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arXiv:cond-mat/9910218 · Physical review. B, Condensed matter

Effect of symmetry-breaking perturbations in the one-dimensional SU(4) spin-orbital model

P. Azaria, E. Boulat, P. Lecheminant

Published May 1, 2000• 47 citations
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Abstract

We study the effect of symmetry-breaking perturbations in the one-dimensional $\mathrm{SU}(4)$ spin-orbital model. We allow the exchange in spin ${(J}_{1})$ and orbital ${(J}_{2})$ channel to be different and thus reduce the symmetry to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2).$ A magnetic field $h$ along the ${S}^{z}$ direction is also applied. Using the formalism developed by Azaria et al. [Phys. Rev. Lett. $83,$ 624 (1999)] we extend their analysis of the isotropic ${J}_{1}{=J}_{2},$ $h=0$ case and obtain the low-energy effective theory near the $\mathrm{SU}(4)$ 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 ${J}_{1}{=J}_{2}<K/4$ 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 $M$ 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 $h$ varies.

DOI
10.1103/physrevb.61.12112

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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 SU(2)\ensuremath⨂SU(2).\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2).SU(2)\ensuremath⨂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 J1\ensuremath≠J2,{J}_{1}\ensuremath{\ne}{J}_{2},J1​\ensuremath=J2​, h\ensuremath≠0.h\ensuremath{\ne}0.h\ensuremath=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−dependenth\ensuremath{-}\mathrm{dependent}h\ensuremath−dependent coupling that may change its sign as hhh varies.
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 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.
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.
Exact diagonalization of finite spin-1/2 chains with periodic boundary conditions is applied to the ground state (GS) of chains with ferromagnetic (F) exchange J1<0{J}_{1}<0J1​<0 between first neighbors, antiferromagnetic (AF) exchange ${J}_{2}=\ensuremath{\alpha}{J}_{1}>0$ between second neighbors, and axial anisotropy $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$. In zero field, the GS is in the Sz=0{S}_{z}=0Sz​=0 sector for the relevant parameters and is doubly degenerate at multiple points ${\ensuremath{\gamma}}_{m}=({\ensuremath{\alpha}}_{m},{\ensuremath{\Delta}}_{m})$ in the $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ plane. Degeneracy under inversion at sites or spin-parity or both leads, respectively, to a bond order wave (BOW), to staggered magnetization, or to vector chiral (VC) order. Exact results up to N=28N=28N=28 spins directly yield order parameters and spin correlation functions whose weak NNN dependencies allow inferences about infinite chains. The high-spin GS at J2=0{J}_{2}=0J2​=0 changes discontinuously at ${\ensuremath{\gamma}}_{1}=(\ensuremath{-}1/4,1)$ to a singlet in the isotropic ($\ensuremath{\Delta}=1$) chain. The transition from high to low spin $S(\ensuremath{\alpha},\ensuremath{\Delta})$ is continuous for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{B}=0.95\ifmmode\pm\else\textpm\fi{}0.01$ on the degeneracy line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. The GS has staggered magnetization between ${\ensuremath{\Delta}}_{A}=0.72$ and ${\ensuremath{\Delta}}_{B}$ and a BOW for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{A}$. When both inversion and spin-parity are reversed at ${\ensuremath{\gamma}}_{m}$, the correlation functions C(p)C(p)C(p) for spins separated by ppp sites are identical. C(p)C(p)C(p) minima are shifted by $\ensuremath{\pi}/2$ from the minima of VC order parameters at separation ppp, consistent with right- and left-handed helices along the zzz axis and spins in the xyxyxy plane. Degenerate GSs of finite chains are related to quantum phase diagrams of extended $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ chains, with good agreement for order parameters along the line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. Degenerate GSs limit a VC phase to intermediate $\ensuremath{\alpha}$ and $\ensuremath{\Delta}$ where $S(\ensuremath{\alpha},\ensuremath{\Delta})$ varies rapidly but continuously, in contrast to many-body treatments in which VC phases extend over a larger range of parameters.
The one-dimensional spin-$S {J}_{1}\ensuremath{-}{J}_{2}\mathrm{XY}$ model is studied within the bosonization approach. Around the two limits ${(J}_{2}{/J}_{1}\ensuremath{\ll}1, {J}_{2}{/J}_{1}\ensuremath{\gg}1)$ where a field theoretical analysis can be derived, we discuss the phases as well as the different phase transitions that occur in the model. In particular, it is found that the chiral critical spin nematic phase, discovered by Nersesyan et al. [Phys. Rev. Lett. 81, 910 (1998)] for S=1/2,S=1/2,S=1/2, exists in the general spin-SSS case. The nature of the effective-field theory that describes the transition between this chiral critical phase and a chiral gapped phase is also determined.
The enhancement of the spin-space symmetry from the usual SU(2)\mathrm{SU}(2)SU(2) to SU(N)\mathrm{SU}(N)SU(N) with N>2N>2N>2 is promising for finding nontrivial quantum spin liquids, but the realization of SU(N)\mathrm{SU}(N)SU(N) spin systems in real materials is still challenging. Although there is a proposal in cold atomic systems, in magnetic materials with a spin-orbital degree of freedom it is difficult to achieve the SU(N)\mathrm{SU}(N)SU(N) symmetry by fine tuning. Here we propose a new mechanism by which the SU(4)\mathrm{SU}(4)SU(4) symmetry emerges in the strong spin-orbit coupling limit. In d1d^1d1 transition metal compounds with edge-sharing anion octahedra, the spin-orbit coupling gives rise to strongly bond-dependent and apparently SU(4)\mathrm{SU}(4)SU(4)-breaking hopping between the Jeff=3/2J_\textrm{eff}=3/2Jeff​=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an SU(4)\mathrm{SU}(4)SU(4)-symmetric Hubbard model, which means that the system has a hidden symmetry in spite of its large spin-orbit coupling. In the strong repulsion limit at quarter filling, as expected in ααα-ZrCl3,_3,3​, the low-energy effective model is the SU(4)\mathrm{SU}(4)SU(4) Heisenberg model on the honeycomb lattice, which cannot have a trivial gapped ground state and is expected to host a gapless spin-orbital liquid. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and correlated together at low temperature due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic "orbitalon" excitations at low temperature, which we have named here in distinction from orbitons in the symmetry-broken Jahn-Teller phases.