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Dynamic Fidelity Susceptibility and its Applications to Out-of-Equilibrium Dynamics in Driven Quantum Systems

Matt Richards

Published Jan 1, 2019• 0 citations
Publisher

Abstract

In this thesis we introduce a new quantity which we call the dynamic fidelity susceptibility (DFS). We show that it is relevant to out-of-equilibrium dynamics in many-particle quantum systems, taking the problem of an impurity in a Bosonic Josephson junction, and the transverse field Ising model, as examples. Both of these systems feature quantum phase transitions in their ground states and understanding the dynamics near such critical points is currently an active area of research. In particular, sweeping a system through a quantum critical point at finite speed leads to non-adiabatic dynamics. A simple theoretical tool for describing such a scenario is the celebrated Kibble-Zurek theory which predicts that the number of excitations is related to the speed of sweep via the phase transition’s critical exponents at equilibrium. Another theoretical tool, useful in describing the static properties of quantum phase transitions, is the fidelity susceptibility. Our DFS generalizes the concept of fidelity susceptibility to nonequilibrium dynamics, reproducing its results in the static limit, whilst also displaying universal scaling properties, akin to those found in Kibble-Zurek theory, in the non-adiabatic regime. Furthermore, we show that the DFS is the same quantity as the time-dependent quantum Fisher information which provides a measure of multi-partite entanglement, as well as being closely related to out-of-time-order correlators (OTOCs).

DOI
10.71548/19871

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In this thesis we introduce a new quantity which we call the dynamic fidelity susceptibility (DFS). We show that it is relevant to out-of-equilibrium dynamics in many-particle quantum systems, taking the problem of an impurity in a Bosonic Josephson junction, and the transverse field Ising model, as examples. Both of these systems feature quantum phase transitions in their ground states and understanding the dynamics near such critical points is currently an active area of research. In particular, sweeping a system through a quantum critical point at finite speed leads to non-adiabatic dynamics. A simple theoretical tool for describing such a scenario is the celebrated Kibble-Zurek theory which predicts that the number of excitations is related to the speed of sweep via the phase transition’s critical exponents at equilibrium. Another theoretical tool, useful in describing the static properties of quantum phase transitions, is the fidelity susceptibility. Our DFS generalizes the concept of fidelity susceptibility to nonequilibrium dynamics, reproducing its results in the static limit, whilst also displaying universal scaling properties, akin to those found in Kibble-Zurek theory, in the non-adiabatic regime. Furthermore, we show that the DFS is the same quantity as the time-dependent quantum Fisher information which provides a measure of multi-partite entanglement, as well as being closely related to out-of-time-order correlators (OTOCs).
In this paper we address the question how the Kibble-Zurek mechanism, which describes the formation of topological defects in quantum systems subjected to a quench across a critical point, is generalized to the same scenario but for driven-dissipative quantum critical systems. In these out of equilibrium systems, the critical behavior is manifested in the steady states rather than in the ground states as in quantum critical models of closed systems. To give the generalization we need to establish what is meant by adiabaticity in open quantum systems. Another most crucial concept that we clarify is the characterization of non-adiabatic excitations. It is clear what these are for a Hamiltonian systems, but question is more subtle for driven-dissipative systems. In particular, the important observation is that the instantaneous steady states serve as reference states. From these the excitations can then be extracted by comparing the distance from the evolved state to the instantaneous steady state. With these issues resolved we demonstrate the applicability of the generalized Kibble-Zurek mechanism to an open Landau-Zener problem and show universal Kibble-Zurek scaling for an open transverse Ising model. Thus, our results support any assumption that non-equilibrium quantum critical behavior can be understood from universal features.
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.
This thesis reports the results obtained during my PhD research in the field of out of equilibrium quantum many-body systems. Chapter 1 consists in a brief introduction of the field and the introduction of concept that are useful for the following chapters. In Chapter 2 the statistics of the work as a tool for characterizing the dynamics of many-body quantum systems is introduced its general features discussed. Then, such a statistics is computed for generic time-dependent protocols (both global and local) in the quantum Ising chain and in the Gaussian field theory, showing, in particular, that in its low-energy part there are features that are independent of the details of the specific chosen protocol. Chapter 3 is devoted to the study of the dynamical phase transition in the O(N)O(N)O(N) quantum vector model in the N→∞N \rightarrow \inftyN→∞ limit, whose critical properties in generic dimensions are characterized. Moreover, a strong connection between such a transition and the statistics of excitations produced in a double quench as a function of the waiting time is showed. The chapter ends by studying the fate of the dynamical transition and the its critical properties when a ramp of finite duration τ\tauτ is applied to the system instead of a sudden quench. In particular, we will show that when τ→∞\tau \rightarrow \inftyτ→∞ the critical point tends to the equilibrium critical point (at zero temperature) in a power-law fashion and that for every finite τ\tauτ the critical properties are always the same (and different from the equilibrium critical properties). Finally in Chapter 4 we will discuss the emergence of a non adiabatic behavior in the dynamics of the order parameter for a low dimensional quantum system driven within a gapped phase by considering in detail the case of a quantum Ising chain subject to a linear variation in time of the transverse field, showing that, no matter how slowly the ramp is performed, such a change leads eventually to the disruption of the order.
We give an overview of the scaling of density of quasi-particles and excess energy (heat) for nearly adiabatic dynamics near quantum critical points (QCPs). In particular we discuss both sudden quenches of small amplitude and slow sweeps across the QCP. We show close connection between universal scaling of these quantities with the scaling behavior of the fidelity susceptibility and its generalizations. In particular we argue that the Kibble-Zurek scaling can be easily understood using this concept. We discuss how these scalings can be derived within the adiabatic perturbation theory and how using this approach slow and fast quenches can be treated within the same framework. We also describe modifications of these scalings for finite temperature quenches and emphasize the important role of statistics of low-energy excitations. In the end we mention some connections between adiabatic dynamics near critical points with dynamics associated with space-time singularities in the metrics, which naturally emerges in such areas as cosmology and string theory.
We give an overview of the scaling of density of quasi-particles and excess energy (heat) for nearly adiabatic dynamics near quantum critical points (QCPs). In particular we discuss both sudden quenches of small amplitude and slow sweeps across the QCP. We show close connection between universal scaling of these quantities with the scaling behavior of the fidelity susceptibility and its generalizations. In particular we argue that the Kibble-Zurek scaling can be easily understood using this concept. We discuss how these scalings can be derived within the adiabatic perturbation theory and how using this approach slow and fast quenches can be treated within the same framework. We also describe modifications of these scalings for finite temperature quenches and emphasize the important role of statistics of low-energy excitations. In the end we mention some connections between adiabatic dynamics near critical points with dynamics associated with space-time singularities in the metrics, which naturally emerges in such areas as cosmology and string theory.
We study the effects of symmetry-breaking perturbations in the out-of-equilibrium quantum dynamics of many-body systems slowly driven by a time-dependent symmetry-preserving parameter, across the quantum critical regime associated with a continuous quantum transition (CQT). For this purpose, we analyze the out-of-equilibrium dynamics arising from generalized Kibble-Zurek (KZ) protocols, within a dynamic renormalization-group framework allowing for finite-size systems. We show that the time dependence of generic observables develops an out-of-equilibrium finite-size scaling (FSS) behavior, arising from the interplay between the timescale ts{t}_{s}ts​ of the parameter variations in the KZ protocol, the size LLL of the system, and the strength hhh of the symmetry-breaking perturbation, in the limit of large ts{t}_{s}ts​ and LLL. Moreover, scaling arguments based on the first-order adiabatic approximation of slow variations in quantum systems allow us to characterize the approach to the adiabatic regimes for some limits of the model parameters (for example, when we take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions of the nonadiabatic behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by numerical analyses for the paradigmatic quantum Ising chain along generalized KZ protocols, with a time-dependent transverse field crossing its CQT, in the presence of a static longitudinal field breaking the Z2{\mathbb{Z}}_{2}Z2​ symmetry.