Quantum Hamiltonian time crystals from covariant quantum Landau--Lifshitz--Gilbert relaxation

Classical Hamiltonian time crystals are constrained energy minima that move along symmetry orbits at frequencies set by Lagrange multipliers. We construct quantum counterparts by minimizing the energy over density matrices with fixed spectrum and charge expectations. A symmetry-covariant quantum Landau--Lifshitz--Gilbert equation selects them as periodic attractors, preserving the spectrum and fixing the multipliers dynamically. A chiral spin trimer motivated by Cu$_3$ molecular magnets realizes this motion at a prepared magnetization, without a Zeeman term.