Wave function for odd frequency superconductors. Hari P. Dahal, E. Abrahams, D. Mozyrsky, Y. Tanaka, A. V. Balatsky New J. Phys. 11 (2009) 065005. spin-triplet: Berezinskii, JETP Lett. 20, 287 (1974) spin-singlet: Balatsky and Abrahams, PRB 45, 13125 (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

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

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

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

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

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

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

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.

Classification

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

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

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

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.

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)

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

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

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.

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 48 7445 (1992)

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, 7445 - 7449 (1992). Elihu Abrahams et al, Phys. Rev. B 52, 1271 (1995).

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) 065005.

Comparison with the BCS case Odd frequency BCS Dahal et al., New J. Phys. 11 (2009) 065005. arXiv:0901.2323

How to get BCS results from the current formulation

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

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

Energy spectrum BCS Odd-frequency qasi-particle excitation energy is not the qasi-particle excitation energy Dahal et al., arXiv:0901.2323 New J. Phys. 11 (2009) 065005.

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

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) 065005. Dahal et al., arXiv:0901.2323

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

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

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

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, 2960 - 2963 (1993). “Odd-frequency pairing in the Kondo lattice”, P. Coleman, and E. Miranda, Phys. Rev. B 49, 8955 - 8982 (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, 174507 (2008). “Nesting, spin fluctuations, and odd-gap superconductivity in NaxCoO2.yH2O.”, Johannes MD, Mazin II, Singh DJ, Papaconstantopoulos DA, Phys. Rev. Lett. 93, 097005(2004). “Quantum transport in a normal metal/odd-frequency superconductor junction”, Jacob Linder, Takehito Yokoyama, Yukio Tanaka, Yasuhiro Asano, and Asle Subdo, Cond-mat/0712.0443. 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, 134510(2007). “Anomalous Josephson Effect between Even- and Odd-Frequency Superconductors“, Yuko Tanaka, Alexander A. Golubov, Satoshi Kashiwaya, and Masahito Ueda, Phys. Rev. Lett. 99, 037005(2007). “Odd-frequency pairing in normal-metal/superconductor junctions.“ Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev. B 76, 054522(2007). “Odd-frequency pairing in binary boson-fermion atom mixture”, Ryan M. Kalas, Alexander Balatsky, and Dimitry Mozyrsky, cond-mat/0806.0419., PRB 78, p 4513 (2008).

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, 7445 - 7449 (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) 065005. – 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.

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, 7445 - 7449 (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) 065005. – 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.

Brief Summary of our Eliashberg calculations for odd frequency SC

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

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)

Hubbard model Triangular lattice K. Shigeta, S. Onari, K. Yada and Y. Tanaka, Phys. Rev. B 79 174507 (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

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

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

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

Conclusion 1) We propose based on the order pqrameter a wave function, New J. Phys. 11 (2009) 065005. 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?