$d_{x^2-y^2}$-wave density wave and $d_{x^2-y^2}$-wave superconducting gap on the extended Hubbard model on a square lattice

The extended Hubbard model with a nearest-neighbor Coulomb repulsion on the square lattice is studied to obtain insight into the phase diagram of cuprate high $T_c$ superconductors (HTS). To pursue the hidden-order scenario proposed in [S. Chakravarty et al., Phys. Rev. B 63, 094503 (2001)], we derive an effective Hamiltonian by using the canonical transformation and develop a mean-field theory. The calculated phase diagrams are qualitatively consistent with the experimental phase diagrams of HTS, and we thus conclude that the pseudogap can be interpreted as the order parameter of the $d_{x^2-y^2}$-wave density wave (DDW) state, and the $d_{x^2-y^2}$-wave superconducting (DSC) rises based on the DDW order. Furthermore, the analytical representation of the density of states is obtained and, near the optimal doping of the DSC, the van Hove singular point of the density of states is located at the Fermi level.

The extended Hubbard model with a nearest-neighbor Coulomb repulsion on the square lattice is studied to obtain insight into the phase diagram of cuprate high Tc superconductors (HTS). To pursue the hidden-order scenario proposed in [S. Chakravarty et al., Phys. Rev. B 63, 094503 (2001)], we derive an effective Hamiltonian by using the canonical transformation and develop a mean-field theory. The calculated phase diagrams are qualitatively consistent with the experimental phase diagrams of HTS, and we thus conclude that the pseudogap can be interpreted as the order parameter of the d x 2 −y 2 -wave density wave (DDW) state, and the d x 2 −y 2 -wave superconducting (DSC) rises based on the DDW order. Furthermore, the analytical representation of the density of states is obtained and, near the optimal doping of the DSC, the van Hove singular point of the density of states is located at the Fermi level.

I. INTRODUCTION
Since cuprate high T c superconductors (HTS) were discovered in 1986 1 , the mechanism of HTS has been one of the most unsolved issues in condensed matter physics. [2][3][4][5][6] In this study, we examine the square-lattice extended Hubbard Hamiltonian with the nearest-neighbor (NN) Coulomb repulsion to obtain the physical properties of HTS. The reasons for introducing this Hamiltonian are as follows.
The second reason is as follows. The pseudogap phase, 35 which have been one of the most incomprehensible parts in the phase diagrams of HTS, 4,36-39 is observed in the phase diagrams of HTS, and the d x 2 −y 2 -wave density wave (DDW) has been considered one of the strong candidates for explaining the pseudogap phase. 27,[40][41][42][43][44][45] S. Chakravarty et al. proposed that this pseudogap is characterized by a hidden broken symmetry of d x 2 −y 2 -type, which is DDW order, and is an actual gap in the one particle excitation spectrum. 40 S. Chakravarty et al. also indicated that the point-contact tunneling measurement shows the considerable size of the tunneling gap, the d x 2 −y 2 -wave-like gap persists above the superconducting T c , and many other experiments are consistent with the presence of DDW. 40,46 Therefore, considering the aforementioned, we assume that the pseudogap derives from the DDW order. Finally, we derive the effective Hamiltonian using the canonical transformation to discuss the d x 2 −y 2 -wave superconducting (DSC) condensation in DDW metal.
Previously, based on the CDW region on the extended Hubbard Hamiltonian with NN Coulomb repulsion V and using the canonical transformation, when the hole is doped and the system deviates from the half filling, we found that d xy -wave superconductivity coexists with the CDW order. 26 In this study, considering the AF region, we investigate whether DSC condensation occurs even in DDW instead of CDW metal. Our obtained phase diagrams prove to be qualitatively consistent with the experimental phase diagrams of HTS 47 and show that a metal-metal quantum transition point exists under the DSC dome, which had been proposed by S. Chakravarty et al. 40 The remainder of this paper is organized as follows. In Sect. II, the extended Hubbard Hamiltonian is defined and the mean-field effective Hamiltonian is derived using the canonical transformation. In Sect. III, we introduce the DDW order parameters and consider the DDW phase in the underdoped region. In addition, we obtain the selfconsistent equations for the DDW order parameters. In Sect. IV, we study the DSC phase by introducing the DSC order parameters, and then derive the self-consistent equations for these parameters. In Sect. V, we obtain the phase diagrams by numerically solving the derived self-consistent equations. Furthermore, we obtain the Fermi surface and density of states, and then discuss the physical properties of the DDW state. In Sect. VI, we summarize the results of this study.

II. HAMILTONIAN
We begin with the Hamiltonian, defined as: where is the electron number operator at the ith site, c † i,σ (c i,σ ) is the electron creation (annihilation) operator, and n i = n i,↑ + n i,↓ . The summation runs over the square lattice and i, j is the NN pair of sites. We divide the square lattice into the A and B sublattices.
For the half-filled case, we have two types of regions. One is the AF region, which is composed of up and down spins in the A and B sublattices, respectively, as shown in FIG. 1(a). Another is the CDW region, which is composed of two-electrons-occupied sites in the A sublattice and vacant sites in the B sublattice, as shown in FIG. 1(b  The arrows represent the electrons whose spin direction is that of the arrow. The sublattice composed from (a) up spins for the AF region and (b) two-electronsoccupied sites for the CDW region is called the A sublattice. By contrast, the sublattice composed from (a) down spins for the AF region and (b) no occupied sites for the CDW region is called the B sublattices.
(CDW) region in the case of U > 4V (U < 4V ). The mother substances of HTS show an AF ground state. Therefore, in this study, we consider the AF region of U > 4V .
As soon as holes are doped in the mother substances, the electrons begin to transfer from site to site. Therefore, we introduce the Hamiltonian of the transition of the electrons as: where δ = ±e x , ±e y are NN vectors, and we assume that t ≪ U throughout this study.
To describe the AF region explicitly, we introduce hole creation and annihilation operators and then rewrite c † i,↑ → b i , c i,↓ → a i for i ∈ A sublattice and c † l,↓ → d l , c l,↑ → c l for l ∈ B sublattice. Hereafter, for convenience, we refer to these notations as hole representations. Substituting the hole representation into Eqs. (1) and (2), we obtain and where Here, we divide H 1 into two parts written as: with where H 1 (2, 0)|AF = 0 is satisfied and H 1 (2, 0) cannot perturb the AF region. By contrast, H 1 (1)|AF = 0 is satisfied and H 1 (1) can change the AF region. Note that the numbers in parentheses in Eqs. (6) and (7) where S satisfies From Eq. (9), we obtain: We obtain S in the hole representation by neglecting those terms consisting of the product of more than four Fermi operators, such as n a i n b i n c l and a i b i n c l n d l , because n is a small quantity in the underdoped region (see Appendix A). Finally, S is written as: where Therefore, substituting Eq. (11) into Eq. (8), the effective Hamiltonian is written as: which is adopted as an equation up to the first order of S. In addition, the other terms, which are a second or greater order of S, are omitted because these consist of the product of more than four operators. We calculate Eq. (20) by using the approximations (I) and (II), described as follows.
(I) We neglect the terms consisting of the product of more than four operators, which were already used in the process of deriving Eq. (11).
(II) The hopping terms that are farther than the NN lattice points, such as a † i a i+ex+ey , are neglected because we expect that the short-range interactions yield more essential effects compared with the long-range interactions. For the same reason, next NN interactions, such as n a i n a i+δ+η , and the terms with three or more different types of suffixes, such as b i c i+δ n d i+η and a i b i d j c l , are all neglected. Therefore, we consider the interactions only between the NN lattice points.
We obtain the effective Hamiltonian after the straightforward calculations (see Appendix B).
n a n b = n δ + n a n a n c n c n d = n δ + n c where the quantities b i c i+δ and a i d i+δ can be represented by c † i,↑ c i+δ,↑ and c i,↓ c † i+δ,↓ , respectively, in the electron representations. These quantities refer to the charge densities. Furthermore, we omit terms such as , which give rise to the Cooper pair of holes, and terms such as The point contact tunneling experiment shows that the d x 2 −y 2 -wave-like tunneling gap is observed in the normal state above the superconducting state. 46 Therefore, we introduce the DDW order parameters as: and By using the mean-field approximation, we obtain the mean-field Hamiltonian of Eqs. (23), (25), and (26) as: where we use n a = n c and n b = n d . From Eqs. (35) and (40)-(42), we finally obtain the mean-field total Hamiltonian in the DDW phase: where Furthermore, performing the Fourier transformation of a i , b i , c l , and d l : and using the symmetry relation ǫ a = ǫ c and ǫ b = ǫ d , we obtain: where with µ 1 = µ 2 . From Eq. (52), we derive self-consistent equations for ∆ and ∆ ′ : where and k and two types of solutions, which are given by: In Appendix C, we describe in detail the derivation of the self-consistent equations.
In the case of ∆ = ∆ ′ , the DDW of electrons with up spin is in phase with that of electrons with down spin. Therefore, we can detect the DDW. From Eqs. (60) or (61), and using Eq. (53), we obtain the magnitude of the DDW as: where where x = V /U . In the AF region (i.e., x < 1/4), v dw is positive definite. The integrand of Eq. (66) is also positive definite. Therefore, Eq. (66) may have a nontrivial solution ∆ dw = 0. By contrast, in the case of ∆ = −∆ ′ , the DDW of electrons with up spin is out of phase with that of electrons with down spin. Therefore, we cannot detect this type of DDW even if it exists. The magnitude of this type of DDW is given by: and Eq. (72) is obtained by substituting Eqs. (53) and (65) into Eq. (62). Furthermore, Eq. (71) is rewritten as: .
) and the integrand of Eq. (70) is always positive, Eq. (70) has no solutions and we have ∆ = −∆ ′ = 0 when U/t < 8.64. Fortunately, many substances of HTS satisfy this condition. Thus, we need not consider the case of ∆ = −∆ ′ . Therefore, in the following, we consider only the case of ∆ = ∆ ′ = i∆ dw .
We must obtain the ground state energy to examine whether the solution of ∆ = ∆ ′ is stable. From Eq. (43), we obtain: where We note that Eq. (75) depends only on n δ because n a = 1−n δ 2 and n b = 1+n δ 2 . It is determined only by Eq. (76) whether the DDW is stable. As previously mentioned, we obtained v dw > 0 and ∆ dw = 0. Therefore, we can conclude that the DDW is always stable.
We derive the equation for obtaining the DDW order parameter ∆ dw in the nonmagnetic phase (m = 0), as later it is proved that the magnetic phase is allowed only at n δ = 0 in our approximation. From Eqs. (C.11) and (C.18), we obtain: where we have utilized Eqs. (53), (62), (64), and (C.33). Therefore, from Eqs. (77) and (78), the hole concentration n δ is obtained by: From Eqs. (57) and (58), we derive: where v m = U + 16t(2I 0 − 3I 1 − 3I 2 ), and we used Eqs. (27)- (31) and (34). Furthermore, from Eqs. (78)-(80), the magnetization m is self-consistently given by: and the Neel temperature T N , which is defined by m = 0, is given by: where and Eq. (83) Finally, we conduct the diagonalization of Eq. (52) and write as: where Substituting Eq. (85) into Eq. (52) and replacing we obtain and omit the constant term. Furthermore, the canonical transformation is applied: where Substituting these transformation into Eq. (89), we finally obtain the diagonal mean-field Hamiltonian: In the nonmagnetic phase, we set m = 0 and obtain: where The superconducting dome of HTS is located in the pseudogap phase, which is considered the DDW phase in the phase diagram. 40 Therefore, we consider the superconductivity with the DDW order parameters. Furthermore, the experiments show that the Cooper pair is the bound state of holes and the superconductivity is a d x 2 −y 2 type in the nonmagnetic phase. 47-49 Therefore, we consider the DSC phase, which is established based on the DDW order parameters. First, we introduce the DSC order parameters: and Next, we obtain the DSC part of the Hamiltonian. Performing the mean-field approximation in the Hamiltonian Eq. (21) by using Eqs. (99) and (100), the mean-field Hamiltonian of the DSC part is written as: Furthermore, performing the Fourier transformation, we can rewrite Eq. (101) as: where Therefore, we obtain the total Hamiltonian by adding the previous Hamiltonian Eq. (52) to Hamiltonian Eq. (102): where we have utilized Eqs. (85), (86), and (88). The experimental results suggest that superconductivity is established on the nonmagnetic phase. 47 Therefore, H MF tot with m = 0 is the Hamiltonian of the DSC phase, which coexists with the DDW phase. Therefore, substituting m = 0 (ǫ a + ǫ b = 0) into Eqs. (94) and (95), θ k = −π/4 is obtained and Eqs. (90)-(93) are written as: Using Eqs. (99),(100), and (107)-(110), we obtain: and assume that two types of solutions are possible. One of them is given by ∆ ds = ∆ ′ ds and the other is ∆ ds = −∆ ′ ds . When ∆ ds = ∆ ′ ds , Appendix D shows that this case can be permitted only in the region with 4V < U < , that is, roughly speaking, 0 < V /t < 1 and U/t < 4. Therefore, when 4 < U/t, considering this solution is not necessary.
As discussed in the following, we consider only the ∆ ds = −∆ ′ ds case. Therefore, from Eqs. (103) and (104), we obtain: Substituting m = 0 and Eqs. (107)-(110) and (113) into Eq. (106), then, H MF tot with m = 0 can be considered as the Hamiltonian of the DSC phase, which is written as: where Here, we have treated Λ as a real number because its phase can be absorbed into operators, and have omitted the constant term. Furthermore, the canonical transformation is applied: where Substituting these transformation into Eq. (114), we finally obtain the diagonal mean-field Hamiltonian: Note that, from Eqs. (111) and (112), our assumption ∆ ds = −∆ ′ ds is equivalent to the relation C † k D k = A † k B k . However, from Eqs. (116)-(119) and (122), we find that C † k D k = A † k B k = 0, which satisfies the aforementioned relation. This result means that ∆ ds = −∆ ′ ds is self-consistently satisfied. Substituting Eq. (116)-(119) into Eq. (111), the self-consistent equation for ∆ ds is given as: where Because the integrand of Eq. (123) is always positive, the condition for Eq. (123) to have a solution is v ds < 0, which is written as . This means that t 2 mainly contributes to the emergence of the DSC. Furthermore, using Eqs. (12), (14), and (30), we can rewrite Eq. (124) as:

V. RESULTS AND DISCUSSION
In this section, we derive the phase diagrams by solving the self-consistent equations numerically. Furthermore, we obtain the Fermi surface, the density of states, and discuss the physical properties in the DDW state.

A. Phase diagrams
We numerically solve the self-consistent Eqs. (66) and (123) to obtain the DDW and DSC order parameters (∆ dw and ∆ ds ) in the nonmagnetic phase, and herein describe the phase diagrams. For convenience sake, we can write the concrete form of the self-consistent equations as follows.
From Eqs. (66) and (80), ∆ dw are obtained by: where The integration is carried out over the first Brillouin zone of the square lattice, and ǫ F = (ǫ a − ǫ b )/4 is determined to be the effective chemical potential. Furthermore, from Eq. (81), the magnetizations are obtained by and from (79), the hole concentration n δ is written as: First, we consider when n δ = 0 at a zero temperature, where the AF ground state is expected. In this case, we have f (E − k,1 ) = 1, f (E + k,1 ) = 0 for all k, and ǫ F = 0 because of E − k,1 < E + k,1 . Therefore, Eq. (128) is rewritten as: and we notice that Eq. (130) is satisfied only in the case of ∆ dw = 0 and m = 1. Thus, we find that the mother substance of HTS is the AF and that it does not have ∆ dw . This result is consistent with our first assumption that the ground state of the system at zero doping is the AF region. However, m = 1 indicates that no quantum fluctuation exist, which may have been caused by neglecting the interaction terms that include more than three sites. However, we do not discuss more details about this discrepancy. Next, we considered when n δ = 0 and m = 0, which are satisfied by Eqs. (126) and (128). Because the integrands of Eqs. (126) and (128) are both positive definite, and (cos k x − cos k y ) 2 ≤ 4, we have: However, we find that the requirement to satisfy Eq. (131) is U < 4V , which is not our first assumption of 4V < U . This contradiction has come from the assumption of m = 0 when Eq. (128) is derived. Therefore, we conclude that m = 0 for any n δ = 0 because Eq. In FIG. 3, we show the calculated ground-state phase diagrams of cases (a) and (b), where we set the inverse temperature to β = 15. Note that, if the value of β is taken as larger, a fine structure appears around the phase boundaries. However, we do not discuss this detail. To focus on investigating qualitatively the structure of the phase diagrams, we regard the obtained results at β = 15 as those of the ground state. When n δ = 0, we have already obtained ∆ dw = 0 and m = 1. However, as soon as we start doping holes, the DDW order parameters ∆ dw emerge and decrease from the maximum value ∆ dw ≃ 0.12 at n δ = +0 for the two cases. Thus, n δ = 0 is a singular point in our approximation. As n δ increase, ∆ dw decrease and vanish at n δ ≃ 0.152 and n δ ≃ 0.150 for the two cases, respectively. By contrast, the DSC order parameters ∆ ds emerge at n δ ≃ 0.085 and 0.100 and vanish at n δ ≃ 0.238 and 0.180 for the two cases, respectively. We find that ∆ ds has a strong dependence on |v ds |. As |v ds | decreases, both the size of the DSC dome and the maximum value of ∆ ds decrease. The maximum values of ∆ ds are ∆ (max) ds ≃ 0.054 and 0.039 for the two cases, respectively. At n δ = 0.1358 and 0.1336, which are near the optimal dopings of the DSC, the van Hove singular point in the density of states coincides with the Fermi level, as described later.
We note that our obtained phase diagrams of FIG. 3 are very similar to the experimental phase diagrams of HTS (see Figure 1b of Ref. 47) if we assume that the magnitudes of ∆ dw and ∆ ds are proportional to the transition temperatures. We thus conclude that of the pseudogap phase can be interpreted as the DDW phase, which has been proposed by Chakravarty  that the hole whose concentration is n δ that the hole whose concentration is n δ  . 3 shows that the optimal doping is near or slightly to the left of the metal-metal transition point. In the following section, we discuss the details in the case of (a).
To understand further the phase diagram in FIG. 3 (a), we calculate the energies of the normal (∆ dw = ∆ ds = 0), DDW (∆ dw = 0, ∆ ds = 0), and DSC (∆ ds = 0, ∆ dw = 0 or ∆ ds = 0, ∆ dw = 0) states. Substituting m = 0, U/t = 6.0, and V /t = 1.2 into Eqs. (75), (76), and (101), we obtain: where ǫ F = ǫ a /2 and we have omitted the constant term and set t = 1 for simplicity. The energies of the normal, DDW, and DSC states are written, respectively, as: Note that ǫ F (n δ ) here is satisfied by Eq. (129) with m = 0 and ∆ dw = 0, and ǫ F (n δ , ∆ dw ) is satisfied by both Eqs. (126) and (129)   When ∆ dw is large (n δ is small), the Fermi surfaces are small ellipses, and as ∆ dw becomes small (n δ is large), they expand to large Fermi surfaces (see FIGs. 5(a)→(b)→(c)). As shown in FIG. 5(c), at n δ = 0.1358 (∆ dw = 0.0572), four Fermi surfaces are in contact with each other near (k x , k y ) = (0, ±π) and (±π, 0). The inset in FIG. 5(c) shows that the Fermi surfaces of the first and second quadrants are in contact with each other near (k x , k y ) = (0, π). At n δ = 0.1358, where the topology of the Fermi surface changes, the van Hove singular point in the density of states coincides with the Fermi level. In addition, as shown in FIG. 5(d), immediately after the four Fermi surfaces come into contact, they are connected to each other and produce small pockets at (k x , k y ) = (0, ±π) and (±π, 0). FIG. 5(e) shows that the united large Fermi surface emerges for more than n δ ≃ 0.1359.
C. Density of states in the d x 2 −y 2 -wave density wave state The density of states per site in the DDW state is defined as: where ǫ k (= −E − k,1,m=0 ) is the dispersion relation of the quasi-particle (see Eq. (83)), p = v dw ∆ dw , and we set t = 1. From Eq. (E.4) of Appendix E, the density of states must be calculated on the two regions depending on ω.

VI. SUMMARY
In this study, the extended Hubbard Hamiltonian with NN Coulomb repulsion on the square lattice was studied by applying the hole picture to the AF region to obtain the phase diagrams of the DDW and DSC phases. The phase diagrams, which were obtained by regarding the order parameter as the transition temperature, were qualitatively consistent with the phase diagrams observed in experiments of the HTS. 47 By overlooking both phase diagrams, we conclude that the pseudogap can be interpreted as the DDW order parameter.
In the underdoped regions, the DDW order parameter decreased with increasing hole doping and vanished at the new metal-metal quantum critical transition point under the DSC dome, which was predicted by Chakravarty et al. 40 The DSC states were established based on the DDW state. As the hole doping increased, the shape of the Fermi surface with the DDW order changed from four small to large ellipses. When the topology of the Fermi surface changed, the van Hove singular point in the density of states was located at the Fermi energy. In addition, we stress that we could obtain the theoretical value of p = v dw ∆ dw if we experimentally observed both the value of energy at which the density of states appeared and the value of the van Hove singular point, as these are written by ω = −ǫ a and ω 0 = 4p √ 1+p 2 − ǫ a , respectively. By contrast, in the overdoped region, the DDW collapsed and the system became a normal metal and the Fermi surface became the large contours. The DSC was suppressed simultaneously. The ground state was as follows: the AF region was found only at n δ = 0 in our approximation and as n δ increased, the ground state became the DDW, DDW+DSC, DSC, and normal metal states.
We conclude that the hidden order proposed by Chakravarty et al. 40 can be obtained from our effective Hamiltonian by using the canonical transformation. We expect that our obtained results assist in the qualitative explanation of the experimental results if the DDW order is present.

Acknowledgment
This work was supported by JSPS KAKENHI Grant Numbers JP17J05190 and JP17K05519.