J / t n Δ$_{s}$ Δ$_{d}$ Δ$_{s}$ / Δ$_{d}$ 0.768 0 . $^{0107}$− 0 . 0042 i − 0 . 0256 + 0 . 0100 i -0.416 0.776 0 . $^{0101}$− 0 . 0040 i 0 . 0261 + 0 . 0103 i -0.386 0.785 0 . $^{0096}$− 0 . 0037 i − 0 . 0266 + 0 . 0103 i -0.360 0.4 0.794 0 . $^{0091}$− 0 . 0032 i 0 . 0270 + 0 . 0100 i -0.336 0.804 0 . $^{0085}$− 0 . 0033 i − 0 . 0270 + 0 . 0105 i -0.314 0.814 0 . $^{0080}$− 0 . 0031 i 0 . 0272 + 0 . 0106 i -0.295 0.720 0 . $^{0147}$− 0 . 0052 i − 0 . 0375 + 0 . 0134 i -0.391 0.722 0 . $^{0145}$− 0 . 0053 i 0 . 0378 + 0 . 0133 i -0.383 0.725 0 . $^{0143}$− 0 . 0050 i − 0 . 0382 + 0 . 0133 i -0.375 0.741 0 . $^{0129}$− 0 . 0045 i 0 . 0398 + 0 . 0139 i -0.325 0.8 0.757 0 . $^{0116}$− 0 . 0046 i − 0 . 0401 + 0 . 0160 i -0.289 0.800 0 . $^{0091}$− 0 . 0037 i 0 . 0419 + 0 . 0168 i -0.217 0.811 0 . $^{0085}$− 0 . 0034 i − 0 . 0421 + 0 . 0169 i -0.203 0.833 0 . $^{0076}$− 0 . 0030 i 0 . 0422 + 0 . 0169 i -0.179 0.842 0 . . 0029 i 0 . 0419 + 0 . 0168 i -0.171 p$_{x}$ ( p$_{y}$ $^{)}$− wave superconducting state. For a physically realistic super-exchange coupling J and a hopping constant t (such as J / t = 0 . 4), a spin-triplet p$_{x}$ ( p$_{y}$ $^{)}$−wave superconducting state coexists with charge and spin density wave order for dopings up to around δ ∼ 0 . 18, with δ ≡ 1 − n . The occurrence of such a spin-triplet p$_{x}$ ( p$_{y}$ $^{)}$−wave superconducting component is unexpected, although this would, in our opinion, have been anticipated from the presence of the Nagaoka’s ferromagnetic state [23]. In addition, Dagotto and Riera [8] observed im- portant ferromagnetic and anti-ferromagnetic correlations in this regime from the ED of the t − J model on a small cluster, which may be properly interpreted as the precursor of a spin- triplet p$_{x}$ ( p$_{y}$ $^{)}$−wave pairing state, coexisting with charge and spin density wave order. Note that the coexisting charge and spin density wave order forms a pattern, as displayed in Fig. 1 (right panel). In this phase, the translational invariance under one-site shifts, and four-fold rotation symmetry, as well as the U (1) symmetry in the charge sector and the S U (2) symmetry in the spin sector, are spontaneously broken. ing of the spin-singlet and spin-triplet channels arises from the spin-rotation symmetry breaking: spin is not a good quantum number . In Fig. 3, the magnitude of the spin-triplet p$_{x}$ ( p$_{y}$ $^{)}$−wave su- perconducting order parameter Δ$_{p}$ , along with those of the charge and spin density wave order parameters, are plotted for both J / t = 0 . 4 and J / t = 0 . 8, which have been evaluated with the truncation dimension D = 6. In this phase, all the order parameters are evaluated for two sites A and B in the unit cell, with the subscripts A and B as their labels, except for the charge density wave order parameter that is defined as half the di ff erence between the densities at two sites A and B (see Fig. 1, right panel). The spin-singlet superconducting phase with extended $^{s}$−wave symmetry. For a fixed 0 . 1 < J / t < 0 . 95, if one keeps on increasing doping until the $^{d}$− wave component Δ$_{d}$ van- ishes, then only an s -wave order parameter Δ$_{s}$ survives, re- sulting in a spin-singlet superconducting phase with extended $^{s}$− wave symmetry. For J / t = 0 . 4 and J / t = 0 . 8, the $^{s}$− wave pairing order parameter Δ$_{s}$ is plotted in Fig. 3. A peculiar fea- ture of the $^{s}$− wave order parameter is that it is almost linearly increasing with increasing doping, until it reaches its maxi- mum, and then it monotonically decreases. The mixing of the spin-singlet d + $^{s}$−wave and spin-triplet p$_{x}$ ( p$_{y}$ $^{)}$− wave superconductivity in the presence of an anti- ferromagnetic background. As shown in Fig. 2, there is a superconducting phase with mixed spin-singlet d + $^{s}$− wave and spin-triplet p$_{x}$ ( p$_{y}$ $^{)}$−wave symmetries in the presence of an anti-ferromagnetic background. For J = 0 . 4, it occurs for fillings from n = 0 . 818 to n = 0 . 765. For J = 0 . 8, it occurs for fillings from n = 0 . 877 to n = 0 . 665. In this phase, the trans- lational invariance under one-site shifts, and four-fold rotation symmetry, as well as the U (1) symmetry in the charge sector and the S U (2) symmetry in the spin sector, are spontaneously broken. Note that the d -wave, s -wave and p$_{x}$ ( p$_{y}$ $^{)}$−wave com- ponents are homogeneous. Although this phase is perhaps only of academic interest due to its unphysical nature of large dopings for cuprate su- perconductors, we believe it is important to clarify its pairing symmetry. Note that a small portion located between 2 . 0 ≤ J / t ≤ 3 . 4367 has been identified as a spin-singlet supercon- ducting phase with $^{s}$− wave symmetry in Refs. [9, 10, 15, 26]. In Fig. 3, the magnitudes of the $^{d}$− wave, $^{s}$− wave and p$_{x}$ ( p$_{y}$ $^{)}$−wave order parameters, Δ$_{d}$ , Δ$_{s}$ , and Δ$_{p}$ (upper panel), along with that of the anti-ferromagnetic order parameter (lower panel), are plotted for the $^{t}$−J model with J / t = 0 . 4 and J / t = 0 . 8 which have been evaluated with the truncation di- mension D = 6. One observes that the $^{d}$− wave component Δ$_{d}$ vanishes at electron fillings n$_{c}$$_{1}$ : n$_{c}$$_{1}$ = 0 . 765 and n$_{c}$$_{2}$ = 0 . 818 for J / t = 0 . 4 and n$_{c}$$_{1}$ = 0 . 665 and n$_{c}$$_{2}$ = 0 . 877 for J / t = 0 . 8. We also list the numerical values of the $^{d}$− wave and $^{s}$− wave components, Δ$_{d}$ and Δ$_{s}$ , for both J / t = 0 . 4 and J / t = 0 . 8 at di ff erent fillings in Table I, together with their ratio Δ$_{s}$ / Δ$_{d}$ . The fact that the pairing symmetry (in the spin-singlet chan- nel) is of the d + s -wave nature manifests itself in that the ratio Δ$_{s}$ / Δ$_{d}$ is always real. On the other hand, the symmetry mix- Doping-induced quantum phase transitions. Now we turn to doping-induced quantum phase transitions (QPTs) for the In Fig. 3, the magnitude of the spin-triplet $^{p}$− wave order parameter, together with that of the ferromagnetic order pa- rameter, are plotted for both J / t = 0 . 4 and J / t = 0 . 8 with the truncation dimension D = 6. Here, the p$_{x}$ ( p$_{y}$ $^{)}$−wave order parameter is identical at all sites, thus it is homogeneous. The spin-triplet superconducting phase with p$_{x}$ ( p$_{y}$ $^{)}$− wave symmetry in a ferromagnetic background. For large dopings, a spin-triplet superconducting phase with p$_{x}$ ( p$_{y}$ $^{)}$−wave sym- metry in a ferromagnetic background occurs. The existence of such a spin-triplet superconducting phase with p$_{x}$ ( p$_{y}$ $^{)}$−wave symmetry has been pointed out by Kagan and Rice [26]. Therefore, our simulation results are consistent with their an- alytical analysis. TABLE I: The numerical values of the s -wave and d -wave supercon- ducting order parameters, Δ$_{s}$ and Δ$_{d}$ , and their ratio Δ$_{s}$ / Δ$_{d}$ for both J / t = 0 . 4 and J / t = 0 . 8 at di ff erent fillings in the DSW + PW + AF phase. 3