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 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.
Abstract We point out that in the first-order time-dependent perturbation theory, the transition probability may behave nonsmoothly in time and have kinks periodically. Moreover, the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is a simple equality about the sinc function sinc x ≡ sin x / x . These physical effects appear in many systems with approximately equally spaced spectra, and are also robust for larger amplitude coupling beyond the domain of perturbation theory. We use a one-dimensional periodically driven tight-binding model to illustrate these effects, both within and outside the perturbative regime.
Bloch oscillations (BOs) represent a paradigmatic manifestation of force-driven band dynamics in lattices, conventionally characterized by continuous quasimomentum evolution and smooth periodic trajectories. Here we show that non-Hermiticity qualitatively changes this paradigm, introducing nonsmooth BOs characterized by cusp-like real-space trajectories and abrupt momentum jumps. Reversing the force switches the dynamics between smooth and nonsmooth BOs, enabling strongly nonreciprocal dynamics. We derive a universal relation for the critical force that governs the nonsmooth-to-smooth dynamical transition in general one-dimensional lattices, which is induced by a pitchfork bifurcation in momentum space. Our work establishes a unified framework for non-Hermitian BOs and uncovers a fundamental bifurcation mechanism that can generate nonsmooth wave transport in entirely linear systems.
We point out that in the first order time-dependent perturbation theory, the transition probability may behave nonsmoothly in time and have kinks periodically. Moreover, the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is a simple equality about the sinc function $\sinc x \equiv \sin x / x$. These physical effects appear in many systems with approximately equally spaced spectra, and is also robust for larger-amplitude coupling beyond the domain of perturbation theory. We use a one-dimensional periodically driven tight-binding model to illustrate these effects, both within and outside the perturbative regime.
We investigate the quantum dynamics of a one-dimensional tight-binding lattice driven by a spatially quadratic and time-periodic potential. Both Hermitian (
J1=J2) and non-Hermitian (
J1=J2) hopping regimes are analyzed. Within the framework of Floquet theory, the time-dependent Hamiltonian is mapped onto an effective static Floquet Hamiltonian, enabling a detailed study of the quasi-energy spectrum as function of the driving frequency
ω. By applying a gauge transformation, we find that critical frequencies
ωc emerge, at which nearly equidistant quasi-energy ladders appear, as revealed by a pronounced minimum in the normalized variance
Δ(ω) of the level spacings. This spectral regularity leads to robust periodic revivals and Bloch-like oscillations in the time evolution. Numerical simulations confirm that such coherent oscillations persist even in the non-Hermitian regime, where the periodic driving stabilizes an almost real and uniformly spaced quasi-energy ladder.
We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-
21 chain with equal
xx and
yy couplings and subject to a periodically varying magnetic field in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period
2q, where
q is a positive integer. For a linear quench of the strength of the magnetic field (or chemical potential) at a rate $1/\ensuremath{\tau}$ across a quantum critical point, we find that the density of defects thereby produced scales as $1/{\ensuremath{\tau}}^{q/(q+1)}$, deviating from the $1/\sqrt{\ensuremath{\tau}}$ scaling that is ubiquitous in a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum
k undergoing a nonlinear quench as well as by performing numerical simulations. We also show that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of $\ensuremath{\tau}$, although it may exhibit a crossover at intermediate values of $\ensuremath{\tau}$. Finally, for the case where a
zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of
q and the interaction strength.