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

Abstract

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 (\n $$E_\\mathrm{g}$$\n ) as a function of anisotropy $$\\Delta $$\n . 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\n $$\\rightarrow $$\n AF) transition and the Haldane-XY (H\n $$\\rightarrow $$\n XY) phase transition. From the gap scaling formulae, $$E_\\mathrm{g}(\\mathrm{H} \\rightarrow \\mathrm{AF}) = A(\\Delta _{c1} - \\Delta )^{z\\nu }$$\n and $$E_\\mathrm{g}( \\mathrm{H} \\rightarrow \\mathrm{XY}) = a \\exp (- bz / \\sqrt{\\Delta -\\Delta _{c2}}) $$\n where $$z, z $$\n are dynamical critical exponents, $$\\nu $$\n is a correlation length critical exponent, $$\\Delta _{c1(2)}$$\n 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.