Systematic Improvement of Hamiltonian Truncation Effective Theory

TL;DR

This paper develops a systematic, order-by-order improvement method for Hamiltonian Truncation Effective Theory using EFT techniques, validated through explicit calculations in 1+1D $ ext{lambda} ext{phi}^4$ theory, demonstrating improved error scaling and consistency.

Contribution

It introduces a rigorous, systematic framework for improving Hamiltonian truncation results with EFT methods, including nonlocal corrections, and validates the approach through explicit 1+1D calculations.

Findings

Error scales as 1/E_max^4 with nonlocal corrections included

Method is consistent with EFT power counting

Estimates critical coupling and confirms scale separation

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.