Quantum Chromodynamics (QCD) --- the (3+1)-d asymptotically free SU(3) gauge theory that describes the strong interaction --- is usually formulated in terms of fundamental quark and gluon fields. CP(N-1) models in (1+1)-d have a global SU(N) symmetry and share many features with QCD. They are also asymptotically free, have a non-perturbatively generated mass gap, and non-trivial theta-vacuum states. CP(N-1) models can be regularized unconventionally by using discrete SU(N) quantum spins forming a (2+1)-d spin ladder that consists of n transversely coupled quantum spin chains. The (1+1)-d asymptotically free CP(N-1) fields then emerge from dimensional reduction when n is increased. Even n leads to the vacuum angle theta = 0, while odd n leads to theta = pi. In a similar way, gluon fields emerge naturally from the dimensional reduction of (4+1)-d quantum links, which are discrete gauge variables that generalize quantum spins. In this formulation quarks arise as domain wall fermions. In contrast to the usual quantum fields, quantum spins and quantum links realize asymptotically free field theories with finite-dimensional local Hilbert spaces. This is advantageous in the context of quantum simulation experiments. Both CP(N-1) models and QCD can be quantum simulated with ultra-cold alkaline-earth atoms in optical super-lattices. When CP(N-1) models are studied at non-zero chemical potential, non-trivial condensed matter physics arises in these quantum field theories. In particular, there are Bose-Einstein condensates, with or without ferromagnetism.
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