FFLO and related phases

FFLO and related phases
Daniel Agterberg – University of Wisconsin - Milwaukee
Thanks to Paolo Frigeri, Nobuhiko Hayashi, Manfred Sigrist (ETH-Zurich),
Huiqiu Yuan, Myron Salamon (UIUC), Akihisha Koga (Osaka University) ,
Raminder Kaur, Shantanu Mukherjee, and Zhichao Zheng (UWM)
Lecture 4:
Broken Parity Superconductors - continuation
FFLO and Helical Phases
Fractional Vortices in FFLO
Making a charge 4e superconductor in 2D

Superconductivity
Broken Inversion appears through:
First derive the gap equation within weak-coupling
assuming Δ

Spin Susceptibility

Spin-Susceptibility-Triplet: Sr 2 RuO 4
Spin triplet: Sr 2 RuO 4
The spin-susceptibility is
unchanged when Hod=0
K. Ishida et al., Nature 396, 658 (1998).
J.A.Duffy et al. , PRL 85, 5412 (2000).

Spin Susceptibility
singlet
Note α>>Δ for CePt 3 Si
L.P. Gorkov and E.I. Rashba, PRL 87 036004 (2001).
P.A. Frigeri, DFA, M. Sigrist, New J.Phys. 6 115 (2004).
triplet
Spin susceptibility becomes the same for singlet and triplet

Experimental Results
CeIrSi 3 : Rashba
Mukuda et al PRL 2010

Strong SO: Two Band System

CePt 3 Si
• The protected spin-triplet state for
CePt 3 Si is:
• Symmetry requires that g(k) vanishes
at points on the Fermi surface:

CePt 3 Si – line nodes
• The protected spin-triplet state has point
nodes for k x =k y =0.
• Line nodes have been observed
Bonalde, PRL 2005 Izawa, PRL 2005

CePt 3 Si
A small s-wave mixing changes the point-nodes
of the protected d(k) to line nodes.
A similar story applies to Li 2 Pt 3 B and Li 2 Pd 3 B

FFLO, PDW, and single-q
(helical) states

Singlet pairing in a Zeeman field: FFLO
Key Point of FF and LO: Pairing between fermions with different
Fermi surfaces leads to “finite momentum” pairing states (FFLO).
(FF)
(LO)
These phases sensitive to disorder
The general class of such states in known as pair density wave
(PDW) states.

Broken Parity Case
Broken parity and magnetic field: no symmetries left to ensure k
and –k both lie on the Fermi surface.
Will focus microscopic theory on Rashba case with field in plane.
Observation of simultaneous ferromagnetism and superconductivity
at LaAlO 3 and SrTiO 3 provides a 2D application of this problem.

Parity Broken Case
k+q
-k+q
field
No ASOC With ASOC With ASOC and
Zeeman Field
Can pair every state on one Fermi surface.
two bands prefer opposite q vectors.
The

Phase Diagram
Multiple q-phase
N 1 =N 2 , δN=0
Multiple q-phase
δN=0.05
H
Single-q phase
H
Uniform phase
Single-q (helical) phase
T/Tc
T/Tc
Single-q phase:
Multiple-q phase:
Two different values of q appear – why?

Ginzburg Landau origin of helical Phase
In broken parity superconductors – new terms appear in
the GL theory (Lifschitz invariants)
Induces a “single q” solution in a uniform magnetic field
with
Lifschitz invariant origin suggests helical phase will exist
even with impurities. Helical phase has no super current.

Single Phase Tc Results for Rashba
Barzykin and Gorkov, PRL (2002); DFA, Physica C (2003);
Kaur, DFA, Sigrist PRL (2004); DFA and Kaur PRB (2007);
Dimitrova and Feigel’man , PRB (2007); Samokhin, PRB (2008);
Mineev and Samokhin (2008).
Michaeli, Potter, Lee, cond-mat (2011)
(LaAlO 3 /SrTiO 3 interfaces)
Survives to high Fields in disordered 2D Rashba

Ginzburg-Landau-Wilson
Theory of FFLO states

Theory of ψ Qx and ψ -Qx
FFLO-like phase:
symmetry!
General feature:
Gauge invariance
Translational invariance

Standard Vortex
The free energy has a U(1) gauge invariance.
Consider a line-defect in the wave function:
Far from the vortex core
φ
The flux contained by a vortex is quantized

FFLO “Vortices”
(n,m) vortices. Consider (1,0) vortex in a phase where the
magnitudes of the components far from the vortex core are
equal
then
(1,1) vortex is usual Abrikosov vortex with flux Φ 0

LO Phase ½ vortices
U(1)xU(1) symmetry implies Ψ -Q and Ψ Q single valued
PDW displacement “phonon” field
usual phase field
Single valued :
Charge Density :
Edge dislocation in charge density from ½ vortex
Uniform SC (charge 4):

2D superfluid: thermal
excitation of vortices
• Free energy governed by phase
fluctuations:
• The energy of a vortex is:
• The entropy is:
above which vortices proliferate
below Tc

Thermal Fluctuations in LO superfluids
Energy of ½ vortex :
Energy of single vortex :
Energy of single dislocation :
CDW
PDW
Disordered
4SF

Superconductors
Summary: 2D LO Phase
A screens 1/r leading to a finite energy vortex
Dislocations still cost log(R/a) energy.
CDW order survives while SC order does not. (Agterberg and Tsunetsugu 2008)
Rotationally invariant superfluids
δQ screens 1/r leading to a finite energy dislocation, vortices cost log(R/a) energy.
Charge 4 superfluid order survives while CDW order does not. (Radzihovsky and Vishwanath
PRL 2009)
Berg, Fradkin, Kivelson argue charge 4e SC without rotational invariance (Nature Physics
2009) – last slide