1. Wave function for odd frequency superconductors.
Hari P. Dahal, E. Abrahams, D. Mozyrsky, Y. Tanaka, A. V. Balatsky
New J. Phys. 11 (2009)
spin-triplet: Berezinskii, JETP Lett. 20, 287 (1974)
spin-singlet: Balatsky and Abrahams, PRB 45, (1992)
New Class of superconductor
ave
Order parameter
Wave function
Ginzburg Landau
Mechanism of pairing
Hamiltonian
New Properties
Composite boson condensate
Kokusai workshop Nagoya, Sep4 2009

2. Outline
Introduction of the Odd-frequency order parameters using symmetry arguments.
2) Why composite paring?
3) Introduce wave function for odd frequency sc.
4) Minimization of energy, gap equation, excitation spectrum, density of states
5) Experimental consequences of odd-frequency superconductivity
6) Conclusion

3. To set the stage let me prove that there are
At least two classes of superconductors with
distinct condensates yet same quantum numbers:
2e charge and total spin S

4. Explicit proof of existence of two classes of SC
Let me start with the wave function. Prove that there are at least two classes of
superconductors with distinct condensates yet same quantum numbers:
2e charge and total spin
assume that there is a spin boson excitation in addition to fermions:
s- band
and d-band electrons that are forming spin
There is a usual, conventional BCS 2e copper pair operator
In addition consider a composite 2e boson that would have same quantum numbers
S can be

5. Coherent state
Coherent state that describes condensate 2e and spin singlet

6. Explicit proof of existence of a new class of SC:
Both states are spin singlet S =0 yet they have zero overlap with BCS wf for macrosopic state.
This proves that there is a spin singlet state that is qualitatively different then
The conventional BSC state! It is an odd frequency S = 0 superconductor with the
wave function

7. Assume that <S+> =0
Otherwise we always can replace
Thus reducing the whole discussion back to conventional BSC
singlet and triplet pairing

8. Now if we agree that there is more then one class: what does it mean
Now if we agree that there is more then one class: what does it mean? It was a wave function for the odd- frequency superconductor
Hamiltonian will be derived later. Situation is
Similar to the case of
BCS where the wave function captured the physics and mena field Hamiltonian was written to accommodate pairing.
Another example: QHE. Wave function was guessed to
accommodate qp quantum numbers. There is no simple Hamitonian for which Laughlin wave function in QHE would be an exact ground state.

9. Classification

11. Proof of PT = +1 for S=0 SC
Same for Berezinskii sate (S = 1): PT= -1

12. Berezinskii state
Berezinskii
(1974)
Spin-triplet s-wave

13. Symmetry of the pair amplitude
+ symmetric, - anti-symmetric
Frequency
（time)
Spin
Orbital
Total
BCS
- (singlet)
+(even)
PT=1
BCS
+(even)
Cuprate
+(even)
3He
A,BW…
+ (triplet)
-(odd)
PT=-1
Sr2RuO4
-(odd)
Berezinskii
+ (triplet)
+(even)
PT=-1
OSO-AB
-(odd)
- (singlet)
-(odd)
PT=1
OSO (Odd-frequency spin-singlet odd-parity) Abrahams Balatsky

14. If there is a state, there is a wave function that describes the state
If there is a state, there is a wave function that describes the state. What is it?
Questions:
What is the order parameter for odd frequency SC
(∆)?
What is the wave function
(ψ)?
Is there a condensate and Off Diagonal Long Range Order (ODLRO)
<ψ(1) ψ(2)> -> < ψ(1)>< ψ(2)>
If there is a SC state, there is Wave Function, Order Parameter,GL:
b)There is a mean field Hamiltonian that captures the physics of odd frequency SC.

15. Odd frequency Even frequency Singlet (PT=+1) Triplet (PT=-1)
Order parameters and their symmetries in even and odd frequency superconductors
Odd frequency
Even frequency
Equal time anomalous Green’s function
gives order parameter
Equal time slope of the anomalous Green’s function gives order parameter
Singlet (PT=+1)
Triplet (PT=-1)
Singlet (PT=+1)
Triplet (PT=-1)

16. Order parameter in BCS SC
Pairing and coherence are captured at the lowest order: easiest “boson” to create:

17. Composite boson condensate
Three body bound state
In Cooper problem
Ck+q/2
Bq/2
C-k+q/2

18. Composite condenate and wave function for odd frequency SC
Why bother the the wave function?
One might ask: If the state appears as some odd time correlation how we
can talk about wave function? How to capture correlations that are inherent
to superconducting state is equal time anomaous function vanishes?
there is no order parameter, no wave function.
This line of thinking is not satisfactory to some of us.
Questions:
What is the order parameter for odd frequency SC
(∆)?
What is the wave function
(ψ)?
Is there a condensate and Off Diagonal Long Range Order (ODLRO)
<ψ(1) ψ(2)> -> < ψ(1)>< ψ(2)>
If there is a SC state, there is Wave Function, Order Parameter,GL:
If there is a state, there is a wave function that describes the state.
There is an order parameter and GL description
There is a mean field Hamiltonian that captures the physics of odd frequency SC.
The wave function is missing and has long been searched at least by some.

19. Equal-time operator (odd-frequency)
(t-J model case)
OSO (odd-frequency spin-singlet odd-parity)
Berezinskii (odd-frequency spin-triplet even-parity)
Balatsky Bonca PRB (1992)

20. Order parameter in time and frequency domain
Define odd time ∆(τ) = K’τ
Define odd frequency ∆(ω) = Kω
We find that K’ ~ K
V. Balatsky, and J. Bonca, Phys. Rev. B 48, (1992).
Elihu Abrahams et al, Phys. Rev. B 52, 1271 (1995).

21. Hamiltonian and wave function
Abrahams et al.,PRB 52, 1271(1995)
We propose the coherent state wave function
Important Properties of this wave function
Describes a composite condensate of a Cooper pair of spin-0 and a magnon of spin-1.
Describes the condensation of charge 2e.
Breaks U1 symmetry of the electrodynamics.
c) It is a mean field wave function for the composite condensate
New J. Phys. 11 (2009)

22. Comparison with the BCS case
Odd frequency
BCS
Dahal et al., New J. Phys. 11 (2009)
arXiv:

23. How to get BCS results from the current formulation

24. Gap equation BCS Odd frequency C Arbitrary small coupling leads to
BCS superconductivity
Odd frequency superconductivity
requires critical coupling

25. Selfconsistency condition: BCS case
Odd frequency case
Integral over momentum corresponding to this extra diagram integrates over the BCS log-divergence.

26. Energy spectrum BCS Odd-frequency qasi-particle excitation energy
is not the qasi-particle excitation energy
Dahal et al., arXiv:
New J. Phys. 11 (2009)

27. Density of States Odd-frequency BCS Gapped excitation
Diverging DOS at the gap edge
Gap is filled in
DOS at the gap edge is reduced

28. Density of States: Gapless
In practice magnon momentum cut-off is bigger than the Fermi momentum.
Odd frequency state is gapless
Consistent with previous
Greens functions calculations.
Dahal et al
New J. Phys. 11 (2009)
Dahal et al., arXiv:

29. Ginzburg Landau functional for odd frequency superconductor
Critical coupling ~1
Positive stiffness
Elihu Abrahams et al, Phys. Rev. B 52, 1271 (1995).

30. Experimental consequences:
1) NMR experiment:
Absence of Hebel-Slichter peark in 1/T1
Absence of exponential decay of 1/T1 below Tc
Presence of a finite 1/T1 at T=0K.
2) Change in Ulrasonic attenuation:
a) Enhanced attenuation near Tc due to the change in sign of coherence factor compared to the BCS case (Journal of Superconductivity, 7, 501(1994)).
b) Absence of exponential decay below Tc
c) Finite attenuation at T=0K.
3) Electromagnetic absorption:
a) Finite absorption at E < delta
b) Finite absorption at E=0.
c) Detail needs to be worked out

31. Odd-frequency Pair amplitude not pair potential ) is generated in ferromagnet junctions
Odd frequency spin-triplet s-wave pair
spin-singlet s-wave pair + _
Ferromagnet
Superconductor
Bergeret, Efetov, Volkov, (2001)
Eschrig, Tanaka, Buzdin,Golubov, Kadigrobov,Fominov, Radovic…
Generation of the odd-frequency pair amplitude
in ferromagnet

32. References of more works in odd frequency superconductivity
“Realization of Odd-Frequency p-Wave Spin–Singlet Superconductivity Coexisting with Antiferromagnetic Order near Quantum Critical Point” Yuki Fuseya, Hiroshi Kohno, and Kazumasa Miyake,J. Phys. Soc. Jpn. 72, 2914(2003)
“Possible realization of odd-frequency pairing in heavy fermion compounds”, P. Coleman, E. Miranda, and A. Tsevelik, Phys. Rev. Lett. 70, (1993).
“Odd-frequency pairing in the Kondo lattice”, P. Coleman, and E. Miranda, Phys. Rev. B 49, (1994)
”Identifying the odd-frequency pairing state of superconductors by a field-induced Josephson effect” Jacob Linder, Takehito Yokoyama, and Asle Sudbo, Phys. Rev. B 77, (2008).
“Nesting, spin fluctuations, and odd-gap superconductivity in NaxCoO2.yH2O.”, Johannes MD, Mazin II, Singh DJ, Papaconstantopoulos DA, Phys. Rev. Lett. 93, (2004).
“Quantum transport in a normal metal/odd-frequency superconductor junction”, Jacob Linder, Takehito Yokoyama, Yukio Tanaka, Yasuhiro Asano, and Asle Subdo, Cond-mat/
4) “Manifestation of the odd-frequency spin-triplet pairing state in diffusive ferromagnet/superconductor junctions.“ T. Yokoyama, Y. Tanaka, and A. A. Golubov, Phys. Rev. B 75, (2007).
“Anomalous Josephson Effect between Even- and Odd-Frequency Superconductors“, Yuko Tanaka, Alexander A. Golubov, Satoshi Kashiwaya, and Masahito Ueda, Phys. Rev. Lett. 99, (2007).
“Odd-frequency pairing in normal-metal/superconductor junctions.“ Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev. B 76, (2007).
“Odd-frequency pairing in binary boson-fermion atom mixture”, Ryan M. Kalas, Alexander Balatsky, and Dimitry Mozyrsky, cond-mat/ , PRB 78, p 4513 (2008).

33. Summary of previous works which use the idea of the composite pairing
1) New class of singlet superconductors which break the time reversal and parity,
Alexander Balatsky, and Elihu Abrahams, PRB 45, 13125(1992).
Proposal of the odd-frequency spin-singlet p-wave superconductivity. e-ph interaction is tried and realized that it can not mediate the odd frequency superconductivity
2) Interactions for odd-ω-gap singlet superconductors
Elihu Abrahams, Alexander Balatsky, J. R. Schrieffer, and Philip B. Allen, PRB 47, 513(1993)
Proposal of a spin dependent electron-electron interaction. Does not face the problem that e-ph interaction faces.
3) Odd Frequency pairing in Superconductors,
J. R. Schrieffer, Alexander Balatsky, Elihu Abrahas, and Douglas J. Scalapino, Journal of Superconductiviy 7, 501(1994).
Using equation of motion it is shown that the condensate of the odd-frequency pairing consists of two fermions and a boson (composite).
4) Even- and odd-frequency pairing correlations in the one-dimensional t-J-h model: A comparative study, A. V. Balatsky, and J. Bonca, Phys. Rev. B 48, (1993).
Binding of Cooper pairs with magnetization fluctuations naturally appears in this model.
5) Properties of odd-gap superconductors,
Elihu Abrahams, Alexander Balatsky, D. J. Scalapino and J. R. Schrieffer, Phys. Rev. B 52, 1271 (1995).
6) H. Daha et al, New J. Phys. 11 (2009) – wave function and QP energy and DOS calculation
Using BCS like Hamiltonian for composite boson condensate it is shown that the superconductivity requires critical coupling. Meissner effect, Josephson effect between odd and BCS superconductors, 1/T1 are studied.

34. Summary of previous work which use the idea of the composite pairing
Odd Frequency pairing in Superconductors,
J. R. Schrieffer, et al Journal of Superconductiviy 7, 501(1994).
Using equation of motion it is shown that the condensate of the odd-frequency pairing consists of two fermions and a boson (composite).
Even- and odd-frequency pairing correlations in the one-dimensional t-J-h model: A comparative study, A. V. Balatsky, and J. Bonca, Phys. Rev. B 48, (1993).
Binding of Cooper pairs with magnetization fluctuations.
Properties of odd-gap superconductors,
Elihu Abrahams, Alexander Balatsky, D. J. Scalapino and J. R. Schrieffer, Phys. Rev. B 52, 1271 (1995).
H. Daha et al, New J. Phys. 11 (2009) – wave function and QP energy and DOS calculation
Using BCS like Hamiltonian for composite boson condensate it is shown that the superconductivity requires critical coupling. Meissner effect, Josephson effect between odd and BCS superconductors, 1/T1 are studied.

35. Brief Summary of our Eliashberg calculations for odd frequency SC

36. Why electron-phonon interaction does not work?
Abrahams et al., PRB 47, 513(1993)
If we use phonon Hamiltonian
Positivity of the phonon spectral function
If only l=0 and 1 components are non zero then
But for
has to be greater than 1.
No solution if we use electron-phonon interaction

37. The solution is the spin dependent Hamiltonian
Abrahams et al., PRB 47, 513(1993)
Spin-dependent interaction
Eliashberg equation in the spin singlet l-wave channel
The change of sing of the spin part of the interaction provides the possibility of density and spin couplings adding in the pairing channel yet opposing each other in the normal self-energy channel, so that self-energy remains of the order of 1 .
Example of high-Tc materials:
Hubbard model (within RPA)

38. Hubbard model Triangular lattice K. Shigeta, S. Onari, K. Yada
and Y. Tanaka, Phys. Rev. B (2009).
AF order is suppressed by
geometrical frustration.
Nesting condition is not good
any more.
Hubbard model
On site Coulomb interaction
Kinetic energy term
wn～0.3
wn～0.75
Triangular lattice
-t1
-t2 U

39. Energy dependence of odd-frequency gap
wn～0.3
wn～0.75

40. Classification of all equal time composite(K) SC states∆
Expansion in odd powers
N = 1, 3, 5
Expansion in even powers
N = 0, 2, 4

41. More then one way to extend Cooper pairing as a paradigm for SC
Conventional discussion
Boson attached to pair ∆

42. Conclusion
1) We propose based on the order pqrameter a wave function,
New J. Phys. 11 (2009)
describes Cooper pair of spin-0 and magnon of spin-1.
2) Odd frequency superconductivity requires critical coupling.
We derived the GL theory for odd frequency SC.
We derive quasi-particle excitation from trial wave function which is gapless.
There are non trivial experimental consequences such as in nuclear magnetic resonance, ultrasonic attenuation and electromagnetic absorption experiments.
How does a composite order forms in FM/SC junctions and vortes cores?