NLO Renormalization in the Hamiltonian Truncation

TL;DR

This paper advances Hamiltonian Truncation by implementing a next-to-leading-order (NLO) renormalization scheme that accounts for high-energy tails via tail-states within a Rayleigh-Ritz-inspired framework. The key result is a variationally improved effective correction $\,\Delta \widetilde{H}=\Delta H_2({\cal E}_*)\big(\Delta H_2({\cal E}_*)-\Delta H_3({\cal E}_*)\big)^{-1}\Delta H_2({\cal E}_*)$, combined with a practical construction using finite-volume $(\phi^4)_2$ theory to demonstrate faster, smoother convergence than raw HT or LO renormalization. The study provides detailed numerical benchmarks for $E_T$-dependence, finite-volume scaling, and the critical coupling $g_c$, including Ising-model-like finite-volume predictions at criticality. These results push HT toward reliable, scalable non-perturbative analyses of strongly coupled QFTs and offer a pathway to applying renormalized truncation techniques in higher spacetime dimensions. The work also highlights computational strategies and potential future refinements, such as bilocal operators and hybrid approaches combining high-cutoff raw data with tail-based corrections.

Abstract

Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is a numerical technique for solving strongly coupled QFTs, in which the full Hilbert space is truncated to a finite-dimensional low-energy subspace. The accuracy of the method is limited only by the available computational resources. The renormalization program improves the accuracy by carefully integrating out the high-energy states, instead of truncating them away. In this paper we develop the most accurate ever variant of Hamiltonian Truncation, which implements renormalization at the cubic order in the interaction strength. The novel idea is to interpret the renormalization procedure as a result of integrating out exactly a certain class of high-energy "tail states". We demonstrate the power of the method with high-accuracy computations in the strongly coupled two-dimensional quartic scalar theory, and benchmark it against other existing approaches. Our work will also be useful for the future goal of extending Hamiltonian Truncation to higher spacetime dimensions.

Figures (17)

• Figure 1: The vacuum energy (left) and the physical mass (right) for $L=10$, plotted as a function of $E_T$ for the three methods: raw HT, local LO renormalized HT, and NLO-HT. The top (bottom) plots refer to $g=1$ ($g=2$).
• Figure 2: Convergence rate of NLO-HT vs local LO renormalized HT. See the text.
• Figure 3: The vacuum energy density ${\cal E}_0/L$ and the physical mass ${\cal E}_1-{\cal E}_0$ as functions of $L$ for three representative values of $g$.
• Figure 4: Left: the vacuum energy density as a function of $g$. The dashed line joining the points is not a fit; it is included to guide the eye. We also show errors divided by $g^2$. Right: the physical mass as a function of $g$. The line is a fit described in the text. We also show fit residuals divided by $g^2$.
• Figure 5: Comparison of energy levels at $g=g_c$ with CFT predictions.