Exact models for trimerization and tetramerization in spin chains

We present exact models for an antiferromagnetic $S=1$ spin chain describing trimerization as well as for an antiferromagnetic $S=3/2$ spin chain describing tetramerization. These models can be seen as generalizations of the Majumdar-Ghosh model. For both models, we provide a local Hamiltonian and its exact threefold or fourfold degenerate ground state wave functions, respectively. We numerically confirm the validity of both models using exact diagonalization and discuss the low-lying excitations.

Figure 1

Phase diagram of Hamiltonian (1) as a function of $\theta$.

Figure 2

Illustration of one of the trimer ground states on a chain with $N=3\mu$ sites, where three neighboring spins are antisymmetrically coupled.

Figure 3

Spectrum of the spin 1 model (6) for trimerization on a chain with $N=18$ sites. The zero-energy ground states (labeled by thick lines in the spectrum) are at $k=0,\frac{2\pi }{3},\frac{4\pi }{3}$ in the Brillouin zone.

Figure 4

(a) Possible triplet excitation as a domain wall on a chain with $3\mu +1$ sites. (b) Possible triplet excitation as a domain wall on a chain with $3\mu +2$ sites. The excitations consist of two antisymmetrically coupled spins on adjacent sites.

Figure 5

The gap size of a chain with certain length $N$ is plotted vs the inverse chain length. Data points for $N=6$, 9, 12, 15, and 18 yield the fitted value of $\Delta =4.56\pm 0.04$.

Figure 6

Illustration of a tetramer state on a chain with $N=4\mu$ sites, where four neighboring spins are antisymmetrically coupled.

Figure 7

Spectrum of the spin 3/2 model (16) for tetramerization on a chain with $N=12$ sites. The zero-energy ground states (labeled by thick lines in the spectrum) are at $k=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2}$ in the Brillouin zone.

Figure 8

Quadruplet excitation serving as a domain wall between two of the tetramer ground states on a spin $S=\frac{3}{2}$ chain with $4\mu +3$ sites.