Hamiltonian Truncation Effective Theory

Abstract

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 H_0H0 with eigenvalues below some energy cutoff E_\text{max}Emax. 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}Emax. 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}1/Emax. The effective Hamiltonian is non-local, with the non-locality controlled in an expansion in powers of H_0/E_\text{max}H0/Emax. The effective Hamiltonian is also non-Hermitian, and we discuss whether this is a necessary feature or an artifact of our definition. We apply our formalism to 2D \lambda\phi^4λϕ4 theory, and compute the the leading 1/E_\text{max}^21/Emax2 corrections to the effective Hamiltonian. We show that these corrections nontrivially satisfy the crucial property of separation of scales. Numerical diagonalization of the effective Hamiltonian gives residual errors of order 1/E_\text{max}^31/Emax3, as expected by our power counting. We also present the power counting for 3D \lambda \phi^4λϕ4 theory and perform calculations that demonstrate the separation of scales in this theory.