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Physical review. B./Physical review. B

Nonequilibrium dynamics of a system with two kinds of fermions after a pulse

Andrei A. Zvyagin

Published Feb 13, 2017• 12 citations
Publisher

Abstract

The nonequilibrium evolution of the system of two kinds of fermions under the action of a pulse of the external field has been studied. The number of fermions of each kind oscillates (with beats and decaying) as a function of the duration of the pulse about the value determined by the magnitude of the pulse, and as a function of the magnitude of the pulse. For low-dimensional systems those oscillations can serve as a non-zero-temperature manifestation of dynamical quantum phase transitions. The response of a Fermi gas or liquid in a tilted magnetic field, an edge state of a topological insulator, a quantum wire with spin-orbit coupling, and a dimerized spin-1/2 chain to the pulse can manifest such dynamical oscillations, which can be observed in experiments.

DOI
10.1103/physrevb.95.075122

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The nonequilibrium evolution of the system of two kinds of fermions under the action of a pulse of the external field has been studied. The number of fermions of each kind oscillates (with beats and decaying) as a function of the duration of the pulse about the value determined by the magnitude of the pulse, and as a function of the magnitude of the pulse. For low-dimensional systems those oscillations can serve as a non-zero-temperature manifestation of dynamical quantum phase transitions. The response of a Fermi gas or liquid in a tilted magnetic field, an edge state of a topological insulator, a quantum wire with spin-orbit coupling, and a dimerized spin-1/2 chain to the pulse can manifest such dynamical oscillations, which can be observed in experiments.
The evolution of the system of fermions with pairing under the action of a pulse (or a quench) of the external potential has been studied. In the dynamical regime the number of fermions oscillates (with beats) as a function of the duration of the pulse (or as a function of time for the quench) about the value determined by the magnitude of the pulse. Oscillations depend on the values of the potential and the pulse. The parameters of those oscillations permit one to determine characteristics of the dynamical quantum phase transitions, such as Fisher's zeros in one-dimensional systems. The effect is absent for zero pairing. In particular, the response of a topological superconductor to the pulse manifests dynamical oscillations, most pronounced for the range of parameters in which the topological superconductivity can occur. The response of Majorana edge states to the pulse is analyzed. The results are generalized for the bosonic system with pairing under the action of the pulse (quench).
Exact results for time evolution of a many-body quantum system (a transverse field Ising spin chain) under the action of several pulses of an external field are obtained. Unlike the situation with a single pulse, the magnetization depends on time for the sequence of two pulses. For the two-pulse case the magnetization does not depend on the value of the time spacing between pulses, in sharp contrast with the standard spin echo cases. Modulated oscillations of magnetization with time do not decay if the energy of eigenstates of the system before pulses is kkk independent (k(k(k is the quasimomentum), while for kkk-dependent initial eigenstates the modulated oscillations of magnetization decay (without inclusion of the ``external'' relaxation) to the steady-state value, depending on the parameters of pulses; i.e., the oscillations reveal time irreversibility.
In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.
The spin–orbit-coupled degenerate Fermi gas provides a new platform for realizing topological superfluids and related topological excitations. However, previous studies have been mainly focused on the topological properties of the stationary ground state. Here, we investigate the quench dynamics of a spin–orbit-coupled two-dimensional Fermi gas in which the Zeeman field serves as the major quench parameter. Three post-quench dynamical phases are identified according to the asymptotic behaviour of the order parameter. In the undamped phase, a persistent oscillation of the order parameter may support a topological Floquet state with multiple edge states. In the damped phase, the magnitude of the order parameter approaches a constant via a power-law decay, which may support a dynamical topological phase with one edge state at the boundary. In the overdamped phase, the order parameter decays to zero exponentially although the condensate fraction remains finite. These predictions can be observed in the strong-coupling regime. Spin–orbit-coupled Fermi gases have been shown to support exotic topological quantum states. Here, the authors investigate quench dynamics of these gases in two dimensions, and find that three different phases with different topological properties emerge depending on the behaviour of the order parameter.
Spin-orbit-coupled bosons can exhibit rich equilibrium phases at low temperature and in the presence of particle-particle interactions. In the case with a 1D synthetic spin-orbit interaction, it has been observed that the ground state of a Bose gas can be a normal phase, stripe phase, or magnetized phase in different parameter regimes. The magnetized states are doubly degenerate and consist of a many-particle two-state system. In this work, we investigate the nonequilibrium quantum dynamics by switching on a simple one-dimensional optical lattice potential as external perturbation to induce resonant couplings between the magnetized phases, and predict a quantum spin dynamics which cannot be obtained in the single-particle systems. In particular, due to particle-particle interactions, the transition of the Bose condensate from one magnetized phase to the other is forbidden when the external perturbation strength is less than a critical value, and a full transition can occur only when the perturbation exceeds such critical strength. This phenomenon manifests itself a dynamical phase transition, with the order parameter defined by the time-averaged magnetization over an oscillation period, and the critical point behavior being exactly solvable. The thermal fluctuations are also considered in detail. From numerical simulations and exact analytic studies we show that the predicted many-body effects can be well observed with the current experiments.