Semiclassical Equations Of Motion Research Articles

Intrinsic anomalous Hall conductivity of Dirac band in presence of the normal magnetic field

We have studied the intrinsic contribution to anomalous Hall conductivity in presence of the normal magnetic field. A semiclassical approach for calculating the Hall conductivity in presence of the nontrivial Berry phase is used. This method is based on the semiclassical equations of motion with anomalous velocity correction and the Boltzmann transport equation. We present the formulation of anomalous Hall conductivity in this approach. We perform explicit calculations for a two-dimensional system which is described by Dirac Hamiltonian. At zero magnetic field, our results coincide with previous results which are obtained both using the semiclassical method and Kubo formalism. For the strong magnetic field, Hall conductivity decreases as a power-law. The exponent of the power-law behavior for the infinite energy band is 1/3 and for the finite energy band, there is a crossover from 1/3 to1.

Chiral Anomaly in Non-Relativistic Systems: Berry Curvature and Chiral Kinetic Theory

Chiral anomaly and the novel quantum phenomena it induces have been widely studied for Dirac and Weyl fermions. In most typical cases, the Lorentz covariance is assumed and thus the linear dispersion relations are maintained. However, in realistic materials, such as Dirac and Weyl semimetals, the nonlinear dispersion relations appear naturally. We develop a kinetic framework to study the chiral anomaly for Weyl fermions with nonlinear dispersions using the methods of Wigner function and semi-classical equations of motion. In this framework, the chiral anomaly is sourced by Berry monopoles in momentum space and could be enhanced or suppressed due to the windings around the Berry monopoles. Our results can help understand the chiral anomaly-induced transport phenomena in non-relativistic systems.

Absence of Hall effect due to Berry curvature in phase space

Transverse current due to Berry curvature in phase space is formulated based on the Boltzmann equations with the semiclassical equations of motion for an electron wave packet. It is shown that the Hall effect due to the phase space Berry curvature is absent because the contributions from “anomalous velocity” and “effective Lorentz force” are completely cancelled out.

Theory of wave-packet transport under narrow gaps and spatial textures: Nonadiabaticity and semiclassicality

We generalise the celebrated semiclassical wavepacket approach from the adiabatic to the non-adiabatic regime. A unified description covering both of these regimes is particularly desired for systems with spatially varying band structures where band gaps of various sizes are simultaneously present, e.g. in moir\'{e} patterns. For a single wavepacket, alternative to the previous derivation by Lagrangian variational approach, we show that the same semiclassical equations of motion can be obtained by introducing a spatial-texture-induced force operator similar to the Ehrenfest theorem. For semiclassically computing the current, the ensemble of wavepackets based on adiabatic dynamics is shown to well correspond to a phase-space fluid for which the fluid's mass and velocity are two distinguishable properties. This distinction is not inherited to the ensemble of wavepackets with the non-adiabatic dynamics. We extend the adiabatic kinetic theory to the non-adiabatic regime by taking into account decoherence, whose joint action with electric field favours certain form of inter-band coherence. The steady-state density matrix as a function of the phase-space variables is then phenomenologically obtained for calculating the transport current. The result, applicable with a finite electric field, expectedly reproduces the known adiabatic limit by taking the electric field to be infinitesimal, and therefore attains a unified description from the adiabatic to the non-adiabatic situations.

Quantum transport in topological semimetals under magnetic fields (II)

We review our recent works on the quantum transport, mainly in topological semimetals and also in topological insulators, organized according to the strength of the magnetic field. At weak magnetic fields, we explain the negative magnetoresistance in topological semimetals and topological insulators by using the semiclassical equations of motion with the nontrivial Berry curvature. We show that the negative magnetoresistance can exist without the chiral anomaly. At strong magnetic fields, we establish theories for the quantum oscillations in topological Weyl, Dirac, and nodal-line semimetals. We propose a new mechanism of 3D quantum Hall effect, via the "wormhole" tunneling through the Weyl orbit formed by the Fermi arcs and Weyl nodes in topological semimetals. In the quantum limit at extremely strong magnetic fields, we find that an unexpected Hall resistance reversal can be understood in terms of the Weyl fermion annihilation. Additionally, in parallel magnetic fields, longitudinal resistance dips in the quantum limit can serve as signatures for topological insulators.

Berry electrodynamics: Anomalous drift and pumping from a time-dependent Berry connection

The Berry curvature of a Bloch band can be interpreted as a local magnetic field in reciprocal space. This analogy can be extended by defining an electric field analog in reciprocal space which arises from the time-dependent Berry connection. We explore the term in the semi-classical equation of motion that gives rise to this phenomenon, and show that it can lead to anomalous drift in wave packet motion. A similar effect arises from changes in the band population due to periodic driving, where the resulting drift depends on the nature of the drive and can be expressed in terms of a shift vector. Finally, these effects can be combined to build a pump with a net anomalous drift during a cyclic evolution in momentum space.

Semiclassical dynamics, Berry curvature, and spiral holonomy in optical quasicrystals

We describe the theory of the dynamics of atoms in two-dimensional quasicrystalline optical lattices. We focus on a regime of shallow lattice depths under which the applied force can cause Landau-Zener tunneling past a dense hierarchy of gaps in the quasiperiodic energy spectrum. We derive conditions on the external force that allow for a "semiadiabatic" regime in which semiclassical equations of motion can apply, leading to Bloch oscillations between the edges of a pseudo-Brillouin-zone. We verify this semiclassical theory by comparing to the results of an exact numerical solution. Interesting features appear in the semiclassical dynamics for the quasicrystal for a particle driven in a cyclic trajectory around the corner of the pseudo-Brillouin-zone: The particle fails to return to its initial state, providing a realization of a "spiral holonomy" in the dynamics. We show that there can appear anomalous velocity contibutions, associated with nonzero Berry curvature. We relate these to the Berry phase associated with the spiral holonomy, and show how the Berry curvature can be accessed from the semiclassical dynamics. Finally, by identifying the pseudo-Brillouin-zone as a higher genus surface, we show that the Chern number classification for periodic systems can be extended to a quasicrystal, thereby determining a topological index for the system.

Tunable axial gauge fields in engineered Weyl semimetals: semiclassical analysis and optical lattice implementations

In this work, we describe a toolbox to realize and probe synthetic axial gauge fields in engineered Weyl semimetals. These synthetic electromagnetic fields, which are sensitive to the chirality associated with Weyl nodes, emerge due to spatially and temporally dependent shifts of the corresponding Weyl momenta. First, we introduce two realistic models, inspired by recent cold-atom developments, which are particularly suitable for the exploration of these synthetic axial gauge fields. Second, we describe how to realize and measure the effects of such axial fields through center-of-mass observables, based on semiclassical equations of motion and exact numerical simulations. In particular, we suggest realistic protocols to reveal an axial Hall response due to the axial electric field , as well as axial cyclotron orbits and chiral pseudo-magnetic effect due to the axial magnetic field .

Optical and atomic stochastic resonances in the driven dissipative Jaynes-Cummings model

In this work we study the stochastic resonance (SR) effect in a driven dissipative Jaynes-Cummings model. The SR effect is systematically studied in the semiclassical and full quantum frameworks and in both cases we find that SRs simultaneously occur for the optical and atomic freedoms. In particular, at zero temperature quantum SR can be induced merely by vacuum fluctuations. The qualitative features of semiclassical SR and quantum SR are similar, but the parameter region of quantum SR are shifted from the semiclassical region due to the widely-used factorization in obtaining semiclassical equations of motion. Our results provide a theoretical basis for experimentally observing and studying the SR phenomenon of the Jaynes-Cummings model in the quantum regime.

Negative Magnetoresistance without Chiral Anomaly in Topological Insulators.

An intriguing phenomenon in topological semimetals and topological insulators is the negative magnetoresistance (MR) observed when a magnetic field is applied along the current direction. Aprevailing understanding to the negative MR in topological semimetals is the chiral anomaly, which, however, is not well defined in topological insulators. We calculate the MR of a three-dimensional topological insulator, by using the semiclassical equations of motion, in which the Berry curvature explicitly induces an anomalous velocity and orbital moment. Our theoretical results are in quantitative agreement with the experiments. The negative MR is not sensitive to temperature and increases as the Fermi energy approaches the band edge. The orbital moment and g factors also play important roles in the negative MR. Our results give a reasonable explanation to the negative MR in 3D topological insulators and will be helpful in understanding the anomalous quantum transport in topological states of matter.

Imbert-Fedorov shift in Weyl semimetals: Dependence on monopole charge and intervalley scattering

The Imbert-Fedorov (IF) shift in optics describes the transverse shift of light beams at the reflection interface. Recently, the IF shift of Weyl fermions at the interface between Weyl semimetals (WSMs) with single monopole charge has been studied. Here, we study the IF shift at the interface between two WSMs, each of which is carrying an arbitrary integer monopole charge. We find a general relation between the monopole charges of the two WSMs and the IF shift. In particular, the IF shift is proportional to the monopole charge if both WSMs have the same one. Our results can be used to infer the topology of the materials by experimentally measuring their IF shift. Furthermore, we consider the possibility that the Weyl fermions are scattered to other Weyl cones during the reflection, which results in qualitatively different behavior of the IF shift. While we use a quantum mechanical approach to solve the problem, semiclassical equations of motion and the conservation of total angular momentum can help us intuitively interpret our results in special cases.

Derivation of one-particle semiclassical kinetic theory in the presence of non-Abelian Berry curvature

In quantum mechanics it is often required to describe in a semiclassical approximation the motion of particles moving within a given energy band. Such a representation leads to the appearance of an analogues of fictitious forces, associated with the Berry curvature, in the semiclassical equations of motion. The purpose of this paper is to derive systematically the kinetic Boltzmann equations displaying these effects in the case that the band is degenerate, and as such the Berry curvature is generically non-Abelian. We use the formalism of phase-space quantum mechanics to derive the results.

Quantum–classical correspondence in chaotic dynamics of laser-driven atoms

This paper is a review article on some aspects of quantum–classical correspondence in chaotic dynamics of cold atoms interacting with a standing-wave laser field forming an optical lattice. The problem is treated from both (semi)classical and quantum points of view. In both approaches, the interaction of an atomic electic dipole with the laser field is treated quantum mechanically. Translational motion is described, at first, classically (atoms are considered to be point-like objects) and then quantum mechanically as a propagation of matter waves. Semiclassical equations of motion are shown to be chaotic in the sense of classical dynamical chaos. Point-like atoms in an absolutely deterministic and rigid optical lattice can move in a random-like manner demonstrating a chaotic walking with typical features of classical chaos. This behavior is explained by random-like ‘jumps’ of one of the atomic internal variable when atoms cross nodes of the standing wave and occurs in a specific range of the atom-field detuning. When treating atoms as matter waves, we show that they can make nonadiabatic transitions when crossing the standing-wave nodes. The point is that atomic wave packets split at each node in the same range of the atom-field detuning where the classical chaos occurs. The key point is that the squared amplitude of those semiclassical ‘jumps’ equal to the quantum Landau–Zener parameter which defines the probability of nonadiabatic transitions at the nodes. Nonadiabatic atomic wave packets are much more complicated compared to adiabatic ones and may be called chaotic in this sense. A few possible experiments to observe some manifestations of classical and quantum chaos with cold atoms in horizontal and vertical optical lattices are proposed and discussed.

Electromagnetic instability induced by neutrino interaction

We consider the generation and evolution of magnetic field in a primordial plasma at temperature [Formula: see text][Formula: see text]MeV in the presence of asymmetric neutrino background, i.e. the number densities of right-handed and left-handed neutrinos are not the same. Semi-classical equations of motion of a charged fermion are derived using the effective low-energy Lagrangian. We show that the spin degree of freedom of the charged fermion couples with the neutrino background. Using this kinetic equation, we study the collective modes of the plasma. We find that there exist an unstable mode. This instability is closely related with the instability induced by chiral-anomaly in high temperature [Formula: see text][Formula: see text]TeV plasma where right and left-handed electrons are out of equilibrium. We find that at the temperatures below the neutrino decoupling this instability can produce magnetic field in the universe. We discuss cosmological implications of the results.

WKB Approxiation of the Dirac Equation with a Supersymmetric Extension

A general scheme of the self-consistent construction of a semiclassical approximation for the Dirac equation in an external gauge field in which the standard Dirac operator is replaced by the Dirac operator with a supersymmetric extension is presented. It is shown that in contrast to the usual WKB method, here the expansion must be carried out over half-integer powers of the Planck constant ħ. The first four terms of the semiclassical expansion of the wave function are obtained in explicit form. It is shown that generalization of the initial Dirac operator leads to the appearance of new additional terms in the semiclassical equation of motion for the spin of a particle in an external field, which thus requires a modification of the Lagrangian of the spinning particle. The result so obtained is used to construct mappings between two Lagrangian descriptions of a classical color-charged spinning particle, one of which possesses local supersymmetry, and the other not. It is shown that in order for the mappings to be one-to-one it is necessary to add new additional terms to the Lagrangian without supersymmetry, obtained within the framework of the semiclassical approximation of the Dirac operator with supersymmetry.

Slow-roll inflation in loop quantum cosmology of scalar–tensor theories of gravity

The slow-roll inflation of scalar–tensor theories (STTs) of gravity in the context of loop quantum cosmology (LQC) is investigated in this paper. After deriving the effective Hamiltonian, we obtain the semiclassical equations of motion for the background variables in both Jordan frame and Einstein frame of STTs. Then we apply these equations in the slow-roll limit and derive the LQC corrections to the scalar spectral index [Formula: see text] and the tensor-to-scalar ratio [Formula: see text] in the two frames of STTs. Finally, we take two special sectors of STTs as specific examples, namely the Starobinsky model and the non-minimally coupled scalar field model (with the coupling function [Formula: see text] and the potential [Formula: see text]). We derive the detailed expressions of the LQC corrections to [Formula: see text] and [Formula: see text] in terms of the [Formula: see text]-folding number for these two models in both frames.

Geometric quantum noise of spin.

The presence of geometric phases is known to affect the dynamics of the systems involved. Here, we consider a quantum degree of freedom, moving in a dissipative environment, whose dynamics is described by a Langevin equation with quantum noise. We show that geometric phases enter the stochastic noise terms. Specifically, we consider small ferromagnetic particles (nanomagnets) or quantum dots close to Stoner instability, and investigate the dynamics of the total magnetization in the presence of tunneling coupling to the metallic leads. We generalize the Ambegaokar-Eckern-Schön effective action and the corresponding semiclassical equations of motion from the U(1) case of the charge degree of freedom to the SU(2) case of the magnetization. The Langevin forces (torques) in these equations are strongly influenced by the geometric phase. As a first but nontrivial application, we predict low temperature quantum diffusion of the magnetization on the Bloch sphere, which is governed by the geometric phase. We propose a protocol for experimental observation of this phenomenon.

Nonlinear electron transport in superlattice driven by a terahertz field and a tilted magnetic field

Vertical electron transport in semiconductor superlattice has been the focus of science and technology during the past two decades due to the potential application of superlattice in terahertz devices. When driven by electromagnetic field, many novel phenomena have been found in superlattice. Here we study the chaotic electron transport in miniband superlattice driven by dc+ac electric fields along the growth axis (z-axis) and a magnetic field tilted to z-axis using semiclassical equations of motion in the preflence of dissipation. We calculate the electron momentum by changing the magnetic field or amplitude of the terahertz field. It is shown that the momentum py(t) of miniband electron exhibits complicated oscillation modes while changing the control parameters. Poincaré bifurcation diagram and power spectrum are adopted to analyze the nonlinear electron states. Poincaré bifurcation diagram is obtained by plotting pym = py(mTac) (with m = 1, 2, 3,… and Tac the period of ac terahertz field) as functions of ac amplitude E1 after the transients decay. The periodic and aperiodic regions can be distinguished from each other since there are a large number of points in the chaotic regions. When the magnetic field is increased from 1.5 to 2 T, the Poincaré bifurcation diagram changes dramatically due to the strong effect of magnetic field on electron motion. The oscillating state of py(t) may be changed between periodic and chaotic syates. Power spectra of electron momentum py for different values of E1 (= 2.06, 2.18, 2.388, and 2.72) are calculated for a deep insight into the nonlinear oscillating mode. It is found that the power spectra of n-periodic states show peaks at frequencies ifac/n (with i = 1, 2, 3,…); the power spectra of chaotic states are very irregular with a large number of peaks. We demonstrate that the dissipation and resonance between Bloch oscillation frequency and cyclotron frequency play an important role in the electron transport process. We attribute the emerging of periodic and chaotic states in a superlattice to the interaction between terahertz radiation and internal cooperative oscillating mode related to Bloch oscillation and cyclotron oscillation. In the case of ωB≠iωc, the time-dependent electron motion is chaotic in most regions of the parameter space. Results of the preflent paper are useful for designing terahertz devices based on the semiconductor superlattices.

Topological aspects in the photonic-crystal analog of single-particle transport in quantum Hall systems

We present a perturbative approach to derive the semiclassical equations of motion for the two-dimensional electron dynamics under the simultaneous presence of static electric and magnetic fields, where the quantized Hall conductance is known to be directly related to the topological properties of translationally invariant magnetic Bloch bands. In close analogy to this approach, we develop a perturbative theory of two-dimensional photonic transport in gyrotropic photonic crystals to mimic the physics of quantum Hall systems. We show that a suitable permittivity grading of a gyrotropic photonic crystal is able to simulate the simultaneous presence of analog electric and magnetic field forces for photons, and we rigorously derive the topology-related term in the equation for the electromagnetic energy velocity that is formally equivalent to the electronic case. A possible experimental configuration is proposed to observe a bulk photonic analog to the quantum Hall physics in graded gyromagnetic photonic crystals.

Connection between wave functions in the Dirac and Foldy-Wouthuysen representations

When the Foldy-Wouthuysen (FW) transformation is exact and the particle energy is positive, upper spinors in the Dirac and FW representations differ only by a constant factor, and lower spinors in the FW representation are zero. Deducing FW wave eigenfunctions directly from Dirac wave eigenfunctions allows one to use the FW representation to calculate expectation values of needed operators and to derive quantum and semiclassical equations of motion.