1. MANYBODY PHYSICS Lab Effective Vortex Mass from Microscopic Theory June Seo Kim

2. MANYBODY PHYSICS Lab Contents Vortex Motion through a Type-II Superconductor 1. Vortex Motion through a Type-II Superconductor 2. Vortex Dynamics 3. Self-consistent Field Method 4. Energy Spectra and Effective Mass

3. MANYBODY PHYSICS Lab Vortex Motion through a Type-II Superconductor Imagine a small magnet with its north/south poles on either side of a thin slab of type-II superconductor. On dragging the magnet the vortex moves too. Force needed to execute the motion is (m+M) a m = mass of magnet M = effective mass of vortex M can be calculated within Caldeira-Leggett theory 2e e S N

4. MANYBODY PHYSICS Lab Vortex Dynamics Hamiltonian including pairing of superconductivity is represented in second quantization. : Energy of the excitation In real space, and are quasi-particle operators.

5. MANYBODY PHYSICS Lab The effect of magnetic field, Bogoliubov-de Genne equation

6. MANYBODY PHYSICS Lab Put and, then We have to transform one more time. Ignoring and putting, then

7. MANYBODY PHYSICS Lab Put be half-odd integers by periodic boundary condition.

8. MANYBODY PHYSICS Lab Self-consistent Field Method How can we solve this coupled differential equation? First of all, we have to treat the energy gap. If the energy gap is Absent, then we can find solution of these equations. Where J is a Bessel function and R is a boundary.

9. MANYBODY PHYSICS Lab and we can find and as combination of Bessel functions. Inserting into BdG equation and using orthogonality of Bessel functions, Therefore we have to know and.

10. MANYBODY PHYSICS Lab Self-consistency requires that the r-dependent gap function obey the relation for a given choice of the pairing interaction strength V. Put as initial condition. In iteration process, we can find exact gap function at zero temperature and finite temperatures. Moreover, and are change by the energy gap is changed. Therefore, we can calculate exact value of and. It is a self-consistent field method.

11. MANYBODY PHYSICS Lab Energy Spectra and Effective Mass Gap profile

12. MANYBODY PHYSICS Lab Energy Gap

13. MANYBODY PHYSICS Lab Energy Spectrum

14. MANYBODY PHYSICS Lab Mass Equation and Transition Matrix Elements Transition matrix element between localized and extended states are non-zero due to vortex motion. Second-order perturbation theory gives effective mass. core-to-core core-to-extended extended-to-extended Energy

15. MANYBODY PHYSICS Lab Effective Mass At zero temperature,

16. MANYBODY PHYSICS Lab Summary We calculate the effective mass of a single quantized vortex in the BCS superconductor at finite temperature. Based on self-consistent numerical diagonalization of the BDG equation we find the effective mass per unit length of vortex at finite temperatures. The mass reaches a maximum value at and decreases continuously to zero at.