Ground-state phase diagram of the two-dimensional t − J model Sheng-Hao Li, 1 Qian-Qian Shi, 1 and Huan-Qiang Zhou 1 $^{1}$Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China The ground-state phase diagram of the two-dimensional $^{t}$− J model is investigated in the context of the tensor network algorithm in terms of the graded Projected Entangled-Pair State representation of the ground-state wave functions. There is a line of phase separation between the Heisenberg anti-ferromagnetic state without hole and a hole-rich state. For both J = 0 . 4 t and J = 0 . 8 t , a systematic computation is performed to identify all the competing ground states for various dopings. It is found that, besides a possible Nagaoka’s ferromagnetic state, the homogeneous regime consists of four di ff erent phases: one phase with charge and spin density wave order coexisting with a p$_{x}$ ( p$_{y}$ $^{)}$−wave superconducting state, one phase with the symmetry mixing of d + $^{s}$− wave superconductivity in the spin-singlet channel and p$_{x}$ ( p$_{y}$ $^{)}$−wave superconductivity in the spin-triplet channel in the presence of an anti-ferromagnetic background, one superconducting phase with extended $^{s}$− wave symmetry, and one superconducting phase with p$_{x}$ ( p$_{y}$ $^{)}$−wave symmetry in a ferromagnetic background. PACS numbers: 74.20.-z, 02.70.-c, 71.10.Fd Since the discovery of high temperature superconductiv- ity [1], significant progress has been made in our understand - ing of strong correlation physics. It was Anderson [2] who re- alized the importance of Mott-Hubbard insulators and put for- ward the resonating valence bond picture as a promising route towards the understanding of an electron pairing mechanism responsible for an unprecedented high transition temperature, which are observed for copper oxides (cuprates). Actually, the detailed analysis of electronic states deduced from exper- iments shows that the undoped parent compound is a Mott- Hubbard insulator and the hole doping is mainly on oxygen sites, with its e ff ective low energy physics described by the two-dimensional t − J model [3]. Although a lot of e ff orts have been made to gain a full pic- ture of the underlying physics of the two-dimensional t − J model (see, e.g., [4–17]), no consensus has been achieved as to the question whether or not the two-dimensional $^{t}$−J model superconducts. On the one hand, the variational Monte Carlo (VMC) method clearly indicates that the d$_{x}$ $^{2}$− y 2 − wave super- conductivity is stable at absolute zero temperature for a phys- ically realistic coupling parameter regime [18], and field t he- oretical slave-boson approximation yields qualitatively many peculiar phenomenological features of cuprate superconduc- tors [19]. On the other hand, exact diagonalization (ED) of the t − J model on a small cluster and quantum Monte Carlo (QMC) simulation of the two-dimensional Hubbard model do not produce convincing evidence supporting the existence of superconductivity in the t − J model within a physically rele- vant regime [8, 20]. In this paper, we attempt to investigate the ground-state phase diagram of the two-dimensional $^{t}$− J model by exploit- ing a newly-developed tensor network algorithm [21] in terms of the graded Projected Entangled-Pair State (gPEPS) repre- sentation of the ground-state wave functions (for an ungraded version, see [22]). It is found that, the algorithm yields reli- able results for the two-dimensional t − J model at and away from half filling, with the truncation dimension up to 6. We are able to locate a line of phase separation (PS) between the Heisenberg anti-ferromagnetic state without hole and a hole- rich state, which qualitatively agrees with the results based on the high-temperature expansions (HTE) [11], the VMC [13], and the density-matrix renormalization method (DMRG) [14]. In the homogeneous regime, the two-dimensional t − J model exhibits very rich physics. Away from half filling, the regim e is divided into four di ff erent phases: one phase with charge and spin density wave order coexisting with a p$_{x}$ ( p$_{y}$ $^{)}$−wave superconducting state (CDW + SDW + PW), one phase with the symmetry mixing of d + $^{s}$− wave superconductivity in the spin-singlet channel and p$_{x}$ ( p$_{y}$ $^{)}$−wave superconductiv- ity in the spin-triplet channel in the presence of an anti- ferromagnetic background (DSW + PW + AF), one supercon- ducting phase with extended $^{s}$− wave symmetry (SW), and one superconducting phase with p$_{x}$ ( p$_{y}$ $^{)}$−wave symmetry in a ferromagnetic background (PW + FM), besides a possible Na- gaoka’s ferromagnetic state (FM) [23]. Ground-state phase diagram of the two-dimensional t − J model. The two-dimensional t − J model is described by the Hamiltonian [8] consisting of a hopping term and a super- exchange interaction: where S$_{i}$ are spin 1 / 2 operators at site i $^{on a square lattice,}$P is the projection operator excluding double occupancy, and t and J are, respectively, the hoping constant and anti-ferromagnetic coupling between the nearest neighbor sites 〈 $^{i j}$〉 . The model is simulated by exploiting the gPEPS tensor net- work algorithm [21], with the truncation dimension up to 6. Here, we stress that, although the algorithm is variational in nature, an assumption has to be made on the choice of the unit cell of the gPEPS representation of the ground-state wave functions. For the t − J model, we have chosen the plaquette as the unit cell (see Fig. 1, left panel). In Fig. 2, we plot the ground-state phase diagram for the two-dimensional t − J model, with the vertical and horizon- tal axes, n and J / t , denoting the number of electrons per site H = − t ∑ 〈 $^{ij}$〉 σ $^{[}$P( $_{c}$† i $_{σ}$c$_{j}$$_{σ}$ + H . c . $^{)}$P] + J ∑ 〈 $^{ij}$〉$^{(}$$^{S$_{i}$}$ · $^{S$_{j}$}$− 1 4 n$_{i}$ n$_{j}$ ) , (1) arXiv:1001.3343v1[cond-mat.supr-con]19 Jan 2010