Antiferromagnetic Spin-$S$ Chains with Exactly Dimerized Ground States

We show that spin $S$ Heisenberg spin chains with an additional three-body interaction of the form $({\mathbf{S}}_{\mathbf{i}\mathbf{-}\mathbf{1}}·{\mathbf{S}}_{\mathbf{i}})({\mathbf{S}}_{\mathbf{i}}·{\mathbf{S}}_{\mathbf{i}+\mathbf{1}})+\mathrm{H}.\mathrm{c}.$ possess fully dimerized ground states if the ratio of the three-body interaction to the bilinear one is equal to $1/[4S(S+1)-2]$. This result generalizes the Majumdar-Ghosh point of the $J_1-J_2$ chain, to which the present model reduces for $S=1/2$. For $S=1$, we use the density matrix renormalization group method to show that the transition between the Haldane and the dimerized phases is continuous with a central charge $c=3/2$. Finally, we show that such a three-body interaction appears naturally in a strong-coupling expansion of the Hubbard model, and we discuss the consequences for the dimerization of actual antiferromagnetic chains.

Figure 1

Spectrum of the Hamiltonian ${\mathcal{H}}_i$ [Eq. (7)] on three adjacent sites of the chain as a function of $S(S+1)$. The solid (green) line indicates the energy of the dimerized eigenstate on this three-site system, $E=-S(S+1)/2$.

Figure 2

Dimerization as a function of $J_3/J_1$ for different system sizes up to $L=250$ sites in the vicinity of the phase transition $J_3/J_1\approx 0.11$. The insets show the size dependence at $J_3/J_1=0.1$, 0.111, and 0.12, respectively.

Figure 3

Correlation length as a function of $J_3/J_1$ for different system sizes. Inset: Spin and quadrupolar correlation functions of Eq. (9) at the critical point $J_3/J_1=0.111$.

Figure 4

Fit of the Calabrese and Cardy formula [Eq. (8), continuous line] to the DMRG results (dots) for the block entropy at $J_3/J_1=0.111$.

Figure 5

Phase diagram of the $J_1\mathrm{\text{−}}{J}_3$ chain of Eq. (5) for $S=1$.