Thus, the meaning of ϕ n is the phase difference for state |n and state |n + 1 in the angle variables of the MMST mapping formalism. This explains why numerically one can also uniformly sample the angle φ n in the MMST mapping, which is equivalent to the uniform sampling of ϕ n . Moreover, it is well-known that the MMST Hamiltonian (Eq. 118) can also be expressed as the action-angle variables (Eq. 122 or Eq. 123) as