  Noncommuting conserved quantities have recently launched a subfield of
quantum thermodynamics. In conventional thermodynamics, a system of interest
and a bath exchange quantities -- energy, particles, electric charge, etc. --
that are globally conserved and are represented by Hermitian operators. These
operators were implicitly assumed to commute with each other, until a few years
ago. Freeing the operators to fail to commute has enabled many theoretical
discoveries -- about reference frames, entropy production, resource-theory
models, etc. Little work has bridged these results from abstract theory to
experimental reality. This paper provides a methodology for building this
bridge systematically: We present an algorithm for constructing Hamiltonians
that conserve noncommuting quantities globally while transporting the
quantities locally. The Hamiltonians can couple arbitrarily many subsystems
together and can be integrable or nonintegrable. Special cases of our
construction have appeared in quantum chromodynamics (QCD). Our Hamiltonians
may be realized physically with superconducting qudits, with ultracold atoms,
with trapped ions, and in QCD.
