Symmetry-protected topological phases in a two-leg SU(N) spin ladder with unequal spins

Abstract

Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective $\mathrm{SU}(N)$ group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with $N>2$. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of $\mathrm{SU}(N)$. We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg $\mathrm{SU}(N)$ spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for $N=3$ and 4 to show that all the possible symmetry-protected topological phases with projective $\mathrm{SU}(N)$ symmetry appear in this simple ladder model.