Momentum dependence of pseudo-gap and superconducting gap in variation theory

To consider the origin of a pseudo-gap and a superconducting gap found in the high-Tc cuprates, the momentum dependence of the singlet gap parameter and the superconductivity correlation function are evaluated in the t–J model by using an optimization variational Monte Carlo method. In the underdoped regime, the singlet gap is significantly modified from the simple dx2- y2-wave gap (∝cos kx-cos ky) by the contributions of long-range pairings. Its angular dependence along the Fermi surface is qualitatively consistent with those experimentally observed in both hole- and electron-doped cuprates. This singlet gap will correspond to the pseudo-gap and its doping dependence agrees with that of the pseudo-gap. On the other hand, the superconductivity correlation function is dominant in the nearest-neighbor pairing and its Fourier transform preserves the original simple dx2- y2-wave form. We argue that this superconductivity correlation function is closely related to the coherent superconductivity gap appearing below T c in the ‘Fermi arc’ region. Its doping dependence is also consistent with the recent experimental observations.


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(∝ cos k x − cos k y ) in some cases. In Bi2212, for example, the angular dependence of the gap becomes concave near the nodal point [13,14], which is different from the linear dependence in the simple d x 2 −y 2 -wave function. In the electron-doped systems, the modification is much more significant. It is found that the maximum of the gap is located midway between the Brillouinzone boundary (π, 0) and the zone diagonal (π/2, π/2) [15,16], which again differs markedly from the simple d x 2 −y 2 -wave. Although it is unclear whether SC or PG mainly contributes to these deformed gaps, it is important to elucidate the origin of these modifications in the momentum space.
With this in mind, we discuss in this paper the difference of the momentum dependence of two kinds of physical quantities, i.e. the singlet gap parameter, k , optimized in a correlated SC state or a projected Bardeen, Cooper and Schriefer (BCS) wave function [17], and the Fourier transform of SC correlation function P(τ ). We consider the t-J model on the square lattice with second-and third-neighbor hoppings (t-t -t -J model) to be fit for the Fermi surface observed by ARPES [18]. This model represents the strongly correlated regime of the Hubbard model and enables us to study the two-gap features from a different viewpoint. In the t-J model, we study variational ground states, using an optimization variational Monte Carlo (VMC) method that accurately gives expectation values of many-body functions. Although finite-temperature behavior is very difficult to discuss, it will be important to study the ground state in a reliable numerical method that can be applied even to the strongly correlated systems. We consider that the singlet gap parameter, k , represents the pseudo-gap PG [19] since k is the excitation gap of spinons in the slave-boson mean-field theory. This excitation gap can survive above T c when the holons become incoherent. On the other hand, we expect that the coherent SC gap, SC , appearing below T c is determined from the SC correlation function, P(τ ), although it is not explicitly proved.
In the following, we introduce the long-distance singlet pairs in k and show that the optimized gap parameter k reproduces the angular dependence of gap functions observed experimentally both for the hole-and electron-doped cuprates. In contrast, the Fourier transform of the SC correlation function, P(τ ), looks very similar to the simple d x 2 −y 2 -wave order parameter. Furthermore, the doping-dependences of k and P(τ ) are consistent with the recent experimental observations of PG and SC .

Model Hamiltonian and variational wave functions
As a model for the CuO 2 plane in high-T c cuprates, we consider the t-J model, which is the most reasonable simplified model for the cuprates [18,20,21]: where P G is the Gutzwiller projection operator, P G = j (1 − n j↑ n j↓ ), excluding doubly occupied states from the wave function, and t i j are first-, second-and third-nearest neighbor hoppings between i and j sites [21]. The superexchange interaction, J , is between the nearestneighbor Cu spins, which acts when the doped carriers (forming Zhang-Rice singlet [20]) are not located on both sites of i and j. It has been shown that the t-J -type model has a superconducting ground state with d x 2 −y 2 -wave symmetry in the various approaches, such as VMC simulations [22]- [25], exact diagonalization methods [26] and high-temperature 4 expansion studies [27,28]. (Note that the exact treatments like the quantum Monte Carlo methods have not been successful in the two-dimensional t-J model.) For the hole-doped case or less-than-half-filling case, we use J/t = 0.3 (t being the nearest-neighbor hopping), t /t = −0.16 (second-nearest-neighbor hopping), and t /t = 0.20 (third-nearest-neighbor hopping) as typical parameterization for YBa 2 Cu 3 O 7−δ (YBCO), and t /t = −0.10 and t /t = 0.10 for LSCO [21]. We find that it is crucial for the present problem to include the t term. For the electron-doped case or more-than-half-filled case, we apply a particle-hole transformation, c † jσ → exp (iQ · r j )h jσ , with Q = (π, π). Using this transformation, we can study the electron-doped case by the less-than-half-filled model with t → −t and t → −t . For example, we use t /t = 0.16 and t /t = −0.20 for Nd 2−x Ce x CuO 4 (NCCO).
To this model, we apply an optimization VMC method [29], which accurately treats the local constraint due to P G [24,25]. As a variational wave function, we extend the Gutzwillerprojected BCS wave function as follows [17,19,24,25]: where N e is the electron number and ϕ k is the ratio of BCS coefficients: withε k = −2t (cos k x + cos k y ) − 4t cos k x cos k y − 2t (cos 2k x + cos 2k y ).
Variational parameters to be optimized are a singlet gap parameter, k , chemical potential, ζ , and band renormalization parameterst /t andt /t. Note thatt /t andt /t are optimized independently of t /t and t /t in the Hamiltonian equation (1). These renormalizations are important [30] since they deform the Fermi surface and stabilize d x 2 −y 2 -wave SC or antiferromagnetic (AF) long-range ordered state, by retrieving the nesting condition of (π, π) [30]- [34].
In this paper, we extend the previous VMC calculations [24,25] by introducing long-range Cooper pairings in k . Preserving the d x 2 −y 2 -wave symmetry, we extend the variational singlet gap k as where (  optimize these gap parameters independently to allow the difference of the signs. Note that ext.d k is reduced to the simple d x 2 −y 2 -wave pairing when (n) d = 0 for n = 2, 3, . . . , 8. Generally speaking, even if the Hamiltonian has only nearest-neighbor interactions, its ground-state wave function can have long-range (n) d , which is in sharp contrast to the mean-field theory. One might think that the long-range Coulomb repulsion can have strong effects on (n) d . However, near the half-filling where only a small number of holes exist, the charge fluctuation is suppressed due to the Gutzwiller projection, and the long-range Coulomb repulsion has little effect. Note also that, in the present calculation, we do not take account of the coexistence of antiferromagnetism with SC observed experimentally [35], which will be important in the very vicinity of halffilling [36].
In the present VMC method, a simple linear optimization of each parameter with fixing the other parameters is employed according to the optimization VMC procedure [29]. In one round of iteration, every parameter is once optimized in one dimension, with 2-5 × 10 4 Monte Carlo samples. Then we average the results of the 10-20 rounds after convergence, and determine the optimized variational parameters and the variational energy. Therefore, our data are substantially the averages of 2-10 × 10 5 samples, and the statistical errors in energy are suppressed down to the order of 10 −4 t. Physical quantities are calculated using the optimized parameters with 2-5 × 10 4 samples. The systems used are of N s = L × L sites with periodic-antiperiodic boundary conditions. We study the systems with L = 12-14 to check the system-size dependence. Figure 2 shows the optimized gap parameters (n) d (n = 1-8) as a function of doping concentration, δ, for the three parameter sets discussed above. In all cases, the magnitude of the nearest-neighbor gap parameter, (1) d , is always by far larger than the other (n) d , supporting the simple d x 2 −y 2 -wave assumption as a first approximation [22,23]. (1) d increases monotonically as δ decreases except for the very vicinity of half-filling, 0 δ 0.05, in the hole-doped cases (figures 2(a) and (b)). The overall δ-dependence is similar to that of the observed pseudo-gap [2,3,4,6]. In figure 2, we also compare the gap parameter with spin structure factor, S(q), at the AF wave number q = Q = (π, π). For all the parameter sets, S(Q) behaves similarly to (1) d , suggesting a close relationship between the AF correlation and the formation of nearestneighbor singlet pairings. Incidentally, the decrease of (1) d in the very vicinity of half-filling is probably caused by the retrieval of nesting condition. For example, (1) d for the hole-doped case of t /t = −0.16 and t /t = 0.20 ( figure 2(b)) is smoothly connected to that for the electrondoped case of the identical parameter sets, namely t /t = 0.16 and t /t = −0.20 ( figure 2(c)).

Singlet gap parameters and SC correlation functions
Let us discuss the long-range gap parameters for the two hole-doped cases shown in figures 2(a) and (b). In these cases, the long-range (n) d have small but finite values. In particular, (2) d has appreciable values of 0.1-0.2. Furthermore, (3) d , (5) d and (6) d are also finite in the range of δ 0.2 for t /t = −0.10 and t /t = 0.10 ( figure 2(a)), and δ 0.3 for 7 t /t = −0.16 and t /t = 0.20 ( figure 2(b)). The energy gain introduced by these long-range components with respect to the pure d x 2 −y 2 -wave form is about 0.0004t ± 0.0002t for most of the parameters we used. For example, the variational energies with and without long-range components are E/t = −0.5979 and −0.5975, respectively, for a typical hole-doped case with J/t = 0.3, t /t = −0.16, t /t = 0.20 and δ = 0.139 in the 12 × 12-site system. Although the energy gain is small, it is always larger than the statistical errors.
The long-range components (n) d deform the simple d x 2 −y 2 -wave gap function. Actually, figure 3 shows angular dependences of the optimized ext.d k along the Fermi surface for several values of doping concentration. Here, the Fermi surface is determined from the tight-binding bandε k with the optimized band-renormalization parameters,t /t andt /t, for each parameter set. Obtained Fermi surfaces for a typical doping rate, δ = 0.139, are shown in the insets of figure 3. As seen in figures 3(a) and (b), the angular dependence in the two hole-doped cases becomes concave for small φ, namely near the nodal point, in contrast to the linear behavior of the simple d x 2 −y 2 -wave symmetry indicated by the dotted lines. These deformations are significant especially for the regime near half-filling (δ → 0). This behavior is qualitatively consistent with experimentally observed angular dependences in Bi2212 [13,14].
Next, let us discuss the electron-doped case. As shown in figure 2(c), the longer-range gap parameters are again meaningful. In this case, however, (2) d is relatively small, but  [15,16]. Some theoretical studies have also discussed the nonmonotonic behavior of ext.d k in the electron-doped cases [37,38]. They showed that the position of maximum ext.d k almost coincides with a hot spot.
The different behavior of ext.d k between the hole-and electron-doped cases will be understood from the difference of AF correlation. In the electron-doped case of the present model, equation (1), AF correlation is dominant owing to the nesting condition [39,40]. As a result, (2) d is suppressed while (3) d is enhanced because parallel spin configurations are encouraged in the (2, 0)-position, while antiparallel configurations are favored in the (3, 0)position. This tendency is apparent if we check the real-space spin correlation function, S z i S z j , as shown in figure 4. In the electron-doped case, the AF correlation is dominant and S z i S z j has alternating signs until the correlation length of about six lattice constants. In contrast, the AF correlation in the hole-doped case is much weaker than in the electron-doped case. This is consistent with the above result that not only (2) d but also the longer-range (n) d have finite values in the hole-doped cases.
As mentioned in section 1, the optimized singlet gap parameter ext.d k represents the excitation gap of spinons without coherence of holons, but generally not a SC gap in the sense of slave-boson mean-field theory. Actually, ext.d k has finite values even at half-filling (δ = 0) where a Mott insulator is realized. Hence, as a quantity representing the SC, we calculate a  long-distance value of SC correlation function: τ is the length of the pair, and r M = (L/2, L/2) is the longest distance the pair can jump. In figure 5, P(τ )'s for various pair length are shown as a function of δ for the three parameter sets. First, one can notice a marked asymmetry between the hole-doped cases (figures 5(a) and (b)) and the electron-doped case (figure 5(c)). P(1) of the hole-doped case is considerably large and extended, compared to that of the electron-doped case. Furthermore, among the hole-doped cases, P(1) is considerably larger and extended for the case of t /t = −0.16 and t /t = 0.20 (YBCO) than that for the case of t /t = −0.10 and t /t = 0.10 (LSCO). From figure 5, one can also see that P(τ )'s for τ 2 are almost zero both for hole-and electron-doped cases, namely, the SC correlation is dominant only in the nearest neighbor pairing even though the singlet gap parameter, ext.d k , has contributions from farther sites. Therefore, the Fourier transform of P(τ ) looks just as the simple d x 2 −y 2 -wave symmetry.

Summary and discussions
To elucidate the origin of the gaps found in the high-T c cuprates, we evaluated the momentum dependence of the singlet gap parameter, ext.d k , and the SC correlation functions both with the d x 2 −y 2 -wave symmetry in the t-t -t -J model using an optimization VMC method. For small doping (δ 0.2), the singlet gap is significantly deformed from the simple d x 2 −y 2 -wave due to the contributions of the long-range pairings. In the hole-doped cases, the angular dependence along the Fermi surface becomes concave near the nodal direction, whereas in the The present results indicate that the excitation spectra observed experimentally can be naturally understood from the behavior of the singlet gap parameter, ext.d k , in the t-t -t -J model. Since the gap parameters are included in the wave function of equation (2), they can be regarded as an excitation gap in the spin degrees of freedom (or spinons) in the sense of the slave-boson mean-field approximation [18]. Although the present VMC method cannot be applied to finite temperatures, let us speculate on the pseudo-gap behavior above T c in view of the present results. In the slave-boson mean-field theory, the state above T c (or pseudo-gap state) is regarded as a resonating valence bond (RVB) state with spinon pairing but without Bose condensation of holons [18]. Therefore, we expect that the wave function is similar to equation (2) but without coherence of holons. In this case, the singlet excitation gap will be similar to ext.d k in figure 3. Thus the pseudo-gap above T c can be understood from ext.d k with some incoherence due to disordered holons. Actually the obtained doping-dependence of ext.d k is consistent with that of the pseudo-gap in the hole-doped cases. However, note that the pseudogap behavior is not completely understood because the pseudo-gap seems to vanish in a sizable area near the nodal points called a 'Fermi arc'. To pursue this phenomenon, other factors may have to be introduced, such as the charge inhomogeneity, etc [8].
As for the coherent SC gap, recent experiments suggest that it appears below T c inside the 'Fermi arc' region. It is tempting to assume that this coherent SC gap is caused by the SC correlation P(τ ) discussed in the present VMC calculation. Actually, P(τ ) represents the true SC correlation that involves charge degrees of freedom, while the gap parameter, ext.d k , represents the spinon RVB gap. In fact, due to the Gutzwiller projection, P(τ ) in equation (6) vanishes at half-filling in which the Cooper pairs cannot have kinetic energy. In the Gutzwiller approximation [19,41], one can show that P(τ ) is proportional to δ for small values of δ. In this sense, P(τ ) is finite only when the holons (or charge degrees of freedom) are coherent, which anticipates a close relationship to the SC gap. It is to be noted that the behavior of P(τ ) agrees with v discussed in the Hubbard model [11], which represents the anomalous velocity parallel to the Fermi surface at the nodal point. It was discussed that v is proportional to the spectral weight of the quasiparticle at the nodal point. This corresponds to the coherent motion of holons in the present t-J model. The obtained doping-dependence of P(τ ) in figure 5 is consistent with that for the coherent SC gap. It also agrees with the recent claim that the coherent gap correlates with T c .
In summary, the doping dependence and angular dependence of ext.d k and SC correlation function seem to be consistent with the pseudo-gap and the coherent SC gap inside the 'Fermi arc'. However, in order to establish these correspondences, it is still necessary to investigate these quantities in other methods.