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arXiv:cond-mat/9809094 · Physical Review Letters

Elementary Excitations of the Symmetric Spin-Orbital Model: The XY Limit

Frédéric Mila, Beat Frischmuth, Andreas Deppeler, Matthias Troyer

Published May 3, 1999• 26 citations
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Abstract

The elementary excitations of the 1D, symmetric, spin-orbital model are studied in two anisotropic cases: the pure $\mathrm{XY}$ and the dimerized $\mathrm{XXZ}$ 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 $\mathrm{XY}$ 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.

DOI
10.1103/physrevlett.82.3697

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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 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.
We study a one-dimensional spin-orbital model with Heisenberg symmetry for spins and XY\mathrm{XY}XY symmetry for orbital degrees of freedom. We establish the ground-state phase diagram of this model. The gapless phases of the model consist of fully polarized ferromagnetic spin states together with the filled Jordan-Wigner Fermi sea for the orbitals. Two such phases, with ferromagnetic and antiferromagnetic XY\mathrm{XY}XY correlations for the orbitals, are found. All other phases are found to have a spin gap with degenerate ground states. Two types of correlated spin and orbital dimerizations are observed: (i) spin singlet with orbital singlet in alternating rungs and (ii) spin singlet with orbital triplet in the same rung. The phase boundaries, energy gaps, and order parameters are evaluated using the density matrix renormalization group method.
The low-energy effective Hamiltonian of three coupled spin chains with periodic boundary conditions (spin tube) is expressed, in the limit of strong interchain coupling, in terms of XXZ\mathrm{XXZ}XXZ chains coupled by biquadratic exchange interaction. A similar effective model was also proposed to describe the coupling of spins to orbital degrees of freedom in materials such as NaV2O5.{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}.NaV2​O5​. We investigate the effective model by means of bosonization and renormalization-group techniques, and find that the generic phase diagram comprises a gapless region and gapped regions consisting of a spin liquid phase and various antiferromagnetic phases. We discuss the properties of the spin liquid phase, in particular the nature of the ground state and of the elementary excitations above it. We then study the effect of a magnetic field, and conclude that a strong enough magnetic field can suppress the dimerized phase leading to a two component Luttinger liquid. The critical exponents at the gapful-gapless transition are calculated and shown to be nonuniversal in the spin-tube case or the generic spin- orbital problem.
The excitation spectrum of the one-dimensional spin-orbital model in a magnetic field is studied, using a recently developed dynamical density-matrix renormalization-group technique. The method is applied to chains with up to 80 sites, and examined for test cases such as the one-dimensional spin-1/2 Heisenberg antiferromagnet whose excitation spectrum is known exactly from the Bethe ansatz. In the spin-orbital chain, the characteristic dynamical response depends strongly on the model parameters and the applied magnetic field. The coupling between the spin and orbital degrees of freedom is found to influence the incommensuration at finite magnetizations. In the regions of the phase diagram with only massive spin and orbital excitations, a finite field is required to overcome the spin gap. An orbital mode is found to become massless in this partially spin-polarized regime, indicating a strong coupling between the two degrees of freedom. In the critical region with three elementary gapless excitations, a prominent particle-hole excitation is observed at higher energies, promoted by the biquadratic term in the model Hamiltonian of the spin-orbital chain.
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.
We study the effect of symmetry-breaking perturbations in the one-dimensional SU(4)\mathrm{SU}(4)SU(4) spin-orbital model. We allow the exchange in spin (J1){(J}_{1})(J1​) and orbital (J2){(J}_{2})(J2​) channel to be different and thus reduce the symmetry to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2).$ A magnetic field hhh along the Sz{S}^{z}Sz direction is also applied. Using the formalism developed by Azaria et al. [Phys. Rev. Lett. 83,83,83, 624 (1999)] we extend their analysis of the isotropic J1=J2,{J}_{1}{=J}_{2},J1​=J2​, h=0h=0h=0 case and obtain the low-energy effective theory near the SU(4)\mathrm{SU}(4)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{J}_{1}{=J}_{2}<K/4J1​=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 MMM 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 hhh varies.