Higher-Order Structure of Hamiltonian Truncation Effective Theory

TL;DR

This paper advances Hamiltonian truncation methods for 2D $ ext{phi}^4$ theory by systematically including higher-order corrections, improving the accuracy of effective theories through all-order local matching and next-to-next-to-local corrections.

Contribution

It introduces all-order local matching corrections and next-to-next-to-local corrections in Hamiltonian truncation, enhancing the precision of effective theories for quantum field models.

Findings

All-order local matching corrections derived

Next-to-next-to-local corrections computed at $ ext{O}(E_{ m max}^{-4})$

Rich operator basis needed for accurate beyond-leading-order descriptions

Abstract

We study the Hamiltonian truncation for the two-dimensional $\lambda\phi^4$ theory within the framework of Hamiltonian truncation effective theory, where truncation artifacts are mitigated through a systematic inclusion of corrective terms organized in inverse powers of the ultraviolet energy cut-off $E_{\rm max}$. Building on the leading-order matching program, we develop two complementary extensions. First, we derive compact all-order expressions for the local matching corrections to the mass and quartic coupling by resumming infinite classes of diagrams sharing fixed topologies within the local approximation. Second, we extend the non-local sector by computing the next-to-next-to-local corrections contributing at $\mathcal{O}(E_{\rm max}^{-4})$, following a continuum-first matching procedure, in which the effective corrections are computed in infinite volume and the spatial direction is subsequently re-compactified to obtain a separable Hilbert-space basis on which the truncated operator construction is implemented. Our results show that an increasingly rich operator basis is necessary to describe the theory beyond leading order.