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