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arXiv:1605.05403 · Europhysics Letters (EPL)

Sudden jumps and plateaus in the quench dynamics of a Bloch state

J. M. Zhang, Hua-Tong Yang

Published Oct 1, 2016• 23 citations
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Abstract

We take a one-dimensional tight-binding chain with periodic boundary condition and put a particle in an arbitrary Bloch state, then quench it by suddenly changing the potential of an arbitrary site. In the ensuing time evolution, the probability density of the wave function at an arbitrary site jumps indefinitely between plateaus . This phenomenon adds to a former one in which the survival probability of the particle in the initial Bloch state shows cusps periodically, which was found in the same scenario ( Zhang J. M. and Yang H.-T. , EPL , 114 (2016) 60001). The plateaus support the scattering wave picture of the quench dynamics of the Bloch state. Underlying the cusps and jumps is the exactly solvable, nonanalytic dynamics of a Luttinger-like model, based on which, the locations of the jumps and the heights of the plateaus are accurately predicted.

DOI
10.1209/0295-5075/116/10008

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We take a one-dimensional tight-binding chain with periodic boundary condition and put a particle in an arbitrary Bloch state, then quench it by suddenly changing the potential of an arbitrary site. In the ensuing time evolution, the probability density of the wave function at an arbitrary site jumps indefinitely between plateaus . This phenomenon adds to a former one in which the survival probability of the particle in the initial Bloch state shows cusps periodically, which was found in the same scenario ( Zhang J. M. and Yang H.-T. , EPL , 114 (2016) 60001). The plateaus support the scattering wave picture of the quench dynamics of the Bloch state. Underlying the cusps and jumps is the exactly solvable, nonanalytic dynamics of a Luttinger-like model, based on which, the locations of the jumps and the heights of the plateaus are accurately predicted.
We report some nonsmooth dynamics of a Bloch state in a one-dimensional tight binding model with the periodic boundary condition. After a sudden change of the potential of an arbitrary site, quantities like the survival probability of the particle in the initial Bloch state show cusps periodically, with the period being the Heisenberg time associated with the energy spectrum. This phenomenon is a nonperturbative counterpart of the nonsmooth dynamics observed previously (Z hang J. M. and H aque M., arXiv:1404.4280) in a periodically driven tight binding model. Underlying the cusps is an exactly solvable model, which consists of equally spaced levels extending from − ∞ to + ∞ , between which two arbitrary levels are coupled to each other by the same strength.
We study the scenario of quenching an interaction-free Fermi sea on a one-dimensional lattice ring by suddenly changing the potential of a site. From the point-of-view of the conventional Friedel oscillation, which is a static or equilibrium problem, it is of interest what temporal and spatial oscillations the local sudden quench will induce. Numerically, the primary observation is that for a generic site, the local particle density switches between two plateaus periodically in time. Making use of the proximity of the realistic model to an exactly solvable model and employing the Abel regularization to assign a definite value to a divergent series, we obtain an analytical formula for the heights of the plateaus, which turns out to be very accurate for sites not too close to the quench site. The unexpect relevance and the incredible accuracy of the Abel regularization are yet to be understood. Eventually, when the contribution of the defect mode is also taken into account, the plateaus for those sites close to or on the quench site can also be accurately predicted. We have also studied the infinite lattice case. In this case, ensuing the quench, the out-going wave fronts leave behind a stable density oscillation pattern. Because of some interesting single-particle property, this dynamically generated Friedel oscillation differs from its conventional static counterpart only by the defect mode.
We study the slow quenching dynamics (characterized by an inverse rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$) of a one-dimensional transverse Ising chain with nearest neighbor ferromagentic interactions across the quantum critical point (QCP) and analyze the Loschmidt overlap measured using the subsequent temporal evolution of the final wave function (reached at the end of the quenching) with the final time-independent Hamiltonian. Studying the Fisher zeros of the corresponding generalized ``partition function,'' we probe nonanalyticities manifested in the rate function of the return probability known as dynamical phase transitions (DPTs). In contrast to the sudden quenching case, we show that DPTs survive in the subsequent temporal evolution following the quenching across two critical points of the model for a sufficiently slow rate; furthermore, an interesting ``lobe'' structure of Fisher zeros emerge. We have also made a connection to topological aspects studying the dynamical topological order parameter $[{\ensuremath{\nu}}_{D}(t)]$ as a function of time (t)(t)(t) measured from the instant when the quenching is complete. Remarkably, the time evolution of ${\ensuremath{\nu}}_{D}(t)$ exhibits drastically different behavior following quenches across a single QCP and two QCPs. In the former case, ${\ensuremath{\nu}}_{D}(t)$ increases stepwise by unity at every DPT (i.e., $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=1$). In the latter case, on the other hand, ${\ensuremath{\nu}}_{D}(t)$ essentially oscillates between 0 and 1 (i.e., successive DPTs occur with $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=1$ and $\mathrm{\ensuremath{\Delta}}{\ensuremath{\nu}}_{D}=\ensuremath{-}1$, respectively), except for instants where it shows a sudden jump by a factor of unity when two successive DPTs carry a topological charge of the same sign.
We point out that by the ``smoothness means fast decay'' principle in Fourier analysis, it is possible to infer the smoothness (or nonsmoothness) of the autocorrelation function from a mere glimpse of the initial state. Specifically, for a generic system with smooth eigenstates, the smoother an initial state is, the faster its decomposition coefficients with respect to the eigenstates of the system decay, and in turn the smoother the autocorrelation function is. The idea is illustrated with three increasingly smooth functions in an infinite square well. By using the Mellin transform, we also find that the nonsmoothness or singularity of the initial state affects the short-time behavior of its autocorrelation function. In particular, a sufficiently nonsmooth or singular initial state could decay in a nonquadratic power law, with the exponent continuously tunable. We now understand the periodic cusps of the autocorrelation function in the quench dynamics of a Bloch state, which was observed previously [Zhang and Yang, EPL 114, 60001 (2016)].
We consider finite-time quantum quenches in the interacting Tomonaga-Luttinger model, for example time-dependent changes of the nearest-neighbor interactions for spinless fermions. We use the exact solutions for specific protocols including the linear and cosine ramps (or, more generally, periodic pumping). We study the dynamics of the total and kinetic energy as well as the Green's functions during as well as after the quench. For the latter we find that the light-cone picture remains applicable; however, the propagating front is delayed as compared to the sudden quench. We extract the universal behavior of the Green's functions and in particular provide analytic, nonperturbative results for the delay applicable to quenches of short to moderate duration but arbitrary time dependency.
We study quench dynamics of the Bose-Hubbard model by exact diagonalization. Initially, the system is at thermal equilibrium and of a finite temperature. The system is then quenched by changing the on-site interaction strength UUU suddenly. Both the single-quench and double-quench scenarios are considered. In the former case, the time-averaged density matrix and the real-time evolution are investigated. It is found that though the system thermalizes only in a very narrow range of the quenched value of UUU, it does equilibrate or relax well into a much larger range. Most importantly, it is proven that this is guaranteed for some typical observables in the thermodynamic limit. In order to test whether it is possible to distinguish the unitarily evolving density matrix from the time-averaged (thus time-independent), fully decohered density matrix, a second quench is considered. It turns out that the answer is affirmative or negative depending on whether the intermediate value of UUU is zero or not.