Finite-size study of the one-dimensional spin-(1/2) dimerized Heisenberg chain

We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter {δ} ({}{δ}{}{≤}1) at zero temperature. This system undergoes a transition at {δ}=0, and a gap opens in the spectrum of elementary excitations. The exponent {ν}, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N=18). We investigate two finite-size-scaling hypotheses, assuming a pure power-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate {ν}, we use the derivative of the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), {ν}=0.71±{}0.01, in agreement with previous estimates, and in case (2), {ν}=0.668±{}0.001, in very good agreement with the value {ν}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form {}{δ}ˣ with x=1.34±{}0.02. However, results for the derivative of this quantity show a different functional dependence upon {δ}, at least for {δ}{}0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as {δ} approaches 1.