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arXiv:cond-mat/9910109 · Physical review. B, Condensed matter

Phase diagram of a one-dimensional spin-orbital model

Chigak Itoi, Shaojin Qin, Ian K. Affleck

Published Mar 1, 2000• 101 citations
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Abstract

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 $\mathrm{SU}(4)$ symmetry. These indicate the existence of a gapless phase in an extended region of the phase diagram, missed in previous studies. This phase is $\mathrm{SU}(4)$ 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.

DOI
10.1103/physrevb.61.6747

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The thermodynamic properties of a one-dimensional model describing spin dynamics in the presence of a twofold orbital degeneracy are studied numerically using the transfer-matrix renormalization group. The model contains an integrable SU(4)-symmetric point and a gapless phase which is SU(4) invariant up to a rescaling of the velocities for spin and orbital degrees of freedom which allows detailed comparison of the numerical results with conformal field theory. We pay special attention to the correlation lengths which show an intriguing evolution with temperature. We find that the model shows an intrinsic tendency towards dimerization at finite temperature even if the ground state is not dimerized.
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 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 numerically study the SU(2)⊗\otimes⊗SU(2) symmetric spin-orbit coupled model as a lower symmetric generalization of the SU(4) exchange model. On the symmetric line with respect to the spin and orbit, our result shows the essentially singular gap formation in consistent with the analytic approach, which is different from the previous numerical calculation. Furthermore, we find new critical phases around the SU(4) point, surrounding the previously known gapless symmetric line. In these novel phases either spin or orbital excitations around momentum q=πq=πq=π form massless continua which split from the excitations belonging to the [22][2^2][22] irreducible representation at the SU(4) point. On the critical symmetric line, the additional coupled spin-orbit excitations around q=π/2q=π/2q=π/2 originating from [21122^11^22112] become critical, too.
The one-dimensional SU(n)\mathrm{SU}(n)SU(n) Hubbard model is investigated numerically for n=2n=2n=2, 3, 4, and 5 at half-filling and 1∕n1∕n1∕n filling using the density-matrix renormalization-group method. The energy gaps and various quantum information entropies are calculated. In the half-filled case, finite spin and charge gaps are found for arbitrary positive UUU if n>2n>2n>2. Furthermore, it is shown that the transition to the gapped phase at Uc=0{U}_{c}=0Uc​=0 is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd nnn. In the 1∕n1∕n1∕n-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for U>Uc=0U>{U}_{c}=0U>Uc​=0, the spin sector remains gapless, and the ground state is nondimerized.
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