Highest weight state description of the isotropic spin-1 chain

We introduce an overcomplete highest weight state basis as a calculational tool for the description of the isotropic spin-1 chain with bilinear exchange coupling $J_1$ and biquadratic coupling $J_2$. The ground state can be expressed exactly at the three special points in the phase diagram where the Hamiltonian corresponds to a sum of nearest neighbor total spin projection operators ($J_1=0>J_2$, $J_1=-J_2<0$, and $J_1=-J_2∕3>0$). In particular, at the phase transition point $J_1=-J_2<0$, it is possible to exactly compute the ground states, excited states, expectation values, and correlation functions by using the new total spin basis.

Figure 1

(Left) Phase diagram of the spin-1 chain as function of $\Theta$: $A\left(0\right)$, pure Heisenberg chain; $B(-\arctan 1∕3)$, AKLT point (Refs. 3, 4); $C(-\pi ∕4)$, Sutherland model (Ref. 5); $D(-\pi ∕2)$, phase transition (Ref. 19); $E(3\pi ∕4)$, phase transition (Refs. 20, 21, 22); $F(\pi ∕2)$, exactly solvable (Refs. 6, 7, 8); and $G(\pi ∕4)$, $SU{\left(2\right)}_2$ integrable point (Refs. 23, 24, 25). (Right) Phase diagram of the spin-$\frac{1}{2}$ chain as function of $\vartheta$: $a\left(0\right)$, Heisenberg chain (Ref. 26); $b(\arctan 1∕2)$, Majumdar-Ghosh point (Ref. 17); $c(\sim 0.24)$, critical point (Ref. 27); $d(-\pi ∕2)$, phase transition; $g(\pi ∕2)$, two independent chains, $SU{\left(2\right)}_1\times SU{\left(2\right)}_1$ integrable; and $e(\pi -\arctan 1∕4)$, phase transition (Ref. 28). The gaps are generically denoted by ▵. The points $B$, $D$, and $E$ are projection points treated in this paper.