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arXiv:cond-mat/9408025 · Physical review. B, Condensed matter

Microscopic analysis of the nondissipative force on a line vortex in a superconductor: Berry’s phase, momentum flow, and the Magnus force

Frank Gaitan

Published Apr 1, 1995• 30 citations
arXivHTMLPDF

Abstract

A microscopic analysis of the nondissipative force ${\mathbf{F}}_{\mathrm{nd}}$ acting on a line vortex in a type-II superconductor at T=0 is given. All work presented assumes a charged BCS superconductor. We first examine the Berry phase induced in the BCS superconducting ground state by movement of the vortex and show how this phase enters into the hydrodynamic action ${\mathit{S}}_{\mathrm{hyd}}$ of the superconducting condensate. Appropriate variation of ${\mathit{S}}_{\mathrm{hyd}}$ give ${\mathbf{F}}_{\mathrm{nd}}$ and variation of the Berry phase term is seen to contribute the Magnus or lift force of classical hydrodynamics to ${\mathbf{F}}_{\mathrm{nd}}$. This analysis, based on the BCS ground state of a charged superconductor, confirms in detail the arguments of Ao and Thouless within the context of the BCS model. Our Berry phase, in the limit e\ensuremath{\rightarrow}0, is seen to reproduce the Berry phase determined by these authors for a neutral superfluid. We also provide a second, independent, determination of ${\mathbf{F}}_{\mathrm{nd}}$ through a microscopic derivation of the continuity equation for the condensate linear momentum. This equation yields the acceleration equation for the superflow and shows that the vortex acts as a sink for the condensate linear momentum. The rate at which momentum is lost to the vortex determines ${\mathbf{F}}_{\mathrm{nd}}$ in this second approach and the result obtained agrees identically with the previous Berry phase calculation. The Magnus force contribution to ${\mathbf{F}}_{\mathrm{nd}}$ is seen in both calculations to be a consequence of the vortex topology and motion. A preliminary discussion is given regarding finite temperature extensions of the Berry phase calculation, with emphasis on its relevance for the sign anomaly occurring in Hall-effect experiments on type-II superconductors in the flux flow regime.

DOI
10.1103/physrevb.51.9061

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A microscopic analysis of the nondissipative force Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ acting on a line vortex in a type-II superconductor at T=0 is given. All work presented assumes a charged BCS superconductor. We first examine the Berry phase induced in the BCS superconducting ground state by movement of the vortex and show how this phase enters into the hydrodynamic action Shyd{\mathit{S}}_{\mathrm{hyd}}Shyd​ of the superconducting condensate. Appropriate variation of Shyd{\mathit{S}}_{\mathrm{hyd}}Shyd​ give Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ and variation of the Berry phase term is seen to contribute the Magnus or lift force of classical hydrodynamics to Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​. This analysis, based on the BCS ground state of a charged superconductor, confirms in detail the arguments of Ao and Thouless within the context of the BCS model. Our Berry phase, in the limit e\ensuremath{\rightarrow}0, is seen to reproduce the Berry phase determined by these authors for a neutral superfluid. We also provide a second, independent, determination of Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ through a microscopic derivation of the continuity equation for the condensate linear momentum. This equation yields the acceleration equation for the superflow and shows that the vortex acts as a sink for the condensate linear momentum. The rate at which momentum is lost to the vortex determines Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ in this second approach and the result obtained agrees identically with the previous Berry phase calculation. The Magnus force contribution to Fnd{\mathbf{F}}_{\mathrm{nd}}Fnd​ is seen in both calculations to be a consequence of the vortex topology and motion. A preliminary discussion is given regarding finite temperature extensions of the Berry phase calculation, with emphasis on its relevance for the sign anomaly occurring in Hall-effect experiments on type-II superconductors in the flux flow regime.
Adiabatic effective action for vortices in neutral and charged superfluids at zero temperature are calculated using the topological Landau-Ginzburg theory recently proposed by Hatsuda, Yahikozawa, Ao and Thouless, and vortex dynamics are examined. The Berry phase term arising in the effective action naturally yields the Magnus force in both neutral and charged superfluids. It is shown that in neutral superfluid there is only one degree of freedom, namely the center of vorticities, and the vortex energy is proportional to the sum of all vorticities so that it is finite only for the vanishing total vorticity of the system. On the other hand the effective mass and the vortex energy for a vortex in charged superfluids are defined individually as expected. The effects of the vortex core on these quantities are also estimated. The possible depinning scenario which is governed by the Magnus force and the inertial mass is also discussed.
The forces on the vortex, transverse to its velocity, are considered. In addition to the superfluid Magnus force from the condensate (superfluid component), there are transverse forces from thermal quasiparticles and external fields violating the Galilean invariance. The forces between quasiparticles and the vortex originate from interference of quasiparticles with trajectories on the left and on the right from the vortex like similar forces for electrons interacting with the thin magnetic-flux tube (the Aharonov-Bohm effect). These forces are derived for phonons from the equations of superfluid hydrodynamics, and for BCS quasiparticles from the Bogolyubov--de Gennes equations. The effect of external fields breaking Galilean invariance is analyzed for vortices in the two-dimensional Josephson junction array. The symmetry analysis of the classical equations for the array shows that the total transverse force on the vortex vanishes. Therefore the Hall effect which is linear in the transverse force is absent also. This means that the Magnus force from the superfluid component exactly cancels with the transverse force from the external fields. The results of other approaches are also brought together for discussion.
We analyze the evolution of the dynamics of a neutral sss-wave superconductor between the BCS and BEC regimes. We consider 2d case when a BCS-BEC crossover occurs already at weak coupling as a function of the ratio of the two scales---the Fermi energy EF{E}_{F}EF​ and the bound state energy for two fermions in a vacuum, E0{E}_{0}E0​. The BCS and BEC limits correspond to ${E}_{F}\ensuremath{\gg}{E}_{0}$ and ${E}_{F}\ensuremath{\ll}{E}_{0}$, respectively. The chemical potential $\ensuremath{\mu}={E}_{F}\ensuremath{-}{E}_{0}$ changes the sign between the two regimes. We use the effective action approach, derive the leading terms in the expansion of the effective action in the spatial and time derivative of the slowly varying superconducting order parameter $\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})$, and express the action in terms of a derivatives of the phase $\ensuremath{\phi}(r,\ensuremath{\tau})$ of $\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})=\mathrm{\ensuremath{\Delta}}{e}^{i\ensuremath{\phi}(r,\ensuremath{\tau})}$. The action contains ${(\ensuremath{\nabla}\ensuremath{\phi})}^{2}$ and ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}}^{2}$ terms, which determine the dispersion of collective phase fluctuations, and $i\ensuremath{\pi}A\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}$ term. For continuous $\ensuremath{\phi}(r,\ensuremath{\tau})$, the latter reduces to the contribution from the boundary and does not affect the dynamics. We show that this long-wavelength action does not change through the BCS-BEC crossover. We apply our approach to a moving vortex, for which $\ensuremath{\phi}$ is singular at the center of the vortex core, and $i\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}$ term affects vortex dynamics. We find that this term has two contributions. One comes from the states away from the vortex core and has Avort,1=n/2{A}_{\mathrm{vort},1}=n/2Avort,1​=n/2, where nnn is the fermion density. The other comes from electronic states inside the vortex core and has ${A}_{\mathrm{vort},2}=\ensuremath{-}{n}_{0}/2$, where n0{n}_{0}n0​ is the fermion density at the vortex core. This last term comes from the continuous part of the electronic spectrum and has no contribution from discrete levels inside the core; it also does not change if we add impurities. We interpret this term as the contribution to vortex dynamics in the continuum limit, when the spacing between energy levels $\ensuremath{\omega}$ is set to zero, while fermionic lifetime $\ensuremath{\tau}$ can be arbitrary. The total ${A}_{\mathrm{vort}}=(n\ensuremath{-}{n}_{0})/2$ determines the transversal force acting on the vortex core, $\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}$ is the velocity of the vortex core and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ a unit vector perpendicular to the 2d sample. The difference $(n\ensuremath{-}{n}_{0})/2$ changes through the BEC-BCS crossover as n0{n}_{0}n0​ nearly compensates nnn in the BCS regime, but vanishes in the BEC regime.
Starting from the time-dependent version of the Feynman-Hellmann theorem, the Magnus force acting on a vortex in fermion superfluid is expressed via the adiabatic curvature over the space of vortex positions. With use of the Bogoliubov--de Gennes approximation, the Magnus force in a homogeneous superfluid at T=0 is shown to originate from virtual transitions between the lowest quasiparticle core bound states. Nonadiabatic corrections to the curvature are obtained to second order in vortex velocity. The adiabatic approximation is shown to break down at a critical velocity equal to the vortex velocity in the first Landau level. The effect of elastic scattering on the Magnus force is discussed in terms of the relaxation-time approximation. It is suggested that this approximation is appropriate only for a large-scale vortex motion. In this case, the effective Magnus force is drastically reduced when the elastic-scattering rate exceeds the core excitation frequency. We conjecture that quantum vortex tunneling is governed by a local Magnus force obtained from the Berry phase approach.
We show how to lift the problem of calculating the force acting on a topological defect in a superfluid from the microscopic to the semiclassical level: Starting from the microscopic kinetic equations for a clean superconductor, we derive a Boltzmann equation for the quasiparticle distribution function in and around the defect. The velocity $\mathrm{q\ifmmode \dot{}\else \.{}\fi{}}$ and force $\mathrm{p\ifmmode \dot{}\else \.{}\fi{}}$ appearing in this Boltzmann equation are given through the Hamiltonian equations $\mathrm{q\ifmmode \dot{}\else \.{}\fi{}}={\ensuremath{\partial}}_{p}{E}_{n}(p,q)$ and $\mathrm{p\ifmmode \dot{}\else \.{}\fi{}}=\ensuremath{-}{\ensuremath{\partial}}_{q}{E}_{n}(p,q),$ where En(p,q){E}_{n}(p,q)En​(p,q) denotes the (nth(n\mathrm{th}(nth branch in the) spectrum of the quasiparticles in the vicinity of the defect. Second, we reformulate the microscopic expression for the force acting on the defect in terms of the total momentum transfer of the quasiparticles from the heat bath to the vortex core. We illustrate our result with an application to vortices in s-wave superconductors, where we derive the vortex equation of motion and identify the Magnus, Hall, and dissipative forces.
The nondissipative transverse force acting on one moving vortex under the influence of another vortex is discussed in fermionic superfluid systems, where the relative velocity between the vortices is finite. On the basis of detailed numerical solutions of the Bogoliubov-de Gennes equation, the Berry phase for an adiabatic motion of the vortex line is examined for a two-vortex system. It is found that the detailed electronic structure of a vortex core can affect the transverse force, without abandoning the previous discussions of the robust Magnus force on a single vortex.