Effective t-J model of pairing: singlet versus triplet

The t-J model is regarded as a canonical model of spin-singlet pairing induced by the {\em kinetic exchange interaction} responsible also for an antiferromagnetic ordering in the strongly correlated narrow-band systems. In the orbitally degenerate systems both ferromagnetic and antiferromagnetic kinetic exchange interactions occur. I review briefly the analogy between the singlet and triplet types of pairing, as well as draw some general conclusions about the pairing induced by these exchange interactions. The general discussion is also illustrated with a concrete case of a two-dimensional lattice with the spin triplet pairing.


I. INTRODUCTION
The t-J model contains a very attractive idea of the spin-singlet pairing induced by electron correlations and taking place in the high-temperature superconducting cuprates [1,2], although some questions remain whether the actual pairing should include explicitly also the hybrid 2p − 3d pairing induced by the interactions of the Kondo-type [3,4]. What is fundamental in this approach is the crucial role of a strong Coulomb repulsion between electrons in these narrow-band systems, which leads to the antiferromagnetic intersite interaction, yielding in turn the effective pairing in real space. Even the effect of the atomic disorder leading to an enhancement of the pairing on local scale can be accounted for semiquantitatively [5]. The question of the pseudogap appearance in the underdoped systems is still under debate, but it may be related to the residual antiferromagnetic correlations in the strongly disordered medium, induced by both the doping and the correlations of charge and spin type (stripes). In general, the nature of the normal state is understood to a lesser extent than the condensed superconducting state in the two spatial dimensions. The role of the third dimension (coupling between the planes), as well as that of electron-lattice coupling, remains still at a further distance.
On the other hand, the first unambiguous defection of spin-triplet superconductivity in the ferromagnetic phase [6,7] did not spark corresponding modeling of the spin-triplet pairing based on an analogous approach, as it deserved (see however [8,9]). The systems such as UGe 2 or URhGe may not be as strongly correlated as the cuprates or heavy fermions, but still it is important to explore the generalized t-J model for orbitally degenerate systems. This topic will be overviewed briefly here. In connection with this one should mention the just discovered Fe-pnictide superconductors, which are orbitally degenerate systems, so they may be good candidates for a singlet-triplet superconducting transition as a function of composition (i.e. concentration of 3d electrons). The presence of such a transition would provide an unambiguous evidence for the role of kinetic exchange interactions in real-space pairing in orbitally degenerate narrow-band systems. The purpose of this paper is first to comment of two nonstandard features of the t-J model and then, to turn attention to an effective t-J model for orbitally degenerate systems.

II. T-J MODEL
The t-J model with inclusion of antiferromagnetic kinetic exchange and the three-site hopping has been derived over 3 decades ago [9,10]. Starting from the Hubbard model it has the following form for electron moving in the lower Hubbard band The first term represents the hopping between the neighbors i and j (t ij = 0 for i = j only), the second the so-called kinetic exchange interaction with the fermionic representation of the spins: Possible hopping process in the lower Hubbard subband: a) virtual hoping between singly occupied sites that leads to an antiferromagnetic (kinetic) exchange interaction between the neighboring sites; b) two-site hopping between empty and singly occupied sites (single-particle hopping); and c) between the singly occupied and empty sites via a singly occupied site.
The whole dynamics can be expressed in the closed form by introducing projected fermionic operators: and then the Hamiltonian takes the following closed form: As the projected operators {b † iσ } and {b iσ } obey non-fermion anticommutation relations from one side and the hopping term does not commute with the second order (∼ t 2 /U ) part, the dynamics of this model is highly nontrivial. Physically, this is because the kinetic-energy part vanishes in the limit of the Mott insulator (< ν i >= 1).
An important comment can be made at this point. Namely, in the low-dimensional systems the Coulomb interaction screening is less effective (particularly for small number of holes in the Mott insulator). Therefore, the intersite interaction part can be regarded as important, particularly near the Mott-insulator limit. In deriving the effective Hamiltonian the coupling constant for the second-order terms is then ∼ t ij t jk /(U − K ij ), which means that the effective spin-spin interaction becomes stronger. This is not so good for the real space pairing, as one has to add to the effective Hamiltonian (3) the intersite repulsion (1/2) K ij ν i ν j and if K ij > 4t 2 ij /(U − K ij ), then the spin-spin coupling is not sufficient to produce the attractive interaction, i.e. to produce bound pairs in the ordinary sense. To see this more explicitly, we introduce the intersite spin-singlet pairing operators as follows and hence (3) takes the most compact form The destructive role of the intersite Coulomb repulsion is usually ignored (see also below). However, taken literally, it simply can preclude any bound-state formation associated with the real-space pairing, i.e. when < B ij > = 0. To overcome this intersite repulsion a local lattice-distortion induced attraction can play an important role. An elementary argument for the influence of the distortion on the effective attractive interaction is briefly analyzed below [10,11].

III. BOND DISTORTION AND THE EFFECTIVE ATTRACTIVE INTERACTION IN T-J MODEL
In the narrow-band limit the hopping integral diminishes roughly exponentially with the increasing inter-atomic distance R ij . On the contrary, in this asymptotic limit the intersite long-range Coulomb repulsion can be approximated to the first order by its classical expression K ij ≈ e 2 /R ij . Therefore the change of the system energy δH under the influence of the classical bond distortion δR ij is mainly influenced by the change of the intersite Coulomb energy via (δK ij /δR ij ) · δR ij , i.e. as We can write down the total system energy with inclusion of the lattice distortion in the form (we omit three-site terms for simplicity): where W ij ≃ K ij ≃ e 2 /(2R ij ), γ 1 and γ 2 are the diagonal and off-diagonal elastic constants, respectively. Minimizing the energy with respect to classical bond distortions δR ij , one obtains [11] the following effective Hamiltonian including both strong correlations and the attractive interaction induced by the particle motion (e.g. holes in the Mott insulator). Assuming that we have an isotropic distortion accompanying the hole motion in the planar case, we obtain the following generalized t-J Hamiltonian (up to an irrelevant term): or equivalently,H where J ij expresses the sum in the bracket in (8). This is the spin-dependent version of an attractive density-density interaction [12], which still expresses the complementary character of antiferromagnetic spin-spin correlations and the intersite singlet pairing (note that ν iσ ν jσ = b † iσ b † jσ b jσ b iσ ). The essence of the above argument is as follows. The inclusion of a bond distortion associated with the particle density-density correlation (< ν i ν j > = 0) results in an effective t-J model of the type (9). The bond-distortion contribution can be large as the ratio of the bare intersite Coulomb interaction (W ij ) to γ 1 R 2 ij can be large and therefore, together with the kinetic-exchange contribution (∼ t 2 ij ), can overcome the intersite repulsion ∼ K ij between the particles with the opposite spins. In this manner, the t-J of model looks as a model of real-space pairing in a single narrow band, in which the Cooper pairs are formed as bound states in an attractive potential. In the absence of the bond distortion, the Cooper-pair and the condensed states are formed purely by the electronic correlations induced by the repulsive Coulomb interactions.
The magnetism of strongly correlated electrons started with pioneering introduction of Hubbard model by Anderson [13,14] in the context of antiferromagnetic ordering of transition-metal oxides. In this treatment the role of the filled 2p states due to oxygen (O 2− ) is passive, i.e. to mediate the effect of d − d interaction between the magnetic ions. The role of oxygen is not only passive in e.g. metal-insulator transition [15]. The question is what happens if the state is metallic and the p bands becomes partially filled and the p − d hybridization starts playing an active role in the dynamics in the metallic phase. The passive role of the p electrons has been assumed by Zhang and Rice [16] in their intuitive justification of the t-J model by assuming that the Kondo-like (Zhang-Rice) singlet is so tightly bound that the p − d antiferromagnetic correlations lead to effective d − d interactions. The situation seems very likely for the cuprates (see however e.g. [17]). However, it is certainly not the case for the heavy fermions, where the hybridization between the strongly correlated 4f states (of e.g. Ce 3+ ion) and the 5d − 6s valence states destabilizes the 4f localized moments, in effect leads to the heavy-fermion behavior, and can become the source of the hybrid pairing at the same time [8,10]. Parenthetically, the situation may be different in heavy-fermion uranium compounds such as UPt 3 , where the U 4+ (approximately 5f 2 ) configuration may lead, in conjunction with the Hund's rule coupling among 5f electrons, to the triplet pairing. This type of situation will be considered in the next section.

V. REAL-SPACE PAIRING IN THE DOUBLY DEGENERATE BAND
A. The physical situation and the model As we have seen in the two foregoing Sections, the real space pairing in the orbitally nondegenerate case leads, even in the two-band situation, to the singlet correlations and the corresponding bound Cooper pairs [19]. We discuss now the situation in the orbitally degenerate Hubbard model and include the Hund's rule coupling directly, as well as highlight its role in the strong-correlation limit, where both ferro-and antiferro-magnetic kinetic exchange interactions appear at a proper band filling. The purpose of this discussion is to show that the real-space spin-triplet pairing is theoretically as feasible as is the singlet pairing in the single-band case.
The starting Hamiltonian for such a two-band case has the following form where l, l ′ = 1, 2 are the orbital indices, the first and the second terms are respectively the hopping and the Hubbard parts, whereas the last three terms describe the interorbital Coulomb repulsion, the ferromagnetic Hund's rule and the pair-electron-hopping terms, respectively (both have the same amplitude J H ). Without a loss of generality, one may assume that U ′ = U − 2J H which holds at least for 3de g orbitals. To simplify the discussion, we assume that the hopping is diagonal and the same, i.e. t ll ′ ij = t ij δ ll ′ to avoid the complications associated with the interorbital hopping part (with l = l ′ ) and the corresponding exchange.
The model in the strong-correlation limit U ≫ J H ≫ |t ij | leads to ferromagnetic Mott insulator (with possible orbital ordering) for quarter-filled band (n = 1) and to the ordinary antiferromagnetic Mott insulator at half filling (n = 2) [20][21][22]. The reason for ferromagnetism near n ∼ 1 is due to the fact, that the dominant virtual hopping process between two neighboring sites takes place to the local spin-triplet state in the intermediate state. This intermediate on-site spin-triplet state does not appear in an orbitally nondegenerate model. On the other hand, for the case of two electron per atom (n = 2) only the virtual-hopping processes leading to the spin-singlet pair configuration in the intermediate state are allowed, so the kinetic exchange interaction is antiferromagnetic, in a direct analogy to that in a single-band case. This gradual transition from paramagnetism through ferromagnetism to antiferromagnetism has been observed in the series of isostructural compounds FeS 2 -CoS 2 -NiS 2 and their mixtures, as then a doubly degenerate band of e g symmetry is being filled when we go from FeS 2 (n = 0), through CoS 2 (n = 1), to NiS 2 (n = 2). At an intermediate composition 1 < n < 2 a spin-glass-like state appears and is caused by the competing ferro-and antiferro-magnetic interactions [23].

B. Spin-triplet paired states
The model of the real-space pairing is built in a direct correspondence to the theory of kinetic exchange. We consider the case of identical orbitals, for which the nearest hopping integrals are the same t ≡ t <ij> . The relevant virtual hopping processes are depicted in Fig. 2. The effective t-J-J H model takes the form for n ≤ 1 [24] The pairing operators with −1, 0, +1 are the projected interorbital pairing operators for the spin-triplet states with S z = 1, −1, and 0, respectively, i.e.
The operators C are the spin-singlet pairing operators which we call the orbital pairing operators. This because they are defined as follows The pairing operators C † ijm are both intraorbital (m = ±1) and interorbital (m = 0) character. We see that the spin-triplet term is dominant for n ≤ 1. One should also note, that the intersite projected pairing operators (12) replace their intraatomic correspondant [25,26], which appear in the limit of low and intermediate correlations and are related then directly to the intraatomic ferromagnetic Hund's rule coupling. Therefore, our work shows that the Hund's rule coupling can lead not only to ferromagnetism (e.g. when the Stoner criterion is fulfilled), but also to spin-triplet superconductivity in both moderate-and strong-correlation regimes. It is tempting to propose that the spin-triplet superconductivity may appear i pure samples of e.g. CoS 2 under pressure, when ferromagnetism is suppressed. The question of a coexistence of the spin-triplet superconductivity and ferromagnetism is a separate issue.

VI. PHASE DIAGRAM OF SPIN-TRIPLET PAIRED STATES
Finally, we provide an overall phase diagram of the spin-triplet phases and leave the details to a separate paper [27]. We consider a two-dimensional square lattice case with the isotropic gap (∆ m = ∆). The superconducting order parameter is chosen in the form ∆ mk = ∆ f cos k x + e iα cos k y .
In the slave-boson language, developed first for t-J model [28] this fermionic gap amplitude ∆ f is multiplied additionally by the slave boson occupancy ∆ b =< b † i b j >≈ b 2 , and the superconducting state is achieved when both the fermionic parameter ∆ f ∼< B † ijm > and the bosonic amplitude ∆ b are simultaneously different from zero. The solutions then are of extended s-wave type (if α = 0), s + id type (when α = π/2), and of d type (if α = π). In general, when α = (0, π/2, π) the state is called s − d mixed state. The phase diagram representing the stable phases in the ground state is shown in Fig. 3 on the plane δ ≡ 1 − n versus J H /U . The other parameter is |t|/U = 0.2. For δ = 0 the system is a ferromagnetic insulator, so the paired states evolve from a ferromagnetic Mott insulator . Note the presence of a usual "dome" of d-wave superconductivity, here located in the interval 0.04 ≤ δ ≤ 0.13. In the regime of low electron concentration δ ≥ 0.20 an ordinary gapless (s-wave) type of pairing takes place. It should be underlined that in order to determine stability of the phases we have to solve the system of equations for ∆ f , ∆ b , and the chemical potential µ, for given phase factor α. The details of this cumbersome analysis will be discussed elsewhere [27].

VII. OUTLOOK
In this brief overview we have first discussed the influence of the classical bond distortion in deriving an effective form of the spin-singlet pairing for strongly correlated electrons in a nondegenerate band. The inclusion of the lattice allows to draw a distinction between the pairing achieved by the effective attractive interaction and that solely by the correlations induced by the purely repulsive Coulomb interaction. The role of hybrid 2p − 3d or 5d − 4f correlations in deriving respectively the effective spin-singlet pairing for high temperature and heavy-fermion systems is noted in passing. The main message of our paper is though to indicate, that an analogous type of pairing can take place for electrons in orbitally degenerate and strongly correlated systems. Namely, we have discussed the hole pairing in the doped ferromagnetic Mott insulator , with the doping δ = 1 − n > 0. It would be interesting to see if such situation is