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Physical Review B

Quantum vortex dynamics from the microscopic Hamiltonian

Ari Mizel

Published May 4, 2006• 3 citations
Publisher

Abstract

Starting with a microscopic Hamiltonian, we derive the Schr\"odinger equation of a superconductor with a vortex. The analysis describes the many-body state of a superconductor as a delocalized superposition of Bogoliubov-de Gennes mean-field vortex states. Taking a suitable one-band extended Hubbard model as the microscopic Hamiltonian of electrons in high-${T}_{c}$ compounds, solutions to the vortex Schr\"odinger equation are computed. We compute the vortex mass. Tuning the Hamiltonian parameters, we obtain solutions for the case of an $s$-wave order parameter and for the $d$-wave case and consider consequences of order parameter symmetry for vortex directional motion.

DOI
10.1103/physrevb.73.174502

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Starting with a microscopic Hamiltonian, we derive the Schr"odinger equation of a superconductor with a vortex. The analysis describes the many-body state of a superconductor as a delocalized superposition of Bogoliubov-de Gennes mean-field vortex states. Taking a suitable one-band extended Hubbard model as the microscopic Hamiltonian of electrons in high-Tc{T}_{c}Tc​ compounds, solutions to the vortex Schr"odinger equation are computed. We compute the vortex mass. Tuning the Hamiltonian parameters, we obtain solutions for the case of an sss-wave order parameter and for the ddd-wave case and consider consequences of order parameter symmetry for vortex directional motion.
A full quantum-mechanical treatment of the Bogoliubov-de Gennes equation for a single vortex in a d -wave superconductor is presented. First, we find low-energy states extended in four diagonal directions, which have no counterpart in a vortex of s -wave superconductors. The four-fold symmetry is due to " quantum effect ", which is enhanced when p F ξ is small. On the other hand, when p F ξ is large, the four-fold symmetry is suppressed. Second, for p F ξ ∼ 1, a peak with a large energy gap E 0 ∼ Δ is found in the density of states, which is due to the formation of the lowest bound states.
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.
We calculate the effective mass of a single quantized vortex in the Bardeen--Cooper--Schrieffer superconductor at finite temperature. Based on effective action approach, we arrive at the effective mass of a vortex as integral of the spectral function $J(\ensuremath{\omega})$ divided by ${\ensuremath{\omega}}^{3}$ over frequency. The spectral function is given in terms of the quantum-mechanical transition elements of the gradient of the Hamiltonian between two Bogoliubov--deGennes (BdG) eigenstates. Based on self-consistent numerical diagonalization of the BdG equation we find that the effective mass per unit length of vortex at zero temperature is of order $m{({k}_{f}{\ensuremath{\xi}}_{0})}^{2}$ (kf=Fermi{k}_{f}=\text{Fermi}kf​=Fermi momentum, ${\ensuremath{\xi}}_{0}=\text{coherence}$ length), essentially equaling the electron mass displaced within the coherence length from the vortex core. Transitions between the core states are responsible for most of the mass. The mass reaches a maximum value at $T\ensuremath{\approx}0.5{T}_{c}$ and decreases continuously to zero at Tc{T}_{c}Tc​.
We investigate generalizations of coherent states as a means of representing the dynamics of excitations of the superconducting ground state. We also analyse the propagation of generalized coherent state wave packets under the Bogoliubov-de Gennes Hamiltonian. The excitations of the superconducting ground state are superpositions of electron and hole quasi-particles described by the Bogoliubov-de Gennes equations, that can only exist at energies outside the band gap. A natural generalization relevant to the excitations of the superconducting ground state is the tensor product of canonical and spin coherent states. This state will quickly become de-localized on phase space under evolution by the Bogoliubov-de Gennes Hamiltonian due to the opposite velocities of the quasi-spin components. We therefore define the electron-hole coherent states which remain localised on phase space over longer times. We show that the electron-hole coherent states though entangled retain many defining features of coherent states. We analyse the propagation of both product and electron hole coherent states in a superconductor with a spatially homogeneous superconducting band gap. The dispersion relation indicates that wavepackets defined on the band gap have a zero group velocity, but we will show that interference effects can create states on the band gap that propagate at the Fermi velocity. We also consider the two semiclassical, short wavelength regimes, hbar->0$ and the large Fermi energy limit mu->infinity. In general these limits produce behaviour analogous to the canonical coherent states except for isolated cases. Finally we analyse the dynamics of the Andreev Reflection of a Gaussian wavepacket incident on a discontinuous normal-superconducting interface. We show that restricting the energy bandwidth of the incident state inside the superconducting band gap precludes the wavepacket from fully entering the superconducting region. We again consider the two semiclassical regimes.
Making use of the Bogoliubov-de Gennes equation, we study the quasi-particle spectrum and the vortex core structure of a single vortex in quasi 2D s-wave superconductors for small pFξ0, where pF is the Fermi momentum and ξ0 = vF/∆0 is the coherence length (¯h = 1). During our numerical calculation the particle number is conserved for each pFξ0. In particular, we find that there are only 1 or 2 bound states for pFξ0 = 1. Also, for pFξ0 = 1, the Kramer-Pesch effect ceases to exist at around T/Tc ≃ 0.3. There are renewed interests in vortex structure since the discovery of high Tc cuprate superconductors. It is possible that the superconductivity of high Tc cuprates can be characterized as d-wave superconductivity 1), 2) and is very close to the quantum
Making use of the Bogoliubov-de Gennes equation, we study the quasi-particle spectrum and the vortex core structure of a single vortex in quasi 2D s-wave superconductors for small pFξ0, where pF is the Fermi momentum and ξ0=vF/Δ0 is the coherence length (ħ=1). During our numerical calculation, the particle number is conserved for each pFξ0. In particular, we find that there are only 1 or 2 bound states for pFξ0=1. Also, for pFξ0=1, the Kramer-Pesch effect ceases to exist at around T/Tc≃0.3.