We study 1-dimension (1-d) and 2-dimensional (2-d) quantum anti-ferromagnets. Following AKLT's theory of valent bond solid (VBS) states, we find a family of topologically nontrivial 1-d VBS states for spin systems and provide a method to construct their parent Hamiltonians. Then we generalize the fermionic representation of S = 1/2 to arbitrary spin, and find an important difference between integer spins and half-odd-integer spins. Depending on different particle number constrains we develop different versions of mean field theory for spin models. We find that the mean field with restored particle-hole symmetry can reproduce Haldane's conjecture for 1-d Heisenberg models. Applying the mean field theory to 2-d anti-ferromagnetic models, we find that S = 1 spin liquids may exist in triang...[ Read more ]

We study 1-dimension (1-d) and 2-dimensional (2-d) quantum anti-ferromagnets. Following AKLT's theory of valent bond solid (VBS) states, we find a family of topologically nontrivial 1-d VBS states for spin systems and provide a method to construct their parent Hamiltonians. Then we generalize the fermionic representation of S = 1/2 to arbitrary spin, and find an important difference between integer spins and half-odd-integer spins. Depending on different particle number constrains we develop different versions of mean field theory for spin models. We find that the mean field with restored particle-hole symmetry can reproduce Haldane's conjecture for 1-d Heisenberg models. Applying the mean field theory to 2-d anti-ferromagnetic models, we find that S = 1 spin liquids may exist in triangle lattice. Comparing with the experimental results of NiGa₂S₄, we argue that the antiferro-nematic f-wave spin liquid is a possible ground state. On the other hand, to check whether the material ZnCu₃(OH)₆Cl₂ is really an U(1) Dirac spin liquid, we derive the Raman scattering operators of different symmetry channels for Hubbard model on Kagome lattice. Then we numerically calculate the Raman cross-section according to the mean field spin liquid state. The low energy continuum part of our results qualitatively match the experimental data.