2. Mass without Mass “PHASING-OUT” of (BARE) MASS FROM PHYSICAL EQUATIONS Newton’s 2 nd law: F=ma Newton’s gravity: F = GMm/r 2 Einstein’s law: E=mc 2 Quantum field theory: Lagrangians with zero mass, but calculated mass is finite! Interaction generates mass Superconductivity: Meissner effect due to photons acquiring finite mass Electrons in periodic potential: Effective mass can be vastly different from bare mass of 0.5 MeV Phonon renormalization of mass, etc. mass is given mass is generated/destroyed dynamic mass generation In solid state physics

3. Calculating the Effective Mass – Diagrammatic Approach A bare propagator for a quasiparticle is characterized by bare mass: Interaction gives rise to self-energy. Real part of self-energy is the mass renormalization.

4. Calculating the Effective Mass - Caldeira-Leggett Approach A foreign object moving through a medium experiences friction and mass renormalization due to interaction with the medium Write down the Lagrangian for the (object)+(medium) Integrate out the degrees of freedom of the (medium) Effective action for the (object) contains interaction effects RENORMALIZATION CLOUD

5. 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

6. Vortex Structure from BdG equation self-consistent solution of BdG equation for a single vortex Gap profile

7. 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 mass correction core-to-core core-to-extended extended-to-extended Energy

8. Effective Mass for Pure Superconductor Vortex Mass ~ Mass of electrons occupying a cylinder of radius at T=0 Rises to a maximum at T ~ 0.5 T c Falls to zero at T= T c

9. Effective Mass for Impure Superconductor Impurities can wash out localized core levels. Core-to-core contribution to mass vanishes, and effective mass is greatly reduced.