Critical Exponents for Antiferromagnetic Spin Chains Obtained From Bosonisation

Abstract

The Heisenberg spin 1/2 chain is revisited in the perturbative RG approach with special focus on the transition of the critical exponents. We give a compact review that first order RG in the couplings is sufficient to derive the exact transition from ν = 1 to ν = 2/3, if the boson radius obtained in the bosonization procedure is replaced by the exact radius obtained in the Bethe approach. We explain the fact, that from the bosonization procedure alone, the critical exponent can not be derived correctly in the isotropic limit Jz → J. We further state that this fact is important if we consider to bosonize the antiferromagnetic super spin chain for the quantum Hall effect.

Emergent dimerization and localization in disordered quantum chains

“…6, obtained with r c = 0.135, z = z clean = 1 and ν = 2, show that our DMRG data are compatible with Eq. (15). [Close to the critical point the rescaling of the data by (1 − r) 2 becomes irrelevant.]…”

“…but other functions having appreciable values only for |x| < 1 yield similar results. The Luttinger parameter K has an initial value which varies with the anisotropy ∆ of the XXZ chain according to 14,15…”

“…In the thermodynamic limit, the ground state of the uniform (clean) system (J i ≡ J) is critical and low-energy excitations are described as a spin (Luttinger) liquid with a dynamical critical exponent z clean = 1. It is perturbatively unstable against dimerization (i.e, alternating couplings J i ≡ 1 + 1 2 (−1) i δ J, with a dimerization strength δ), which produces an energy gap ∆E ∼ |δ| above the ground state and a finite correlation length diverging as ξ ∼ |δ| −ν for δ → 0 with a critical exponent 14,15 ν = 2 (π − arccos ∆) /(3π−4 arccos ∆).…”