Fluctuation Length Scales in Random Singlet Phases

For any disorder strength, the ground state of the random-bond spin-1/2 AFM Heisenberg chain flows to an infinite-randomness fixed point and a random singlet (RS) state forms on long length scales.\footnote{D. S. Fisher, PRB \textbf{50}, 3799 (1994).} This state can be characterized by its valence-bond entanglement entropy,\footnote{F. Alet, \emph{et al.}, PRL \textbf{99}, 117204 (2007).} defined to be $\overline{\langle n_L\rangle}$, the average number of valence bonds leaving a block of $L$ spins, as well the fluctuations of this number, $\sigma_{L}^2 = \overline{\langle n_L^2\rangle - \langle n_L\rangle^2}$, (angle brackets denote amplitude weighted averages over valence-bond states in the ground state, and overbar denotes disorder average). For large $L$, $\overline{\langle n_L \rangle}$ scales logarithmically, indicating a power-law distribution of valence-bond lengths, while $\sigma^2_L$ {\it saturates} at a crossover length scale $\xi$, beyond which the valence bonds ``lock" into a particular RS configuration.\footnote{H. Tran and N. E. Bonesteel, arXiv:0909.0038.} Using valence-bond Monte Carlo, we have studied the dependence of $\xi$ on disorder strength in the limit of weak disorder for both the Heisenberg chain and the critical random transverse-field Ising model. We compare our results with previous calculations of related crossover length scales in these models.\footnote{N. Laflorencie \emph{et al.}, PRB \textbf{70}, 054430 (2004).}