Haldane Gap of the Three-Box Symmetric $\mathrm{SU}(3)$ Chain

Abstract

Motivated by the recent generalization of the Haldane conjecture to SU(3) chains [M. Lajkó et al., Nucl. Phys. B 924, 508 (2017)] according to which a Haldane gap should be present for symmetric representations if the number of boxes in the Young diagram is a multiple of 3, we develop a density matrix renormalization group algorithm based on standard Young tableaus to study the model with 3 boxes directly in the representations of the global SU(3) symmetry. We show that there is a finite gap between the singlet and the symmetric [3 0 0] sector ∆ [3 0 0] /J = 0.040 ± 0.006 where J is the antiferromagnetic Heisenberg coupling, and we argue on the basis of the structure of the low energy states that this is sufficient to conclude that the spectrum is gapped.

Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach

“…It also agrees for N = 3 with the recent numerical investigation of the Heisenberg model (1) in the 3box symmetric representation. 65 A non-degenerate phase with a very small gap ∆ ≃ 0, 04J has been reported while a critical behavior was more likely in previous numerical studies. 29,66 The edge states were also found to belong to the adjoint representation of SU(3) as expected from the underlying AKLT construction.…”