Valence bond description of the long-range, nonfrustrated Heisenberg chain
Abstract
The Heisenberg chain with antiferromagnetic, powerlaw exchange has a quantum phase transition separating spin liquid and Neel ordered phases at a critical value of the powerlaw exponent alpha. The behaviour of the system can be explained rather simply in terms of a resonating valence bond state in which the amplitude for a bond of length r goes as r-alpha for alpha < 1, as r-(1+alpha)/2 for 1 < alpha < 3, and as r-2 for alpha > 3. Numerical evaluation of the staggered magnetic moment and Binder cumulant reveals a second order transition at alphac = 2.18(5), in excellent agreement with quantum Monte Carlo. The divergence of the magnetic correlation length is consistent with an exponent nu = 2/(3-alphac) = 2.4(2).
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