Order Parameter to Characterize Valence-Bond-Solid States in Quantum Spin Chains

We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin chains, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the number of valence bonds (broken valence bonds) at the boundary for periodic (open) systems. This allows us to determine the phase transition point in between different VBS states. We demonstrate this theory in the successive dimerization transitions of the bond-alternating Heisenberg chains, using the quantum Monte Carlo method.

Figure 1

Generalized VBS states with $(m,n)=(4,2)$ for (a) a periodic chain and for (b) an open chain with even $L$. Broken ovals mean the symmetrization of $S=1/2$ variables at each site, and solid triangles denote isolated bosons.

Figure 2

$\delta$ dependence of $z_L^{(1)}$ of the $L=64$ periodic BAHC with $S=\frac{1}{2}$, 1, $\frac{3}{2}$, and 2, obtained by the QMC calculation. The horizontal lines indicate $z_L$ for the VBS states calculated by the MP method [Eq. (15)].

Figure 3

System size dependence of the successive dimerization transition points ${\delta }_{\mathrm{c}}(L)$ of the periodic BAHC with $S=\frac{1}{2}$, 1, $\frac{3}{2}$, and 2. The extrapolation to the $L\to \infty$ limit is done by ${\delta }_{\mathrm{c}}(L)={\delta }_{\mathrm{c}}(\infty )+A/L^2+B/L^4+C/L^6$ (solid lines).