Systematic Improvement of Hamiltonian Truncation Effective Theory

Abstract: Hamiltonian Truncation Effective Theory is a framework that aims to improve the results of Hamiltonian truncation in a systematic, order-by-order fashion using Effective Field Theory methodology. The result is a truncated effective Hamiltonian with corrections that result from a matching procedure. We establish the rigor of this method by calculating nontrival next-to-leading order corrections in a $1/E_{\rm max}$ expansion, where $E_{\rm max}$ is our effective theory cutoff. We illustrate this explicitly using 1+1D $\lambda \phi^4$ theory, calculating corrections up to order $1/E_{\rm max}^3$. At this order, novel nonlocal contributions to the matching conditions must be incorporated. We show that by including these nonlocal terms, the error scales as $1/E_{\rm max}^4$, as expected from the Effective Field Theory power counting, providing a nontrivial check that this method is consistent and robust. We also estimate the critical coupling at which this theory flows to the 2D Ising conformal field theory and confirm that separation of scales, an essential feature of Effective Field Theory, persists at this order. These results establish Hamiltonian Truncation Effective Theory as a generic, systematic framework for improving convergence in Hamiltonian truncation and lay the groundwork to apply this method to more complex systems in higher dimensions.