$SU\left(3\right)$ quantum spin ladders as a regularization of the $CP\left(2\right)$ model at non-zero density: From classical to quantum simulation

Quantum simulations would be highly desirable in order to investigate the finite density physics of QCD. $(1+1)$-d $CP(N-1)$ quantum field theories are toy models that share many important features of QCD: they are asymptotically free, have a non-perturbatively generated massgap, as well as $\theta$-vacua. $SU\left(N\right)$ quantum spin ladders provide an unconventional regularization of $CP(N-1)$ models that is well-suited for quantum simulation with ultracold alkaline-earth atoms in an optical lattice. In order to validate future quantum simulation experiments of $CP\left(2\right)$ models at finite density, here we use quantum Monte Carlo simulations on classical computers to investigate $SU\left(3\right)$ quantum spin ladders at non-zero chemical potential. This reveals a rich phase structure, with single- or double-species Bose–Einstein "condensates", with or without ferromagnetic order.