Quantum spin chains and the conformal anomaly

Abstract

The conformal anomaly c determines the universality class of a model system in statistical mechanics. The value of c characterizes both 2D classical models and their 1D quantum counterparts. The conformal anomaly may therefore be determined numerically for quantum spin chains using the relation: E0(N)≂E0(∞)−(NΔE/12)c(1/N2), where E0 (N) is the ground-state energy of an N-spin finite system, E0 (∞) is the ground-state energy in the thermodynamic limit, and ΔE is the energy gap between the ground state at k=0 and the first excited state of the dispersion curve at k=2π/N. The numerical approach is highly successful when tested on the integrable s= 1/2 Heisenberg antiferromagnetic XXZ chain and the integrable s=1 SU(2) model. The method gives c=1 to within 2% accuracy for the s=1 and (3)/(2) XY chains, placing them in the universality class of the 2D XY model. The result c=1 (2% accuracy) is obtained for the s= (3)/(2) Heisenberg antiferromagnetic chain, in agreement with the Haldane prediction. The s=1 pure antiferromagnetic biquadratic chain and the s=1 XXZ model with uniaxial anisotropy in the vicinity of the critical point Δ=Δ2 ∼1.15 −1.18 have also been studied.