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arXiv:cond-mat/9908237 · Journal of the Physical Society of Japan

Numerical Study of the One-Dimensional Spin-Orbit Coupled System with SU(2) ⊗ SU(2) Symmetry

Yasufumi Yamashita, Naokazu Shibata, Kazuo Ueda

Published Jan 1, 2000• 49 citations
arXivHTMLPDF

Abstract

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 $q=π$ form massless continua which split from the excitations belonging to the $[2^2]$ irreducible representation at the SU(4) point. On the critical symmetric line, the additional coupled spin-orbit excitations around $q=π/2$ originating from [$2^11^2$] become critical, too.

DOI
10.1143/jpsj.69.242

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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.
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 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.
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.
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.
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 $\ensuremath{\sim}\mathrm{cos}[(\ensuremath{\pi}{/2a}_{0})x]{x}^{\ensuremath{-}3/2}$. 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.