The one-dimensional spin-orbital model is studied by means of Abelian bosonization. We derive the low-energy effective theory which enables us to study small deviations from the SU(4) symmetric point. We show that there exists a massless region with algebraically decaying correlation functions $\ensuremath{\sim}\mathrm{cos}[(\ensuremath{\pi}{/2a}_{0})x]{x}^{\ensuremath{-}3/2}$. When entering the massive phase, the system displays an approximate SO(6) enlarged symmetry with a dimerization type of ordering consisting in alternating spin and orbital singlets.
We study the effect of symmetry-breaking perturbations in the one-dimensional
SU(4) spin-orbital model. We allow the exchange in spin
(J1) and orbital
(J2) channel to be different and thus reduce the symmetry to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2).$ A magnetic field
h along the
Sz direction is also applied. Using the formalism developed by Azaria et al. [Phys. Rev. Lett.
83, 624 (1999)] we extend their analysis of the isotropic
J1=J2, h=0 case and obtain the low-energy effective theory near the
SU(4) point in the generic case ${J}_{1}\ensuremath{\ne}{J}_{2},$ $h\ensuremath{\ne}0.$ In zero magnetic field, we retrieve the same qualitative low-energy physics as in the isotropic case. In particular, the isotropic massless behavior found on the line
J1=J2<K/4 extends in a large anisotropic region. We discover, however, that the anisotropy plays its trick in allowing nontrivial scaling behaviors of the physical quantities. For example, the mass gap
M has two different scaling behaviors depending on the anisotropy. In addition, we show that in some regions, the anisotropy is responsible for anomalous finite-size effects and may change qualitatively the shape of the computed critical line in a finite system. When a magnetic field is present the effect of the anisotropy is striking. In addition to the usual commensurate-incommensurate phase transition that occurs in the spin sector of the theory, we find that the field may induce a second transition of the KT type in the remaining degrees of freedom to which it does not couple directly. In this sector, we find that the effective theory is that of an SO(4) Gross-Neveu model with an $h\ensuremath{-}\mathrm{dependent}$ coupling that may change its sign as
h varies.
We study a one-dimensional spin-orbital model using both analytical and numerical methods. Renormalization group calculations are performed in the vicinity of a special integrable point in the phase diagram with
SU(4) symmetry. These indicate the existence of a gapless phase in an extended region of the phase diagram, missed in previous studies. This phase is
SU(4) invariant at low energies apart from the presence of different velocities for spin and orbital degrees of freedom. The phase transition into a gapped dimerized phase is in a generalized Kosterlitz-Thouless universality class. The phase diagram of this model is sketched using the density matrix renormalization group technique.
A generalized one-dimensional $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ spin-orbital model is studied by means of Schwinger boson mean-field theory (SBMFT). We focus on exploring the dimer phases and clarify how to capture properly the low-temperature properties of the system. The phase diagrams for the cases of
S=T=1∕2 and of
S=1 and
T=1∕2 are presented. Three dimer phases, the orbital valence bond state, spin valence bond state and spin-orbital valence bond state, are found to arise in proper parameter regions. The results indicate that the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ type of spin-orbit coupling can serve as both the spin-Peierls and orbital-Peierls mechanisms which are responsible for the formations of spin singlet and orbital singlet, respectively. We have also calculated the static spin and pseudospin susceptibilities in the SBMFT scheme to characterize and distinguish the three types of dimer phases.
To describe the spin-orbital energy gap formation in the one-dimensional symmetrically coupled spin-orbital model, we propose a simple mean-field theory based on an SU(4) constraint fermion representation of spins and orbitals. A spin-orbital gapped phase is formed due to a marginally relevant spin-orbital valence bond pairing interaction. The energy gap of the spin and orbital excitations grows extremely slowly from the SU(4)-symmetric point up to a maximum value and then decreases rapidly. By calculating the spin, orbital, and spin-orbital tensor static susceptibilities at zero temperature, we find a crossover from coherent to incoherent magnetic excitations as the spin-orbital coupling decreases from large to small values.
Effect of quantum fluctuations concerned with the orbital degrees of freedom is discussed for the model with SU(4) symmetry in one dimension. An effective Hamiltonian is derived from the orbitally degenerate Hubbard model at quarter filling. This model is equivalent to the Bethe soluble SU(4) exchange model. Quantum numbers of the ground state and the lowest branch of excitations are determined. The spin-spin correlation functions are obtained numerically by the density matrix renormalization group method. It shows a power-law decay with oscillations of the period of four sites. The period originates from the interference between the spin and orbital degrees of freedom. The exponent of the power-law decay estimated from the finite size data is consistent with the prediction by the conformal field theory.