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 Δ . 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 ( Δ c 1 - Δ ) z ν and E g ( H → XY ) = a exp ( - b z ′ / Δ - Δ c 2 ) where z , z ′ are dynamical critical exponents, ν is a correlation length critical exponent, Δ c 1 ( 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.