Antiferomagnetism and triplet superconductivity in Bechgaard salts Daniel Podolsky (Harvard and UC...
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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
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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
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Bechgaard salts
Stacked molecules form 1d chains Jerome, Science 252:1509 (1991)
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Evidence for triplet superconductivity in Bechgaard salts
•Strong suppression of Tc by disorderChoi 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)
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P-wave superconductor without nodes
--
--
++
+
+
Order parameter
px
py
Specific heat in (TMTSF)2PF6
Garoche et al., J. Phys.-Lett. 43:L147 (1982)
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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
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Spin waves in triplet superconductors
Spin wave:d-vector rotatesIn space
Dispersion of spin waves
Easy axis anisotropyFull spin symmetry
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Spin anisotropy of the triplet superconducting order parameter
Spin anisotropy in the antiferromagnetic state:
Easy direction for the superconducting order parameter is along the b axis
For Bechgaard salts we estimate
Assuming the same anistropy in the superconducting state
Dumm et al. (2000)
Torrance et al. (1982)
Spin z axis points along the crystallographic b axis.
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Experimental regime of parameters
Contribution of spin waves to 1/T1
Moriya relation: -- nuclear Larmor frequency
Creation or annihilation ofspin waves does not contribute to T1
-1
Scattering of spin wavescontributes to T1
-1
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Contribution of spin waves to 1/T1
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
(1) (2)
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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 nuclearspin relaxation can be checked in experiments
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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 doesnot benefit from the Zeeman energy.
For the order parameter flops into the xy plane.
This state can benefit from the Zeeman energy withoutsacrificing the pairing energy.
For Bechgaard salts we estimate
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Field and direction dependent Knight shift in UPt3
Tau et al., PRL 80:3129 (1998)
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Competition of antiferomagnetism and triplet superconductivity in Bechgaard salts
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Coexistence of superconductivity and magnetismVuletic et al., EPJ B25:319 (2002)
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Interacting electrons in 1d
L’ R’ L’ L’ L’ L’
g1 g2 g4 g4
R L R R L L R R
R’ R’
Interaction Hamiltonian
K
g1
SDW(CDW)
TSC(SS)
CDW CDW(SS)
SS(CDW)
SS
1/2 1 2
SDW/TSC transition at K=1.
This corresponds to
2g2 = g1
Phase diagram
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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
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SO(4)=SO(3)SxSO(3)I symmetry.Unification of antiferromagnetism and
triplet superconductivity.
transforms as a vector under spin and isospin rotations
isospinspin
Order parameter for antiferromagnetism:
Order parameter for triplet superconductivity:
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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
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Role of interchain hopping
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Ginzburg-Landau free energy
SO(4) symmetry requires
SO(4) symmetric GL free energy
Weak coupling analysis
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GL free energy. Phase diagram
Unitary TSC for . TSC order parameter
First order transitionbetween AF and TSC
r1
AFunitary TSC
r2
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Unitary TSC and AF. Thermal fluctuations
AF
Unitary TSC
r2
r1
Extend spin SO(3) to SO(N). Do large N analysis in d=3
• First order transition between normal and triplet superconducting phases (analogous result for 3He: Bailin, Love, Moore (1997))
• Tricritical point on the normal/antiferromagnet boundary
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Triplet superconductivity and antiferromagnetism.Phase diagram
First order transition becomes a coexistence region
AF TSC
NormalT
P
TSC
AF
Normal
T
“V”
Phase diagram ofBechgaard salts
Vuletic et al., EPJ B25:319 (2002)
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Operator Charge Spin Momentum
0 1 2kF
2 1 0
2 0 2kF
operator rotates between AF and TSC orders
-mode should appear as a sharp resonance in the TSC phase
Energy of the mode softens at the first order transitionbetween superconducting and antiferromagnetic phases
Experimental test of quantum SO(4) symmetry
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• 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)
Conclusions