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arXiv:cond-mat/9807270 · Physical Review Letters

Alternating Spin and Orbital Dimerization and Spin-Gap Formation in Coupled Spin-Orbital Systems

Swapan Kumar Pati, Rajiv R. P. Singh, Daniel I. Khomskii

Published Dec 14, 1998• 131 citations
arXivHTMLPDF

Abstract

We discuss a novel mechanism for obtaining a spin-gap state through the creation of a coherent spin-orbital structure in systems with orbital degeneracy. Using the density matrix renormalization group, we calculate the phase diagram for a coupled spin-orbital model. We find that, in addition to ferromagnetic and power-law antiferromagnetic phases for spin and orbital degrees of freedom, this model has a gapless line extending from the ferromagnetic phase to the Bethe ansatz solvable SU(4) critical point, and a gapped phase with doubly degenerate ground states which form alternating spin and orbital singlets. The relevance to several recently discovered materials is discussed.

DOI
10.1103/physrevlett.81.5406

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We discuss a novel mechanism for obtaining a spin-gap state through the creation of a coherent spin-orbital structure in systems with orbital degeneracy. Using the density matrix renormalization group, we calculate the phase diagram for a coupled spin-orbital model. We find that, in addition to ferromagnetic and power-law antiferromagnetic phases for spin and orbital degrees of freedom, this model has a gapless line extending from the ferromagnetic phase to the Bethe ansatz solvable SU(4) critical point, and a gapped phase with doubly degenerate ground states which form alternating spin and orbital singlets. The relevance to several recently discovered materials is discussed.
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.
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)\mathrm{SU}(4)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)\mathrm{SU}(4)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.
In order to clarify a key role of the orbital degree of freedom in the spin S=1S=1S=1 Haldane system, we investigate ground-state properties of the t2g{t}_{2g}t2g​-orbital degenerate Hubbard model on the linear chain by using numerical techniques. Increasing the Hund's rule coupling in multiorbital systems, in general, there occurs a transition from an antiferromagnetic to a ferromagnetic phase. We find that the antiferromagnetic phase is described as the Haldane system with spin gap, while in the ferromagnetic phase, there exists the gapless excitation with respect to the orbital degree of freedom. Possible relevance of the present results to actual systems is also discussed.
We investigate the spin and orbital correlations of a superexchange model with spin S=1S=1S=1 and orbital L=1L=1L=1 relevant for 5d45{d}^{4}5d4 transition-metal Mott insulators [O. N. Meetei et al., Phys. Rev. B 91, 054412 (2015)], using exact diagonalization and density matrix renormalization group (DMRG). For spin-orbit coupling $\ensuremath{\lambda}=0$, the orbitals are in an entangled state that is decoupled from the spins. We find two phases with increasing $\ensuremath{\lambda}$: (I) the S2S2S2 phase with two peaks in the structure factor for $\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{c1}\ensuremath{\approx}0.34J$ where JJJ is the ferromagnetic exchange; and, (II) the S1S1S1 phase for ${\ensuremath{\lambda}}_{c1}<\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{c2}\ensuremath{\approx}1.2J$ with emergent antiferromagnetic correlations. Both S1S1S1 and S2S2S2 phases are shown to exhibit power-law correlations, indicative of a gapless spectrum. Upon increasing $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c2}$ leads to a product state of local spin-orbital singlets that exhibit exponential decay of correlations, indicative of a gapped phase. We obtain insights into the phases from the well-known Uimin-Lai-Sutherland model in an external field that provides an approximate description of our model within mean-field theory.
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.