Valence-Bond Monte Carlo Study of Random-Singlet Phase Formation

In valence-bond Monte Carlo (VBMC)\footnote{A. Sandvik, PRL {\bf 95}, 207203 (2005).} the ground state of a quantum spin system is sampled directly from the valence-bond (VB) basis --- a useful basis for visualizing the properties of singlet ground states. For example, the ground state of the uniform AFM spin-$\frac{1}{2}$ Heisenberg chain is characterized by strongly fluctuating bonds with power-law length distribution, while in the random-singlet phase (RSP) of a {\it random} Heisenberg chain these bonds, while still having a power-law length distribution, lock into a particular VB state on long length scales.\footnote{D. S. Fisher, PRB \textbf{50}, 3799 (1994).} We use VBMC to directly probe the formation of a RSP by calculating both the average number of bonds $n_L$ leaving a block of $L$ spins (the VB entanglement entropy\footnote{F. Alet, et al., PRL \textbf{99}, 117204 (2007).}), and its {\it fluctuations}, $\sigma_{n_L}^2 = \left\langle\langle n_L^2\rangle- \langle n_L\rangle^2\right\rangle$. For the uniform chain they have been calculated exactly\footnote{J. L. Jacobsen and H. Saleur, PRL \textbf{100}, 087205 (2008).} and shown to grow logarithmically with $L$ --- signaling the strong bond fluctuations. For random chains while $n_L$ grows logarithmically with $L$, we find $\sigma_{n_L}^2$ {\it saturate} for large $L$, signaling the ``freezing" of the bonds into a particular random singlet state.