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arXiv:1604.01995 · Physical review. B./Physical review. B

Transient phases and dynamical transitions in the post-quench evolution of the generalized Bose-Anderson model

Dmitry V. Chichinadze, Pedro Ribeiro, Yulia E. Shchadilova, Alexey N. Rubtsov

Published Aug 1, 2016• 7 citations
arXivHTMLPDF

Abstract

The exact description of the time evolution of open correlated quantum systems remains one of the major challenges of condensed matter theory, especially for asymptotic long times where most numerical methods fail. Here, the post-quench dynamics of the $N$-component Bose-Anderson impurity model is studied in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. The equilibrium phase diagram is similar to that of the Bose-Hubbard model in that it contains local versions of Mott and Bose phases. Using a numerically exact procedure, we are able to study the real-time evolution including asymptotic long-time regimes. The formation of long-lived transient phases is observed for quench paths crossing foreign phases. For quenches inside the local Bose condensed phase, a dynamical phase transition is reported that separates the evolution towards a new equilibrium state and a regime characterized at large times by a persistent phase rotation of the order parameter. We explain how such nondecaying modes can exist in the presence of a dissipative bath. We discuss the extension of our results to the experimental relevant finite-$N$ case and their implication for the existence of nondecaying modes in generic quantum systems in the presence of a bath.

DOI
10.1103/physrevb.94.054301

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The exact description of the time evolution of open correlated quantum systems remains one of the major challenges of condensed matter theory, especially for asymptotic long times where most numerical methods fail. Here, the post-quench dynamics of the NNN-component Bose-Anderson impurity model is studied in the N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ limit. The equilibrium phase diagram is similar to that of the Bose-Hubbard model in that it contains local versions of Mott and Bose phases. Using a numerically exact procedure, we are able to study the real-time evolution including asymptotic long-time regimes. The formation of long-lived transient phases is observed for quench paths crossing foreign phases. For quenches inside the local Bose condensed phase, a dynamical phase transition is reported that separates the evolution towards a new equilibrium state and a regime characterized at large times by a persistent phase rotation of the order parameter. We explain how such nondecaying modes can exist in the presence of a dissipative bath. We discuss the extension of our results to the experimental relevant finite-NNN case and their implication for the existence of nondecaying modes in generic quantum systems in the presence of a bath.
Critical properties of the dynamical phase transition in the quenched generalized Bose-Anderson impurity model are studied in the mean-field limit of an infinite number of channels. The transition separates the evolution toward ground state and toward the branch of stable excited states. We perform numerically exact simulations of a close vicinity of the critical quench amplitude. The relaxation constant describing the asymptotic evolution toward ground state, as well as asymptotic frequency of persistent phase rotation and number of cloud particles at stable excited state are power functions of the detuning from the critical quench amplitude. The critical evolution (separatrix between the two regimes) shows a non-Lyapunov power-law instability arising after a certain critical time. The observed critical behavior is attributed to the irreversibility of the dynamics of particles leaving the cloud and to memory effects related to the low-energy behavior of the lattice density of states.
We study the dynamic response of an sss-wave BCS-BEC (atomic-molecular) condensate to detuning quenches within the two-channel model beyond the weak-coupling BCS limit. At long times after the quench, the condensate ends up in one of three main asymptotic states (nonequilibrium phases), which are qualitatively similar to those in other fermionic condensates defined by a global complex order parameter. In phase I the amplitude of the order parameter vanishes as a power law, in phase II it goes to a nonzero constant, and in phase III it oscillates persistently. We construct exact quench phase diagrams that predict the asymptotic state (including the many-body wave function) depending on the initial and final detunings and on the Feshbach resonance width. Outside of the weak-coupling regime, both the mechanism and the time dependence of the relaxation of the amplitude of the order parameter in phases I and II are modified. Also, quenches from arbitrarily weak initial to sufficiently strong final coupling do not produce persistent oscillations in contrast to the behavior in the BCS regime. The most remarkable feature of coherent condensate dynamics in various fermion superfluids is an effective reduction in the number of dynamic degrees of freedom as the evolution time goes to infinity. As a result, the long-time dynamics can be fully described in terms of just a few new collective dynamical variables governed by the same Hamiltonian only with ``renormalized'' parameters. Combining this feature with the integrability of the underlying (e.g., the two-channel) model, we develop and consistently present a general method that explicitly obtains the exact asymptotic state of the system.
Using the non-perturbative numerical renormalisation group (NRG) we analyse purely fermionic as well as more complex Bose-Fermi quantum impurity systems on their equilibrium properties and beyond: We present an extensive exploration of real-time dynamics for various quenches using the time-dependent NRG. For the purely fermionic system, we use the Single-impurity Anderson model (SIAM) in which the impurity is a localised orbital hybridising with a surrounding conduction band. The model enables charge scattering with a pseudo-gap scattering rate and features a quantum critical point separating the local moment (LM) phase with an unscreened spin moment from a symmetric strong coupling (SSC) phase with a fractionally screened spin moment. The real-time dynamics of the local double occupancy nicely thermalises for quenches regarding the SSC phase. Whereas for quenches within or into the LM phase the double occupancy systematically deviates from thermal equilibrium for long times. A comparison with results of a recently published time-dependent Gutwiller approach reveals clear discrepancies since this approach restricts the formation of the local moment only on the impurity site. We extend the system by an additional bosonic bath leading to the Bose-Fermi Anderson model (BFAM) and study the influence of the additional bosonic degrees of freedom on the real-time dynamics. In equilibrium the bosonic bath features an additional quantum critical point leading to a localised phase once the coupling to a sub-ohmic bosonic bath exceeds a critical value. In general the coupled bosonic bath leads to a renormalisation of the Coulomb interaction. We show that the dynamics is influenced by the specific choice of the bath exponents. For a F-type combination the non-equilibrium dynamics can be completely reproduced by an effective purely fermionic SIAM while for a B-type combination the nature of the bosonic bath and the coupling strength has a profound impact on the electron dynamics which can only be partially understood by an effective SIAM. Furthermore, we investigate the spin dynamics of an initially polarised impurity spin which is coupled to a fermionic and a bosonic bath leading to the Bose-Fermi Kondo model (BFKM). Through the coupling to the fermionic bath the polarisation vanishes with time. We show that in analogy to the spin-boson model (SBM) an initially coupling to the bosonic bath affects the real-time dynamics fundamentally leading to a remaining polarisation for long times.
In the context of out of equilibrium dynamics of isolated interacting quantum many-body systems, we study the mean-field discretized and nonlinear Gross-Pitaevskii equation (GPE) of motion for a fully connected (or all-to-all coupled) Bose-Hubbard model describing a large population of bosons on a lattice of generic dimension V, with potential experimental applications ranging from ultra-cold atomic gases in optical traps to systems of Josephson junctions. Considering a quench on the system Hamiltonian, we describe the time evolution of a system initialized to the Mott insulating phase and taken in the strong superfluid regime, finding a sharp change in the dynamical behaviour when varying the quench intensity across a critical value, that is a dynamical phase transition. Our results generalize the study presented in some previous papers focusing on the particular cases of V = 2, V = 3 and 1D lattice, and place particular emphasis on the relaxation of some dynamical order parameters in the large V limit. On the side of the dynamical phase transition corresponding to the strong superfluid regime, we find such relaxation to consist of a π-synchronization of the mean field bosonic phases, that we find to compete against the eventual site-dependent disorder of the Hamiltonian in what, in analogy with the notorious Kuramoto model for nonlinear coupled oscillators, can be referred to as a synchronization transition. Additionally, in support of the validity of the mean field approximation far from the transition, we report a study beyond mean field based on the Bogolioubov-de Gennes method.
The Bose-Hubbard model provides a foundational paradigm for exploring quantum phase transitions in many-body systems. This thesis presents a systematic investigation into this model and its extensions, designed to provide a clear path for understanding both the rich underlying physics and the sophisticated numerical methods required for its study. Our approach is sequential: we begin with the standard one-dimensional Bose-Hubbard model, introduce complexity via the extended model with long-range interactions, and conclude by analyzing the impact of dissipation in a cavity-coupled open quantum system.To navigate these regimes, we implement and apply a suite of powerful computational techniques, including Density Matrix Renormalization Group (DMRG), finite-temperature Quantum Monte Carlo (QMC), and Mean-Field Theory (MFT). A significant component of this work is the detailed exposition of the implementation and logic of these numerical methods. Through their comparative application, we precisely characterize the equilibrium phase diagrams and benchmark the capabilities of each method. Crucially, our analysis of the open-system dynamics reveals that dissipation replaces the equilibrium phase diagram with one governed by non-equilibrium steady states (NESS), where the system evolves to either a stable solution or a persistent limit cycle.As such, this thesis is intended to serve a dual purpose: it is both a rigorous physical study of quantum phase transitions in a condensed matter system and a selfcontained pedagogical guide for researchers beginning to explore the phenomenology of quantum phase transitions through numerical schemes.
The Bose-Hubbard model is the simplest model of interacting bosons on a lattice. It has recently been the focus of much attention due to the realization of this model with cold atoms in an optical lattice. The ability to tune parameters in the Hamiltonian as a function of time in cold atom systems has opened up the possibility of studying out-of-equilibrium dynamics, including crossing the quantum critical region of the model in a controlled way. In this paper, I give a brief introduction to the Bose Hubbard model, and its experimental realization and then give an account of theoretical and experimental efforts to understand out-of-equilibrium dynamics in this model, focusing on quantum quenches, both instantaneous and of finite duration. I discuss slow dynamics that have been observed theoretically and experimentally for some quenches from the superfluid phase to the Mott insulating phase and the picture of two timescales, one for fast local equilibration and another for slow global equilibration, that appears to characterize this situation. I also discuss the theoretical and experimental observation of the Lieb-Robinson bounds for a variety of quenches and the Kibble-Zurek mechanism in quenches from the Mott insulator to superfluid. I conclude with a discussion of open questions and future directions.