Spatial variation of d-density-wave order in the presence of impurities

Transcription

1 PHYSICAL REVIEW B 69, (2004) Spatial variation of d-density-wave order in the presence of impurities Amit Ghosal Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 Hae-Young Kee Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 (Received 20 January 2004; published 30 June 2004) Effect of finite density of nonmagnetic impurities on a coexisting phase of d-density-wave (DDW) order and d-wave-superconducting (DSC) order is studied using Bogoliubov-de Gennes (BdG) method. The spatial variation of the inhomogeneous DDW order due to impurities has a strong correlation with that of density, which is very different from that of DSC order. The length scale associated with DDW is of the order of a few lattice spacing, and the nontrivial inhomogeneities are shown to make DDW order much more robust to the repulsive potential, while DSC order becomes very sensitive to them. The effect of disorder on the density of states is also discussed. DOI: /PhysRevB PACS number(s): k, z I. INTRODUCTION One of the recent proposals in the context of high temperature cuprates is that a true broken symmetry state dubbed as d-density-wave state (DDW) is responsible for the pseudogap phenomena. 1 This phase was first suggested in relation to the excitonic insulators, 2 and it was found as one of the ground states of the t-j type model. 3 The DDW is a particle hole condensate with angular momentum 2. The ordered state can be characterized by the circulating current arranged in an alternating pattern on a square lattice, which can be detected as a Bragg scattering signal in neutron scattering measurements. 4,5 But the neutron scattering experiments 6 8 in cuprates remain undetermined. Thus, the definite conclusion on the relevance of the DDW order to the cuprates requires more precise experiments on various doping concentration of cuprates, and further theoretical studies on the properties of this order. Especially, the effect of the nonmagnetic impurities on DDW order is an important subject to investigate, since any well-prepared cuprate sample contain an intrinsic disorder, minimally from nonstoichiometry. The simplest possible description of the impurity effect is the self-consistent T-matrix approximation (SCTMA). 9 This mean field picture excludes not only the freedom of the ordered patterns, but also the interference of the impurities. Within this approximation, the thermodynamics were found to be identical to those of a d-wave BCS superconductor (DSC) in the unitary limit. 10 From the density of states, one can see that electrons are localized close to the Fermi energy, and the change in the transition temperature is given by the Abrikosov-Gorkov formula known in BCS superconductors. 10 Within the standard noncrossing approximation, the similarity between the DDW and DSC is based on the d-wave symmetry of the gap. In this paper, we study the effect of impurities on DDW order and for the case where DDW coexists with DSC using the Bogoliubov-de Gennes (BdG) technique. This method is the mean field approximation, but it allows spatial inhomogeneity in order parameter. In the case of the disordered DSC with a short coherence length, it was shown that the superfluid stiffness is significantly larger than that predicted by the SCTMA, due to the nontrivial spatial structures of the order parameter. 11 We found that the DDW order is more robust than the DSC order to the repulsive potential, which cannot be understood within the conventional T-matrix approach. It was also found that there is a striking similarity in spatial variations of DDW order and density in the presence of impurities. The spatial correlation between the DDW order and density is based on our finding that the length scales of DDW order is only a few lattice spacing. The paper is organized as follows. In Sec. II, we describe the two-dimensional model Hamiltonian, t-j with next nearest neighbor interaction, and the BdG method to investigate the effect of impurities on DDW and DSC orders. The evolution of DDW and DSC orders as a function of impurity concentration are summarized in Sec. III. The spatial variations of DDW and DSC orders are presented in Sec. IV, and compared with the density modulation due to impurities. In Sec. V, the local density of states is studied to confirm the results obtained in the previous section: the DDW order is more robust than the DSC order to the repulsive potential. 12 Then we summarize our findings and its physical grounds in the last section. II. MODEL We model two-dimensional disordered DSC and DDW order by the following Hamiltonian: H = t c i c j + h.c. + V i n i ij, + J S i S j n i n j /4 W n i n j. ij ij,, The first term is the kinetic energy which describes electrons, with spin at site i created by c i, hopping between nearest neighbors ij on a square lattice. The disorder potential V i i /2004/69(22)/224513(6)/$ The American Physical Society

2 AMIT GHOSAL AND HAE-YOUNG KEE PHYSICAL REVIEW B 69, (2004) in the second term is an independent random variable at each site which is either +V 0 (repulsive potential), with a probability n imp (impurity concentration), or zero, and is the chemical potential. The last, interaction term 13 is chosen to lead to a coexisting DSC and DDW order ground state in the disorder-free system, where S i and n i are the spin and density operators, respectively. The mean field decomposition of the earlier Hamiltonian leads to following BdG equations: 14,15 ˆ ˆ 2 ˆ * ˆ* u n v n = E n u n v n, where ˆ and ˆ are defined as follows: ˆu n j = t + j; e iq r j u n j + + V j j u n j 3 ˆ u n j = j; u n j +, where =±xˆ,±ŷ. The local DSC pairing, j; and DDW order, j; =Im j;, defined on a bond j, j+ are given by j; = J + W 4 c j+ c j + c j c j+, j; = J 2W 4 c j+ c j c j c j+ e iq r j. 4 The inhomogeneous Hartee and Fock shifts are obtained as j= + J/4+W n j+ and Re j;, respectively. We numerically solve for the BdG eigenvalues E n 0 and eigenvectors u n,v n on a lattice of N sites with periodic boundary conditions. We then calculate the d-wave-pairing amplitude j;, and the DDW order and Fock shift as the imaginary and real parts of j;, and the density n j at T=0 which are given by j; = V DSC u n j + v * n j + u n j v * n j +, n j; = V DDW e j iq r v * n j v n j + u n j u * n j +, n n j =2 v n j 2, 5 n where V DSC = J+W /4 and V DDW = J 2W /4. These are fed back into the BdG equation, and the process iterated until self consistency 16 is achieved for each of the (local) variables defined on the sites and bonds of the lattice. The chemical potential is chosen to obtain a given average density n = i n i /N. We define the site dependent order parameters in terms of the bond variables as FIG. 1. The evolution of / 0 and / 0 are plotted as a function of n imp, where 0 and 0 are the DSC and DDW order values in pure systems, respectively. All lines except (c) are plotted for the potential, V 0 =100 (close to unitary limit), J=1.15, and W=0.6 in units of t=1. (a) The DDW order at the half filling where the DSC order is zero in the pure system. (b) The DSC order at n =0.95 where DDW order, is forced to be zero. (c) The DDW order obtained by the SCTMA for a unitary limit at half-filling which is lifted from the Fig. 1 of Ref. 10. (d) The DDW order at n =0.95 where the DSC order is finite (but smaller than ) within the model Hamiltonian. (e) The DSC order at n =0.95 with finite. For system with only DDW (a) or DSC (b), impurity is rather insensitive compared to the T-matrix result (c). However, when both order coexist at n =0.95, becomes significantly more robust to impurities while is affected severely. j = j; +xˆ j; +ŷ + j; xˆ j; ŷ /4 6 and similarly for j. We have studied the model at T=0 for a range of parameters and lattice sizes up to Here we focus on J =1.16, and W=0.6, in units of t=1, with n =0.95 on systems of typical size For these parameters, and n imp =0, the maximum DSC order is max =0.16 and the maximum DDW order is max =0.31. In the pure system our calculations reproduce a phase diagram of max and max as functions of filling similar to Ref. 17. For the impurity potential we choose (repulsive) V 0 =100, close to the unitary limit. The results are averaged over different realizations of the random potential. 18 III. EFFECT OF IMPURITY ON DDW AND DSC ORDERS We summarize our main results in Fig. 1, where we plot the disorder dependence of different orders (normalized to n imp =0 values). Let us first look at the line (a) that represents the behavior of as a function of n imp at half filling n =1. At this

3 SPATIAL VARIATION OF d-density-wave ORDER PHYSICAL REVIEW B 69, (2004) filling DDW is the stable order and DSC order in fact vanishes for the pure system (see Fig. 1 in Ref. 17 by Zhu et al.). It is worth comparing this result with the results obtained from SCTMA calculations. The line (c) represents the behavior of the DDW order as a function of n imp, which is taken from Fig.1 of Ref. 10 by Dora et al. The result is obtained in the unitary nonmagnetic impurity limit, where we convert the x axis to n imp from using the relation, =2n imp / N 0 where N 0 is the density of states at the chemical potential. It is clearly shown that the DDW order is more robust to impurities than predicted by SCTMA. The monotonic decrease of DDW order shown in (a) is true only at half filling because of the following. If the average density of electrons is at half filling, adding impurities would make the local density away from half filling everywhere near the repulsive impurity sites, electron density is almost depleted, whereas far from impurity n 1. Because DDW order decreases away from n=1 (see the pure phase diagram), DDW order would be suppressed everywhere. On the other hand, away from half filling when we force =0 in BdG equations, DSC becomes the surviving order and the n imp dependence of is given by the (b) line, which in fact is very similar to in the (a) curve. Such robustness of the DSC order to impurity compared with the SCTMA result shown in (c), had been studied before, 11 and it is attributed primarily to the fact that each impurity affects superconductivity rather inhomogeneously by destroying SC order within a small region (of size determined by coherence length ) around it. However, a similar study for the coexisting phase of DSC+DDW order at n =0.95 reveals surprising results. The line (d) represents the DDW order as a function of n imp when DDW order coexists with DSC order. Please notice that the DDW and DSC orders are rescaled by 0 and 0 in Fig. 1, where 0 and 0 are the amplitudes of order parameter in the absence of impurities. Since the values of 0 and 0 depend on the doping concentration, the comparison of the absolute values of cannot be made. However, the trends of line (d) as a function of impurity strength show that the DDW order becomes even more robust to impurities away from half filling. In fact for low n imp, even increases with impurity. The reason of the nonmonotonic behavior of DDW order as a function of n imp is the following. The average density of electrons is fixed (at n=0.95) in the system (in our model) at any impurity concentration. As a result, for low n imp, when the local density of electrons near the repulsive impurity sites goes to zero, the average density of electrons away from the impurities would have to increase to keep the average density fixed. If there are large regions away from impurities where the electronic density approaches half filling n=1, the DDW order at those regions strengthens, which results into overall increase of DDW order in the system at the level of mean field picture, for low n imp. As impurity concentration increases further, the system becomes completely inhomogeneous and the density of electrons on the sites away from impurities increases further, so that the local density at those regions increases beyond half filling. So a local DDW order in those regions (away from impurities) decreases. This causes a decrease of averaged DDW order, but note that it occurs at high n imp, while the DSC order monotonically decreases even for any small amount of n imp. This argument for the robustness of DDW order does not depend on the average density of electrons (which is 0.95 for our data), and would remain valid for the whole of the coexistence region within our model; as density increases toward half filling, the DDW (DSC) gets stronger (weaker) in the coexistence regime of our pure model. We have found such evidence for n=0.925 as well, and further decrease of the electron density would make the DDW order unstable even in the pure system. On the contrary, the superconducting order is severely affected by disorder in the coexisting phase. Line (e) represents the DSC order as a function of n imp at the same density n =0.95 as line (d). It is expected that the superconducting order is suppressed by impurities, 11,15 however the effect of impurities in the coexisting phase is much more severe than the DSC phase without DDW order. This is due to the interplay of the existence of DDW order and impurities. The impurities suppress locally (of size of coherence length ) both DDW order and DSC order, and decreases the electron density near impurity sites. However, since the density is conserved in the system, there are other regions away from impurity sites, where the local density exceeds the average value of density. In this region, the DDW order becomes strong (consistent with the earlier argument for DDW), which suppresses the DSC order. Therefore, the DSC order is suppressed almost everywhere. To get further insight, we investigate the spatial structures of the order parameter on the lattice for each impurity configuration in the following section. IV. SPATIAL STRUCTURES OF ORDER PARAMETERS In Fig. 2, we present a Grey-scale plot of strength of orders, and on a typical lattice at n imp =0.06 for a given realization of scatterers. Data for each of the panels are normalized by the largest value of the corresponding order on the lattice. The dark (light) regions represent larger (smaller) values of orders. Figure 2(a) shows a Grey-scale plot of when the DDW order coexists with the DSC order. We then plot the spatial distribution of density, n 1 in Fig. 2(b) to see whether there is any spatial correlation between the DDW order and density. Comparing the structure of Fig. 2(a) with Fig. 2(b) for the same n imp, we see that is large in space where local density is close to 1 (half filling). The strong spatial correlation between these two panels is striking, although it is not exact; the scale of modulation of is somewhat larger than that of density. The spatial structure of without coexisting DSC order is very similar to Fig. 2(a) and, hence, is not plotted separately. However, the strong tie of local n and suggests that the length scale of fluctuation of would be governed by that of n, which is rather small (of the order of k F 1 ). This can be understood as follows. The length scale associated with the DDW order is determined by, DDW 1/. When impurity is introduced, the bond current attached to the impurity site is forced to be zero. However, the bond current near the impurity site is reconstructed to satisfy the current conservation, and one can

4 AMIT GHOSAL AND HAE-YOUNG KEE PHYSICAL REVIEW B 69, (2004) FIG. 2. Grey scale plot of normalized DDW and DSC orders on lattice for n imp =0.06 and for a particular configuration of unitary impurities. The dark (light) region indicates large (small) values of the variables on given locations. Panel (a) and (b) are the and n 1, respectively, for a system with coexisting DDW+DSC order at n =0.95. Note the spatial correlation between (a) and (b). is shown in (c) with a finite, which is severely affected under the same conditions of impurity and the density. (d) is the profile of, which is rather large at various locations, when is forced to zero everywhere. The comparison of (c) and (d) shows the effect of existence of DDW on DSC order show some examples of the current reconstruction where the healing length is of order of a few lattice spacing. How the magnitude of the reconstructed bond current is determined? This magnitude is strongly related to the local density. The electron density depletes close to impurities and increases at locations far from it, to keep the average at the desired value. Since at low disorder, a large number of sites attain n i 1, increases at those sites; the DDW order is most stable near half filling, where perfect nesting occurs for our model. As a result average increases initially for small n imp as shown in line (d) of Fig. 1. At very large n imp, local density would be either much larger or smaller than 1, and average of would eventually decrease for large n imp. This argument can be substantiated by looking into our results for each configuration of impurities. For n =1, the introduction of impurity makes the local density only deviate from half filling. As a result decreases monotonically as found in Fig. 1(a). The earlier argument for the behavior of DDW order with impurity is independent of the coexisting DSC order and we also found similar trend in as in Fig. 1(d) for n 1 even in the absence of DSC order, which is consistent with our picture. This shows that DDW order responds to the density fluctuations due to impurities, which allows us to understand its response to impurities based on the phase diagram as a function of density in the pure system. On the contrary, the DSC order in the presence of impurities is not related to the local density fluctuations as DDW is, even though the length scale, DSC 1/, which is of the order of a few lattice spacing for high temperature superconductors under considerations. The behavior of in the presence of impurities is shown to be related to the electron-hole mixing in the real space. 15 In fact, in the regions with large electrons density the electrons are trapped in disorder valleys, so their numbers are fixed. This results in no contribution to pairing due to the random phase. In the regions where density is two small, there are not enough electrons to form cooper pair, so DSC order would be small there too. The DSC order would be large only in the region where density is moderate and disorder potential is small so that electrons delocalize and particle-hole mixing is large. 15 However, such length scale for DSC order is related to, which is normally larger than k F 1. For short coherence length HTSC under consideration, it is hard to distinguish these length scales numerically, though the behavior of independent and in Figs. 1(a) and 1(b) indicates that the impurity length scale for is somewhat smaller than that of (as a result the respective order is somewhat larger for for the same n imp ). This is also seen directly in the spatial structures of our calculations, e.g., comparing the impurity length scales in Figs. 2(a) and 2(d). Figures 2(c) and 2(d) present the spatial structure of on lattice with the same n imp configuration in the presence and absence of DDW order. We clearly see that the DSC is strongly suppressed by the impurities when coexisting with DDW order [as also observed in Figs. 1(b) and 1(e)]. Aswe have described in the previous section, the existence of DDW strongly affects the strength of the DSC, because away from the impurities there are regions where the density is near half filling, hence, the DDW becomes strong. The reason why the DSC is suppressed near half filling, where the DDW is strong in pure system, is because strong DDW allows significant weight of, scattering that mixes the + and lobes of the DSC order and thereby DSC becomes weak. The regions of small density do not contribute to DSC order as well, due to the absence of enough electrons for pairing. Thus, in the inhomogeneous media, when DSC order coexists with DDW order, it is suppressed everywhere. This is clearly illustrated in Fig. 2(c) whose spatial structures neither correlate with Fig. 2(a) nor with Fig. 2(b). It is important to note that the the dark regions in Fig. 2(c) imply only relatively larger values of on lattice, the actual DSC order is very small due to normalization (discussed earlier in this section), and also clear from Fig. 1(e). To get further insight on our numerical results, we study the averaged density of states in the next section. V. AVERAGED DENSITY OF STATES Let us now study the (impurity) averaged density of states (DOS): N = 1 u n i 2 E n + v n i 2 + E n, 7 N n,i where we broaden the delta functions with a width comparable to average level spacing. A large number of states is

5 SPATIAL VARIATION OF d-density-wave ORDER PHYSICAL REVIEW B 69, (2004) FIG. 3. Panel (a): Disorder averaged density of states N on a system with only DDW order (at n =1) on unit cells each of which are of size N= For n imp =0, there are two coherence peaks, and at its center, =, the DOS vanishes, where is 0 for half filling. With increasing n imp low lying impurity resonances produce an impurity band with enhanced weight at the particle side of the spectrum. The energy difference between coherence peaks, however, is not much affected showing the robustness of DDW order to impurities. Panel (b): Similar to panel (a), but with coexisting DDW+DSC phases (at n =0.95). Note that for a pure systems n imp =0, the DOS vanishes at =0, because there are only four points on Fermi surface (nodes in the d-wave superconductor) which contributes to the DOS at =0. With increasing n imp, the DOS at =0 is filled out, but two coherence peaks which represent the DDW order emerges. For n imp =0.03, the DOS is smilier to that of DDW phase. necessary to produce a reasonably smooth N. With the aim of getting a much better statistics, we work with an effective large system that is made out of unit cells, each of which is of dimension This is done by using a repeated zone scheme, details of which can be found in Ref. 19. For the range of n imp reported here, the method would work well, because the wave function is still extended. The effect of a single impurity on the DDW order is already studied within T-matrix formalism in Ref. 20 and using BdG calculations in Ref. 17. T-matrix works rather well for single impurity case, where the overlap between impurity states are naturally absent. We could reproduce previous results within our formalism. The extension to the case for a finite density of Unitary scatterer is presented in Fig. 3. In Fig. 3 we plot N as a function of for different n imp for the case of only DDW order [panel (a)] and coexisting DDW+DSC order [panel (b)]. For the pure system with only DDW order, N is the standard d-wave DOS, except that the DOS vanishes at = where =0 for half filling. With increasing n imp we see that the gap-edge singularities get rounded off and a small accumulation of states is produced at the particle side of spectrum close to =0. This is in accordance with our previous results. A small negative overall shift of N is due to the fact that the chemical potential,, gets shifted in the presence of the impurities to keep n at half filling. The accumulation of electrons around a single impurity effectively provides impurity screening, which will produce enhanced states at the particle side of the spectrum. 20 Such resonances from each impurity contribute to the average N and produce a broadband which is reflected as a bump in Fig. 3(a). However, the strength of the DDW order is not affected much (given by the relative location of the two coherence peaks). At this point, we should emphasize that the DOS structure for the impure DSC state is very different, where coherence peaks get strongly suppressed and a thin gap persists at =0, 11,21 so that N 0 =0 for all n imp. From our results with DDW order, we find that N 0 n imp, which is in disagreement with the prediction of T-matrix result N 0 nimp 9,10. N for a system with coexisting DSC+DDW order at n =0.95 is presented in Fig. 3(b). For the pure system, the quasiparticle spectrum in the coexisting state is given by E 1k = k 2 + k k 2 and E 2k = k 2 + k k 2, where k = max /2 cos k x cos k y, and k = max /2 cos k x cos k y. Therefore, the DOS is given by 22 N = k u 2 1k E 1k + v 2 1k + E 1k + u 2 2k E 2k where + v 2 2k + E 2k, u 2 1k = k + 2 k /E 1k, u 2 2k = k + 2 k + /E 2k, with u 2 ik +v 2 ik =1 for i=1,2. Therefore, it is expected that there are only four points on the Fermi surface which will contributes to the DOS at =0 for the pure system. This is what we found in the top figure in Fig. 3(b). With increasing n imp, the DOS at =0 filled and by n imp =0.03, N looks very similar for Figs. 3(a) and 3(b). The overall shift for the later case is due to the particle-hole asymmetry, and different chemical potential for Figs. 3(a) and 3(b). This demonstrates in a different way our main result, that, the DSC order is very sensitive to impurity whereas DDW order is robust in the coexisting phase. VI. SUMMARY AND DISCUSSION We studied the effect of nonmagnetic impurity on the DDW ordered state using the BdG technique. While the standard SCTMA indicates that the effect of impurity on DDW is similar to that on DSC, we found that the spatial variation of the DDW order has a strong correlation with that of density [Figs. 2(a) and 2(b)], and it is very different from that of DSC order [Figs. 2(a) and 2(c)]. We discussed that this occurs because the length scale associated with the DDW order is of order of a few lattice spacing 1/k F, which suggests

6 AMIT GHOSAL AND HAE-YOUNG KEE PHYSICAL REVIEW B 69, (2004) that the spatial variation of DDW order is related to the density fluctuation, while the DSC order is related to particlehole mixing. Therefore, the effect of impurity on DDW order is very different from that of DSC order, which cannot be obtained from the standard SCTMA method. When DSC and DDW coexist, it turns out that DDW order does not care about the existence of DSC and it still follows the density profile in the presence of impurity, and hence, the spatial inhomogeneity can be understood from the phase diagram as a function of density. However, DSC order would vanish almost everywhere [see Fig. 2(c)]. This is because in the region of larger density it is destroyed by DDW, and in the region of smaller density it is destroyed by disorder. Thus, in the inhomogeneous media both DDW and impurity are acting to suppress the DSC order. Our current picture brings out the unexpected results and their understanding at the mean field level; if the DDW phase exists in cuprates, the Bragg signal would be detected in neutron scattering measurements even in the presence of strong nonmagnetic impurity, while the width of the Bragg peaks depends on strength of impurity. However, the definite answer for its relevance to the cuprates requires the understanding of the role of strong correlation, and interplay between different competing orders, 1,5 which warrants further studies. ACKNOWLEDGMENTS The authors would like to thank Y. B. Kim and A. Vishwanath for illuminating discussions. They acknowledge SHARCNet computational facilities at McMaster University where most of the calculations were carried out. This work is supported by SHARCNet fellowship (A.G.), NSERC of Canada (H.-Y.K.), Canada Research Chair (H.-Y.K.), Canadian Institute for Advanced Research (H.-Y.K.), and Alfred P. Sloan Fellowship. 1 S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B 63, (2001). 2 B. I. Halperin and T. M. Rice, in Solid State Physics, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic Press, New York, 1968), Vol. 21, p I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774 (1988); J.B. Marston and I. Affleck, ibid. 39, (1989). 4 T. C. Hsu, J. B. Marston, and I. Affleck, Phys. Rev. B 43, 2866 (1991). 5 S. Chakravarty, H.-Y. Kee, and C. Nayak, Int. J. Mod. Phys. B 15, 2901 (2001). 6 H. A. Mook et al., Phys. Rev. B 64, (2001); 66, (2002). 7 Y. Sidis et al., Phys. Rev. Lett. 86, 4100 (2001). 8 C. Stock et al., Phys. Rev. B 66, (2002). 9 For a review on SCTMA results see K. Maki in Lectures on the Physics of Highly Correlated Electron System, edited by F. Mancini, AIP Conf. Proc. No. 438 (AIP, New York, 1998), and references therein. 10 B. Dora, A. Virosztek, and K. Maki, Phys. Rev. B 66, (2002); cond-mat/ A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. B 63, 20505R (2000). 12 We present our results for only repulsive potential because the repulsive potential is more realistic for materials such as high T c cuprates. However, we checked that our main conclusion see the conclusion section: the DDW order responds to the density fluctuations due to impurities so that it allows us to understand the effect of impurities based on the phase diagram of the pure system also applies to the attractive potential. Based on the phase diagram of the pure system, the DDW order is more sensitive to the attractive potential than that of DSC order. 13 Note that either of the spin exchange term or the extended Hubbard term of H int could produce both the DSC or the DDW orders. However, we keep both terms for stabilizing DDW phase [see J.-X. Zhu, W. Kim, C. S. Ting, and J. P. Carbotte, Phys. Rev. Lett. 87, (2001); C. Wu and W. V. Liu, Phys. Rev. B 66, (R) (2002)] and also to be able to tune both the orders independently. 14 P. G. de Gennes, Superconductivity in Metals and Alloys (Benjamin, New York, 1966). 15 A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. Lett. 81, 3940 (1998); ibid. 65, (2001). 16 In order to achieve accelerated convergence on multivariable space we use the Broyden method [see, e.g., W. E. Pickett, Comput. Phys. Rep. 9, 115(1989)]. We have checked that the same self-consistent solution is obtained for different initial guesses. 17 J.-X. Zhu, W. Kim, C. S. Ting, and J. P. Carbotte, Phys. Rev. Lett. 87, (2001); C. Wu and W. V. Liu, Phys. Rev. B 66, (R) (2002). 18 We found that the quantities presented in the Figs. 1 and 3 are reasonably self-averaging, so that averaging over realizations is sufficient. Similar number of configurations were considered in the A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. B 63, 20505R (2000); Phys. Rev. Lett. 81, 3940 (1998); Phys. Rev. B 65, (2001). 19 A. Ghosal, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 66, (2002). 20 D. K. Morr, Phys. Rev. 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