2 FIG. 1: (color online) Left panel: The unit cell of the graded Pro- jected Entangled-Pair State representation of the ground-state wave functions. Right panel: The pattern of the coexisting charge and spin density wave order. Here, the radius of the circles are proportional to the fillings at sites, whereas the arrows inside the circle s represent the directions and magnitudes of the spin density wave order param- eters. This represents a vertical commensurate stripe state, which breaks the four-fold rotation symmetry and the translation symmetry in the horizontal direction for charge density wave order and in both directions for spin density wave order. FIG. 2: (color online) The proposed ground-state phase diagram of the two-dimensional t − J model. First, for J / t ≥ 0 . 95, there is a line of phase separation (PS), whereas for J / t 0 . 95, no PS oc- curs. Second, the homogeneous regime is divided into four di ff erent phases: one phase with charge and spin density wave order coexist- ing with a p$_{x}$ ( p$_{y}$ $^{)}$−wave superconducting state (CDW + SDW + PW), one phase with the symmetry mixing of d + $^{s}$− wave superconduc- tivity in spin-singlet channel and p$_{x}$ ( p$_{y}$ $^{)}$−wave superconductivity in spin-triplet channel in the presence of an anti-ferromagnetic back- ground (DSW + PW + AF), one superconducting phase with an ex- tended $^{s}$− wave symmetry (SW), and one superconducting phase with a $^{p}$− wave symmetry (PW + F), besides a possible Nagaoka’s ferro- magnetic state (FM). Here, a systematic computation has been per- formed for both J / t = 0 . 4 and J / t = 0 . 8 with the truncation dimen- sion D = 6, whereas the dash lines separating di ff erent phases are a guide for the eyes. For J / t = 0 . 4, the DSW + PW + AF phase occurs for fillings from n = 0 . 818 (denoted as point a) to n = 0 . 765 (de- noted as point b), the SW phase occurs for fillings from n = 0 . 765 to n = 0 . 087 (denoted as point c), and the CDW + SDW + PW and PW + FM phases occur for fillings from n = 1 to n = 0 . 818 and from n = 0 . 087 to n = 0, respectively. For J / t = 0 . 8, the DSW + PW + AF phase occurs for fillings from n = 0 . 877 (denoted as point d) to n = 0 . 665 (denoted as point e), the SW phase occurs for fillings from n = 0 . 665 to n = 0 . 051 (denoted as point f), and the CDW + SDW + PW and PW + FM phases occur for fillings from n = 1 to n = 0 . 877 and from n = 0 . 051 to n = 0, respectively. and the ratio between the anti-ferromagnetic coupling and the hoping constant, respectively. At half filling, n = 1, the t − J model reduces to the two-dimensional Heisenberg spin 1 / 2 model. Therefore, a long range anti-ferromagnetic order ex- ists [24]. In this case, our algorithm yields the ground-state energy per site, e = − 1 . 1675 J , for the truncation dimension D = 4, and e = 1 . 1683 J , for the truncation dimension D = 6. This is comparable to the best QMC simulation re- sult: e = − 1 . 1694 J [24, 25], although the anti-ferromagnetic N´eel order moment is 0 . 36 versus 0 . 31. Away from half fill- ing, the model exhibits quite di ff erent behaviors for small and large values of the anti-ferromagnetic coupling J . For J / t ≥ 0 . 95, there is a line of PS between the Heisenberg anti-ferromagnetic state without hole and a hole-rich state, whereas for J / t ≤ 0 . 95, no PS occurs. This agrees qual- itatively with the results based on the HTE [11], the VMC method [13], and the DMRG [14]. Note that our result for the transition point J$_{c}$ = 3 . 45 t at low electron density is quite close to the exact value J$_{c}$ = 3 . 4367 t [15]. Here, the simula- tion has been performed for the truncation dimension D = 6. We point out that a discrepancy exists concerning the PS boundary of the t − J model. In Ref. [9], a combination of analytic and numerical calculations is used to establish the ex- istence of PS at all super-exchange interaction strengths. Al- though the simulation based on the Green’s function Monte Carlo method supports this scenario [10], many others [11– 14] found that the model phase separates only for J / t larger than some finite critical value around J / t ∼ 1. That is, PS oc- curs only outside the physically realistic parameter region of the t − J model. With the observation that no significant shift with the truncation dimension D increasing from D = 4 [21] to D = 6 is found, we conclude that PS does not occur for J / t ≤ 0 . 95. In the homogeneous regime, the two-dimensional t − J model exhibits very rich physics. Away from half filling, bot h the d + $^{s}$− wave superconductivity in the spin-singlet channel Charge and spin density wave order coexisting with a and the p$_{x}$ ( p$_{y}$ $^{)}$−wave superconductivity in the spin-triplet channel, and / or charge and spin density wave order, occur in di ff erent doping regimes for a fixed J / t . Here, a super- conducting state is characterized by a superconducting order parameter Δ ≡ 〈 ˆ $^{Δ}$〉 , with ˆ Δ defined as follows: For s -wave, ˆ Δ$_{s}$ = 1 / $_{(4}$√$_{2)}$ c + i$_{x}$ , $^{i$_{y}$}$↑ [ c + i$_{x}$ + 1 , $^{i$_{y}$}$↓ + c + $^{i$_{x}$}$− 1 , $^{i$_{y}$}$↓ + c + i$_{x}$ , i$_{y}$ + $^{1}$↓ + c + i$_{x}$ , $^{i$_{y}$}$− $^{1}$↓ $^{]}$−$^{[}$↑↔↓ ]; for d -wave, ˆ Δ$_{d}$ = 1 / $_{(4}$√$_{2)}$ c + i$_{x}$ , $^{i$_{y}$}$↑ [ c + i$_{x}$ + 1 , $^{i$_{y}$}$↓ + c + $^{i$_{x}$}$− 1 , $^{i$_{y}$}$↓ − c + i$_{x}$ , i$_{y}$ + $^{1}$↓ − c + i$_{x}$ , $^{i$_{y}$}$− $^{1}$↓ $^{]}$−$^{[}$↑↔↓]; for p$_{x}$ -wave, ˆ Δ$_{p}$$_{x}$ = ˆ $^{Δ$_{p}$$_{x}$$_{+}$}$− ˆ $^{Δ$_{p}$$_{x}$}$− , with ˆ $^{Δ$_{p}$$_{x}$}$± = 1 / 2 ( c + i$_{x}$ , $^{i$_{y}$}$↑ c + $^{i$_{x}$}$± 1 , $^{i$_{y}$}$↑ , [ c + i$_{x}$ , $^{i$_{y}$}$↑ c + $^{i$_{x}$}$± 1 , $^{i$_{y}$}$↓ + c + i$_{x}$ , $^{i$_{y}$}$↓ c + $^{i$_{x}$}$± 1 , $^{i$_{y}$}$↑ ] / √$_{2}$$_{,}$$_{c}$ + i$_{x}$ , $^{i$_{y}$}$↓ c + $^{i$_{x}$}$± 1 , i$_{y}$ $_{↓}$) , and a similar definition of ˆ Δ$_{p}$$_{y}$ for p$_{y}$ -wave. 0.0 1.7 3.4 0.00 0.25 0.50 0.75 1.00 J/t n PS CDW+SDW+PW FM DSW+PW+AF SW PW+F a e d c b f A B C D A B C D