. 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.

DOI: 10.1103/physrevlett.83.624

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\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.

\sim \cos(π/{2a_0} 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.

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.

\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.

S=T=1∕2

and of

S=1

and

T=1∕2

are presented. Three dimer phases, the orbital valence bond state, spin valence bond state and spin-orbital valence bond state, are found to arise in proper parameter regions. The results indicate that the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ type of spin-orbit coupling can serve as both the spin-Peierls and orbital-Peierls mechanisms which are responsible for the formations of spin singlet and orbital singlet, respectively. We have also calculated the static spin and pseudospin susceptibilities in the SBMFT scheme to characterize and distinguish the three types of dimer phases.

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.