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arXiv:cond-mat/9401044 · Physical Review Letters

From Feynman’s wave function to the effective theory of vortex dynamics

Q. Niu, Ping Ao, D. J. Thouless

Published Mar 14, 1994• 64 citations
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Abstract

We calculate the overlap between two many-body wave functions for a superfluid film containing a vortex at shifted positions. Comparing the results to phenomenological theories, which treat vortices as point particles, we find that the results are consistent if the point-particle vortices are considered as under the action of the Magnus force and in weak interaction with sound waves of the superfluid. We are then able to resolve the disagreement concerning the effective mass of vortices, showing it is finite.

DOI
10.1103/physrevlett.72.1706

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We calculate the overlap between two many-body wave functions for a superfluid film containing a vortex at shifted positions. Comparing the results to phenomenological theories, which treat vortices as point particles, we find that the results are consistent if the point-particle vortices are considered as under the action of the Magnus force and in weak interaction with sound waves of the superfluid. We are then able to resolve the disagreement concerning the effective mass of vortices, showing it is finite.
We give a general review of recent developments in the theory of vortices in superfluids and superconductors, discussing why the dynamics of vortices is important, and why some key results are still controversial. We discuss work that we have done on the dynamics of quantized vortices in a superfluid. Despite the fact that this problem has been recognized as important for forty years, there is still a lot of controversy about the forces on and masses of quantized vortices. We think that one can get unambiguous answers by considering a broken symmetry state that consists of one vortex in an infinite ideal system. We argue for a Magnus force that is proportional to the superfluid density, and we find that the effective mass density of a vortex in a neutral superfluid is divergent at low frequencies. We have generalized some of the results for a neutral superfluid to a charged system.
We investigate the dynamics of quantized vortices in a model two-dimensional supersolid. Starting from an effective action that captures the dynamics of the superfluid condensate and its coupling to the lattice displacements, we integrate out the low-energy Goldstone modes-the phonons of the solid and the superfluid condensate-to arrive at an effective action for the vortices in the condensate. In the low-velocity limit we calculate the effective inertial mass for the vortices, and we find that the mass has a logarithmic frequency dependence, similar to the inertial mass found in superfluid vortices. The vortex dynamics also includes a Magnus force term in the equation of motion that arises from the Berry phase in the effective action.
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.
Vortices and vortex arrays have been used as a hallmark of superfluidity in rotated, ultracold Fermi gases. These superfluids can be described in terms of an effective field theory for a macroscopic wave function representing the field of condensed pairs, analogous to the Ginzburg-Landau theory for superconductors. Here we establish how rotation modifies this effective field theory, by rederiving it starting from the action of Fermi gas in the rotating frame of reference. The rotation leads to the appearance of an effective vector potential, and the coupling strength of this vector potential to the macroscopic wave function depends on the interaction strength between the fermions, due to a renormalization of the pair effective mass in the effective field theory. The mass renormalization derived here is in agreement with results of functional renormalization-group theory. In the extreme Bose-Einstein condensate regime, the pair effective mass tends to twice the fermion mass, in agreement with the physical picture of a weakly interacting Bose gas of molecular pairs. Then we use our macroscopic-wave-function description to study vortices and the critical rotation frequencies to form them. Equilibrium vortex state diagrams are derived and they are in good agreement with available results of the Bogoliubov--de Gennes theory and with experimental data.
The transverse force on an isolated and moving vortex in a neutral superfluid at rest is evaluated at finite temperature in the case of noninteracting phonons. Using the Thouless, Ao, Niu (TAN) [Phys. Rev. Lett. 76, 3758 (1996)] general theory, we show that the transverse force is exactly equal to the superfluid Magnus force. We extend this theory in the case of a slowly moving vortex on a circular trajectory, and find an additional contribution coming from the centrifugal reaction. This term gives a negative vortex mass due to the phonons and diverges logarithmically at low frequency. The friction force is also evaluated for zero and finite frequencies, and compared with the scattering theory.
We derive an effective action for the vortex-position degree of freedom in a superfluid by integrating out condensate phase- and density-fluctuation environmental modes. When the quantum dynamics of environmental fluctuations is neglected, we confirm the occurrence of the vortex Magnus force and obtain an expression for the vortex mass. We find that this adiabatic approximation is valid only when the superfluid droplet radius RRR, or the typical distance between vortices, is very much larger than the coherence length $\ensuremath{\xi}$. We go beyond the adiabatic approximation numerically, accounting for the quantum dynamics of environmental modes and capturing their dissipative coupling to condensate dynamics. For the case of an optical-lattice superfluid, we demonstrate that vortex motion damping can be adjusted by tuning the ratio between the tunneling energy JJJ and the on-site interaction energy UUU. We comment on the possibility of realizing vortex-Landau-level physics.