Exact Results of Dimerization Order Parameter in $SU(n)$ Antiferromagnetic Chains

Abstract

After a proper definition of the dimerization order parameter for a spin-$S$ system, I show that this order parameter in the $SU(n)$ ($n=2S+1$) antiferromagnetic chains (or equivalently the $SU(2)$ spin-$S$ chains with Hamiltonians which project out singlet states) is, in the thermodynamic limit, directly proportional to the staggered-magnetization in the corresponding spin-$1\over2$ $XXZ$ chains which had already been mapped onto the $SU(n)$ chains.