Unified phase diagram of antiferromagnetic SU(N) spin ladders

Motivated by near-term experiments with ultracold alkaline-earth atoms confined to optical lattices, we establish numerically and analytically the phase diagram of two-leg SU($N$) spin ladders. Two-leg ladders provide a rich and highly non-trivial extension of the single chain case on the way towards the relatively little explored two dimensional situation. Focusing on the experimentally relevant limit of one fermion per site, antiferromagnetic exchange interactions, and $2\leq N \leq 6$, we show that the phase diagrams as a function of the interchain (rung) to intrachain (leg) coupling ratio $J_\perp/J_\Vert$ strongly differ for even vs. odd $N$. For even $N=4$ and 6, we demonstrate that the phase diagram consists of a single valence bond crystal (VBC) with a spatial period of $N/2$ rungs. For odd $N=3$ and 5, we find surprisingly rich phase diagrams exhibiting three distinct phases. For weak rung coupling, we obtain a VBC with a spatial period of $N$ rungs, whereas for strong coupling we obtain a critical phase related to the case of a single chain. In addition, we encounter intermediate phases for odd $N$, albeit of a different nature for $N=3$ as compared to $N=5$. For $N=3$, we find a novel gapless intermediate phase with $J_\perp$-dependent incommensurate spatial fluctuations in a sizeable region of the phase diagram. For $N=5$, there are strong indications for a narrow potentially gapped intermediate phase, whose nature is not entirely clear.