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Physical review. B, Condensed matter

Phases of thet-Jmodel from variational Monte Carlo studies: Occurrence of time-reversal symmetry breaking

G. J. Chen, Robert Joynt, Fu‐Chun Zhang, Claudius Gros

Published Aug 1, 1990• 80 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.42.2662

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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 use a variational Monte Carlo technique to study the ground state of the two-dimensional Hubbard and t-J models on a square lattice. We use a trial wave function that allows a continuous description of the paramagnetic, antiferromagnetic, and superconducting phases, as well as the coexistence of these phases, with no a priori constraint on double occupancy. The phase diagram of both models is given for intermediate coupling (U=10t). We show that the apparent discrepancies between the two models, that appeared in previous variational Monte Carlo studies, are not present when this sufficiently good variational wave function is used. The two models are in good qualitative agreement. We find that except at half-filling a pure antiferromagnetic phase does not exist but is always in coexistence with a d-wave superconducting phase, and is followed by a pure d-wave superconducting phase at doping up to \ensuremath{\delta}\ensuremath{\sim}0.3. The staggered magnetization and the superconducting gap are measured, and a comparison with other analytical or numerical results is made.
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 investigate numerically the ground state of two holes with dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 symmetry in the two-dimensional t-J model, on lattices of significantly larger sizes than the ones studied so far. The Green's-function Monte Carlo method, in conjunction with suitable initial states and guidance functions, is used to estimate the ground-state energy of one and two holes, as well as to compute the two-hole distribution function. Our results show a significant decrease of the two-hole binding energy as the size of the lattice is increased from 4\ifmmode\times\else\texttimes\fi{}4. A critical value Jc{\mathit{J}}_{\mathit{c}}Jc​\ensuremath{\sim}0.27t is found such that d-wave hole binding no longer occurs for JJc{\mathit{J}}_{\mathit{c}}Jc​.
The authors have numerically evaluated the energy of the staggered flux phase state (SFP), the d-wave resonating valence bond state, and the mixed SFP and d-wave state (SFP+d) of the extended t-J model at various hole densities ( delta ). The flux per plaquette and the size of the d-wave gap are taken as variational parameters. The calculation includes the kinetic energy, spin energy, the effective hole-hole interaction and the three-site pair-hopping term. They found that the SFP is stable against the projected Fermi liquid up to delta approximately=20%. For moderately large J, J/t>or=0.4, the SFP+d state may be favoured against the pure d-wave state but only at very low hole density ( delta approximately 20%). These calculations show that the pure SFP state is never the ground state at hole densities of experimental interest.
We investigate the ground-state phase diagram of the two-dimensional Hubbard model based on the optimization variational Monte Carlo method. We use a wave function that is an off-diagonal type given as ψ=exp⁡(−λK)PGψ0ψ=\exp(-λK)P_Gψ_0ψ=exp(−λK)PG​ψ0​, where ψ0ψ_0ψ0​ is a one-particle state, PGP_GPG​ is the Gutzwiller operator, KKK is the kinetic operator, and λλλ is a variational parameter. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We examine the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion UUU, the band parameter t′t't′ and the electron density nen_ene​. We show a phase diagram that includes superconducting and antiferromagnetic phases and that t′=0t'=0t′=0 is most favorable for superconductivity.