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Physical review. B, Condensed matter

Excitations in the anisotropic spin-orbital model

Jan Bała, Andrzej M. Oleś, G. A. Sawatzky

Published Mar 5, 2001• 7 citations
Publisher

Abstract

We study the elementary excitations of the ferromagnetic spin-orbital model which represents the low-energy excitations for strongly correlated electrons described by the Hubbard Hamiltonian with a twofold degeneracy. At most two and four spin-and-orbital bound states have been found in one and two dimensions, respectively. For a completely degenerate system, the spin-and-orbital bound states are the lowest-lying elementary excitations in most of the Brillouin zone and for Ising-like interactions they can have much lower energy than the other excitations of the model and might thus be responsible for the instability of the ground state. This might lead to quantum disordered states in both spin and orbital space, as indicated by exact solutions obtained for small clusters.

DOI
10.1103/physrevb.63.134410

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We study the elementary excitations of the ferromagnetic spin-orbital model which represents the low-energy excitations for strongly correlated electrons described by the Hubbard Hamiltonian with a twofold degeneracy. At most two and four spin-and-orbital bound states have been found in one and two dimensions, respectively. For a completely degenerate system, the spin-and-orbital bound states are the lowest-lying elementary excitations in most of the Brillouin zone and for Ising-like interactions they can have much lower energy than the other excitations of the model and might thus be responsible for the instability of the ground state. This might lead to quantum disordered states in both spin and orbital space, as indicated by exact solutions obtained for small clusters.
The Hubbard model for electrons with orbital degeneracy is shown to have an underlying SUd(4){\mathrm{SU}}_{d}(4)SUd​(4) symmetry of spin-orbital double. A hidden charge SUc(4){\mathrm{SU}}_{c}(4)SUc​(4) symmetry is exposed and an extended Lieb-Mattis transformation, which maps these two symmetries into each other, is given. On the basis of elementary degenerate perturbative theory, it is shown that the system with strong repulsive coupling is equivalent to a SO(6)\mathrm{SO}(6)SO(6) Heisenberg magnet at half-filling and a SU(4)\mathrm{SU}(4)SU(4) one at quarter-filling. The band is half-filled at all temperature for $\ensuremath{\mu}=3U/2.$ The features of ground-state and low-lying excitations in one dimension are indicated according to exact solutions.
The elementary excitations of the 1D, symmetric, spin-orbital model are studied in two anisotropic cases: the pure XY\mathrm{XY}XY and the dimerized XXZ\mathrm{XXZ}XXZ models. While they preserve the symmetry between spin and orbital degrees of freedom, these models allow for a picture of the low-lying excitations. In the pure XY\mathrm{XY}XY case, a phase separation takes place between two phases with free-fermion-like gapless excitations. In the dimerized case the low-energy effective Hamiltonian reduces to the 1D Ising model with gapped excitations. All the elementary excitations involve simultaneous flips of the spin and orbital degrees of freedom: a clear indication of the breakdown of the traditional mean-field theory.
We use perturbative series expansions about a staggered dimerized ground state to compute the ground-state energy, triplet excitation spectra, and spectral weight for a one-dimensional model in which each site has an S=12S=\frac{1}{2}S=21​ spin Si{\mathbf{S}}_{i}Si​ and a pseudospin Ti,{\mathbf{T}}_{i},Ti​, representing a doubly degenerate orbital. An explicit dimerization is introduced to allow the study of the confinement of spinon excitations. The elementary triplet represents a bound state of two spinons and is stable over much of the Brillouin zone. A special line is found in the gapped spin-liquid phase, on which the triplet excitation is dispersionless. The formation of triplet bound states is also investigated.
A Hubbard-like model with SU(4) symmetry for electrons with twofold orbital degeneracy is studied extensively. Exact solution in one dimension is derived by means of Bethe ansatz, where the sites are supposed to be occupied by at most two electrons. The features of ground state and excited states for repulsive coupling are shown. For finite NNN number of electrons, the configurations of quantum numbers are given explicitly and the spectra of excitations are obtained by solving Bethe-ansatz equation numerically. For infinite N,N,N, the ground state and various kinds of low-lying excitations are obtained on the basis of thermodynamics limit.
Effective spin Hamiltonians with t 2g orbital degeneracy have been investigated theoretically. We assume that a d electron occupies one of doubly degenerate d z x and d y z orbitals which have the lowest energy in atomic level. The orbital degrees of freedom are represented as Ising iso-spin operators because of the symmetry of the orbital characters. We apply the Hamiltonians to a ladder and a chain lattices and investigate the models using the exact diagonalization. For the ladder model, a spin gap opens above a finite value of the spin exchange coupling on rung, which is qualitatively different from the ladder model with single orbital. For the chain model, the spin gap behavior is observed in the uniform spin correlation at low temperature. Origin of the spin gap is a spin-dimer triggered by the occurrence of the ferromagnetic orbital ordering, which suggests a new mechanism of the spin gap formation.
We studied ferromagnetism in the one-dimensional Hubbard model with doubly degenerate atomic orbitals by means of the density-matrix renormalization-group method and obtained the ground-state phase diagrams. It was found that ferromagnetism is stable from low to high (0<n<1.75)(0<n<1.75)(0<n<1.75) electron density when the interactions are sufficiently strong. The quasi-long-range order of triplet superconductivity coexists with the ferromagnetic order for a strong Hund coupling region, where the interorbital interaction ${U}^{\ensuremath{'}}\ensuremath{-}J$ is attractive. At quarter-filling (n=1),(n=1),(n=1), the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities (n<1),(n<1),(n<1), ferromagnetism occurs owing to the ferromagnetic exchange interaction caused by virtual hoppings of electrons, the same as in the quarter-filled system. This comes from separation of the charge and spin-orbital degrees of freedom in the strong-coupling limit. This ferromagnetism is fragile against variation of band structure. For high densities (n>1),(n>1),(n>1), the phase diagram of the ferromagnetic phase is similar to that obtained in infinite dimensions. In this case, the double exchange mechanism is operative for stabilizing the ferromagnetic order and this long-range order is robust against variation of the band dispersion. A partially polarized state appears in the density region $1.68\ensuremath{\lesssim}n\ensuremath{\lesssim}1.75$ and phase separation occurs for n just below the half-filling (n=2).(n=2).(n=2).