Valence bond description of the long-range, nonfrustrated Heisenberg chain

TL;DR

This paper investigates the quantum phase transition in a long-range Heisenberg chain using valence bond theory, identifying critical behavior and phase boundaries through numerical and analytical methods.

Contribution

It introduces a valence bond framework to describe the phase transition in long-range Heisenberg chains and provides numerical evidence for the critical point and exponents.

Findings

Identifies the critical alpha value as 2.18(5) for the phase transition.

Shows the valence bond amplitude scales as r^{-alpha} or related forms depending on alpha.

Numerical results agree with quantum Monte Carlo simulations.

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 alpha_c = 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-alpha_c) = 2.4(2).