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Physical review. B, Condensed matter

Energy-level ordering in the one-dimensionalt-Jmodel: A rigorous result

Tao Xiang, N. d’Ambrumenil

Published Jul 1, 1992• 8 citations
Publisher

Abstract

Two theorems about the spin properties of the ground state and the ordering of the energy levels for different spins of the one-dimensional t-J model are stated and proved. It is shown that when the number of electrons N=4n+2 (n an integer) with periodic boundary conditions or N=4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet.

DOI
10.1103/physrevb.46.599

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Two theorems about the spin properties of the ground state and the ordering of the energy levels for different spins of the one-dimensional t-J model are stated and proved. It is shown that when the number of electrons N=4n+2 (n an integer) with periodic boundary conditions or N=4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet.
A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N=4n+2 (n an integer) with periodic boundary conditions or N=4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.
A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N = 4n + 2 (n an integer) with periodic boundary conditions or N = 4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.
Ground states of the two-dimensional t - t '- J model with one and two holes in 4×4 sites (with periodic boundary conditions) are studied numerically. Using a Lanczos algorithm, we calculate the ground-state energy and momentum distribution, and the phase diagram in the plane of J / t and t '/ t is presented. The momentum distribution for J / t =0.25 is compared with the Fermi surface obtained in the mean-field theory. It is shown that the t '/ t dependences of the momentum distribution are consistent with the mean-field approximation, although there are quantitative differences between them.
The spin-gap phase of the one-dimensional t-J-J' model is studied by the level-crossing of the singlet and the triplet excitation spectra. The phase boundary obtained between the Tomonaga-Luttinger and the spin-gap phases is remarkably consistent with the analytical results at the J,J′→0J,J'\to 0J,J′→0 and the low-density limits discussed by Ogata et al. The spin-gap phase has a single domain in the phase diagram even if the spin gap opens at half-filling. The phase boundary coincides with the Kρ=1K_ρ=1Kρ​=1 line where the Tomonaga-Luttinger liquid behaves as free electrons, in the low-density region. The relation between our method and the solution of the two-electron problem is also discussed.
A new phase of a gas of pairs of electrons bounded in a singlet state is found in the one-dimensional t-J model for J>2t and the density of electrons less than 0.2. This phase was conjectured in the study of the diagonalization of small lattices [Phys. Rev. Lett. 66, 2388 (1991)]. The existence of this new phase for much larger lattice sizes is demonstrated by a combination of two numerical methods, the variational Monte Carlo and the power method. A trial wave function for this phase is proposed and shown to be in good agreement with the ground state obtained by the power method.