We develop an analytical approach for the study of the quench dynamics of the anisotropic Heisenberg model (XXZ model) on the infinite line. We present the exact time-dependent wave functions after a quench in an integral form for any initial state and for any anisotropy Δ by means of a generalized Yudson contour representation. We calculate the evolution of several observables from two particular initial states: starting from a local Néel state we calculate the time evolution of the antiferromagnetic order parameter-staggered magnetization; starting from a state with consecutive flipped spins (1) we calculate the evolution of the local magnetization and express it in terms of the propagation of magnons and bound state excitations, and (2) we predict the evolution of the induced spin currents. These predictions can be confronted with experiments in ultracold gases in optical lattices. We also show how the "string" solutions of Bethe ansatz equations emerge naturally from the contour approach.
We develop an analytical approach for the study of the quench dynamics of the anisotropic Heisenberg model (XXZ model) on the infinite line. We present the exact time-dependent wave functions after a quench in an integral form for any initial state and for any anisotropy Δ by means of a generalized Yudson contour representation. We calculate the evolution of several observables from two particular initial states: starting from a local Néel state we calculate the time evolution of the antiferromagnetic order parameter-staggered magnetization; starting from a state with consecutive flipped spins (1) we calculate the evolution of the local magnetization and express it in terms of the propagation of magnons and bound state excitations, and (2) we predict the evolution of the induced spin currents. These predictions can be confronted with experiments in ultracold gases in optical lattices. We also show how the "string" solutions of Bethe ansatz equations emerge naturally from the contour approach.
The recent advance of techniques in controlling ultra-cold gases in optical lattice provides a ideal experimental platform to study quantum many-body physics. The almost isolated and highly tunable systems created in these experimental setups are perfect for studying coherent time evolution of closed quantum many-body systems. In this thesis I present an analytical approach on the time evolution of the anisotropic Heisenberg spin chain (XXZ model), which is a fundamental model but has rich dynamics. This approach is developed through generalizing the Yudson contour representation. With this approach I obtain the exact time-dependent wave functions after a quantum quench in an integral form for generic initial states and for any anisotropy ∆. I reformulate the integrals and show their physical interpretation as propagating magnons and their bound states, which were predicted by the “string” solutions of Bethe Ansatz equations. To apply the new approach to specific problems, I calculate the quench dynamics of various observables from some particular initial states: starting from a local N ́eel state I calculate the time evolution of the antiferromagnetic order parameter–staggered magnetization; starting from a state with consecutive flipped spins I calculate the evolution of the local magnetization, and predict the evolution of the induced spin currents. These results and predictions can be confronted with recently experiments in ultra-cold gases in optical lattices, and are expected to contribute to the understanding of non-equilibrium physics of isolated many-body systems.
The steady state after a quantum quench from the Néel state to the anisotropic Heisenberg model for spin chains is investigated. Two methods that aim to describe the postquench non-thermal equilibrium, the generalized Gibbs ensemble and the quench action approach, are discussed and contrasted. Using the recent implementation of the quench action approach for this Néel-to-XXZ quench, we obtain an exact description of the steady state in terms of Bethe root densities, for which we give explicit analytical expressions. Furthermore, by developing a systematic small-quench expansion around the antiferromagnetic Ising limit, we analytically investigate the differences between the predictions of the two methods in terms of densities and postquench equilibrium expectation values of local physical observables. Finally, we discuss the details of the quench action solution for the quench to the isotropic Heisenberg spin chain. For this case we validate the underlying assumptions of the quench action approach by studying the large-system-size behavior of the overlaps between Bethe states and the Néel state.
The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e. by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic Neel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.
Abstract The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e. by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic Néel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.
Recent experimental achievements in controlling ultracold gases in optical lattices open a new perspective on quantum many-body physics. In these experimental setups, it is possible to study coherent time evolution of isolated quantum systems. These dynamics reveal new physics beyond the low-energy properties that are usually relevant in solid-state many-body systems. In this paper, we study the time evolution of antiferromagnetic order in the Heisenberg chain after a sudden change of the anisotropy parameter, using various numerical and analytical methods. As a generic result, we find that the order parameter, which can show oscillatory or non-oscillatory dynamics, decays exponentially except for the effectively non-interacting case of the XX limit. For weakly ordered initial states, we also find evidence for an algebraic correction to the exponential law. The study is based on numerical simulations using a numerical matrix product method for infinite system sizes (iMPS), for which we provide a detailed description and an error analysis. Additionally, we investigate in detail the exactly solvable XX limit. These results are compared to approximative analytical approaches including an effective description by the XZ model as well as by mean-field, Luttinger-liquid and sine-Gordon theories. The comparison reveals which aspects of non-equilibrium dynamics can, as in equilibrium, be described by low-energy theories and which are the novel phenomena specific to quantum quench dynamics. The relevance of the energetically high part of the spectrum is illustrated by means of a full numerical diagonalization of the Hamiltonian.
This thesis studies the quench dynamics of strongly correlated quantum systems described by one dimensional integrable Hamiltonians. We develop the Yudson approach for such systems in terms of contour integrals allowing the expansion of arbitrary states in terms of the Bethe Ansatz eigenstates which in turn provides the means to calculate the time evolution of such observables as densities or noise correlations from any initial states. As a motivation to the present work, I present the state of art ultracold atom techniques and fundamental questions related to the quench dynamics. Then various Bethe ansatz solutions are discussed. This is followed by an introduction to the Yudson approach where advantages and difficulties are listed. As applications of the Yudson approach, I studied the nonequilibrium dynamics of the Lieb-Liniger model,(bosonic) Gaudin-Yang model, quenched from a Mott state with a super fluid Hamiltonian. Integration contours are specified for various models and different interactions. It is shown that the Yudson approach incorporates all free states and bound states into the contour, separating them apart sheds light on the validity of the String hypothesis. The result is affirmative for Lieb-Linger model, but not for the Gaudin-Yang model. Our calculations for the density and correlation shows that for interacting system, if the pre-quench state has negligible overlap among the particles, the system retains this feature after the quench. Particularly, normalized noise function c(z; -z) at the origin shows different stages. Shortly after the quench, the sign are different for attractive and repulsive interaction. Then they both quickly approaches the value where possibility to find both particles equals zero. Then the value increases gradually for attractive models, while c(0; 0) remains small for repulsive systems.