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arXiv:cond-mat/9603031 · Physical Review Letters

Exact Single Spin Flip for the Hubbard Model in d=∞

Götz Silvester Uhrig

Published Oct 21, 1996• 21 citations
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Abstract

It is shown that the dynamics of a single $\ensuremath{\downarrow}$-electron interacting with a band of $\ensuremath{\uparrow}$-electrons can be calculated exactly in the limit of infinite dimension. The corresponding Green function is determined as a continued fraction. It is used to investigate the stability of saturated ferromagnetism and the nature of the ground state for two generic nonbipartite infinite dimensional lattices. Non-Fermi-liquid behavior is found. For certain dopings the $\ensuremath{\downarrow}$-electron is bound to the $\ensuremath{\uparrow}$-holes.

DOI
10.1103/physrevlett.77.3629

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It is shown that the dynamics of a single \ensuremath↓\ensuremath{\downarrow}\ensuremath↓-electron interacting with a band of \ensuremath↑\ensuremath{\uparrow}\ensuremath↑-electrons can be calculated exactly in the limit of infinite dimension. The corresponding Green function is determined as a continued fraction. It is used to investigate the stability of saturated ferromagnetism and the nature of the ground state for two generic nonbipartite infinite dimensional lattices. Non-Fermi-liquid behavior is found. For certain dopings the \ensuremath↓\ensuremath{\downarrow}\ensuremath↓-electron is bound to the \ensuremath↑\ensuremath{\uparrow}\ensuremath↑-holes.
We consider the question of ferromagnetism in the one-band Hubbard model on the square lattice with infinite on-site repulsion using exact diagonalization techniques. Our results show a nonmonotonic behavior as a function of the number of holes Nh{N}_{h}Nh​ away from half filling. We find that the spin per electron is generally finite but less than its maximum value and, for a fixed hole number, increases with increasing system size. We interpret this as indicating that the inifinite-U Hubbard model is ferromagnetic in the thermodynamic limit for a finite fraction of holes, but that there are nonferromagnetic states which are very close in energy.
The problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ 0 (1) (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ 0(n)<ɛ FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m.
A Hubbard model with infinite Coulomb repulsion is studied in a many-electron operator representation. A picture of the density of states is constructed using expressions for the one-particle Green’s functions of first order in 1/ z . Its behavior is studied near the Fermi level, especially Condon effects. The stability of saturated ferromagnetism is examined. The corresponding critical current carrier densities are found for semielliptical and rectangular seed densities of state, and for square and cubic lattices. These results are compared with earlier work.
The stability of the ferromagnetic state with complete spin alignment against a single spin reversal is studied in the square-lattice Hubbard model with nearest-neighbour hopping. A variational ansatz is used, which is exact in one dimension and by comparison with accurate results for small clusters yields almost exact results in the two-dimensional case. The ferromagnetic region in the (W/U, delta ) phase diagram is mapped out, where W is the bandwidth, U the on-site electron interaction and delta is the number of holes per atom. This region is considerably smaller than in previous variational calculations and it is rigorously concluded that the state of complete spin alignment is unstable when delta >0.29, for all U, and when W/U>0.19, for all delta . The nature of the instability, and of the low-lying excitations in the ferromagnetic state for varying delta and W/V, is discussed.
The infinite-U Hubbard model on finite two-dimensional square lattices is investigated by the use of exact numerical diagonalization. Configurations with one spin flip away from the ferromagnetic state and up to three holes are studied for lattice sizes ranging from 4*4 to 12*12. For all these situations, the state with one spin flip is found to be more stable than the Nagaoka ferromagnetic state. The crystal momentum of the ground state in this spin sector depends on the system size. Systematic comparison with the Gutzwiller wavefunction shows that it reproduces qualitatively the crystal momentum dependence of the energy, but not the exact correlation between the spin flip and the holes.