  Noncommuting conserved quantities have recently launched a subfield of
quantum thermodynamics. In conventional thermodynamics, a system of interest
and an environment 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 a prescription 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. Our Hamiltonians may be
realized physically with superconducting qudits, with ultracold atoms, and with
trapped ions.
