A description of the Hubbard model on a square lattice consistent with its global $SO(3)\times SO(3)\times U(1)$ symmetry

Abstract

In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral t, on-site repulsion U , and N 2 a ≫ 1 sites consistent with its exact global SO(3) × SO(3) × U (1) symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron -rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For U/4t > 0 the occupancy configurations of these objects generate the energy eigenstates that span the one-and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spindensity ground state of one-and two-electron operators. Our description involves three basic objects: charge c fermions, spin-1/2 spinons, and η-spin-1/2 η-spinons. Independent spinons and independent η-spinons are invariant under the above unitary transformation. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the η-spinon (and spinons) that are not invariant under that transformation have η spin 1/2 (and spin 1/2) but are part of η-spinneutral (and spin-neutral) 2ν-η-spinon (and 2ν-spinon) composite ην fermions (and sν fermions) where ν = 1, 2, ... is the number of η-spinon (and spinon) pairs. The occupancy configurations of the c fermions, independent spinons and 2ν-spinon composite sν fermions, and independent η-spinons and 2ν-η-spinon composite ην fermions correspond to the state representations of the U (1), spin SU (2), and η-spin SU (2) symmetries, respectively, associated with the model2 global symmetry. The components of the αν fermion discrete momentum values qj = [qj x1 , qj x2 ] are eigenvalues of the corresponding set of αν translation generators in the presence of fictitious magnetic fields Bαν. Our operator description has been constructed to inherently the αν translation generators ˆ q αν in the presence of the fictitious magnetic field Bαν commuting with the momentum operator, consistently with their component operators qαν x 1 and qαν x 1 commuting with each other. In turn, unlike for the 1D model such generators not commute in general with the Hamiltonian, except for the Hubbard model on the square lattice in the oneand two-electron subspace. Concerning one-and two-electron excitations, the picture that emerges is that of a two-component quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions. The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at x = 0 hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations x > 0. For 0 < x ≪ 1 the latter short-range spin order has an incomensurate-spiral character. Our results are of interest for studies of ultra-cold fermionic atoms on optical lattices ...

The square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions

“…The square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions and the related general rotated-electron description of Ref. 9 are consistent with the global SO(3) × SO(3) × U (1) = [SO(4) × U (1)]/Z 2 symmetry found recently in Ref. 18 for the model on any bipartite lattice.…”

“…In this paper we study the momentum bands, energy dispersions, and velocities of the charge c fermions and spinneutral two-spinon s1 fermions introduced for the model on the square lattice in Ref. 9 . The s1 fermions emerge from the spin-neutral two-spinon s1 bond particles 9,10 through an extended Jordan-Wigner transformation [11][12][13][14] .…”

“…9 . The s1 fermions emerge from the spin-neutral two-spinon s1 bond particles 9,10 through an extended Jordan-Wigner transformation [11][12][13][14] . Our study has as starting point the properties of the Hubbard model on the square lattice in the one-and two-electron subspace defined in Ref.…”

“…Our study has as starting point the properties of the Hubbard model on the square lattice in the one-and two-electron subspace defined in Ref. 9 and profits from the general rotated-electron description introduced in that reference.…”

“…The building blocks of the general description introduced in Ref. 9 are the η-spin-1/2 η-spinons, spin-1/2 spinons, and spinless and η-spinless charge c fermions whose occupancy configurations generate the state representations of the η-spin SU (2) symmetry, spin SU (2) symmetry, and U (1) symmetry, respectively, associated with the model global SO(3) × SO(3) × U (1) = [SU (2) × SU (2) × U (1)]/Z 2 2 symmetry. Such three basic objects are well defined for U > 0.…”