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arXiv:cond-mat/9804182 · Physical review. B, Condensed matter

SU(4) spin-orbit critical state in one dimension

Yasufumi Yamashita, Naokazu Shibata, Kazuo Ueda

Published Oct 1, 1998• 121 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.58.9114

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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.
We study the one-dimensional SU(4) exchange model under magnetic fields, which is the simplest effective Hamiltonian to investigate the orbital quantum fluctuations in coupled spin-orbit systems. The Bethe ansatz approaches and numerical calculations using the density matrix renormalization group method are employed. The main concern of the paper is how the system changes from the SU(4) to the SU(2) symmetric limit as the magnetic field is increased. For this model the conformal field theory predicts an unusual behavior: there is a jump of the critical exponents just before the SU(2) limit. For a finite-size system, however, the orbital-orbital correlation functions approach continuously the SU(2) limit after interesting crossover phenomena. The crossover takes place in the magnetization range 1/3--1/2 for the system with 72 sites studied in this paper.
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 investigate the ground-state property and the excitation gap of a one-dimensional spin-orbital model with isotropic spin and anisotropic orbital exchange interactions, which represents the strong-coupling limit of a two-orbital Hubbard model including the Hund's rule coupling (J)(J)(J) at quarter filling, by using a density-matrix renormalization group method. At J=0J=0J=0, spin and orbital correlations coincide with each other with a peak at $q=\ensuremath{\pi}∕2$, corresponding to the SU(4) singlet state. On the other hand, spin and orbital states change in a different way due to the Hund's rule coupling. With increasing JJJ, the peak position of orbital correlation changes to $q=\ensuremath{\pi}$, while that of spin correlation remains at $q=\ensuremath{\pi}∕2$. In addition, orbital dimer correlation becomes robust in comparison with spin dimer correlation, suggesting that quantum orbital fluctuation is enhanced by the Hund's rule coupling. Accordingly, a relatively large orbital gap opens in comparison with a spin gap, and the system is described by an effective spin system on the background of the orbital dimer state.
We study the effects of a uniform magnetic field on the one-dimensional spin-orbital model in terms of effective field theories. Two regions are examined: one around the SU(4) point (J=K/4)(J=K/4)(J=K/4) and the other with $K\ensuremath{\ll}J$ [see Eq. (1)]. We found that when J< K/4,J<~K/4,J< K/4, the spin and orbital correlation functions exhibit power-law decay with nonuniversal exponents. In the region with J>K/4,J>K/4,J>K/4, the excitation spectrum has a gap but the structure of it changes at different range of the ratio J/K.J/K.J/K. Furthermore, near the SU(4) point, the magnetic field exerts a great influence on the spectrum of the Sz=0{S}^{z}=0Sz=0 sector. When the magnetic field is beyond some critical value, a quantum phase transition occurs. The different structure of the excitation spectrum results in distinct critical behavior of correlation functions around the SU(4) point and the region with $K\ensuremath{\ll}J.$ Especially, the exponents in the latter case are universal.
We investigate low-energy excitations of the one-dimensional half-filled SU(4) Hubbard model with an attractive on-site interaction U<0U<0U<0 using the density-matrix renormalization-group method as well as perturbation theory. We find that the ground state is a charge-density-wave state with long-range order. The ground state is completely incompressible since all the excitations are gapful. The charge gap, which is the same as the four-particle excitation gap, is a nonmonotonic function of UUU, while the spin gap and others increase with increasing $\ensuremath{\mid}U\ensuremath{\mid}$ and have linear asymptotic behaviors.
Motivated by recent experiments on ultracold fermions we study the effect of the spin-orbit interaction in an exactly solvable one-dimensional model with strong local attractive correlations between the fermions. We show that the asymptotes of correlation functions, calculated in the framework of the conformal field theory and finite size corrections of the Bethe ansatz exact solution, are strongly affected by the spin-orbit coupling. In the mixed phase, the correlation functions consist of terms that are the product of a power law of the distance and an oscillating function of the distance. We obtain the critical exponents for superfluidity and density waves. The leading exponents decrease as a function of spin-orbit coupling. In particular, the exponent for superfluidity is the smallest one signaling an instability to a phase where a weak interchain coupling can produce superfluidity of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type. The spatial oscillation of the order parameter is also modified by the spin-orbit parameter, i.e., its modulation is not just given by the spin imbalance.