Peierls substitution method by taking the functional derivative of Hamiltonian can be used to determine the form of current-operator in continuum model (See Bruus-Flensberg) as well as lattice model. There is an alternate way to do it using Heisenberg equation of motion $\frac{dA}{dt} = \frac{i}{\hbar} [H, A] $ and interpreting it as a continuity equation $\frac{\partial \rho}{\partial t} = - \nabla \cdot \mathbf{J} $. But on a lattice the right side of this equation would give something like $-(J_{i+1}-J_{i})$, which does not directly determine $J_i$. Then how should I obtain the expression for spin-current and energy-current operators on lattice using Heisenberg equation of motion?
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