approach. Fortunately, a numerical method exist, which allows to treat unusually large one -dimensional system at very high precision. This method is the density-matrix renormalization group method (DMRG), introduced by Steve White in 1992. Originally limited to the study of static problems, time-dependent DMRG has been developed allowing one to investigate non-equilibrium phenomena in quantum mechanics. In this thesis I present the solution of three conceptionally different problems, which have been addressed using mostly the Krylov-subspace version of the time-dependent DMRG. My findings are directly relevant to recent experiments with ultracold atoms, also carried out at LMU in the group of Prof. Bloch. The first project aims the ultimate goal of atoms in optical lattices, namely, the possibility to act as a quantum simulator of more complicated condensed matter system. The underline idea is to simulate a magnetic model using ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system, which is captured by a two-species Bose-Hubbard model, realizes in a certain parameter range the physics of a spin-1/2 Heisenberg chain, where the spin exchange constant is given by second order processes. Tuning of the superlattice parameters allows one to controlling the effect of fast first order processes versus the slower second order ones. The analysis is motivated by recent experiments, %\cite{Folling2007,Trotzky2008} where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were detected. My project investigates the coherent many-particle dynamics, which takes place after coupling the double well. I provide the theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for non-integrable and Bethe ansatz integrable models. It turns out that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models In the second project I study a many-body generalization of the original Landau-Zener formula. This formula gives the transition probability between the two states of a quantum mechanical two-level system, where the offset between the two levels is varying linearly in time. In a recent experiment this framework has been extended to a many-body system consisting of pairwise tunnel-coupled one-dimensional Bose liquids. It was found that the tunnel coupling between the tubes and the intertube interactions strongly modify the original Landau-Zener picture. After a introduction to the two-level and the three-level Landau-Zener problem I present my own results for the quantum dynamics of the microscopic model and the comparison to the experimental results. I have calculated both Landau-Zener sweeps as well as the time-evolution after sudden quenches of the energy offset. A major finding is that for sufficiently large initial density quenches can be efficiently used to create quasi-thermal states of arbitrary temperatures. The third project is more mathematical and connects the fields of quantum computation and of quantum information. Here, the main purpose is to analyse systematically the effects of decoherence on maximally entangled multi-partite states, which arise typically during quantum computation processes. The bigger the number of entangled qubits the more fragile is its entanglement under the influence decoherence. As starting point I consider first two entangled qubits, whereby one qubit interacts with an arbitrary environment. For this particular case I have derived a factorization law for the disentanglement. Next, I calculate the decrease of entanglement of two , three and four entangled qubits, general W- and general GHZ-state, coupled to a global spin-1/2 bath or several independent spin-1/2 baths , one for each qubit. Although there is no appropriate entanglement measure for three and more qubits, it turns out that this decrease is directly related to the increase of entanglement between the central system and the bath. This implies the formation of a much bigger multipartite entangled network. Thus, using the von Neumann entropy and the Wootters concurrence, I derive a simple upper bound for the bath-induced entanglement breaking power of the initially maximally entangled multi-partite states.
In this work we develop a new numerical method, the adaptive time-dependent density-matrix renormalization-group (adaptive t-DMRG), which turns out to be very well suited to investigate time-dependent phenomena in one-dimensional strongly correlated systems. The method is based on the original density-matrix renormalization group method by White and the time evolving block-decimation procedure by Vidal. We further show the applicability of the new method to different physical systems: bosonic, fermionic, and spin chains. We first discuss the consequences of the presence of an external trapping potential on the static properties of trapped cold bosonic atoms in an optical lattice using the Bose-Hubbard model. It is shown that a properly rescaled one-particle density matrix characterizes superfluid versus insulating states just as in the homogeneous system. We note that the superfluid to Mott-insulating transition is seen most directly in the half width of the interference peak. Then we turn to the evolution of density perturbations in these systems. In particular, we discuss the dependence of the velocity and the decay of the amplitude on density, interaction strength, and the extent and height of the perturbation in a numerically exact way, covering a wide range of interaction and perturbation strengths. By comparing our results for the sound velocity to theoretical predictions, we determine the limits of a Gross-Pitaevskii or Bogoliubov type description and the regime where repulsive one-dimensional Bose gases exhibit fermionic behaviour. As a second application of the adaptive t-DMRG, we investigate the phenomenon of spin-charge separation in the Hubbard model and propose an experimental setup for its observation in cold Fermi gases. We show the robustness of this separation beyond the low-energy regime by studying the time evolution of density wave packets of finite strength and at length scales down to a few lattice spacings. A striking signature of spin-charge separation is found in 1D cold Fermi gases in a harmonic trap using the different propagation properties of the liquid and Mott-insulating phases. The third application relates to the area of spintronics. A simplified model for the magnetization transport between two coupled reservoirs with opposite spin polarization is studied. We investigate whether a simple long time limit for the spin transport exists and if so, how it depends on the properties of the system such as integrability and criticality. Time-scales accessible to us are of the order of 100 units of time measured in 1/J while maintaining insignificant error in the observables.
In this work, we study the non-equilibrium dynamics of low-dimensional strongly correlated quantum systems, which in our case are generated by sudden and time-dependent quantum quenches. Such non-equilibrium dynamics are a particularly difficult problem, because the time evolution of the initial many-body quantum state, which is governed by the post-quench Hamiltonian, is highly non-trivial. On top of that, applying an operator to this many-body quantum state in order to calculate its expectation values, typically creates an intricate set of excitations. Throughout the thesis, we furthermore contrast the results that we have obtained using exact numerical methods for two microscopic models, namely the Heisenberg and Fermi-Hubbard chains, with results derived from the Luttinger liquid theory, which, in equilibrium, describes the universal behavior of the two models.In the second part of the thesis, we reduce the dimensionality even further and employ complementary numerical and analytical tools to examine the unusual non-equilibrium behavior of a ring-shaped quantum impurity hosting interacting spin-less fermions.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.
In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. In Chapter 5 the use of imaginary-time dynamics in the context of finite-temperature response functions is highlighted. As an application, we study an intriguing quantum phase featuring both glassy order and Bose-Einstein condensation. [35] We introduce and validate a model for the role of geometrical frustration in the coexistence of off-diagonal long range order with an amorphous density profile. The exact characterization of the response of the system to an external density perturbation is what allows here to establish the existence of a spin-glass phase. The differences between such a phase and the otherwise insulating "Bose glasses" are further elucidated in the Chapter. In the second part of the Thesis we focus our attention on the dynamics of closed systems out of equilibrium. This is accomplished by both non-stochastic exact methods for the dynamics and the introduction of a novel time-dependent Variational Monte Carlo scheme. In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.
Experimental advances have made ultracold atoms in optical lattices a favourable setting to study out-of-equilibrium phenomena and attracted considerable attention in recent years.These systems are highly versatile in that experimental parameters can be tuned over a wide range of values in real time.This facilitates the study of quantum quenches, in which parameters in the corresponding Hamiltonian are varied in time faster than the system can respond adiabatically.Such protocols open the door to a rich range of many-body physics and have been studied intensely both theoretically and experimentally.The Bose-Hubbard model (BHM) has been shown to describe interacting ultracold bosons in an optical lattice, allowing the opportunity for experiments to probe the out-of-equilibrium dynamics of the model.The BHM is a particularly convenient context for studying quantum quenches as it displays a quantum phase transition between superfluid and Mott-insulator phases.In this thesis, we develop a strong-coupling approach that allows the study of correlations in space and time in both the superfluid and Mott-insulating phases of the BHM.Specifically, we obtain a two-particle irreducible (2PI) effective action within the contourtime formalism that allows for a description of both equilibrium and out-of-equilibrium phenomena.We derive equations of motion for both the superfluid order parameter and the single-particle many-body Green's functions.First, we assess the accuracy of our formalism by studying the equilibrium solution for the homogeneous BHM and comparing our results to existing strong-coupling methods as well as to exact methods where possible.We then consider homogeneous systems that are initially thermalized in the Mott phase, and which are then subjected to quenches.We solve numerically the equations of motion for this scenario and calculate the single-particle density matrix.We demonstrate a Lieb-Robinson-like maximal propagation velocity for the spreading of single-particle correlations in one, two, and three dimensions.We study the dependence of the maximal propagation velocity on the quench protocol, chemical potential, temperature, and dimensionality.We compare our results to exact methods, existing strong-coupling approaches, and experiments where possible.Lastly, we extend our strong-coupling approach to the disordered BHM and derive equations of motion for the disorder-averaged single-particle Green's function.We discuss how these equations of motion can be used to study the phase stability of many-body localization in the disordered BHM for dimensions higher than one.