1. Antiferomagnetism and triplet superconductivity in Bechgaard salts
Daniel Podolsky (Harvard and UC Berkeley)
Timofey Rostunov (Harvard)
Ehud Altman (Harvard)
Antoine Georges (Ecole Polytechnique)
Eugene Demler (Harvard)
References: Phys. Rev. Lett. 93: (2004)
Phys. Rev. B 70: (2004)
cond-mat/

2. Outline New experimental tests of triplet superconductivity
Introduction. Phase diagram of Bechgaard salts
New experimental tests of triplet superconductivity
Antiferomagnet to triplet superconductor transition
in quasi 1d systems. SO(4) symmetry
Implications of SO(4) symmetry for the phase diagram.
Comparison to (TMTSF)2PF6
Experimental test of SO(4) symmetry

3. Bechgaard salts Stacked molecules form 1d chains
Jerome, Science 252:1509 (1991)

4. Evidence for triplet superconductivity in Bechgaard salts
Strong suppression of Tc by disorder
Choi et al., PRB 25:6208 (1982)
Tomic et al., J. Physique 44: C (1982)
Bouffod et al, J. Phys. C 15:2951 (1981)
Superconductivity persists at fields
exceeding the paramagnetic limit
Lee et al., PRL 78:3555 (1997)
Oh and Naughton, cond-mat/
No suppression of electron spin
susceptibility below Tc. NMR Knight
shift study of 77S in (TMTSF)2PF6
Lee et al, PRL 88:17004 (2002)

5. P-wave superconductor without nodes+
Order parameter py px
Specific heat in (TMTSF)2PF6
Garoche et al., J. Phys.-Lett. 43:L147 (1982)

6. Nuclear spin lattice relaxation rate in (TMTSF)2PF6
Lee et al., PRB 68:92519 (2003)
For (TMTSF)2ClO4 similar behavior has been observed
by Takigawa et.al. (1987)
Typically this would be attributed to nodal quasiparticles (nodal line)
This work: T3 behavior of 1/T1 due to spin waves

7. Spin waves in triplet superconductors
d-vector rotates
In space
Dispersion of spin waves
Full spin symmetry
Easy axis anisotropy

8. Spin anisotropy of the triplet superconducting order parameter
Spin anisotropy in the antiferromagnetic state:
Torrance et al. (1982)
Dumm et al. (2000)
Spin z axis points along the crystallographic b axis.
Assuming the same anistropy in the superconducting state
Easy direction for the superconducting order parameter is along the b axis
For Bechgaard salts we estimate

9. Contribution of spin waves to 1/T1
Experimental regime of parameters
Moriya relation: nuclear Larmor frequency
Creation or annihilation of
spin waves does not
contribute to T1-1
Scattering of spin waves
contributes to T1-1

10. Contribution of spin waves to 1/T1
(1)
(2)
is the density of states for spin wave excitations. Using
For we can take where is the dimension
This result does not change when
we include coherence factors

11. Contribution of spin waves to 1/T1
For small fields, T1-1 depends on the direction of the magnetic field
When , we have T3 scaling of T1-1 in d=2
When , we have exponential suppression of T1-1
These predictions of the spin-wave mechanism of nuclear
spin relaxation can be checked in experiments

12. Spin-flop transition in the triplet superconducting state
S=1 Sz=0
S=1 Sx=0
S=1 Sy=0
At B=0 start with (easy axis). For this state does
not benefit from the Zeeman energy.
For the order parameter flops into the xy plane.
This state can benefit from the Zeeman energy without
sacrificing the pairing energy.
For Bechgaard salts we estimate

13. Field and direction dependent Knight shift in UPt3
Tau et al., PRL 80:3129 (1998)

14. Competition of antiferomagnetism and
triplet superconductivity in Bechgaard salts

15. Coexistence of superconductivity and magnetism
Vuletic et al., EPJ B25:319 (2002)

16. Interacting electrons in 1d
Interaction Hamiltonian
Ls’
Ls’
Ls’
Ls’
Rs’
Rs’
Ls’
Rs’ g1 g2 g4 g4 Rs Ls Rs Rs Ls Ls Rs Rs
Phase diagram g1
SDW/TSC transition at Kr=1.
This corresponds to
SDW
(CDW)
TSC
(SS)
1/2 2 1 Kr
2g2 = g1
CDW
(SS) SS
(CDW) SS
CDW

17. Symmetries Isospin SO(3)I symmetry Spin SO(3)S algebra
SO(3)S is a good symmetry of the system
Isospin SO(3)I symmetry
We always have charge U(1) symmetry
When Kr=1, U(1) is enhanced to SO(3)I because

18. Order parameter for triplet superconductivity:
SO(4)=SO(3)SxSO(3)I symmetry. Unification of antiferromagnetism and triplet superconductivity.
Order parameter for antiferromagnetism:
Order parameter for triplet superconductivity:
transforms as a vector under spin and isospin rotations
isospin
spin

19. SO(3)SxSO(4)I symmetry at incommensurate filling
Two separate SO(3) algebras
Isospin group SO(4)I= SO(3)RxSO(3)L
Umklapp scattering reduces SO(4)I to SO(3)I

20. Role of interchain hopping

21. Ginzburg-Landau free energy
SO(4) symmetry requires
SO(4) symmetric GL free energy
Weak coupling analysis

22. GL free energy. Phase diagram
Unitary TSC for
TSC order parameter r1 AF
unitary TSC r2
First order transition
between AF and TSC

23. Unitary TSC and AF. Thermal fluctuations
Extend spin SO(3) to SO(N). Do large N analysis in d=3 r1 AF r2
Unitary TSC
First order transition between normal and triplet superconducting
phases (analogous result for 3He: Bailin, Love, Moore (1997))
Tricritical point on the normal/antiferromagnet boundary

24. Triplet superconductivity and antiferromagnetism. Phase diagram
First order transition becomes
a coexistence region
Phase diagram of
Bechgaard salts AF
TSC
Normal T P
TSC AF
Normal T
“V”
Vuletic et al., EPJ B25:319 (2002)

25. Experimental test of quantum SO(4) symmetry
Q operator rotates between AF and TSC orders
Operator
Charge
Spin
Momentum 1
2kF 2
q-mode should appear as a sharp resonance in the TSC phase
Energy of the q mode softens at the first order transition
between superconducting and antiferromagnetic phases

26. Conclusions
New experimental tests of triplet pairing in Bechgaard salts:
1) NMR for T < 50mK and small fields. Expect strong suppression
of 1/T1
2) Possible spin flop transtion for magnetic fields
along the b axis and field strength around 0.5 kG
3) Microwave resonance in Bechgaard salts at
(For Sr2RuO4 expect such resonance at )
SO(4) symmetry is generally present at the antiferromagnet
to triplet superconductor transition in quasi-1d systems
SO(4) symmetry helps to explain the phase diagram of (TMTSF)2PF6
SO(4) symmetry implies the existence of a new collective mode,
the q resonance. The q resonance should be observable using
inelastic neutron scattering experiments (in the superconducting
state)