Possible Odd-Frequency Superconductivity in Strong-Coupling Electron–Phonon Systems

Abstract

A possibility of the odd-frequency pairing in the strong-coupling electron-phonon systems is discussed. Using the Holstein-Hubbard model, we demonstrate that the anomalously soft Einstein mode with the frequency ω E ≪ ω c (ω c is the order of the renormalized bandwidth) mediates the s-wave odd-frequency triplet pairing against the ordinary even-frequency singlet pairing. It is necessary for the emergence of the odd-frequency pairing that the pairing interaction is strongly retarded as well as the strong coupling, since the pairing interaction for the odd-frequency pairing is effective only in the diagonal scattering channel, (ω n , −ω n ) → (ω n ′ , −ω n ′ ) with ω n ′ = ω n ω E . Namely, the odd-frequency superconductivity is realized in the opposite limit of the original BCS theory. The Ginzburg-Landau analysis in the strong-coupling region shows that the specific-heat discontinuity and the slope of the temperature dependence of the superfluid density can be quite small as compared with the BCS values, depending on the ratio of the transition temperature T c and ω c .

Consequences of bulk odd-frequency superconducting states for the classification of Cooper pairs

“…[17][18][19][20][21][22] Diamagnetic odd-frequency pairs could form a homogeneous superconducting ground state. 19,21,22 Although several theoretical papers have suggested possibilities of odd-frequency superconductivity in strongly correlated electron systems [23][24][25] , no clear experimental evidence of odd-frequency superconductivity has been presented so far. It is difficult to resolve directly the frequency symmetry of pair potentials.…”

Mean-field description of odd-frequency superconductivity with staggered ordering vector

“…However, using mean-field-type theory, it has been shown that the second-order phase transition into the OF superconductivity is unstable in general [27,28]. In this case, the Meissner kernel has the sign opposite to that in the ordinary BCS theory.…”

“…From another point of view, it is also discussed within the path-integral approach that the homogeneous OF superconducting state cannot be stable in a Hermitian mean-field Hamiltonian, but can exist if one assumes the non-Hermitian relations for anomalous Green functions [28,36,37]. Another study has demonstrated that the positive Meissner effect is found using the composite-operator description [38,39].…”

Symmetry of superconducting pairing state in a staggered field

“…After that, Balatsky and Abrahams suggested an odd-ω spin-singlet pairing state 14 in a study on high-T C cuprates while, of course, the consensus is now that a conventional even-frequency (even-ω) spin-singlet d-wave pairing state is realized in the high-T C cuprates. A lot of intensive studies on the odd-ω pairing state follow these proposals in both bulk [15][16][17][18][19][20][21][22] and inhomogeneous systems. [23][24][25][26][27] Since there have been predicted many anomalous novel quantum phenomena in odd-ω pairings generated in inhomogeneous systems, [28][29][30][31][32][33][34][35][36][37][38] it is a challenging issue to pursue the possible bulk odd-ω superconductor from various aspects now.…”