Lecture 3 Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign, USA and Director, Center for Complex Physics Shanghai Jiao Tong University S HANGHAI J IAO T ONG U NIVERSITY L ECTURE

SJTU 3.1 Bose – Einstein Condensation (“BEC”) Recall: for a system of (structureless, spinless) bosons, wave function must be totally symmetric under interchange of coordinates of only two particles: For a gas of noninteracting particles, this leads to Bose-Einstein statistics: for particles whose total number is not conserved (e.g. photons) But if total number is conserved (e.g. He atoms) chemical potential total number of particles

SJTU 3.2  definition of BEC

SJTU 3.3 Two problems with BEC as an explanation of superconductivity: 1.Does not (by itself) explain metastability of supercurrents. 2.Electrons are not bosons but fermions! The problem of supercurrent metastability

SJTU 3.4 “topologically” distinct Let’s form a function which interpolates between these forms: Because of linearity of Schrödinger equation ⇒ no metastability. *with EM field treated as classical.

SJTU 3.5 Stability of supercurrents: C. Topological argument (Analogy: string wound around hula-hoop)

SJTU 3.6 Now we have: Hence

SJTU 3.7 Problem of statistics: the “BEC-BCS crossover”

SJTU 3.8 Apparent answer (from theory and experiment in ultracold Fermi gases) i.e. in many-particle system, onset of 2-particle bound state is just not seen. Partial clue: statements for 2-particle system are valid only in 3D. In 2D or 1D a bound state is formed for arbitrarily weak attraction (but in 2D case, binding energy exponentially small in interaction strength). So: can we regard superconductivity as a sort of BEC of pairs of electrons? nothing!