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arXiv:1001.3343

Ground-state phase diagram of the two-dimensional t-J model

Sheng-Hao Li, Qian‐Qian Shi, Huan-Qiang Zhou

Published Jan 19, 2010• 3 citations
arXivHTMLPDF

Abstract

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=0.4t and J=0.8t, 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 different 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.

DOI
10.48550/arxiv.1001.3343

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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=0.4t and J=0.8t, 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 different 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.
By introducing the concepts and methods of machine learning, the traditional tensor network algorithm is upgraded to the machine learning tensor network algorithm in order to obtain more accurate results in complex systems. Under the context of the virtual time evolution tensor network algorithm established in the graded projection entangled pair state (gPEPS) representation of the two-dimensional strongly correlated electron infinite fermion system based on machine learning, this paper mainly investigates the quantum phase diagram, quantum phase transition and quantum entanglement of the two-dimensional t-J model related to high temperature superconductivity through the ground state energy. The research results showed that the t-J model has a phase separation line between the Heisenberg antiferromagnetic state with no vacancies and the state rich in vacancies, with the upper critical point at Jc=0.95t and the lower critical point at Jc=3.45t. It was found that at J=0.4t, in addition to the potential long-range ferromagnetic state, four different phases were composed in the different doping uniform regions: one wasis the superconducting phase with coexistence of charge and spin density waves and px(py) wave symmetry, one wasis the mixed superconducting phase with d+s wave superconducting pairing and spin triplet px(py) wave superconducting symmetry in the antiferromagnetic background, one wasis the superconducting phase with extended s-wave symmetry, and the other one iswas the superconducting phase with px(py) wave symmetry in the ferromagnetic background.
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We present a density matrix renormalization group (DMRG) study of an extended t−Jt-Jt−J model with hopping to the first and second neighbors -- the one dimensional t1−t2−Jt_1-t_2-Jt1​−t2​−J model. The full phase diagram as a function of the density nnn and exchange strength JJJ, for both positive and negative values of t2t_2t2​, is obtained. For t2=−0.5t_2=-0.5t2​=−0.5 we observe that, in the strongly interacting region, Nagaoka Ferromagnetism (FM) is accompanied by a triplet pair density wave (PDW) upon doping. As the spin exchange JJJ increases, a charge density wave (CDW) phase emerges and then gives way to singlet superconductivity (SC). This phase behaves as a singlet PDW with vanishing spacial average of the order parameter and a spin gap. When t2=0.5t_2=0.5t2​=0.5, the physics is basically reminiscent of the conventional t−Jt-Jt−J model, undergoing a transition from a metallic to a SC phase as a function of JJJ.
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We have numerically evaluated the energy of several kinds of wave functions considered to be candidate ground states of the two-dimensional t-J model at various hole densities. We searched a parameter space which includes d-wave and s-wave superconductivity and spin-density-wave ordering as well as the projected Fermi-liquid state. Coexistence of different orderings, such as the s+id state and d-wave spin-density-wave state, were found to be stable states. We find a phase diagram in the density-t/J plane which has coexistence of antiferromagnetism and superconductivity at very low hole concentrations and superconductivity up to rather high values of density---about 40%. At intermediate concentrations, the time-reversal symmetry-breaking s+id state is found.
Two-dimensional t-J model is studied by a variational Monte Carlo method, using Gutzwiller-Jastrow-type wave functions. Various kinds of superconducting pairing symmetries are compared in order to determine the phase diagram of the ground state in the full J/t-n plane. Near the half filling where the high temperature superconductivity is expected, the d_{x^2-y^2} wave pairing state is always the most stable among various symmetries. The three-site term hardly changes the phase diagram in this regime. In the low electron density, the extended s-type wave becomes a quantitatively good state for large J/t, although the energy gain is small. The Gutzwiller wave function is shown to be the exact ground state in the low-electron-density limit for the supersymmetric case (J/t=2).
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/81/81/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/51/51/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.