Valence-Bond Monte Carlo for Chains of Non-Abelian Quasiparticles

In non-Abelian FQH states, quasiparticles carry quantum numbers (topological charge) which characterize a degenerate Hilbert space. When these quasiparticles are close enough together, the degeneracy of this Hilbert space is lifted and the quasiparticles are said to interact.\footnote{A. Feiguin \textit {et al.}, PRL \textbf{98}, 160409 (2007).} Here we show that the valence-bond Monte Carlo method introduced by Sandvik\footnote{A. W. Sandvik, PRL \textbf{95}, 207203 (2005).} for spin-1/2 systems can be generalized to simulate 1D chains of such interacting non-Abelian quasiparticles. For uniform chains, our Monte Carlo results for the ground state energy agree with known exact values.$^1$ For random chains we confirm numerically that, as expected,\footnote{ N. E. Bonesteel and K. Yang, PRL \textbf{99}, 140405 (2007).} the ground state freezes into a random singlet phase. By suitably generalizing the notion of valence-bond entanglement entropy\footnote{F. Alet, \textit{et al.}, PRL \textbf{99}, 117204 (2007); R. W. Chhajlany \textit{et al.}, PRL \textbf{99}, 167204 (2007).} to the non-Abelian case we also confirm the predicted result$^3$ that in this phase the entropy of a block of length $L$ scales as $S^{\rm VB}_L \simeq \frac{\ln d}{3} \log_2 L$, where $d$ is the quantum dimension of the quasiparticles. Work supported by US DOE.