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Acta Physica Polonica A

S-Wave Bound- and Resonant States of Two Fermions in Simple Cubic Lattice

M. Bąk

Published Apr 1, 2012• 1 citation
Publisher

Abstract

Resonant two-electron states are examined in attractive Hubbard model on simple cubic lattice and exact formula for scattering cross section in the limit of low density (empty lattice) is calculated.S-wave pair is considered by means of lattice Green functions (LGF).Analytical form of these functions found by Joyce is used facilitating calculations, which were greatly hindered before by the necessity of using LGF's tabulated values.It is found that the actual peak of scattering cross-section is formed on the lower band boundary in discrepancy with formulae of the theory of scattering in solids.

DOI
10.12693/aphyspola.121.808

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Resonant two-electron states are examined in attractive Hubbard model on simple cubic lattice and exact formula for scattering cross section in the limit of low density (empty lattice) is calculated.S-wave pair is considered by means of lattice Green functions (LGF).Analytical form of these functions found by Joyce is used facilitating calculations, which were greatly hindered before by the necessity of using LGF's tabulated values.It is found that the actual peak of scattering cross-section is formed on the lower band boundary in discrepancy with formulae of the theory of scattering in solids.
The properties of resonant two-electron pairs are examined in three-dimensional simple cubic extended Hubbard tUW model, in sand dsymmetry channels, in the limit of empty lattice.Lattice Green function technique is applied to calculate the scattering amplitude by means of T matrix for any energy in the band on Γ R line within the Brillouin zone.The analysis shows the persistence of resonant states even for Coulomb interactions so weak that the equation for the resonant states is not formally fullled.The dierence in behavior of resonances controlled by on-site and intersite Coulomb interactions is found.The ranges of applicability of approximate formula for the actual position of resonant peak and its width are shown.
We numerically study the problem of two fermions in a three-dimensional optical lattice interacting via a zero-range Feshbach resonance and display the dispersions of the bound states as a two-particle band structure with unique features compared to typical single-particle band structures. We show that the exact two-particle solutions of a projected Hamiltonian may be used to define an effective two-channel, few-band model for the low-energy, low-density physics of many fermions at arbitrary s-wave scattering length. Our method applies to resonances of any width and can be adapted to multichannel situations or higher-ℓ pairing. In strong contrast to usual Hubbard physics, we find that pair hopping is significantly altered by strong interactions and the presence of the lattice, and the lattice induces multiple molecular bound states.
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We study the effective Bloch-wave scattering of a spinless Fermi gas in one-dimensional (1D) optical lattices. By tuning the odd-wave scattering length, we find multiple resonances of Bloch waves scattering at the bottom (and the top) of the lowest band, beyond which an attractive (and a repulsive) two-body bound state starts to emerge. These resonances exhibit comparable widths in the deep lattice limit, and the finite interaction range plays an essential role in determining their locations. Based on exact two-body solutions, we construct an effective microscopic model for the low-energy scattering of fermions. The model can reproduce not only the scattering amplitudes of Bloch waves at the lowest-band bottom or top, but also the attractive or repulsive bound states within a reasonably large energy range below or above the band. These results lay the foundation for quantum simulating topological states in cold Fermi gases confined in 1D optical lattices.
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A three-body scattering theory previously proposed by one of the present authors is developed to be applied to the saturated ferromagnetic state in the two-dimensional Hubbard model. The single-particle Green's function is calculated by taking account of the multiple scattering between two electrons and one hole. Several limiting cases are discussed and the relation to the variational principle is examined. The importance of the three-body correlation is demonstrated in comparison with the results of the ladder approximation. A possible phase boundary for the Nagaoka ground state is presented for the square lattice, which improves the previous variational results.