Nontrivial quantized berry phases for itinerant spin liquids

Quantized Berry phases used as local order parameters in t-J models are studied. A texture pattern of the local order parameters is topologically stable due to the quantization of non-Abelian Berry phases defined by low-energy states below a spin gap, which exists in the large J/t case with a few electrons. We have confirmed that the itinerant singlets in the wide class of t-J models carry the nontrivial Berry phase pi. In the large J/t case for the one-dimensional t-J model, the Berry phases are uniformly pi when the number of electrons is N = 4n + 2 (n = 0, 1, 2,...).