Two-leg $\mathrm{SU}\left(2n\right)$ spin ladder: A low-energy effective field theory approach

We present a field-theory analysis of a model of two $\mathrm{SU}\left(2n\right)$-invariant magnetic chains coupled by a generic interaction preserving time-reversal and inversion symmetry. Contrary to the SU(2)-invariant case the zero-temperature phase diagram of such two-leg spin ladder does not contain topological phases. Only generalized valence bond solid phases are stabilized when $n>1$ with different wave vectors and ground-state degeneracies. In particular, we find a phase which is made of a cluster of $2n$ spins put in an $\mathrm{SU}\left(2n\right)$ singlet state. For $n=3$, this cluster phase is relevant to ${}^{173}\mathrm{Yb}$ ultracold atoms, with an emergent SU(6) symmetry, loaded in a double-well optical lattice.

Figure 1

Phase diagram of model (9) for $N=4$ by means of a field-theory analysis when $|{\lambda }_1|\gg |{\lambda }_2|$ and a mean-field approach for $|{\lambda }_2|\gg |{\lambda }_1|$. A similar phase diagram is obtained in the even-$n$ case where the plaquette phase is an $n$-site rectangular singlet phase.