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On the interplay of Jahn-Teller physics and Mott physics in cuprates

The extended two-story house model which is now called the Kamimura-Suwa (K-S) model has clarified how the interplay of Mott physics and Jahn-Teller physics plays an important role in determining the superconducting as well as metallic state of underdoped cuprates. In this paper it is first pointed out for underdoped cuprates that Mott physics leads to the existence of local antiferromagnetic order constructed from the localized spins while that the anti-Jahn-Teller effect as a central issue of Jahn-Teller physics leads to the existence of two kinds of orbitals parallel and perpendicular to a CuO2 plane whose states have nearly the same energy. As a result of the interplay of both physics the K-S model has shown that the exchange interactions between the spins of a localized hole and of a carrier hole play an important role in producing the coexistence of superconductivity and antiferromagnetism in underdoped cuprates. The appearance of d-wave superconductivity even in the phonon-involved mechanism is also shown to be due to the interplay of Jahn-Teller physics and Mott Physics. Brief review of these facts as well as the K-S model is given in this paper. More outstanding result in this paper is that the origin of pseudogap in the deeply underdoped regime has been clarified. In this paper it is shown theoretically for the first time that the so-called T* pseudogap observed in ARPES, STM and tunneling experiments below Tc in underdoped cuprates corresponds to the real transition of photo-excited electrons from the occupied states in the originally conduction band below the superconducting gap to a free-electron state above the vacuum level. Thus we conclude that the T* pseudogap in the underdoped cuprates which increases with decreasing the hole concentration is not ‘pseudo’, but a real gap which exists even below Tc.


Introduction
Undoped copper oxide La 2 CuO 4 is an antiferromagnetic Mott insulator, in which an electron correlation plays an important role. Thus we may say that undoped cuprates are governed by Mott physics. In 1986 Bednorz and Müller discovered high temperature superconductivity in copper oxides by doping hole carriers into La 2 CuO 4 [1]. Their motivation was that higher T c could be achieved for copper oxide materials by combining Jahn-Teller (JT) active Cu ions with the structural complexity of layer-type perovskite oxides. In order to investigate the mechanism of high temperature superconductivity, most of models assumed that the doped holes itinerate through an orbital of B 1g symmetry extended over a CuO 2 plane in the systems consisting of the CuO 6 octahedrons elongated by the JT effect. Those models are called "single-component theory", because the orbitals of hole carriers extend only over a CuO 2 plane. This singlecomponent theory, however, met a serious difficulty that, in the presence of anti-ferromagnetic (AF) order constructed from the localized spins, the hole carriers cannot move smoothly. In order to remove this difficulty, RVB model [2], t-J model [3], etc. were proposed.
In 1989, Kamimura and his coworkers showed by first-principles calculations that the apical oxygen in the CuO 6 octahedrons tend to approach towards Cu 2+ ions, when Sr 2+ ions are substituted for La 3+ ions in LSCO, in order to gain the attractive electrostatic energy in ionic crystals such as cuprates [4,5]. As a result the elongated CuO 6 by the JT effect shrink by doping holes. This deformation against the JT distortion is called "anti-Jahn-Teller effect" [6]. By this effect the energy separation between the two kinds of orbital states, the a * 1g anti-bonding orbital state constructed by Cu dz 2 orbital and six surrounding oxygen p orbitals and the b 1g bonding orbital state consisting mainly of four in-plane p σ orbitals, becomes smaller [7,8]. In figure 1 the spatial extension of the a * 1g anti-bonding orbital and the b 1g bonding orbital is shown. antibonding orbital and | b 1g bonding orbital By taking account of the anti-Jahn-Teller effect Kamimura and Suwa proposed that one must consider the a * 1g and b 1g states equally in forming the metallic state of cuprates, and they constructed a metallic state coexisting with the local AF order. This model is now called "K-S model" [9,10]. Since "anti-Jahn-Teller effect" is the central issue of the Jahn-Teller physics, we may say that the K-S model bears important characteristics born from the interplay of Jahn-Teller Physics and Mott Physics. Since the a * 1g anti-bonding orbital extends not only over the CuO 2 plane but also along the direction perpendicular to it, the K-S model is said to represent a prototype of "two-component theory", in the contrast to the single-component theory.
In this paper we will discuss two important results obtained from the interplay of Jahn-Teller Physics and Mott Physics, based on the K-S model. One is concerned with the origin of the pseudogap in the antinodal region for the underdoped regime. On the basis of the K-S model we will show that the pseudogap in the underdoped regime is not "pseudo", but corresponds to a real transition and that the "Fermi arcs" observed in ARPES should be the inner edges of real small Fermi surfaces in the nodal region. This prediction is shown to be consistent with recent ARPES experimental results reported by Tanaka et al [11].
As the second important result obtained from the interplay of Jahn-Teller Physics and Mott Physics, we will show that the mechanism of superconductivity is caused by the interplay of strong electron-phonon interactions and an AF order. It is shown that (1) the characteristic phase difference of wave functions between up-and down-spin carriers in the presence of the AF order leads to the superconducting gap of d x 2 −y 2 symmetry even in the phonon-involved mechanism, (2) the out-of plane phonon modes in LSCO with tetragonal symmetry contribute to the formation of Cooper pairs while the in-plane modes do not, and (3) the calculated holeconcentration dependences of T c and of the isotope effects for LSCO are consistent with recent experimental results.
The organization of the present paper is the following: It consists of three parts. In the first part (Section 2) we will review briefly the K-S model and the key features of the many-bodyeffects included energy bands and Fermi surfaces. In the second part (Section 3) we will discuss the origin of pseudogap and in the final part (Section 4) we will show how the d-symmetry superconducting gap appears from the K-S model. Section 5 is devoted to conclusion and concluding remarks.
2. Brief review of the Kamimura-Suwa (K-S) model 2.1. Anti-Jahn-Teller Effect When Sr 2+ ions are substituted for La 3+ ions in LSCO, one may think intuitively that apical oxygen in the CuO 6 octahedrons tend to approach toward central Cu 2+ ions in order to gain the attractive electrostatic energy. Theoretically it was shown by the first-principles variational calculations of the spin-density-functional approach [4,5] that the optimized distance between apical O and Cu in LSCO which minimizes the total energy of LSCO decreases with increasing Sr concentration. As a result the elongated CuO 6 octahedrons by the Jahn-Teller (JT) interactions shrink by doping holes. We have called this shrinking effect against the Jahn-Teller distortion "anti-Jahn-Teller effect" [6].

The important role of two kinds of multiplets
By this anti-Jahn-Teller effect, the energy separation between the two kinds of orbital states, the a * 1g anti-bonding orbital state and the b 1g bonding orbital state, becomes smaller when spintriplet Hund's coupling interaction and spin-singlet exchange interaction with a localized spin in the antibonding b * 1g orbital, respectively, are taken into account for the a * 1g and b 1g states. Thus, as a many-electron states both the Hund's coupling spin-triplet and the Zhang-Rice spin-singlet appear at nearly the same energy, as shown in figure 2. Figure 2 shows the energy-level landscape starting from the e g and t 2g orbitals of a Cu 2+ ion in a regular CuO 6 octahedron with octahedral symmetry shown at the left column in the case of LSCO. By the JT effect the Cu e g orbital state splits into a 1g and b 1g orbital states, which form antibonding and bonding cluster orbitals of A 1g and B 1g symmetry with in-plane oxygen p σ and apical oxygen p z orbitals in an elongated CuO 6 cluster, respectively. These cluster orbitals are denoted by a * 1g , a 1g , and b * 1g and b 1g , as shown at the middle column. In an undoped case, 9 electrons occupy these cluster orbitals, so that the highest occupied b * 1g state is half filled, resulting in an S = 1/2 state, where a b * 1g orbital has Cu d x 2 −y 2 character. Following Mott physics, we introduce the Hubbard U interaction (U = 10eV) as a strong electron-correlation effect. Then the b * 1g state splits into the lower and upper Hubbard bands denoted by L.H. and U.H. in the figure, and this gives rise to a localized spin around a Cu site. These localized spins form the antiferromagnetic (AF) order by the superexchange interaction via an intervening O 2− ion in undoped La 2 CuO 4 . Now we dope one hole into this CuO 6 cluster embedded in LSCO material. This means that we take out an electron from this system. In this case, there appear two multiplets, as shown at the right column in figure 2. One case is that a dopant hole occupies an antibonding a * 1g orbital. In this case its spin becomes parallel to a localized spin in the b * 1g orbital to get a spin-triplet state, because the Hund's coupling makes the spins of holes in orbitals with different symmetry  parallel. As a result the energy of the a * 1g orbital state for a single electron increases by the Hund's coupling energy of 0.5 eV. Thus this spin-triplet multiplet is called the "Hund's coupling triplet" denoted by 3 The other case is that a dopant hole occupies a bonding b 1g orbital, and its spin becomes antiparallel to the localized spin in the b * 1g orbital, since the dopant and localized holes occupy the orbitals of the same symmetry. As a result the energy of b 1g orbital state for a single electron increases by the spin-singlet exchange interaction of 3.0 eV. This spin-singlet multiplet corresponds to the "Zhan-Rice singlet" in the t-J model [3], and is denoted by 1 By the first-principles cluster calculations which takes into account the Madelung potential due to all the ions surrounding a CuO 6 cluster in LSCO, Kamimura and Eto showed that the lowest-state energies of these two multiplets are nearly equal when both the Hund's coupling and the spin-singlet exchange interaction are taken into account [7,8]. Consequently the energy difference between the highest occupied orbital states a * 1g in 3 B 1g multiplet and b 1g in 1 A 1g multiplet becomes only 0.1eV for the optimum doping (x = 0.15) in La 2−x Sr x CuO 4 , as seen in the right column of figure 2. Thus, when the two CuO 6 clusters with localized up and down spins are placed at the nearest neighbors, these states are easily mixed by the transfer interaction between a * 1g and b 1g orbitals, which is about 0.3eV for LSCO.

2.
3. An extended two-story house model (The K-S model) By using the calculated results of Kamimura and Eto [7,8] and assuming that the localized spins form an AF order in a spin-correlated region, Kamimura  "an extended two-story house model" [6], which is now called "the K-S model" [10]. The physics of the extended two-story house model is sketched in figure 3. In this figure the first story of a Cu house with (yellow) roof is occupied by the Cu localized spins, which form the AF order in the spin-correlated region by the superexchange interaction. The second story in a Cu house consists of two floors, lower a * 1g floor and upper b 1g floor. These second stories between neighboring Cu houses are connected by Oxygen rooms with (blue-color) roof. In the second story a hole-carrier with up spin enters into the a * 1g floor at the left-hand Cu house due to Hund's coupling with Cu localized up-spin in the first story (Hund's coupling triplet), as shown in the extreme left column of the figure. By the transfer interaction marked by a long (red-color) arrow in the figure, the hole is transferred into the b 1g floor at the neighboring Cu house (the second from the left) through the Oxygen room, where the hole with up spin forms a spin-singlet state with a localized down spin at the second Cu house from the left (Zhang-Rice singlet). The key feature of the K-S model is that the hole-carriers in the underdoped regime of LSCO form a metallic state, by taking the Hund's coupling triplet and the Zhang-Rice singlet alternately in the presence of the local AF order without destroying the AF order, as indicated in the figure. In the extended two-story house model, the second story consists of the two floors of different symmetry. Thus this model has been considered as a prototype of a "two-component scenario theory". In the case of the two-component scenario theory, the AF order is preserved when a hole-carrier itinerates. Thus the important feature of the extended two-story huse model is the coexistence of the AF order and a metallic (superconducting) state in the underdoped regime. This feature in the two-component theory is different from that of the single-component theory.

Effective Hamiltonian for the extended two-story house (K-S) model
The following effective Hamiltonian is introduced in order to describe the extended two-story house (K-S) model by Kamimura and Suwa [9]. It consists of four parts: The effective oneelectron Hamiltonian H eff for a * 1g (a * 1 ) and b 1g (b 1 ) orbital states, the transfer interaction between neighboring CuO 6 octahedrons (CuO 5 pyramids) H tr , the superexchange interaction between where ε m (m = a * 1g (a * 1 ) or b 1g (b 1 )) represents the effective one-electron energy of the a * 1g (a * 1 ) and b 1g (b 1 ) orbital states, C † imσ and C imσ are the creation and annihilation operators of a dopant hole with spin σ in the i-th CuO 6 octahedron (i-th CuO 5 pyramid), respectively, t mn the effective transfer integrals of a dopant hole between m-type and n-type orbitals of neighboring CuO 6 octahedrons (CuO 5 pyramids), J the superexchange interaction between the spins S i and S j of d x 2 −y 2 localized holes in the b * 1g (b * 1 ) orbital at the nearest neighbor Cu sites i and j (J > 0 for AF interaction), and K m the exchange integral for the exchange interaction between the spin of a dopant hole s im and the d x 2 −y 2 localized spin S i in the i-th CuO 6 octahedron (i-th CuO 5 pyramid).
The values of the parameters in Eq. (1) for the case of LSCO are: where the values of Hund's coupling exchange constant K a * 1g and Zhang-Rice exchange constant K b 1g are taken from the first principles cluster calculations for a CuO 6 octahedron in LSCO [7,8], and the energy difference of the effective one-electron energies between a * 1g and b 1g orbital states, ε a * 1g −ε b 1g = −2.6, is determined so as to reproduce the energy difference between the 3 B 1g and 1 A 1g multiplets in LSCO in the MCSCF cluster calculations [8], while those of t mn 's are due to band structure calculations [4,5].

Derivation of an effective one-electron type energy band
Here "effective" is meant by the inclusion of many-body-effects in the one-electron type energy bands. That is, the exchange interactions between carrier's and localized spins are included by the mean field approximation. In order to solve the effective Hamiltonian (1), Kamimura and Ushio [12,13] paid attention to the experimental results by Mason et al [14] and Yamada et al [15] that the spin-correlation length λ s in the underdoped region of LSCO increases as the Sr concentration increases from x = 0.05 in LSCO, the onset of superconductivity, and reaches a value of about 50Å or more at the optimum doping (x = 0.15). In this spin-correlated region, the frustrated spins on its boundary change their directions by the fluctuation effect in the 2D Heisenberg AF spin system during the time of τ s defined by τ s ≡ /J with J being the superexchange interaction (∼0.1 eV). In this case the hole-carriers at the Fermi level can move coherently much longer than the observed spin-correlation length, when the traveling time of a hole-carrier at the Fermi level over an area of the spin-correlation length which is given by τ F ≡ λ s /v F is much longer than τ s , where v F is the Fermi velocity of a hole-carrier at the Fermi level. This is the case for underdoped LSCO, because τ s is of the order of 10 −15 sec while τ F is of the order of 10 −14 sec for the underdoped region of x = 0.10 to x = 0.15 in LSCO. In this way the region of a metallic state becomes much wider than the spin-correlation length so as to reduce the increase of the kinetic energy due to the confinement of hole-carriers in the spincorrelated region. Since it is difficult at present to calculate the length of a metallic region in the underdoped cuprate materials from first-principles calculations, we will determine it empirically in subsection 4.3 of Section 4 entitled "Mechanism of High Temperature Superconductivity", where the length of a metallic region 0 in LSCO will be determined so as for the highest T c value of 40 K in LSCO to appear at the optimum doping concentration x o (= 0.15). Although the K-S model and its effective Hamiltonian do not include disposal parameters to fit experimental values, later we introduce two adjustable parameters in order to reproduce the experimental results of pseudogap and T c .
In the point of view that a metallic region should be much expanded compared with the spincorrelation length based on the arguments previously mentioned, Kamimura and Ushio [12,13] treated the fourth term in Eq. (1), the exchange interaction between the spins of a dopant hole and a localized hole spin, in the mean field approximation by replacing S i by its average value S . Thus the effect of the localized hole spin system is dealt with as an effective magnetic field acting on the hole-carrier. As a result they could separate the localized hole spin system in the AF order and the hole-carrier system and calculated the effective one-electron type energy band for a carrier system assuming a periodic AF order [13].
In figure 4, the calculated many-body effect included energy band structure for up-spin (or down-spin) dopant holes for LSCO is shown for various values of wave-vector k and symmetry points in the AF Brillouin zone, where the AF Brillouin zone is shown in the left hand side of the figure. Here one should note that the energy in this figure is taken for electron-energy but not hole-energy, and the Hubband bands for localized b * 1g holes do not appear in this figure.

Energy[eV]
# " 1 G X Figure 4. The many-body-effect included band-structure for up-spin dopant holes, obtained by solving the effective one-electron-type 34 × 34 dimensional Hamiltonian matrixH for an antiferromagnetic unit cell. The highest occupied band is marked by the #1 band. The ∆-point corresponds to (π/2a, π/2a, 0), while the G 1 -point to (π/a, 0, 0). In this figure the Cu-O-Cu distance, a, is taken to be unity In the undoped La 2 CuO 4 , all the energy bands in figure 4 are occupied fully by electrons so that La 2 CuO 4 is an insulator, consistent with experimental results. In this respect the present effective energy band structure is completely different from the ordinary LDA energy bands [16,17]. When Sr are doped, holes begin to occupy the top of the highest band in figure 4 marked by #1 at ∆ point which corresponds to (π/2a, π/2a, 0) in the AF Brillouin zone. At the onset concentration of superconductivity, the Fermi level is located at the energy of E = 9.04 eV just below the top of the #1 band at ∆, which is a little higher than that of the G 1 point.
Here the G 1 point in the AF Brillouin zone lies at (π/a, 0, 0), and corresponds to a saddle point of the van Hove singularity. From figure 3, we can say that a metallic state coexists with the AF order. In the presence of the local AF order due to the localized hole spins in the CuO 2 layer, the hole-carriers can move freely in the CuO 2 plane by taking the characters of a Zhang-Rice spin-singlet state and a Hund's coupled spin-triplet state alternately between neighboring sites without destroying the AF order. By averaging the character of wavefunction for each wave-vector k over the Fermi surface for x = 0.1 in LSCO, we have obtained the mixing ratio of characters of the Hund's coupling triplet state to the Zhang-Rice singlet state in the metallic state at the Fermi level, which is found to be 1 to 5. Thus the existence ratio of the Hund's coupling triplet state in the metallic state at the Fermi level is only 20 %. This value is close to the observed intensity ratio of the E c to the E ⊥ c polarization of x-ray at the Cu L 3 satellite peak for x = 0.1 in LSCO in the polarized x-ray absorption spectra reported by Chen et al [18].
The most important consequence of this mean-field treatment is that, having taken into account the strong spin exchange interactions in the mean field approximation, the thermal, transport, and paramagnetic properties of the deeply underdoped cuprates can be treated within the framework similar to a single particle band structure. In this case, an assumption of an antiferromagnetic lattice is made, but the exchange interactions between dopant holes and this lattice obtained by their first principles calculations are included in the single particle band structure in a mean field sense .
In 1997 Anisimov, Ezhov and Rice calculated the energy band structure of the ordered alloy La 2 Li 0.5 Cu 0.5 O 4 by the LDA+U method [19], and they showed that a fairly modest reduction of the apical Cu-O bond length is sufficient to stabilize the Hund's coupling spin triplet state with dopant holes in both b 1g and a * 1g orbitals. Their calculated result supports the K-S model.   the K-S model is considered not to hold beyond a certain hole concentration x o . As a result FS will change from small ones to a large one beyond x o . In the case of LSCO we think that x o is 0.2 from experimental results of Loram et al [22] and of Nakano et al [23]. We show a sketch of such large FS in figure 7.

Density of states by the K-S model
Ushio and Kamimura calculated the density of states of the # 1 band in figure 4 for LSCO [13,6]. The calculated density of states is shown as a function of energy in figure 8, where the origin of the energy is taken at the top of the # 1 band at the ∆ point. Reflecting the quasitwo dimensionality in the layer-type structure of cuprates, the density of states rises sharply at the top of the # 1 band and then increases steeply. It has a sharp peak, when energy ε reaches G 1 point with (π, 0, 0) in the AF Brillouin zone, because of a modified type of a saddle point singularity at G 1 point in the AF Brillouin zone. The E F corresponding to x ∼ 0.3 in La 2−x Sr x CuO 4 appears at the G 1 saddle point.

Origin of pseudogap
Recently considerable attention has been paid to a pseudogap called T * in the underdoped regime of cuprates, where T * increases with decreasing the hole concentration x in the underdeoped regime and disappears at a certain concentration x p in the overdoped region. In this section, based on the K-S model we discuss the origin of this pseudogap which we call T * pseudogap hereafter.
Following the K-S model, the FS in the underdoped regime is small and it exists in the nodal region as seen in Figs. 5 and 6. Below T c the hole carriers around the Fermi level in the # 1 conduction band form Cooper pairs, contributing to the formation of a superconducting state, and a superconducting gap appears across the Fermi level. As a result the superconducting Fig_9_2 ( Fig.8)

T < Tc
Below % c , superconducting gap appears. Figure 9. Cartoon picture for expressing the change of states below T c in both the nodal and antinodal regions  still exist, because small FS's have existed only around ∆ point in the nodal region for the deeply underdoped regime. Such situation in the antinodal region is shown schematically in the right-hand inset of figure 9, where one should note that AF state still coexists with a modified energy band structure. Based on figure 9, we can predict the features of ARPES spectra below T c in such a way that a coherent peak due to the superconducting density of states appears at the nodal region around ∆ point while a broad hump appears at the antinodal region, in connection with a real transition of photo-excited electrons from the occupied states, say the state A in Fig 9, below the Fermi level ε F in the # 1 energy band to a free-electron state above the vacuum level. The intensity of photo-excited electrons in the antinodal region is proportional to the density of states at the energy corresponding to the sates A. A picture for explaining the above-mentioned argument is shown in figure 10.  When the electrons occupying the state A are excited by x-ray photons, ARPES experiment can measure an energy difference between the A state with energy ε(A) and Fermi level with ε F , that is |ε(A) − ε F |. In case where the electrons occupying the saddle-point state at G 1 point with (π, 0, 0) are excited, we have calculated the doping dependence of the energy difference |ε(G 1 ) − ε F | in LSCO as a function of the hole concentration. The calculated result is shown in figure 11. This theoretical result for LSCO is compared with the results of ARPES experiments for Bi2212 by Tanaka et al [11], where the observed 3 peak positions relative to node in ARPES experiments in the antinodal region (point 16 in the AF zone in reference [11]) are compared with theoretical results. Here we call the energy difference |ε(G 1 ) − ε F | antinodal gap. Since we can show theoretically that the antinodal gap does not depend on cuprate materials such as LSCO, Bi2212, etc., we can say that agreement between theory and experiment seen in figure 11 is remarkably good. Thus it is concluded that a broad hump observed in the aninodal region in ARPES experiment by Tanaka et al [11] below T c can be ascribed to the real transition of photo-excited electrons from occupied states below the conduction band to the free-electron states above the vacuum level. Further in figure 12 we show the calculated temperature dependence of the energy difference |ε(G 1 ) − ε F | in LSCO in the underdoped regime. From this result we conclude that this energy difference does not depend on temperature in the underdoped regime. This theoretical result coincides well with the experimental results by Norman et al [24], as seen in figure 12. Although our calculated results so far mentioned are non-empirical in the sense of not using disposal parameters to fit experiments, the calculated results shown in figure 12 include two parameters; one corresponds to the transition energy from the electronic phase of a large FS to the antiferromagnetic Mott insulator in the undoped case and the other is the hole concentration x o at which FS changes from a small one to a large one, as defined in subsection 2.6. Here we have taken x o to be 0.2. According to calculated results, the feature that the antinodal gap is almost independent of temperature in the underdoped region is not influenced by the parameter value.
From good agreement between theoretical results in Figs 11 and 12 and experimental results by Tanaka et al and Norman et al, we can say that the concept of T * pseudogap is no longer necessary and that the phenomena observed as " T * pseudogap" are explained by the real gap or the real transitions of photo-excited electrons from occupied states well below the conduction band to the free-electron states above the vacuum level.

Mechanism of high temperature superconductivity 4.1. Spin-dependent electron-phonon interaction
In this subsection we describe that the electron-phonon interaction in the K-S model depends on a spin direction of a hole carrier. Because of the page limitation of the present paper, in this section 4 we describe only a stream of theoretical derivation. Therefore, for those who have interests in the derivation of equations, please read chapters 13 and 14 in our Springer text book entitled "Theory of Copper Oxide Superconductors" [6]. On the basis of the K-S model we have shown that the interplay between electron-phonon interaction and the AF order leads to the d-wave phonon-involved mechanism in LSCO [25,26,6]. As seen in figure 13, the wavefunctions of a hole-carrier with up and down spin on the extended two-story house model have the following phase relation: Ψ k↓ (r) = exp(ik · a)Ψ k↑ (r).
From this relation (2), the electron-phonon interaction matrix elements from states k to k with up spin scattered by phonon with wave vector q, V ↑ (k, k ), has the following spindependent property: where K = k − k − q is a reciprocal lattice vector in the AF Brillouin zone and a is a Cu-O-Cu distance. Since a reciprocal lattice vector in the AF Brillouin zone is expressed as K = (nπ/a, mπ/a, 0) with n + m = even in its component form in the k space, the electronphonon interaction may have a different sign depending on a value of K.

effective inter-hole interaction via phonon
From the relation (3) and K = (nπ/a, mπ/a, 0) with n + m = even, the effective interactions of a pair of holes from (k ↑, −k ↓) to (k ↑, −k ↓) is expressed as Since exp(iK · a) = +1 for n = even and exp(iK · a) = −1 for n = odd, the effective interaction for forming a Cooper pair becomes attractive for n = even while repulsive for n = odd, as shown in figure 14. This remarkable result in the K-S model leads to the superconducting gap of d x 2 −y 2 symmetry. Then a hole is scattered by phonon with wavevector q from A to state C on FS. Since state C is transferred to an equivalent state B by the translation of a reciprocal lattice vector (−Q 1 − Q 2 ), the effective interaction from A to B is attractive according to equation (4) with n = 2. On the other hand, the hole at the state A on FS is also scattered from A to state D by phonon with wavevector q. Since the state D is equivalent to state B by the translation of a reciprocal lattice vector (−Q 1 ), the effective interaction is repulsive by equation (4) with n = m = 1. State C is inside an AF Brillouin zone while state D is outside the AF Brillouin zone, one may say that a scattering process from A to B via C is a Normal scattering while a scattering process from A to B via D is Umklapp scattering. Normal and Umklapp scattering have different sign, so that the effective interaction between holes at state A and B becomes attractive or repulsive depending on the strengths of two scatterings. These strengths vary by depending a wave vector k. As an example of calculated results, we present the calculated results of the k and k dependence of the electron-phonon spectral function α 2 F ↑↓ (Ω, k, k ) with spin-singlet for one of out-of-plane modes, A 1g mode (see the lower left corner of the figure 15) in LSCO with tetragonal symmetry in figure 15, where F ↑↓ (Ω, k, k ) is the momentum-dependent density of phonon states and α 2 is the square of the electron-phonon coupling constant [25,6]. From the upper right corner of figure 15 we can see that the momentum-dependent spectral function varies by taking values with + and − sign in each section of FS with 1 to 4 , when the wave vector k changes from the section 1 of small FS to the section 4 while k is fixed. This is clearly a d-wave behavior. By using this spectral function, we can obtain the k dependence of a gap function. As seen in the lower right corner of figure 15, the obtained d-component gap function ∆(k) vary as a function of (cos(k x a) − cos(k y a)) and the wavefunction of a Cooper pair has a spatial extension of d x 2 −y 2 symmetry.  the spectral function for LSCO with tetragonal symmetry, α 2 F ↑↓ (2) (Ω), is shown in figure 16 as a function of phonon frequency Ω. The phonon modes of LSCO with tetragonal symmetry are classified into in-plane modes and out-of-plane modes with regard to a CuO 2 plane. Figure  16 shows that the out-of-plane modes such as the A 1g mode in figure 15 yield positive spectral function so that they contribute to the formation of Cooper pairs by acting as an attractive force while the in-plane modes yield negative spectral function so that these do not contribute to the formation of Cooper pairs by acting as a repulsive force.   Figure 17. The calculated isotope effect α and T c curve (thick line) for tetragonal LSCO as a function of x 4.3. Calculated results of the hole-concentration dependence of T c and of isotope effect As mentioned above, we have shown that even though the electron-phonon interaction is involved in the mechanism of superconductivity, the interplay of the AF order and phonon mechanism in the K-S model creates superconductivity of "d-wave symmetry", when a system is infinite and homogeneous. Now we will present the calculated results of T c and the isotope effect for LSCO with tetragonal symmetry. Kamimura et al [25,26,6] calculated the hole-concentration dependence of T c and the isotope effects for LSCO using a slightly modified McMillan T c -equation [27], and their calculated results of T c and the isotope effect α are shown in figure 17. The result of isotope effect is compared with experimental results by Crawford et al [28] and Ronay et al [29].
Here the isotope effect α is defined by where M denotes the mass of constituent atoms. In their calculation a well-known relation between the Debye frequency and a mass, Here we should make a remark on the calculated results of T c . On the K-S model the size of a metallic state is finite, because a two-story house model (the K-S model) holds in the spincorrelated region of the AF local order, as shown in figure 3, as we have already mentioned above and also in subsection 2.5. In order to obtain the observed value of 40 K for the highest T c of LSCO at the optimum doping (x = 0.15), Kamimura et al [25,26,6] have chosen the size of a metallic region 0 to be 300Å. The T c curve shown in figure 17 is the result calculated for the case of 300Å.
From the calculated results of the isotope effect shown in figure 17, the following conclusions emerge: The isotope effect on T c in LSCO depends on the hole concentration critically, and the isotope constant α is remarkably large near the onset concentration of superconductivity while it is small around the optimum concentration. Recently Bishop et al [30] published a review article on recent and earlier results on isotope effects in cuprates, pointing out that the isotope effect on T c vanishes at optimum doping but increases with decreasing doping level to be substantially larger than the BCS value (α = 0.5) at the border to the AF state. This experimental result is consistent with the theoretical result obtained from the K-S model shown in figure 17.

Conclusion and concluding remarks
The extended two-story house model (the Kamimura-Suwa (K-S) model) has clarified how the interplay of Mott physics and Jahn-Teller physics plays an important role in determining the superconducting as well as metallic state of underdoped cuprates. In this paper it was first pointed out for underdoped cuprates that Mott physics gives rises to the existence of local antiferromagnetic order due to the localized spins while that the anti-Jahn-Teller effect as a central issue of Jahn-Teller physics produces the existence of two kinds of orbitals parallel and perpendicular to a CuO 2 plane which are energetically nearby. As a result of the interplay of both physics the K-S model has shown the appearance of exchange interactions between the spins of a localized hole and of a carrier hole in underdoped cuprates and that these exchange interactions play an important role in producing the coexistence of superconductivity and antiferromagnetism and the appearance of d-wave superconductivity even in the phonon-involved mechanism. Brief review of these facts as well as the K-S model has been given in this paper. More important original result obtained in this paper was the following: In this paper it was shown theoretically for the first time that the so-called T * pseudogap observed in ARPES, STM and tunneling experiments below T c in underdoped cuprates corresponds to the real transition of photo-excited electrons from the occupied states in the originally conduction band below the superconducting gap to a free-electron state above the vacuum level. In this context we conclude that the concept of T * pseudogap in the underdoped cuprates which increases with decreasing the hole concentration is no longer necessary.
At the end of this paper a remark is made on the small Fermi surface in the underdoped regime of cuprates. Following the extended two-story house model by Kamimura and Suwa [9], Kamimura and Ushio predicted small Fermi surfaces for the underdoped regime of cuprates in 1994 [12]. Anomalous results we have described in the present paper such as the appearance of the real antinodal gap and the occurrence of the phonon-involved d-wave superconductivity are all due to the existence of small Fermi surfaces in the underdoped regime. Finally it should be mentioned that Doiron-Leyraud et al reported recently the observation of a Fermi surface of small pockets in underdoped YBa 2 Cu 3 O 6.5 with T c of 57.5K [31]. This report supports the prediction by Kamimura and Ushio [12]. Theoretically Wen and Lee developed a slave-boson theory for the t-J model at finite doping in 1996 and they showed small Fermi pockets at low doping which continuously evolved into the large Fermi surface at high doping concentrations [32]. Although their theoretical model is different from the extended two-story house model (K-S model), it is interesting to find that they obtained the results similar to the prediction by the K-S model in 1994 with regard to the change from small FS to large FS with increasing the