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Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is an efficientnumerical technique to solve strongly coupled QFTs in d=2 spacetime dimensions.Further theoretical developments are needed to increase its accuracy and therange of applicability. With this goal in mind, here we present a new variantof Hamiltonian Truncation which exhibits smaller dependence on the UV cutoffthan other existing implementations, and yields more accurate spectra. The keyidea for achieving this consists in integrating out exactly a certain class ofhigh energy states, which corresponds to performing renormalization at thecubic order in the interaction strength. We test the new method on the stronglycoupled two-dimensional quartic scalar theory. Our work will also be useful forthe future goal of extending Hamiltonian Truncation to higher dimensions d >=3.

High-Precision Calculations in Strongly Coupled Quantum Field Theory with Next-to-Leading-Order Renormalized Hamiltonian Truncation