Closing of the Haldane gap in a spin-1 XXZ chain

We study the energy gaps of a spin-1 XXZ model in one dimension. Using an infinite-size density matrix renormalization group (iDMRG) and a variational uniform matrix product state algorithm (vuMPS), we obtained the energy gaps (E-g) as a function of anisotropy Delta. We found that the closing point of the energy gap changes as we increase the bond dimension. By scaling the energy gaps, we find the critical points for the Haldane-antiferromagnetic (H -> AF) transition and the Haldane-XY (H -> XY) phase transition. From the gap scaling formulae, E-g (H -> AF) = A(Delta(c1) - Delta)(z nu) and E-g(H -> XY) = a exp(-bz'root Delta - Delta(c2)) where z, z' are dynamical critical exponents, nu is a correlation length critical exponent, Delta(c1(2)) is the critical point for each transition and A, a, and b are scaling parameters, we obtained information regarding the critical points and the scaling parameters.