Unified phase diagram of antiferromagnetic SU( N ) spin ladders

Abstract

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 $\mathrm{SU}(N)$ spin ladders. Two-leg ladders provide a rich and highly nontrivial 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 $2\ensuremath{\le}N\ensuremath{\le}6$, we show that the phase diagrams as a function of the interchain (rung) to intrachain (leg) coupling ratio, ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}$, strongly differ for even versus 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 ${J}_{\ensuremath{\perp}}$-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. Our results are based on (i) field theoretical techniques, (ii) qualitative symmetry considerations, and (iii) large-scale density matrix renormalization group (DMRG) simulations keeping beyond a million of states by fully exploiting and thus preserving the $\mathrm{SU}(N)$ symmetry.