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arXiv:cond-mat/0005481 · International Journal of Modern Physics B

VARIATIONAL WAVE FUNCTIONS OF A VORTEX IN CYCLOTRON MOTION

Jian‐Ming Tang

Published May 10, 2001• 6 citations
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Abstract

In two dimensions the microscopic theory, which provides a basis for the naive analogy between a quantized vortex in a superfluid and an electron in an uniform magnetic field, is presented. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with the energy, and scales with the size of the vortex.

DOI
10.1142/s0217979201006082

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cond-mat.mes-hallCross-listedarXiv:2607.21528•Jul 24, 2026
In two dimensions the microscopic theory, which provides a basis for the naive analogy between a quantized vortex in a superfluid and an electron in an uniform magnetic field, is presented. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with the energy, and scales with the size of the vortex.
In two dimensions a microscopic theory providing a basis for the naive analogy between a quantized vortex in a superfluid and an electron in a uniform magnetic field is presented. Following the variational approach developed by Peierls, Yoccoz, and Thouless, the cyclotron motion of a vortex is described by the many-body wave function, which is a linear combination of Feynman wave functions centered at different positions. An integral equation for the weighting functions of the superposition is derived by minimizing the energy functional. The matrix elements of the kernel are the overlaps between any two displaced Feynman wave functions. A numerical study is conducted for a bosonic superfluid based on a Hartree ground state. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with energy, and scales with the size of the vortex. The fluid density is finite at the vortex axis and the vorticity is distributed in the core region. The effective mass of a quantized vortex defined by the inverse of the energy-level spacing is shown to be logarithmically divergent with the size of the vortex.
We investigate the microscopic theory of collective excitations and vortex dynamics in superfluid systems. We derive an effective Gross-Pitaevskii Lagrangian for the vortex and collective excitations and obtain the dynamical equations governing their motion. We solve these equations without assuming adiabaticity of vortex motion and show that the vortex has a natural cyclotron frequency. This frequency allows us to identify the vortex mass which turns out to be finite. We also conclude that this implies the existence of a length scale due to the Berry phase associated to the vortex and that it acts as an infrared cutoff.
Superfluid vortex dynamics on an infinite cylinder differs significantly from that on a plane. The requirement that a condensate wave function be single valued upon once encircling the cylinder means that such a single vortex cannot remain stationary. Instead, it acquires one of a series of quantized translational velocities around the circumference, the simplest being $\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}/(2MR)$, with MMM the mass of the superfluid particles and RRR the radius of the cylinder. A generalization to a finite cylinder automatically includes these quantum-mechanical effects through the pairing of the single vortex and its image in either the top or bottom end of the surface. The dynamics of a single vortex on this surface provides a hydrodynamic analog of Laughlin pumping. The interaction energy for two vortices on an infinite cylinder is proportional to the classical stream function $\ensuremath{\chi}({\mathbit{r}}_{12})$, and it crosses over from logarithmic to linear when the intervortex separation ${\mathbit{r}}_{12}$ becomes larger than the cylinder radius. An appendix summarizes the connection to an earlier study of Ho and Huang for one or more vortices on an infinite cylinder. A second appendix reviews the topologically equivalent planar annulus, where such quantized vortex motion has no offset, but Laughlin pumping may be more accessible to experimental observation.
Quantum mechanically, an electron moving in a uniform magnetic field forms Landau levels. A curious feature is that for states with a negative angular quantum number, the total probability current vanishes, which appears to contradict the classical picture of cyclotron motion. While a geometric interpretation based on classical orbits exists, alternative interpretations remain of interest. In this paper, we examine the probability current density and identify a critical radius that naturally partitions the plane into an inner clockwise-flow region and an outer counterclockwise-flow region. We show that the vanishing total current results from an exact cancellation between these two regions. Furthermore, by defining a partitioned kinetic angular momentum with respect to the critical radius, we reveal an intrinsic competitive structure: the electron simultaneously carries two opposing rotational components. The negative quantum number manifests in the strength of the inner counter-rotation, while the net kinetic angular momentum remains positive. This bidirectional flow picture also provides a dynamical interpretation of the infinite degeneracy of Landau levels.
A theory employing the vortex shape of the electron was presented to resolve the enigma of the wave-particle duality. Conventions such as “particle” and “wave” were used to describe the behavior of quantum objects such as electrons. A superfluid vacuum formed the base to describe the basic vortex structure and properties of the electron, whereas various formulations derived from hydrodynamic laws described the electron vortex circumference, radius, angular velocity and angular frequency, angular momentum (spin) and magnetic momentum. A vortex electron fully explained the associations between momentum and wave, and hydrodynamic laws were essential in deriving the energy and angular frequency of the electron. In general, an electron traveling in space possesses internal and external motions. To derive the angular frequency of its internal motion, the Compton wavelength was used to represent the length of one cycle of the internal motion that is equal to the circumference of the electron vortex. The angular frequency of the electron vortex was calculated to obtain the same value according to Planck’s theory. A traveling vortex electron has internal and external motions that create a three-dimensional helix trajectory. The magnitude of the instantaneous velocity of the electron is the resultant of its internal and external velocities, being equal to the internal velocity reduced by the Lorentz factor (whose essence is presented in a detailed formulation). The wavelength of the helix trajectory represents the distance traveled by a particle along its axis during one period of revolution around the axis, resulting in the same de Broglie wavelength that corresponds to the helix pitch of the helix. Mathematical formulations were presented to demonstrate the relation between the energy of the vortex and its angular frequency and de Broglie’s wavelength; furthermore, Compton’s and de Broglie’s wavelengths were also differentiated.
In a two-dimensional electron gas, a strong perpendicular magnetic field confines electrons to quantized cyclotron orbits, giving rise to Landau levels with discrete orbit radii. Even the smallest Landau orbit, set by the magnetic length, spans tens of nanometers for fields of a few Tesla, imposing an intrinsic nonlocal response to electromagnetic excitations. From a microscopic theory of the nonlocal susceptibility, we derive the Green's function, the central quantity governing all electromagnetic interactions, and evaluate Purcell factors, Lamb shifts, and emission spectra from a proximal dipole emitter beyond the Markov and rotating-wave approximations. Significant nonlocal effects resulting from spatial dispersion of the Landau level response modify the response for experimentally relevant situations up to distances of hundreds of nanometers and, in particular, brighten locally dipole-forbidden transitions due to near-field gradients at multiples of the cyclotron frequency. The relevant length scales are typical of state-of-the-art nanostructured terahertz architectures, and some of our nonlocal features are consistent with recent experiments using Landau level polaritons.