Hamiltonian Truncation Effective Theory

Hamiltonian truncation is a non-perturbative numerical method for calculating observables of a quantum field theory. The starting point for this method is to truncate the interacting Hamiltonian to a finite-dimensional space of states spanned by the eigenvectors of the free Hamiltonian with eigenvalues below some energy cutoff $E_\text{max}$. In this work, we show how to treat Hamiltonian truncation systematically using effective field theory methodology. We define the finite-dimensional effective Hamiltonian by integrating out the states above $E_\text{max}$. The effective Hamiltonian can be computed by matching a transition amplitude to the full theory, and gives corrections order by order as an expansion in powers of $1/E_\text{max}$. The effective theory has a number of unusual features at higher orders, such as non-local interactions and non-Hermiticity of the effective Hamiltonian, whose physical origin we clarify. We apply this formalism to the theory of a relativistic scalar field $\phi$ with a $\lambda \phi^4$ coupling in 2 and 3 spacetime dimensions. We perform numerical tests of the method in 2D, and find that including our matching corrections yields significant numerical improvements consistent with the expected dependence on the $E_\text{max}$ cutoff.