Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity

The two-dimensional layers of cuprate oxides are known to be the systems of strongly repulsive (correlated) electrons as the Mott insulators which have revealed various novel physical properties uniquely different from the conventional low temperature superconductors. They show the antiferromagnetic (AF) infinite-range or long-range order (AFLRO) at and near half-filling. As hole doping increases, the AFLRO diminishes and the short-range (finite-range) AF order takes over with the emergence of d-wave superconductivity. The two-dimensional systems of strongly correlated electrons involved with strong repulsive interactions may favor the spin singlet pairing order (or correlations) of d-wave symmetry over that of s-wave symmetry. Here the spin singlet paring correlations are concerned with the AF spin fluctuations of the shortest possible correlation length among the AF spin fluctuations of all possible correlation lengths which appear in the region of hole doping away from half-filling. In this region of hole doping the cuprate oxides exhibit the novel structure of the high TC phase diagram characterized by the dome-shaped superconducting transition temperature, TC and the monotonously decreasing pseudogap temperature, T∗.

Soon after this study we [5] proposed an improved slave-boson theory which fundamentally differs from these approaches in that a term involving coupling between the spin and charge degrees of freedom or simply spin-charge coupling appears in our rigorous slave-boson treatment of the t-J Hamiltonian. The resulting effective mean field Lagrangian reveals coupling between the spin (spinon) paring order, Δ f and the charge (holon) pairing order, Δ b . As a consequence the Cooper pairing order is satisfactorily seen to be a composite of these two order parameters, Δ f and Δ b to allow for the bose condensation of the Cooper pairs rather than the single-holon bose condensation or the double-holon bose condensation. Accordingly this theory has led to successful reproductions of not only the monotonously decreasing spin gap temperature but also the long-waited dome-shaped structure of the superconducting transition temperature in the phase diagram. Further other important physical observations such as the boomerang behavior of superfluid weight, the peak-dip-hump structure of optical conductivity and both the temperature and doping dependence of spectral functions are reproduced in agreement with observations [6].
For the sake of self-containment we will first review our earlier proposed slave-boson theory [5] of the t-J Hamiltonian which reveals the spin-charge coupling mentioned above. Earlier it was shown by others that inclusion of the t ′ term in the t-J Hamiltonian leads to satisfactory descriptions of the electronic structure of high T C cuprates [7][8][9][10][11] and the enhancement of pairing correlation resulting in an increasing trend of T C in the overdoped region in the phase diagram for the choice of t ′ /t < 0, e.g., t ′ /t = −0.3 [12,13]. It is, thus, of great interest to see how its inclusion affects the entire structure of the phase diagram which includes the pseudogap temperature. At present there has been no study which addresses the role of the diagonal hopping t ′ on the spin gap temperature, T * .S u c h study is needed to find whether there exists any relation between T * and T C or the spin gap phase and the superconducting phase. In this regard we would like to draw attention to the fact that the observed phase diagrams of high T C cuprate samples (e.g., LSCO and BSCCO samples) reveal that higher the T * , higher the T C as earlier discussed by Oda et al. [14] This suggests that the two energy or temperature scales, T * and T C are no longer independent of each other. Thus one of our main objectives is to study how the pseudogap or spin gap temperature, T * and the superconducting transition temperature, T C are correlated and show that such correlation arises owing to the presence of the short-range antiferromagnetic (AF) spin fluctuations of the shortest possible correlation length involved with the spin pairing correlations. For a concerted, self-consistent study, we use a predicted phase diagram to calculate magnetic susceptibility and discuss two important observations made by the inelastic neutron scattering (INS) measurements, namely the temperature dependence of the magnetic resonance peak [17] and the linear scaling behavior between the magnetic peak resonance energy, E res and the superconducting transition temperature [18]. From this study we show that the short-range AF spin fluctuations are directly responsible for the magnetic susceptibility observed by the INS measurements mentioned here.

Theory: U(1) slave boson representation of the t-J Hamiltonian
In the present study we limit ourselves to the derivation of the U(1) slave boson representation of the t-J Hamiltonian. We refer details of its derivation to Appendix A. In Appendix B a brief exposure of the SU(2) approach is made in association with the U(1) representation. Here only a rudimentary description is presented by introducing the next-nearest neighbor hopping or diagonal hopping t ′ term into the t − J Hamiltonian. It is given by, Here ∑ <i,j> denotes summation over the nearest neighbor sites i and j, ∑ <i,j> , the summation over the next-nearest neighbor (diagonal) sites andc iσ (c † iσ ), the electron annihilation(creation) operator with the constraint of no double occupancy at each site i. t is the nearest neighbor hopping integral; t ′ , the next-nearest neighbor hopping integral and J, the Heisenberg coupling constant.
We take the slave-boson representation of electron operator as a composite of spinon ( f )and holon (b), that is, c iσ = f iσ b † i with the single occupancy constraint at each site i. Following a rigorous slave-boson treatment of S i = 1 2 ∑ αβ c † iα σ αβ c iβ with σ αβ , the Pauli spin matrix in the above equation, the resulting U(1) slave-boson representation of the above t − t ′ − J Hamiltonian is given by Here λ i is the Lagrange multiplier field which enforces the single occupancy constraint.
Taking proper Hubbard-Stratonovich transformations and associated algebras by closely following our recently proposed slave-boson theory [5] (see Appendix A for details), one obtains the following effective Lagrangian, for the spin spinon sector and ( 6 ) for the charge (holon) sector. Here μ f (μ b ) is the spinon(holon) chemical potential. χ ij is the hopping order parameter and Δ , the spinon (holon) pairing order parameter; x, the hole doping concentration and J x = J(1 − x) 2 .Th ela stt ermof Eq. 6 reveals the presence of coupling between the spin (spinon) and charge (holon) degrees of freedom, i.e., simply termed as spin-charge coupling, as seen in the form of the product of the spin (spinon) single pairing order, Δ f and the charge (holon) pairing order, Δ b . Thanks to this coupling the Cooper pairing order Δ is, now, properly represented as a composite of these two order parameters, Δ f and Δ b . We point out that the holon (charge) sector, Eq. 6 is coupled with the spinon (spin) sector, Eq. 5 owing to the presence of coupling between the spinon paring order Δ f and the holon pairing order Δ b asshowninthelasttermofEq.6.
The resulting free energy (see derivation in Appendix A) is given by, where , the spinon(holon) quasiparticle energy; x, the hole doping rate;J x = J(1 − x) 2 and N, the total number of sites in a square lattice. Here the spinon and holon energies, ǫ f k and ǫ b k are respectively,  The contribution of the next-nearest neighbor hopping or the diagonal hopping is readily understood from the inspection of Eq.8 by noting that the value of cos k x cos k y is negative at the hot spot (π, 0), zero at the cold spot (π/2, π/2) and positive at (0, 0). From this we see that stabilization (destabilization) of the spinon energy at the hot spot with t ′ < 0(t ′ > 0) is expected to lead to the enhancement (depression) of AF spin (spinon) pairing correlations or the spin singlet pairing order of d-wave symmetry compared to the case of t ′ = 0),t h a ti s , no diagonal hoping. The charge (holon) pairing of s-wave symmetry will be enhanced at the nodal points.

Role of next-nearest neighbor hopping on the structure of phase diagram
Here we explore the role of the next-nearest neighbor hopping, i.e., the diagonal hopping on both the pseudogap temperatureT * and the superconducting transition temperature, T C and the cause of correlation between these two temperature scales or relatedly the spin gap phase and the d-wave superconducting phase. Earlier the negative value of t ′ was shown to match well the observed Fermi surface of the hole doped cuprate oxides while its positive value matches that of the electron doped cuprate oxides [7] as mentioned above.
Choosing the two different cases of the diagonal hopping, one for t ′ < 0(e.g.,t ′ = −0.3t)and the other for t ′ > 0(e.g.,t ′ = 0.3t), we examine the dependence of the phase diagram on t ′ for the hole doped cuprate oxides.
On the other hand, in the overdoped region both T * and T C are seen to simultaneously increase (decrease) for the case of t ′ /t < 0( t ′ /t > 0) with reference to that of t ′ /t = 0. The predicted superconducting transition temperature at optimal doping concentration did not change appreciably despite the considerable variation of t ′ /t as shown in the figure. The simultaneous increase (decrease) of T * and indicates that the two temperature scales, T * and T C or the spin gap phase and the superconducting phase are interrelated. To see the cause of such interplay, below we probe the role of the short-range AF spin fluctuations or the spin paring correlations on the determination of the phase diagram.
For the case of t ′ < 0( t ′ > 0) the spinon energy at the hot spot(π, 0) is lowered (raised) with reference that of t ′ = 0, i.e., no diagonal hopping, as can be readily understood from Eq. 8. Thus the spin (spinon) pairing correlation at the hot spot for t ′ < 0 is energetically more stable than the case for t ′ > 0. It is to be recalled that the spin gap temperature is the temperature at which the spin singlet paring order or (correlations) of d-wave symmetry or the spin pairing correlations emerges. The spin paring correlations will be less prone to change in the underdoped region compared to the case of the overdoped region. This is because owing to lower hole concentrations in this region, chances of electron hopping from site to site are reduced and, consequently, the existing short-range AF order is not easily perturbed. Thus the spin paring correlations or the short-range antiferromagnetic spin fluctuations is expected to remain more robust in the underdoped region compared to the case of the overdoped region. Indeed, the predicted T * and T C is shown to be sensitive to the variation of t ′ , preferentially in the overdoped region. This is displayed in Fig. 1.
It is reminded that the Cooper pairing order can be seen as the composite of the spin (spinon) pairing order Δ f and the charge (holon) pairing order Δ b , which results from the presence of the spin-charge coupling shown in the last term of Eq. 6. As a result of the coupling between the two orders, the superconducting phase transition with its onset temperature, T C may arise owing the short-range AF spin fluctuations involved with the formation of the spin pairing order (correlations) which initiates the onset of the spin gap temperature T * . To put it otherwise, owing to the spin-charge coupling both T * and T C are simultaneously affected or correlated. Indeed, such simultaneous change is seen to appear by exhibiting the simultaneous increase (decrease) of both T * and T C with t ′ /t < 0(t ′ /t > 0) as J increases. Such trend is seen in Fig. 2. Our findings of both the enhancement of the spin pairing correlations and the increasing trend of the superconducting transition temperature for t ′ = −0.3t which appear in the overdoped region agree well with the variational Monte Carlo, mean-field calculations of Lee and coworkers [13]. However, unlike our present study they did not show a study of the spin gap temperature concerned with the role of the spin paring correlations.
For further verification from a different angle we closely examine the predicted structural dependence of the phase diagram on J in Fig.2. Both T * and T C are predicted to simultaneously increase with J as shown in Fig. 2. Needless to say, spin pairing correlations should increase with J. This will, in turn, cause the simultaneous increase of both the spin gap temperature and the superconducting transition temperature with increasing J. Such simultaneous increase with J is predicted as shown in the figure. This clearly demonstrates that the the superconducting transition temperature and the pseudogap temperature or relatedly the spin gap phase and the d-wave superconducting phase are correlated via the spin pairing correlations or the AF spin fluctuations of the shortest possible correlation length. To put it otherwise, the short-range AF spin fluctuations play a key role of causing such inseparable relation between the two temperature scales or relatedly the spin gap phase and the superconducting phase. This finding is consistent with the observed phase diagrams with different cuprate samples which shows higher the T * , higher the T C [14] as mentioned earlier.
It is then assured that the superconducting phase transition will not arise in the absence of the spin gap phase below T * which is initiated by the short-range AF spin fluctuations involved with the spin paring correlations

Magnetic susceptibility based on the U(1) slave-boson representation
The observed high T c phase diagrams of cuprate oxides are characterized by the pseudogap or spin gap phase which exists below the monotonously decreasing T * and the d-wave superconducting phase below the dome shaped T C [15,16]. From their inelastic neutron scattering measurements (INS) of the temperature dependence of magnetic resonance peaks for YB 2 Cu 3 O 6+x (YBCO) Dai et al. [17] reported that the magnetic resonance begins to appear at the pseudogap temperature T * as its onset temperature and continues to exist with an increasing trend of the resonance peak height in the underdoped region as temperature is lowered and that near the optimal doping T ⋆ tends to get closer to T c . On the other hand, He et al. observed from their INS measurements of the doping dependence of the resonance peak energy, E res for Bi 2 Sr 2 CaCu 2 O 8+δ (BSCCO) that in the underdoped region E res increases with increasing hole concentration x up to optimal doping x 0 , and that in the overdoped cuprates it decreases with increasing x, by exhibiting a linear scaling behavior of E res with T c at all hole concentrations [18]. Most recently Stock et al. [19] observed that spin waves decay above the pseudogap of a heavily underdoped YBCO. Using a time-of-flight neutron spectroscopy for the studies of dynamic spin correlations or spin fluctuations in the overdoped La 2−x Sr x CuO 4 (LSCO) sample, Wakimoto et al. [20] showed from their study of the doping dependence of antiferromagnetic spin excitations that the excitations decrease with hole doping above the optimal doping of La 2−x Sr x CuO 4 (LSCO) and disappear at x = 0.3. Here we discuss the magnetic susceptibility [21,22] at the wave vector Q =( π, π) in association with our computed phase diagram and focus our attention to the observed linear scaling behavior of magnetic resonance peak energy E res with the superconducting transition temperature T c .For self-containment we first discuss the U(1) slave-boson representation of irreducible magnetic susceptibility for our calculations of magnetic susceptibility.
Allowing external magnetic field h, we introduce into the effective Lagrangian L eff above the Zeeman coupling term,Ĥ where in the U(1) slave boson representation, The associated free energy is formally, where β = 1/kT and the partition function, Converting the magnetic (spin) susceptibility, into its four momentum space ( q, ω)expression and allowing isotropic response to the applied magnetic field, the RPA form of magnetic susceptibility is obtained to be, [23] χ( q, w)= χ 0 ( q, w) where J( q)=2J(cos q x + cos q y ) and χ 0 ( q, ω) is the irreducible magnetic susceptibility given by where the quasi-spinon energy is /(e βE f k + 1).Inthecomplete expression of the effective Lagrangian Eq. 3, interplay between the two sectors, one for the spinon (spin) sector and the other for holon (charge) sector, namely Eq. 5 and Eq. 6 appears owing to the presence of coupling between the spinon pairing order and holon pairing order as shown in the last term of Eq. 6. Thus it should be noted that the effect of coupling between the two order parameter is embedded in the expression of the above irreducible magnetic susceptibility, Eq. 15, including the effect of the nearest neighbor hopping.

Computed results of magnetic susceptibility
Earlier, with the neglect of the next-nearest neighbor (or diagonal) hopping t ′ term we were able to obtain the generic feature of the dome shaped superconducting transition temperature and the monotonously decreasing pseudogap temperature in the phase diagram [5] in agreement with observaions [15,16]. Now with the inclusion of the diagonal hopping term, such generic feature is, still, well predicted in the computed result of the phase diagram as shown in Fig. 1 and Fig. 2. As a concerted study we use the predicted phase diagram shown in Fig. 3 to calculate the magnetic spin susceptibility of present interest. As in our earlier study of the magnetic susceptibility [21,22], we take the negative value [7,13] of the next-nearest neighbor hopping integral with the choice of t ′ = −0.45 (to conform with the study of Brinckmann and Lee [23]) in the t − t ′ − J Hamiltonian of interest [21] .
In Fig. 4 we display the variation of magnetic susceptibility at (π, π) with temperature T and transfer energy E at a fixed hole doping, x = 0.05. The magnetic resonance peak is shown to  decrease with increasing temperature and disappears at the onset temperature T * .I ts h o w sa steady decrease of the resonance peak peak height with increasing temperature and eventual disappearance at T * in agreement with observation [17]. This indicates that the short-range AF spin fluctuations involved with the spin pairing correlations or the spin (spinon) singlet pairing order disappears at the onset temperature, T * .
He et al. [18] showed from their INS measurements of Bi 2 Sr 2 CaCu 2 O 8+δ (BSCCO) that in the underdoped cuprates the magnetic (spin) resonance peak energy E res (or ω res )i n c r e a s e s with T c showing a linear scaling behavior between the two energy scales, E res and T c , i.e., E res /T c ≃ const. In Fig. 5 we show that the predicted E res with J eff = αJ (where α = 0.4 [23]) monotonously increases with increasing T c , yielding a linear scaling behavior of E res /T c ≃ const. This predicted linear scaling behavior is in agreement with the observations made by He et al. [18]. We note some quantitative differences between the observed value (around 5) and the predicted one (around 3).

Summary
In this study we applied the recently proposed slave-boson theory [5] in which the spin (spinon) paring order and the charge (holon) pairing order are coupled to result in the generic feature of the dome-shaped superconducting transition temperature and the monotonously decreasing spin gap temperature in the phase diagram. From the present study with the inclusion of the diagonal hopping t ′ term we also found that such generic feature still holds, as shown in Fig. 1 through Fig. 3. Further we showed that there exists correlation (or interplay) between the two different temperature scales, T * and T C , resulting in the increasing T C with increasing T * . Relatedly, it can be said that the superconducting phase is correlated with the spin gap phase. We find that such correlation between the two phases is attributed to the short-range AF spin fluctuations involved with spin pairing correlations. The simultaneous increase of the superconducting transition temperature with the spin gap temperature with incrasing J is shown to be consistent with the observed phase diagrams for high T C cuprate samples (e.g., LSCO and BSCCO samples) [15] which shows that the higher T * samples always accompany higher T C . In addition, to achieve a self-consistent, concerted study we used the predicted phase diagram to study the magnetic susceptibility. Specifically, resorting to the computed phase diagram shown in Fig. 3 we found that both the temperature dependence of the magnetic resonace peak and the linear scaling behavior of the magnetic (spin) resonance peak energy E res with the superconducting transition temperature T c agree with the INS measurements [17,18]. We showed that this linear scaling behavior is attributed to the short-range AF spin fluctuations. Although not discussed here, such linear relation is found to be invariant with the Heisenberg coupling constant [22], implying high T C cuprate sample independence. In short, based on the above concerted studies of both the phase diagram and the magnetic susceptibility we find that the short-range (spin dimer) AF spin fluctuations of the shortest possible correlation length involved with the spin pairing correlations are responsible for high T C superconductivity. We argue that this finding is supported by the reproducibility of both the dome-shaped superconducting transition temperature, T C in the phase diagram and the linear scaling behavior between E res and T C ,inbothofwhichtheT C and thus the superconducting phase transition is shown to occur as a result of the short-range AF spin fluctuations in association with the spin-charge coupling.

Acknowledgement
One(SHSS) of us greatly acknowledges the Korean Ministry of Education (HakJin Program) and Pohang University of Science and Technology for financial supports at the initial stage of the present work. He thanks the Korean Academy of Science and Technology for continuous encouragements in science research. We are grateful to Professor Ki-Seok Kim at Pohang University of Science and Technology for his earlier contribution.

Appendix A: Heisenberg interaction term in the U(1) slave-boson representation
The t-J Hamiltonian of interest is given by, and the Heisenberg interaction term is rewritten Here t is the hopping energy and S i , the electron spin operator at site i, S i = 1 2 c † iα σ αβ c iβ with σ αβ , the Pauli spin matrix element. n i is the electron number operator at site i, n i = c † iσ c iσ . μ is the chemical potential.
In the U(1) slave-boson representation [1,2,24,25], with single occupancy constraint at site i the electron annihilation operator c iσ is taken as a composite operator of the spinon (neutrally charged fermion) annihilation operator f iσ and the holon (positively charged boson) creation operator b † i ,andthus,c iσ = f iσ b † i . Rigorously speaking, it should be noted that the expression c iσ = b † i f iσ is not precise since these operators belong to different Hilbert spaces and thus the equality sign here should be taken only as a symbol for mapping. Using c iσ = f iσ b † i and introducing the Lagrange multiplier term (the last term in Eq.(A3)) to enforce single occupancy constraint, the t-J Hamiltonian is rewritten, with the Heisenberg interaction term, The first term represents hopping of a spinon from site j to site i and of a holon (positively charged boson) from site i to site j. In the slave-boson representation a charged fermion (electron or hole) is taken as a composite particle of a 'spinon' and a 'holon'. They can conveniently serve as book-keeping labels to discern physical properties or objects involved with the charge or spin degree of freedom (e.g., spin gap phase, spin singlet pairs, hole pairs, ...). With the single occupancy constraint, electron is allowed to hop from a singly occupied copper site i only to a vacant copper site j. A site of single occupancy in the CuO 2 plane of high T c cuprates physically represents an electrically neutral site (net charge 0) with an electron of spin 1/2 and the vacant site, a site of positive charge +e w i t hn e ts p i n0 . I nt h e slave-boson representation, hopping of an electron (a composite of spinon and holon) from a singly occupied copper site (neutral site) j to an empty site (positively charged site with +e) i results in the annihilation of a spinon (a fermion of charge 0 and spin 1/2) and the creation of a positively charged holon (a boson of charge +e and spin 0) at site j while at the copper site i a composite of a spinon (fermion of charge 0 and spin 1/2) and a negatively charged holon is created. It is of note that as a result of electron hopping the newly occupied copper site i in the CuO 2 plane can, also, be labeled as 'spinon' since this is an electrically neutral (charge 0) site with an electron of spin 1/2 and the vacant site j, 'holon' since this is a positively charged site with a single charge +e and the net spin 0 as mentioned above. Thus in practical sense, there is no distinction between the two different cases above. At times, we will call the singly occupied site as 'spinon' and the vacant (empty) site as 'holon' as long as there is no confusion. This is because any site occupied by a spinon is identified as an electrically neutral site occupied by a single electron with spin 1/2 and the site with a positive holon is a positively charged vacant site with spin 0. Thus physical spin-charge separation is not allowed.
The Heisenberg interaction term, Eq.(A4) shows coupling between the charge and spin degrees of freedom. Physics involved with the charge degree of freedom is manifested by the four holon (boson) operator b i b j b † j b † i in the Heisenberg interaction term. Judging from the intersite charge coupling term J 4 n i n j present in the Heisenberg interaction term H J = J ∑ <i,j> (S i · S j − 1 4 n i n j ), it is obvious that this charge contribution can not be neglected in its slave-boson representation. It is to be noted that the Hubbard Hamiltonian contains repulsive interaction U between charged particles and is mapped into the t-J Hamiltonian H t−J in the large U limit. The Coulomb repulsion, Un i↑ n i↓ = U 4 (n i↑ + n i↓ ) 2 − U 4 (n i↑ − n i↓ ) 2 obviously manifests the presence of both the charge (the first term) and spin (the second term) degrees of freedom. Thus, under mapping the charge part of contribution naturally appears in the Heisenberg interaction term.
Let us now take another look at the importance of the charge contribution. In general, uncertainty principle between the number density (amplitude ) and the phase of a boson order parameter applies. As an example, arbitrarily large fluctuations of the number density fix the phase, or arbitrarily large phase fluctuations fix the number density of the boson. The conventional BCS superconductors of long coherence length meet the former classification, and thus the phase fluctuations of the Cooper pair order parameter are minimal. For charged bosons, e.g., the Cooper pairs, the number density fluctuations refer to charge density fluctuations. For short coherence length superconductors such as the high T c cuprate systems of present interest, local charge density fluctuations exist and cause large phase fluctuations. Thus, both the charge and phase fluctuations need to be taken into account to fully exploit the quantum fluctuations .
Let us now consider the importance of the charge and spin fluctuations. In generally, coupling between physical quantities A and B is decomposed into terms involving fluctuations of A, i.e., (A− < A >) and B, i.e., (B− < B >), separately uncorrelated mean field contribution of < A > and < B > and correlation between fluctuations of A and B,thatis,(A− < A >) and for spin (spinon) contribution, the Heisenberg coupling term, Eq(A4) can be decomposed into terms involving coupling between the charge and spin fluctuations separately, the mean field contributions and coupling (correlation) between fluctuations (charge and spin fluctuations). Using such decomposition of the Heisenberg interaction term for Eq.(A3), we write the partition function,

U(1) mean field Hamiltonian
Noting that [b i , b † j ]=δ ij for boson, the intersite charge (holon) interaction term (the second term) in Eq.(A7) is rewritten, the spinon pairing field. The third term in Eq.(A7) represents the intersite spin (spinon) interaction and is rewritten, x,theuniform hole doping concentration [27]. The fourth term in Eq.(A7) is written, The intersite spinon interaction term in Eq.(A9) is decomposed into the direct, exchange and pairing channels [25], where E f ks is the quasispinon excitation energy, with the spinon pairing energy (gap), Δ f 0 = J p ξ k (τ f )Δ f ,a n dE b ks is the quasiholon excitation energy, where the holon pairing energy, with γ k =( cos k x + cos k y ) and ϕ k =( cos k x − cos k y ). ∑ ′ denotes the summation over momentum k in the half reduced Brillouin zone, and s =+ 1a n d−1 represent the upper and lower energy bands of quasiparticles respectively. Here α ks↑ (α † ks↑ ) and α ks↓ (α † ks↓ ) are the annihilation(creation) operators of spinon quasiparticles of spin up and spin down respectively, and β ks (β † ks ), the annihilation(creation) operators of holon quasiparticles of spin 0. ǫ f ks and ǫ b ks are the kinetic energies for spinons and holons respectively. The minus sign (−Δ 2 ) in the expression of the holon quasiparticle energy (ǫ − μ) 2 − Δ 2 arises as a consequence of the Bose Einstein statistics [28]. From the diagonalized Hamiltonian Eq.(A24), we calculate the total free energy.
Rewriting Eq.(A24) as the partition function is derived to be, Using the above expression, the total free energy is given by The set of uniform phase (θ = 0) for the hopping order parameter, d-wave symmetry (τ f = π/2) for the spinon pairing order parameter and s-wave symmetry (τ b = 0) for the holon pairing order parameter is found to yield a stable saddle point energy for both the underdoped and overdoped regions. There is another set of order parameters which yield the same energy as the above one; 2π-flux phase (θ = π/2) for the hopping order parameter, s-wave symmetry (τ f = 0) for the spinon pairing order parameter and d-wave symmetry (τ b = π/2) for the holon pairing order parameter. In both cases, the d-wave symmetry of the electron or hole (not holon) pairs occurs as a composite of the d-wave (s-wave) symmetry of spinon pairs and s-wave (d-wave) symmetry of holon pairs. Only at very low doping near half filling, the flux phase [25] becomes more stable. Thus, the phase of the order parameters of present interest are θ = 0, τ f = π/2 and τ b = 0. Then the d-wave symmetry of the electron or hole (not holon) pairs is a composite of the d-wave symmetry of spinon pairs and s-wave symmetry of holon pairs. Minimizing the free energy with respect to the amplitudes of the order parameters χ, Δ b and Δ f , we obtain the self-consistent equations for the order parameters,