Superconductivity in a Correlated E e
Jahn-Teller System
Takada GroupThe competition of electron-phonon (e-ph) andelectron-electron (e-e) interactions in the mechanismof superconductivity is an old issue in strongly
correlated systems and it has been investigated mostly
in a single-orbital system, like the Hubbard-Holsteinmodel in which the e-ph interaction enhances charge
fluctuations, inducing an s-wave superconductivity inthe vicinity of a charge density-wave (CDW) phase,whereas the e-e interaction suppresses such charge
fluctuations but enhances spin ones, leading to ad-wave superconductivity near a spin density-wave(SDW) phase. If the effect of the e-ph interaction isabout the same as that of the e-e interaction, there
appears a rather complex nature of the pairing, namely,
the off-site pairing (leading to either the extendeds-wave or the d-wave nature, depending on the latticestructure) composed of not the bare electrons but the(phonon fully-dressed) polarons [1].
Here we add a further complication to thiscorrelated and strongly phonon-coupled system byincluding the orbital degree of freedom. Morespecifically, we consider a two-dimensional (2D)square lattice with each site made of an E⊗eJahn-Teller (JT) center, namely, a site composed of
doubly degenerate orbitals like the eg orbitals in the dbands which are coupled to the doubly-degenerate JTphonons. At each center, we also consider the e-einteraction in an appropriate way to make this JT
crystal as a prototype of the charge-spin-orbital
complexes. Then the Hamiltonian H of this system isgiven by H=H0+He-e+He-ph, where H0 is thenoninteracting part composed of the electron hopping
term characterized by the nearest-neighbor andnext-nearest-neighbor hopping integrals, t and t’,respectively, with keeping the orbital symmetry andthe degenerate-phonon term with the phonon energy
0. The orbital degree of freedom will be described bypseudospin for analogy to spin degree of freedom andthe pseudospin symmetry is conserved throughout thecrystal in this choice of H0. Other terms, He-e and He-ph,consist of local-site terms written with the intra-orbital
Coulomb interaction U, the Hund’s-rule coupling J,
and the JT coupling g.
Due to the SU(2) symmetry in spin space and theconserved symmetry in pseudospin space, the Cooper
pairing state can be specified by three quantum
numbers; S the total spin of the pair, L the total
pseudospin, and Ly its y component, making it possible
to write the anomalous self-energy as ΔSLLy(k), where k
is a combined notation of crystal momentum k andfermion Matsubara frequency iωn=iπT(2n+1) at
temperature T with an integer n. Because of therotational symmetry around the orbital-y axis, Ly=±1
states are degenerate and thus we treat only either
Ly=0 or 1 here. The group theory determines thetransformation property of ΔSLLy(k) in k space; ittransforms in accordance with Γ, one of the irreducible
representation of the point group C4v (A1, A2, B1, B2,or E). The Pauli exclusion principle dictates thatΔSLLy(k) must be antisymmetric under two-electroninterchange, indicating that Γ must be E for (S,L)
equal to either (0, 0) or (1, 1); otherwise Γ must beeither A1, A2, B1, or B2. With including thistransformation property in Γ, we can easily write down
the Eliashberg equation for ΔSLLy(k) at T=Tc with thepairing interaction VSLLy(q) containing the charge, spin,and orbital susceptibilities c(q), s(q) and o(q), all ofwhich are evaluated in the RPA with use of theirreducible susceptibility 0(q).
In Fig. 1, the phase diagram at T=0.02t is plotted in
the U-g plane for the typical case of t’=0.125t, U=8t,
J=t, and 0=0.10t at half filling. Two boundaries,denoted by LI and LII, indicates the lines where χo(q)and χs(q) diverge, respectively. In the close vicinity ofthese boundaries, those fluctuations are enhanced
strongly enough to make the system enter into various
superconducting phases, each labeled by (Γ;S,L,Ly).
Among them, we find (E;0,0,0) which is a novel chiralp-wave pairing state, px(k)±ipy(k), characterized byspin-singlet, orbital-singlet, and odd-parity inmomentum space. This is a state very specific to thedegenerate multi-orbital system and is induced by thecooperative effects of orbital and spin fluctuations thatare, respectively, enhanced by e-ph and e-einteractions [2].

Fig. 1. Phase diagram in the U-g plane at half filling for
U/8=J=t, t’=0.125t, and Ω0=0.10t at T=0.02t. Units of strengths
are so defined as UM=32/9 0(Q,0) and gM2/Ω0=1/ 0(Q,0),
where Q [=( )] is the momentum maximizing both o(q,0)
and s(q,0).

The conservation of the pseudospin symmetry is
assumed in this study, but it is not always the case. Bysome tentative works, we come to know that theperturbation breaking this conservation will enhance
Tc for the iron pnictides, while it reduces Tc very muchfor the vanadium oxides. This is an issue to be studied
further in the future.
References[1] Y. Takada, J. Phys. Soc. Jpn. 65, 1544 (1996).
[2] C. Hori, H. Maebashi, and Y. Takada, J. Supercond.
Nov. Magn. 25, 1369 (2012).
Authors
C. Hori, H. Maebashi, and Y. Takada