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Physical review. B, Condensed matter

Exact-diagonalization study of thet-Jmodel in the low-density limit: Implications for phase separation

V. Waas, Holger Fehske, H. Büttner

Published Sep 15, 1993• 10 citations
Publisher

Abstract

The ground state of the two-dimensional (2D) t-J model is studied by exact diagonalization for seven different unit cells with N=10, 16, 18, 20, 26, 32, and 36 sites on a square lattice. In the limit of low electron densities (up to six electrons) we calculate the lowest eigenvalues and classify the ground state according to wave vector K, total spin S, and point-group symmetry both in the limit J=0 (U=\ensuremath{\infty} Hubbard model) and for finite J>0. The transition to a completely phase-separated state for J>${\mathit{J}}_{\mathrm{PS}}$(N) is examined by calculating the free energy as a function of the electron concentration. We find that ${\mathit{J}}_{\mathrm{PS}}$(N) increases with the lattice size N. An extrapolation of our finite-size results to the thermodynamic limit gives ${\mathit{J}}_{\mathrm{PS}}$(\ensuremath{\infty})/t=4.1 (\ifmmode\pm\else\textpm\fi{}0.1).

DOI
10.1103/physrevb.48.9106

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The ground state of the two-dimensional (2D) t-J model is studied by exact diagonalization for seven different unit cells with N=10, 16, 18, 20, 26, 32, and 36 sites on a square lattice. In the limit of low electron densities (up to six electrons) we calculate the lowest eigenvalues and classify the ground state according to wave vector K, total spin S, and point-group symmetry both in the limit J=0 (U=\ensuremath{\infty} Hubbard model) and for finite J>0. The transition to a completely phase-separated state for J>JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) is examined by calculating the free energy as a function of the electron concentration. We find that JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) increases with the lattice size N. An extrapolation of our finite-size results to the thermodynamic limit gives JPS{\mathit{J}}_{\mathrm{PS}}JPS​(\ensuremath{\infty})/t=4.1 (\ifmmode\pm\else\textpm\fi{}0.1).
Exact many-body states of square clusters up to 18 sites (with periodic boundary conditions) are investigated in order to study the dynamics of one or two holes in the t-J model. Our method takes full advantage of the translation, rotation, and reflection symmetries of the cluster. Using a modified Lanczos algorithm, we calculate the lowest state of all the different space-group representations and its total spin as well as its spin and hole correlations. For a single hole, at intermediate J/t values, the magnetic, kinetic energies, and staggered magnetization follow J/t power-law behaviors with size-dependent exponents. A binding energy of order J between two holes appears for sufficiently large J/t(\ensuremath{\ge}0.25). One-hole and two-hole calculations give different results regarding the stability of the Nagaoka state (against the singlet state) when J/t\ensuremath{\rightarrow}0. Energy expectation values of variational resonating-valence-bond states are calculated exactly and optimized. Significant overlaps with the exact states are found for any symmetry.
The t-t'-J model (strongly interacting limit of a particular Hubbard model) is solved exactly on small clusters of eight sites with periodic boundary conditions for the simple-, body-centered-, and face-centered-cubic lattices and for the two-dimensional square lattice. The symmetry, k vector, and spin of the ground state are studied as functions of crystalline environment, interaction strength, and electron concentration. Phase diagrams are presented for stable solutions, and regions of parameter space that exhibit ferromagnetism and heavy-fermionic behavior are identified.
The spin- and density-correlation functions of the two-dimensional Hubbard model at low electronic density 〈n〉 are calculated in the ground state by using the power method, and at finite temperatures by using the quantum Monte Carlo technique. Both approaches produce similar results, which are in close agreement with numerical and high-temperature-expansion results for the two-dimensional t-J model. Using perturbative approximations, we show that the examination of the density-correlation function alone is not enough to support recent claims in the literature that suggested spin and charge separation in the low electronic density regime of the t-J model.
Using exact diagonalization, we calculate the density of states of the two-dimensional Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 square lattice at U/t=0.5, 4, and 10, and even number of electrons with filling factors n ranging from a quarter filling up to half filling. We compare the ground-state energy and density of states at U/t=0.5 and 4 with second-order perturbation theory in U/t in the paramagnetic phase, and find that while the agreement is reasonable at U/t=0.5, it becomes worse as the perturbatively determined (i.e., using Stoner's criterion) boundary of the paramagnetic to spin-density-wave instability is approached. In the strong coupling regime (U/t=10), we find reasonable agreement between the density of states of the Hubbard and the t-J model especially for low doping fractions. In general, we find that at half filling the filled states are separated from the empty states by a gap. At U/t=10, the density of states shows two bands clearly separated by a Mott-Hubbard gap of order U.
The ground-state properties of the periodic Anderson model on one-dimensional chains from three to six sites are studied by exact diagonalization. An appropriate average over boundary conditions is found to be necessary in order to eliminate finite-size oscillations. In the weak-coupling limit, the results for the short-range correlations suggest a smooth transition from a ferromagnetic state at low electron concentration to an antiferromagnetic state at half-filling, rather than a sudden jump at a critical concentration.
We exactly diagonalize the effective Hamiltonian obtained from the Hubbard model in the strong-coupling limit for a two-dimensional $\sqrt{10}\ifmmode\times\else\texttimes\fi{}\sqrt{10}$ size square lattice. The effective Hamiltonian operates on a restricted Hilbert space containing only states with singly occupied sites, which makes the diagonalization possible within reasonable computational time for the above lattice. The ground-state energy and wave function are obtained for several values of the coupling ratio tU\frac{t}{U}Ut​ and the doping fraction xxx. We find three different phases in the (x,tUx,\frac{t}{U}x,Ut​) phase diagram. The first is characterized by antiferromagnetic order which extends to the longest possible distance in the 10-site system. The second and third phases are characterized by antiferromagnetic and ferromagnetic short-range correlations, respectively. We comment on the possible relevance of the results to the recently discovered high-temperature copper-oxide superconductors.