Lecture 16: Order parameter symmetries
The gap equation: Basis for understanding OP
symmetry
• Qualitatively correct description for all superconductors (comes from
variation of a trial many-body wavefunction).
The gap equation: Basis for understanding OP
symmetry
• Qualitatively correct description for all superconductors (comes from
variation of a trial many-body wavefunction).
• Symmetry of effective potential Vk,k’ determines symmetry of order
parameter Δk.
The gap equation: Basis for understanding OP
symmetry
• Qualitatively correct description for all superconductors (comes from
variation of a trial many-body wavefunction).
• Symmetry of effective potential Vk,k’ determines symmetry of order
parameter Δk.
• Determining V from microscopics is difficult*.
Not just bare Coulomb.
Also lattice (phonons), spin-orbit coupling, spin degrees of freedom,…
* Asking what the effective V is not the same thing as asking what the “glue” is!
For non-s-wave,V can be repulsive!
The gap equation: Basis for understanding OP
symmetry
• Qualitatively correct description for all superconductors (comes from
variation of a trial many-body wavefunction).
• Symmetry of effective potential Vk,k’ determines symmetry of order
parameter Δk.
• Determining V from microscopics is difficult*.
Not just bare Coulomb.
Also lattice (phonons), spin-orbit coupling, spin degrees of freedom,…
This lecture: Solving gap equation for several Vk,k’ ’s and discuss
origin of such interactions for electron-phonon mechanism and
spin fluctuations.
What is the physical meaning of the gap?
• It’s 1. (In momentum space) The momentum-dependent gap in the excitation
spectrum near the Fermi surface. 2. An order parameter.
3. (In real space) The wavefunction of the Cooper pair “molecule”.
What is the physical meaning of the gap?
• It’s 1. (In momentum space) The momentum-dependent gap in the excitation
spectrum near the Fermi surface. 2. An order parameter.
3. (In real space) The wavefunction of the Cooper pair “molecule”.
• It is easiest to solve the gap equation in momentum space.
visualize in real space.
Easier to
Cooper pair wavefunction (real space picture)
• In momentum space (& 2nd quantized), many-body BCS
wavefunction is:
Cooper pair wavefunction (real space picture)
• In momentum space (& 2nd quantized), many-body BCS
wavefunction is:
• In real space, it’s easiest to see that the many-body BCS wavefunction
represents BEC of Cooper pairs:
Cooper pair wavefunction (real space picture)
• In momentum space (& 2nd quantized), many-body BCS
wavefunction is:
• In real space, it’s easiest to see that the many-body BCS wavefunction
represents BEC of Cooper pairs:
• Φ(r1,r2): two-electron Cooper pair wavefunction. Related to the gap:
Cooper pair wavefunction (real space picture)
• In momentum space (& 2nd quantized), many-body BCS
wavefunction is:
• In real space, it’s easiest to see that the many-body BCS wavefunction
represents BEC of Cooper pairs:
• Φ(r1,r2): two-electron Cooper pair wavefunction. Related to the gap:
Gap symmetry = Cooper pair wavefunction symmetry
Solving the gap equation
• Symmetry of effective potential Vk,k’ determines symmetry of order
parameter Δk.
Solving the gap equation
• Symmetry of effective potential Vk,k’ determines symmetry of order
parameter Δk.
• For el.-ph. coupling, effective potential
is isotropic and attractive for electrons
close to Fermi surface; s-wave.
Solving the gap equation
• Symmetry of effective potential Vk,k’ determines symmetry of order
parameter Δk.
• For el.-ph. coupling, effective potential
is isotropic and attractive for electrons
close to Fermi surface; s-wave. • antiferromagnetic spin fluctuations &
bare Coulomb are repulsive and
anisotropic. Gap must be negative
somewhere to solve gap eqn. (d-wave).
Attractive effective interaction: electrons and phonons
• Frohlich hamiltonian:
phonon dispersion; e.g., ωq=cq
electrons
phonons
electron-phonon coupling
Attractive effective interaction: electrons and phonons
• Frohlich hamiltonian:
• Integrate phonons out of partition function to get effective theory for fermions:
Attractive effective interaction: electrons and phonons
• Frohlich hamiltonian:
• Integrate phonons out of partition function to get effective theory for fermions:
• Explicitly,
Attractive effective interaction: electrons and phonons
• Frohlich hamiltonian:
• Integrate phonons out of partition function to get effective theory for fermions:
• Explicitly,
Attractive effective interaction: electrons and phonons
• Time-dependent potential (frequency-dependent Fourier
transforming): Interaction mediated by phonons that propagate with a
finite velocity c: ωq = cq. ω is the characteristic energy scale εk+q-εk
of the electrons upon absorbing the phonon.
• Hence, two regimes:
For electrons close to the Fermi surface,
ω ~ εk+q-εk ≪ cq, interaction is attractive.
For electrons not both close to the Fermi
surface, ω ~ εk+q-εk ≫ cq, interaction is repulsive.
• Simple model of this interaction: attractive, isotropic (s-wave) for
electrons within a Debye energy of the Fermi surface.
Solving the gap equation: s-wave
• Simple model of this interaction: attractive, isotropic (s-wave) for
electrons within a Debye energy of the Fermi surface:
Solving the gap equation: s-wave
• Simple model of this interaction: attractive, isotropic (s-wave) for
electrons within a Debye energy of the Fermi surface:
Solving the gap equation: s-wave
• Simple model of this interaction: attractive, isotropic (s-wave) for
electrons within a Debye energy of the Fermi surface:
Repulsive effective interaction: antiferromagnetic
fluctuations
• Strong AFM correlations
bonds between
and
short-range repulsive interactions on diagonal
spins. (
, spin. susceptibility).
Repulsive effective interaction: antiferromagnetic
fluctuations
• Strong AFM correlations
bonds between
and
short-range repulsive interactions on diagonal
spins. (
, spin. susceptibility).
Repulsive effective interaction: antiferromagnetic
fluctuations
• Strong AFM correlations
bonds between
and
short-range repulsive interactions on diagonal
spins. (
, spin. susceptibility).
• The only way to solve the gap equation with a potential that is every
where positive (i.e., repulsive) is if the gap is negative in some places.
Repulsive effective interaction: antiferromagnetic
fluctuations
• Strong AFM correlations
bonds between
and
short-range repulsive interactions on diagonal
spins. (
, spin. susceptibility).
• The only way to solve the gap equation with a potential that is every
where positive (i.e., repulsive) is if the gap is negative in some places.
Repulsive effective interaction: antiferromagnetic
fluctuations
Gap has nodes where the
repulsive interaction is strongest.
• Strong AFM correlations
bonds between
and
short-range repulsive interactions on diagonal
spins. (
, spin. susceptibility).
• The only way to solve the gap equation with a potential that is every
where positive (i.e., repulsive) is if the gap is negative in some places.
AFM fluctuations: ubiquitous mechanism in
unconventional superconductors?
heavy fermions
Knebel et al, arXiv:0911.5223.
iron-pnictides
Physics 3, 41 (2010)
cuprates
Nature 468,184 (2010)
AFM fluctuations: ubiquitous mechanism in
unconventional superconductors?
heavy fermions
Knebel et al, arXiv:0911.5223.
iron-pnictides
Physics 3, 41 (2010)
cuprates
Nature 468,184 (2010)
No! For cuprates, repulsive Coulomb
interaction is enough!
Summary
• Cooper pairs like molecules.
Symmetry of two-electron Φ(r1,r2)
wavefunction determined by symmetry of gap:
• Symmetry of effective potential Vk,k’ determines symmetry of gap Δk
via gap equation
• For attractive interactions, gap has same structure as interaction:
e.g., spherical
harmonics
• For repulsive interactions, gap must be negative in places.
chosen to avoid repulsive interactions.
Shape