Exact ground state and elementary excitations of the spin tetrahedron chain

We study the antiferromagnetic spin exchange models with $S=1∕2$ and $S=1$ on a one-dimensional tetrahedron chain by both analytical and numerical approaches. The system is shown to be effectively mapped to a decoupled spin chain in the regime of strong rung coupling, and a spin sawtooth lattice in the regime of weak rung coupling with spin $2S$ on the top row and spin $S$ on the lower row. The ground state for the homogeneous tetrahedron chain is found to fall into the regime of strong rung coupling. As a result, the elementary excitation for the spin-$1∕2$ system is gapless whereas the excitation for the spin-1 system has a finite spin gap. With the aid of the exact diagonalization method, we determine the phase diagram numerically and find the existence of an additional phase in the intermediate regime. This phase is doubly degenerate and is characterized by an alternating distribution of rung singlet and rung spin $2S$. We also show that the SU(3) exchange model on the same lattice has completely different kind of ground state from that of its SU(2) correspondence and calculate its ground state and elementary excitation analytically.

Figure 1

(Color online) Schematic pictures of (a) the homogeneous spin tetrahedron chain, (b) the spin tetrahedron chain corresponding to model (2).

Figure 2

Three types of ground state for the spin-$1∕2$ and spin-1 pyrochlore strips. The open and close circles denote the spins with $S=1$ and $S=1∕2$, respectively, for the spin-$1∕2$ pyrochlore strip. For spin-1 pyrochlore strip, the open circles represent the spins with $S=2$ and the close circles represent the spins with $S=1$.

Figure 3

The phase diagram for spin-$1∕2$ tetrahedron chain with variable vertical exchange $J_{\perp }$ in the parameter space of $c=J_{\perp }∕J$. For $c>c_1$, the system is in a decoupled phase whose ground state is a product of rung singlets and the critical spin liquid phase on the horizontal spin-$1∕2$ chain; For $c_2<c<c_1$, the ground state is a double degenerate mixed state with the alternating spin singlet and spin triplet on the rungs; For $c<c_2$, the ground state is a ferrimagnetic state with spin triplet on the rungs.

Figure 4

(Color online) The lowest energies of $E(N,M)∕\mathrm{J}$ as a function of $c=J_{\perp }∕J$ for $N=4,6,8$. The solid, dashed, and dotted lines corresponding to the case with $M=N$, $M=N∕2$, and $M=0$ or (c), (b), (a) as displayed in Fig. 2.

Figure 5

(Color online) The phase boundaries $c_1$ and $c_2$ versus the sizes of the system.

Figure 6

(Color online) The ground state energies of systems (a), (b), and (c) versus the sizes of the system.

Figure 7

The phase diagram for spin-1 tetrahedron chain. For $c>c_1$, the system is in a decoupled phase whose ground state is a product of rung singlets and Haldane phase on the horizontal chain; For $c_2<c<c_1$, the ground state is a double degenerate mixed state with the alternating spin singlet and spin quintet on the rungs; For $c<c_2$, the ground state is a ferrimagnetic state with spin quintet on the rungs.