GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s50>GLYPH<s53> GLYPH<s48>GLYPH<s46> GLYPH<s53>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s55>GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s51> GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s54> GLYPH<s110> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s52> GLYPH<s115> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s52> GLYPH<s100>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s52> GLYPH<s112> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s52> GLYPH<s112> GLYPH<s65>GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s52> GLYPH<s112> GLYPH<s66>GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s110>GLYPH<s61>GLYPH<s48> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s56>GLYPH<s32> GLYPH<s115> GLYPH<s32>GLYPH<s32>GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s56> GLYPH<s100> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s56> GLYPH<s112> GLYPH<s32> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s56> GLYPH<s112> GLYPH<s65> GLYPH<s32> GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46> GLYPH<s56> GLYPH<s112> GLYPH<s66> GLYPH<s32> GLYPH<s110>GLYPH<s61>GLYPH<s49>GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s32> GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s50>GLYPH<s53> GLYPH<s48>GLYPH<s46> GLYPH<s53>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s55>GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s48>GLYPH<s48> GLYPH<s48>GLYPH<s46> GLYPH<s50>GLYPH<s50> GLYPH<s48>GLYPH<s46> GLYPH<s52>GLYPH<s52> GLYPH<s67>GLYPH<s68>GLYPH<s87>GLYPH<s47>GLYPH<s83>GLYPH<s68>GLYPH<s87>GLYPH<s32>GLYPH<s79> GLYPH<s114> GLYPH<s100> GLYPH<s101> GLYPH<s114> GLYPH<s32>GLYPH<s80>GLYPH<s97> GLYPH<s114> GLYPH<s97> GLYPH<s109> GLYPH<s101> GLYPH<s116>GLYPH<s101> GLYPH<s114> GLYPH<s115>GLYPH<s32> GLYPH<s110> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s52>GLYPH<s32>GLYPH<s67> GLYPH<s68> GLYPH<s87> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s52>GLYPH<s32>GLYPH<s83>GLYPH<s68> GLYPH<s87> GLYPH<s65> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s52>GLYPH<s32>GLYPH<s83>GLYPH<s68> GLYPH<s87> GLYPH<s66> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s52>GLYPH<s32>GLYPH<s70> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s52>GLYPH<s32>GLYPH<s65>GLYPH<s70> GLYPH<s32>GLYPH<s110>GLYPH<s61>GLYPH<s48>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s70> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s56>GLYPH<s32>GLYPH<s67> GLYPH<s68> GLYPH<s87> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s56>GLYPH<s32>GLYPH<s83>GLYPH<s68> GLYPH<s87> GLYPH<s65> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s56>GLYPH<s32>GLYPH<s83>GLYPH<s68> GLYPH<s87> GLYPH<s66>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s56>GLYPH<s32>GLYPH<s70>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32> GLYPH<s32>GLYPH<s74>GLYPH<s61>GLYPH<s48>GLYPH<s46>GLYPH<s56>GLYPH<s32>GLYPH<s65>GLYPH<s70> GLYPH<s32>GLYPH<s110>GLYPH<s61>GLYPH<s49>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s65>GLYPH<s70> GLYPH<s32> FIG. 3: (color online) Upper panel: The dependence of the d -wave, s -wave, and p$_{x}$ ( p$_{y}$ $^{)}$−wave superconducting order parameters, Δ$_{d}$ , Δ$_{s}$ and Δ$_{p}$ , on electron filling n . Here, we evaluate Δ$_{s}$ , Δ$_{d}$ and Δ$_{p}$ for J / t = 0 . 4 and J / t = 0 . 8. Lower panel: The dependence of the spin density wave (SDW) and the charge density wave (CDW) order pa- rameters in the CDW + SDW + PW phase, the anti-ferromagnetic N´eel order parameter in the DSW + PW + AF phase, and the ferromagnetic order parameter in the PW + F phase on electron filling n . Note that, in the CDW + SDW + PW 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). eters being read o ff from both one-site and two-site reduced density matrices according to a general scheme advocated in Ref. [28]. In summary, we have investigated the ground-state phase diagram of the two-dimensional t − J model in the context of the tensor network algorithm. The relevance of our simulation results to the high temperature cuprate superconductors will be discussed in a forthcoming publication [29]. This work is supported in part by the National Natural Science Foundation of China (Grant Nos: 10774197 and 10874252). 4 [1] J.G. Bednorz and K.A. M¨uller, Z. Phys. B 64 , 189 (1986). [2] P.W. Anderson, Science 235 , 1196 (1987). [3] F.C. Zhang and T.M. Rice, Phys. Rev. B 37 , 3759 (1988). [4] G. Baskaran and P.W. Anderson, Phys. Rev. B 37 , 580 (1988). [5] G. Baskaran, Z. Zou, and P.W. Anderson, Solid State Comm. 63 , 973 (1987). [6] I. A ffl eck and J.B. Marston, Phys. Rev. B 37 , 3774 (1988). [7] F.C. Zhang, C. Gros, T.M. Rice, and H. Shiba, Supercond. Sci. Tech. 1 , 36 (1988). [8] E. Dagotto and J. Riera, Phys. Rev. Lett. 70 , 682 (1993); for a review, see, E. Dagotto, Rev. Mod. Phys. 66 , 763 (1994). [9] V.J. Emery, S.A. Kivelson, and H.Q. Lin, Phys. Rev. Lett. 64 , 475 (1990). [10] C.S. Hellberg and E. Manousakis, Phys. Rev. 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[19] For a review, see P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78 , 17 (2006). [20] S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E. Gu- bernatis, and R. T. Scalettar, Phys. Rev. B 40 , 506 (1989); E. Dagotto, A. Moreo, F. Ortolani, D. Poilblanc, and J. Riera Phys. Rev. B 45 , 10741 (1992); A. Moreo, Phys. Rev. B 45 , 5059 (1992). [23] The presence of such a ferromagnetic state is assumed as a re- sult of the variation Monte Carlo method, see H. Yokoyama and H. Shiba, J. Phys. Soc. Japan 56 , 3570 (1987), which is [22] J. Jordan, R. Or´us, G. Vidal, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 101 , 250602 (2008). [21] Qian-Qian Shi, Sheng-Hao Li, Jian-Hui Zhao, and Huan-Qiang Zhou, arXiv:0907.5520. two-dimensional t − J model. For J / t = 0 . 4, the model undergoes a QPT from the CDW + SDW + PW phase to the DSW + PW + AF phase at n = 0 . 818, and a QPT from the DSW + PW + AF phase to the SW phase at n = 0 . 765, and again a QPT from the SW phase to the PW + F phase at n = 0 . 087. For J / t = 0 . 8, the model undergoes a QPT from the CDW + SDW + PW phase to the DSW + PW + AF phase at n = 0 . 877, and a QPT from the DSW + PW + AF phase to the SW phase at n = 0 . 665, and again a QPT from the SW phase to the PW + F phase at n = 0 . 051. In addition, a QPT oc- curs between the DSW + PW + AF phase and the PW + F phase. Here, all QPTs are first-order, as one may judge from the be- haviors of the order parameters shown in Fig. 3. In passing, we point out that the above results have been deduced from the ground-state fidelity approach in the context of tensor n et- work representations [27], with superconducting order param-