Relevance of operators in modified lattice Hamiltonians

The modified Hamiltonians with an exact ground state for lattice gauge theory and the nonlinear $\sigma$ model are analyzed in the nonperturbative subtraction scheme. By subtracting operators with lower dimensions, we obtain a subtracted operator $\overline{\Delta H}$ which is divergence-free. For a modified Hamiltonian with an exact ground state, $\overline{\Delta H}$ is in general relevant. The vacuum functional and low-energy spectrum calculations are discussed.