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arXiv:cond-mat/9607126 · Physical review. B, Condensed matter

Nagaoka state and degeneracy in theU=∞ Hubbard model

V. Subrahmanyam, Fajian Shi

Published Oct 1, 1996• 0 citations
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Abstract

The U=\ensuremath{\infty} Hubbard model on a hypercubic lattice with exactly one hole is studied. It is seen that a degeneracy can occur in the maximal-spin sector. It is proved variationally that the Nagaoka (maximal spin) state is not a ground state, if degeneracy exists. We give an explicit construction of a variational state, an incommensurate spin-wave state, with energy lower than that of the Nagaoka state. For the case of more than one hole, a few examples are discussed where a degeneracy exists. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.9011

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The U=\ensuremath{\infty} Hubbard model on a hypercubic lattice with exactly one hole is studied. It is seen that a degeneracy can occur in the maximal-spin sector. It is proved variationally that the Nagaoka (maximal spin) state is not a ground state, if degeneracy exists. We give an explicit construction of a variational state, an incommensurate spin-wave state, with energy lower than that of the Nagaoka state. For the case of more than one hole, a few examples are discussed where a degeneracy exists. \textcopyright{} 1996 The American Physical Society.
We present an extension of Nagaoka's theorem in the SU(nnn) generalization of the infinite-UUU Hubbard model. It is shown that, when there is exactly one hole, the fully polarized states analogous to the ferromagnetic states in the SU(2) Hubbard model are ground states. For a restricted class of models satisfying the connectivity condition, these fully polarized states are the unique ground states up to the trivial degeneracy due to the SU(nnn) symmetry. We also give examples of lattices in which the connectivity condition can be verified explicitly. The examples include the triangular, Kagom\'e, and hypercubic lattices in $d\phantom{\rule{0.28em}{0ex}}(\ensuremath{\ge}\phantom{\rule{-0.16em}{0ex}}2)$ dimensions, among which the cases of d=2d=2d=2 and 3 are experimentally realizable in ultracold atomic gases loaded into optical lattices.
In this paper, we have rigorously shown that, in a thermodynamic limit, there always exists a one-spin-flipped state degenerate with the Nagaoka state in one-band Hubbard model on a two and three-dimensional simple cubic lattice when the number of holes [Formula: see text] (with 0≤α<1, N∧ and N e are the numbers of sites and electrons respectively) for all U (including U=∞). It is also shown that there does not exist a finite critical value U c for which, when U>U c , the Nagaoka state has the ground-state energy of the system for a two-dimensional square lattice when ρ→0.
Abstract We discuss two single spin flip variational wave functions describing spin wave excitations which were proposed earlier by Shastry, Krishnamurthy and Anderson (SKA) and by Basile and Elser (BE), respectively, in order to investigate the instability of the fully polarized ferromagnetic state (Nagaoka state) in the infinite U Hubbard model. We calculate the energy of these variational states for the square lattice and for multiple chains. At the zone boundary in the vicinity of the point (0, π) the spin wave energy is reduced substantially by the binding of the spin up hole to the flipped down spin. For the square lattice this leads to a critical hole density of δcr = 0.407 for the SKA spin wave and of δcr = 0.322 for the BE spin wave which implies remarkable improvements in comparison to the corresponding scattering states investigated previously.
A variational approach has been used to study the stability of the Nagaoka state in the Hubbard model with infinite interaction. In this approach, real Fermi operators are replaced by auxiliary ones; an effective Hamiltonian is constructed in terms of these auxiliary operators. A two-chain (ladder) system is considered as an example. For this model, we show that the Nagaoka state is unstable with respect to the creation of two-phase states: ferromagnetic and singlet spin states. The instability occurs at a finite density of holes.