KINEMATIC SPIN-FLUCTUATION MECHANISM OF HIGH-TEMPERATURE SUPERCONDUCTIVITY
N. M. Plakida'1*, V. S. Oudovenkoh
Joint Institute for Nuclear Research H1980, Dubna, Moscow Region, Russia
b Rutgers University OSS54, New Jersey, USA
Received February 20, 2014
We study d-wave superconductivity in the extended Hubbard model in the strong correlation limit for a large intersite Coulomb repulsion V. We argue that in the Mott-Hubbard regime with two Hubbard subbands, there emerges a new energy scale for the spin-fluctuation coupling of electrons of the order of the electronic kinetic energy W much larger than the exchange energy J. This coupling is induced by the kinematic interaction for the Hubbard operators, which results in the kinematic spin-fluctuation pairing mechanism for V" < W. The theory is based on the Mori projection technique in the equation of motion method for the Green's functions in terms of the Hubbard operators. The doping dependence of the superconductivity temperature Tc is calculated for various values of U and V.
DOI: 10.7868/S0044451014090235
1. INTRODUCTION
Olio of crucial issues in the superconductivity theory is to disclose the mechanism of high-temperature superconductivity (HTSC) in cupratos (see, e.g., fl, 2]). In early studies of the problem, a model of strongly correlated electrons was proposed by Anderson [3], where superconductivity occurs at finite doping in the resonating valence bond state due to the antiferroniag-iiotic (AF) suporoxchango interaction J. However, the intersite Coulomb interaction (CI) V that in cupratos is of the order of J may destroy the resonating valence bond state and superconducting pairing. Recently, a competition of the intersite CI V and pairing induced by the on-site CI U in the Hubbard model [4] or by the intersite CI V was actively discussed. In particular, it was stressed in [5] that a contribution from the repulsive well-screened weak CI in the first order strongly suppresses the pairing induced by contributions of higher orders, and a possibility of superconductivity "from repulsion" was questioned. Using the renormalization group method, the extended Hubbard model with CI V was studied in [6], where superconducting pairing of various symmetries, extended h-,
E-mail: plakida'ffltheor.jinr.ru
and (1-wave typos was found depending on the electron concentration and V. Following the original idea of Kohn Luttinger [7], it was shown in [8] that the />-wave superconductivity exists in the electronic gas at low density with a strong repulsion U and a relatively strong intersite CI V (also see [9] and the references therein). Studies of the phase diagram within the extended Hubbard model in the weak correlation limit have shown that superconducting pairing of different types of symmetry, s, p, dry, and dx2_y2, can occur depending on the CI between the nearest 11 and the next V2 neighbor sites and electron hopping parameters between distant sites in a broad region of electron concentration [10].
However, the Fermi-liquid model was considered in the weak correlation limit U < W in these investigations, while cupratos are Mott Hubbard (more accurately, charge-transfer) doped insulators, where a theory of strongly correlated electronic systems should be applied for U > W. Hero, W ~ At is the electronic kinetic energy for the two-dimensional Hubbard model with the nearest-neighbor hopping parameter t. In the limit of strong correlations, various numerical methods for finite clusters are commonly used. There are many investigations of the conventional Hubbard model (see, e.g., fll 14]), but only a few studies of the extended Hubbard model in which the intersite CI V is taken
into account. In particular, in Rofs. [15 17], the extended Hubbard model was considered in a broad region of U and V. The results in Rofs. [15, 16] show that a strong on-site repulsion U effectively enhances the d-wave pairing, which is preserved for large values of V J. In Ref. [17], using the slave-boson representation, it was found that superconductivity is destroyed at a small value of V = J. We discuss these results in more detail in Sec. 4.3 by comparing them with our findings.
In our recent paper [18], we studied the extended Hubbard model in the limit of strong correlations by-taking the CI V and electron phonon coupling into account. It was found that the high-Tc d-wave pairing is mediated by the strong kinematic interaction of electrons with spin fluctuations. Contributions coming from a weak CI V and phonons turned out to be small since only / = 2 harmonics of the interactions make a contribution to the d-wave pairing.
In this paper, we consider superconductivity in the two-dimensional extended Hubbard model with a large intersite Coulomb repulsion V in the limit of strong correlations to elucidate the spin-fluctuation mechanism of high-temperature superconductivity. We argue that in the two-subband regime for the Hubbard model with U > 6t, a spin electron kinematic interaction results from complicated commutation relations for the Hubbard operators (HOs) [19]. This interaction leads to the weak exchange interaction J = 4t2/U due to interband hopping, and at the same time intraband hopping results in a much stronger kinematic interaction g8j ~ W ,J of electrons with spin excitations. Therefore, the exchange interaction J is not so important for the spin-fluctuation pairing driven by the strong kinematic interaction g8f. We calculate the doping dependence of the superconducting Tc for various values of U and V and show that as long as V does not exceed the kinematic interaction, V < M", the d-wave pairing is preserved. In calculations, we use the Mori-typo projection technique [20] in the equati-on-of-niotion method for thermodynamic Green's functions (GFs) [21] expressed in terms of the HOs. The self-energy in the Dyson equation is calculated in the self-consistent Born approximation (SCBA) as in our previous publications [18,22].
In Sec. 2, the two-subband extended Hubbard model is introduced and equations for the GFs in the Nanibu representation are derived. A self-consistent system of equations for GFs and the self-energy is formulated in Sec. 3. Results and discussion are presented in Sec. 4. Concluding remarks are given in Sec. 5.
2. GENERAL FORMULATION 2.1. Extended Hubbard model
We consider the extended Hubbard model on a square lattice,
11 E '~ I' E Ni +
i=£j7cr i
(1)
i i^j
where are the single-electron hopping parameters, u\a and Uur are the Fermi creation and annihilation operators for electrons with spin <r/2 (<r = ±1 = = (f4), <? = —a) on the lattice site i, U is the on-site CI, and the \ ]j is the intersite CI. Furthermore, Ar, = = ATicr, ATicr = "i,T"„T is the number operator and //. is the chemical potential.
In the strong correlation limit, the model describes the Mott Hubbard insulating state at half-filling (n = (ATi) = 1) when the conduction band splits into two Hubbard subbands. In this case, the Fermi operators u\a and in (1) fail to describe single-particle electron excitations in the system and the Fermi-liquid picture becomes inapplicable to cuprates. The projected-type operators, the HOs, referring to the two subbands, singly occuped a|ff(l — Ni9) and doubly-occupied ul^Nof, must be introduced. In terms of the HOs, model (1) becomes
11 xr + E Af2 + \ E KjNiNj +
i,cr i i=£j
+ E Uj {A'fA'f + X;,T.\f +
+ <r(Af Af + H.c.)}, (2)
where ei = —//. is the single-particle energy- and e-2 = = U — 211. is the two-particle energy. The matrix HO Xf^ = |m){//?| describes transition from the state |/,/?) to the state \i, a) on a lattice site t taking four possible states for holes into account: an empty state (a, 3 = 0), a singly- occupied hole state (a, 3 = a), and a doubly-occupied hole state (a, 3 = 2). The number operator and the spin operators are defined in terms of the HOs
Ni = j2xr + 2x!2> (3)
<7
57 = A7ff, 57 = (<T/2)[Xf - Afff]. (4)
The chemical potential //. is determined from the equation for the average occupation number for holes
n= ! + <■> = (Ni),
(5)
where (...) denotes the statistical average with Haniil-tonian (2).
The HOs obey the completeness relation A'f0 + + Xf + Xf2 = 1, which rigorously preserves the constraint that only one quantum state a can be occupied on any lattice site i. The commutation relations for the HOs
=6
where { 1. 13} = AD + DA, A(t) = .'"'A. . and 9(x) = 1 for x > 0 and 9(x) = 0 for x < 0. The Fourier representation in the (k,u>)-space is defined by the relations
OO
Gyv(f-f') = ^ j tttoxi>[-i(t-t')]Gijir(u}), (11)
] =%(^Afrf±4aA7'j), (6) Gijtr(u>) = - ^oxp[/k-(i-j)]G(T(k,u;). (12)
with the upper sign for Fermi-type operators (such as A'p) and the lower sign for Boso-typo operators (such as Ni in (3) or the spin operators in (4)), result in the so-called kinematic interaction. To demonstrate this, we consider the equation of motion for the HO Xf2 = iiiiiTiillTnl,T in the Hoisonborg representation (h=l):
ijtXf = [Xf,H] = (u-fi+^Vu Nrj Xf2 +
+ {j^itTtr' AT " _ ° Bicro' Af^ —
/,£r'
-YtUiX^iXf + aX?*), (7) i
where , are the Boso-typo operators
i(T(T'
->22 c v-22
r>22 _ ( y22 | y17171 X I Y0"0" Â - —
^ u7(T' --' > (T (T I i U(Tf(T -
= (Ni/2 + <TSi)6trl(r + S?6trllt, (8)
D2^, = (N/2 ■
■trS'i)6„
S; dry
(9)
We see that the hopping amplitudes depend on the number operator in (3) and spin operators (4), which results in the kinematic interaction describing effective scattering of electrons on spin and charge fluctuations. In phenomonological models for cuprates, a dynamical coupling of electrons with spin and charge fluctuations is introduced specified by fitting parameters, while the interaction in Eq. (7) is determined by the hopping en-orgy tij fixed by the electronic dispersion.
2.2. Green's functions
To consider the superconducting pairing in model (2), we introduce the two-time thermodynamic GF [21] expressed in terms of the four-component Nanibu op-orators ,Y„T and Xl = (X2,TXf0
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