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. 93:246402 (2004) Phys. Rev. B 70:224503 (2004) cond-mat/0506548 Outline 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 Bechgaard salts Stacked molecules form 1d chains Jerome, Science 252:1509 (1991)

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: C3-1075 (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/0401611 No suppression of electron spin susceptibility below Tc. NMR Knight

shift study of 77S in (TMTSF)2PF6 Lee et al, PRL 88:17004 (2002) P-wave superconductor without nodes - - py + + px

- Order parameter + + Specific heat in (TMTSF)2PF6 Garoche et al., J. Phys.-Lett. 43:L147 (1982) 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 Spin waves in triplet superconductors Spin wave: d-vector rotates In space Dispersion of spin waves Full spin symmetry Easy axis anisotropy 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 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 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 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 T 1-1 These predictions of the spin-wave mechanism of nuclear spin relaxation can be checked in experiments 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 not benefit from the Zeeman energy. For

this state does 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 Field and direction dependent Knight shift in UPt 3 Tau et al., PRL 80:3129 (1998) Competition of antiferomagnetism and triplet superconductivity in Bechgaard salts

Coexistence of superconductivity and magnetism Vuletic et al., EPJ B25:319 (2002) Interacting electrons in 1d Interaction Hamiltonian L R g1 R L L g2

L L L g4 R g4 L L R R

R R R Phase diagram g1 SDW (CDW) 1/2 CDW CDW

(SS) TSC (SS) SDW/TSC transition at K=1. This corresponds to 2 1 SS (CDW) K

SS 2g2 = g1 Symmetries 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 K=1, U(1) is enhanced to SO(3)I because 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 spin isospin 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 Role of interchain hopping Ginzburg-Landau free energy SO(4) symmetry requires SO(4) symmetric GL free energy Weak coupling analysis GL free energy. Phase diagram Unitary TSC for

. TSC order parameter r1 AF unitary TSC r2 First order transition between AF and TSC 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 Triplet superconductivity and antiferromagnetism. Phase diagram First order transition becomes a coexistence region

Phase diagram of Bechgaard salts T Normal AF TSC P T Normal AF

TSC Vuletic et al., EPJ B25:319 (2002) V Experimental test of quantum SO(4) symmetry operator rotates between AF and TSC orders Operator Charge Spin Momentum

0 1 2kF 2 1 0 2

0 2kF -mode should appear as a sharp resonance in the TSC phase Energy of the mode softens at the first order transition between superconducting and antiferromagnetic phases 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 resonance. The resonance should be observable using inelastic neutron scattering experiments (in the superconducting state)