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Physical review. B, Condensed matter

Superconducting flux phase of the t−J model

Gen Tatara, Tetsuo Matsui

Published Aug 1, 1991• 8 citations
Publisher

Abstract

The condensation pattern of hole pairs in the two-dimensional $t\ensuremath{-}J$ model is studied by use of gap equations. The ground state at small hole concentration is realized in the superconducting flux state of hole pairs. The parity symmetry is conserved.

DOI
10.1103/physrevb.44.2867

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arXiv:0906.5347•Physical Review B•Sep 29, 2009•26 citations
The condensation pattern of hole pairs in the two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model is studied by use of gap equations. The ground state at small hole concentration is realized in the superconducting flux state of hole pairs. The parity symmetry is conserved.
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.
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.
Using the factorization of the wave function in the $t\mathrm{\text{\ensuremath{-}}}J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ model at small exchange couplings, we demonstrate the connection between the existence of a spin gap and an hc/2ehc/2ehc/2e flux periodicity of the ground state energy. We conjecture that all spin-gapped SU(2)-invariant Luttinger liquids have hc/2ehc/2ehc/2e flux periodicity, and that this is connected to the fact that a gapped spin-12\frac{1}{2}21​ chain always breaks translational symmetry by doubling the unit cell.
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 compute the ground-state fidelity and various correlations to gauge the competition between different orders in two-dimensional $t\text{\ensuremath{-}}J$-type models. Using exact numerical diagonalization techniques, these quantities are examined for (i) the plain $t\text{\ensuremath{-}}J$ and $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ models, (ii) for the $t\text{\ensuremath{-}}J$ model perturbed by infinite-range ddd-wave or extended-sss-wave superconductivity inducing terms, and (iii) the $t\text{\ensuremath{-}}J$ model, plain and with a ddd-wave perturbation, in the presence of nonmagnetic quenched disorder. Various properties at low hole doping are contrasted with those at low electron filling. In the clean case, our results are consistent with previous work that concluded that the plain $t\text{\ensuremath{-}}J$ model supports ddd-wave superconductivity. As a consequence of the strong correlations present in the low hole doping regime, we find that the magnitude of the ddd-wave condensate occupation is small even in the presence of large ddd-wave superconductivity inducing terms. In the dirty case, we show the robustness of the ground state in the strongly correlated regime against disorder.