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Journal of Physics Condensed Matter

An exactly soluble one-dimensional quantum mechanical Heisenberg and Nagaoka t-J model

M W Long, R. Fehrenbacher

Published Mar 26, 1990• 15 citations
Publisher

Abstract

For a quasi-one dimensional sawtooth topology the authors show that the quantum mechanical ground state of the Heisenberg Hamiltonian has only nearest neighbour singlet correlations. The ground state is multiply degenerate with precisely as many ground states as atoms in the system. In fact they can associate each ground state with a special site that has a fixed spin. Allowing this special site to become a hole, in a frustrated t-J description, allows them to deduce the ground state for the motion of one hole in the presence of the Heisenberg interactions. When the hopping is unfrustrated they look at the competition between t and J and find a reasonable description for a 'polaron'. The polaron maintains low spin.

DOI
10.1088/0953-8984/2/12/003

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For a quasi-one dimensional sawtooth topology the authors show that the quantum mechanical ground state of the Heisenberg Hamiltonian has only nearest neighbour singlet correlations. The ground state is multiply degenerate with precisely as many ground states as atoms in the system. In fact they can associate each ground state with a special site that has a fixed spin. Allowing this special site to become a hole, in a frustrated t-J description, allows them to deduce the ground state for the motion of one hole in the presence of the Heisenberg interactions. When the hopping is unfrustrated they look at the competition between t and J and find a reasonable description for a 'polaron'. The polaron maintains low spin.
We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one-dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intra-pair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1d and 2d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole motion in the CuO 2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone.
We present a new class of variational wavefunctions that can be used to investigate quantum antiferromagnetic systems. The states are a natural generalization of the nearest-neighbour dimer phases which have been thoroughly studied in the literature. We describe a one-dimensional class of total-spin-singlet wavefunctions, which include as special cases the two nearest-neighbour dimer phases and the exact solution to the total-spin-two projector Hamiltonian. These different solutions are 'smoothly' interpolated by a range of singlets with short-range correlations. Although the states have only short-range order, we can obtain over 98% of the ground-state energy of the nearest-neighbour Heisenberg model. We apply our basis to the J 1 -J 2 Heisenberg model and the single hole in the t 1 -t 2 infinite-U Hubbard model, successfully predicting the behaviour observed in finite-size scaling calculations.
Abstract. We introduce a Hubbard model on a particular class of geometries, and consider the effect of doping the highly spin-degenerate Mott-insulating state with a microscopic number of holes in the extreme strong-coupling limit. The geometry is quite general, with pairs of atomic sites at each superlattice vertex, and a highly frustrated inter-atomic connectivity: the one dimensional realization is a chain of edge-sharing tetrahedra. The sole model parameter is the ratio of intrapair to inter-pair hopping matrix elements. If the intra-pair hopping is negligible then introducing a microscopic number of holes results in a ferromagnetic Nagaoka groundstate. Conversely, if the intra-pair hopping is comparable with the inter-pair hopping then the groundstate is low spin with short-ranged spin correlations. We exactly solve the correlated motion of a pair of holes in such a state and find that, in 1-d and 2-d, they form a bound pair on a length scale that increases with diminishing binding energy. This result is pertinent to the long-standing problem of hole-motion in the CuO2 planes of the high-temperature superconductors: we have rigorously shown that, on our frustrated geometry, the holes pair up and a short-ranged low-spin state is generated by hole motion alone. Submitted to: J. Phys. A: Math. Gen. 1.
We report on recent studies of the spin-half Heisenberg and the Hubbard model on the sawtooth chain. For both models we construct a class of exact eigenstates which are localized due to the frustrating geometry of the lattice for a certain relation of the exchange (hopping) integrals. Although these eigenstates differ in details for the two models because of the different statistics, they share some characteristic features. The localized eigenstates are highly degenerate and become ground states in high magnetic fields (Heisenberg model) or at certain electron fillings (Hubbard model), respectively. They may dominate the low-temperature thermodynamics and lead to an extra low-temperature maximum in the specific heat. The ground-state degeneracy can be calculated exactly by a mapping of the manifold of localized ground states onto a classical hard-dimer problem, and explicit expressions for thermodynamic quantities can be derived which are valid at low temperatures near the saturation field for the Heisenberg model or around a certain value of the chemical potential for the Hubbard model, respectively.
Abstract. We study the influence of on-site disorder on the magnetic properties of the ground state of the infinite-U one-dimensional (1D) Hubbard model. We find that the ground state is not ferromagnetic. This is analysed in terms of the algebraic structure of the spin dependence of the Hamiltonian. A simple explanation is derived for the 1=N periodicity in the persistent current for this model. The Hubbard model is the simplest model one can study to examine the effects of correlations between electrons in narrow energy bands. The Hamiltonian consists of a nearest neighbour hopping term and an electron–electron repulsion, U, which acts only when two electrons are on the same site. The Hubbard model is also the canonical model for the study of itinerant ferromagnetism. The strong coupling regime is of special importance for the study of ferromagnetism, since a theorem by Nagaoka [1] states that in theU D 1 limit, the ground state (GS) is ferromagnetic given some connectivity property of the lattice (which holds in most cases for d> 1). The model is solvable in one dimension and it was shown [2] that for open boundary conditions (BCs) the GS for finiteU is a singlet, i.e. there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. For infinite U, the GS of