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arXiv:cond-mat/9909394 · Physical review. B, Condensed matter

Effects of a magnetic field on the one-dimensional spin-orbital model

Yu-Li Lee, Yu-Wen Lee

Published Mar 1, 2000• 9 citations
arXivHTMLPDF

Abstract

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 $(J=K/4)$ and the other with $K\ensuremath{\ll}J$ [see Eq. (1)]. We found that when $J<~K/4,$ the spin and orbital correlation functions exhibit power-law decay with nonuniversal exponents. In the region with $J>K/4,$ the excitation spectrum has a gap but the structure of it changes at different range of the ratio $J/K.$ Furthermore, near the SU(4) point, the magnetic field exerts a great influence on the spectrum of the ${S}^{z}=0$ 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.

DOI
10.1103/physrevb.61.6765

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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≪JK\ensuremath{\ll}JK\ensuremath≪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≪J.K\ensuremath{\ll}J.K\ensuremath≪J. Especially, the exponents in the latter case are universal.
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.
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.
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.
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.
We study the dynamics and thermodynamics of one-dimensional spin-orbital models relevant for transition-metal oxides. We show that collective spin, orbital, and combined spin-orbital excitations with infinite lifetime can exist, if the ground state of both sectors is ferromagnetic. Our main focus is the case of effectively ferromagnetic (antiferromagnetic) exchange for the spin (orbital) sector, and we investigate the renormalization of spin excitations via spin-orbital fluctuations using a boson-fermion representation. We contrast a mean-field decoupling approach with results obtained by treating the spin-orbital coupling perturbatively. Within the latter self-consistent approach we find a significant increase of the linewidth and additional structures in the dynamical spin structure factor as well as Kohn anomalies in the spin-wave dispersion caused by the scattering of spin excitations from orbital fluctuations. Finally, we analyze the specific heat c(T)c(T)c(T) by comparing a numerical solution of the model obtained by the density-matrix renormalization group with perturbative results. At low temperatures TTT we find numerically c(T) Tc(T)~Tc(T) T pointing to a low-energy effective theory with dynamical critical exponent z=1z=1z=1.
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, ∼cos⁡(π/2a0x)x−3/2\sim \cos(π/{2a_0} x) x^{-3/2}∼cos(π/2a0​x)x−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.