Krylov spaces for truncated spectrum methodologies

Abstract

We propose herein an extension of truncated spectrum methodologies, a nonperturbative numerical approach able to elucidate the low energy properties of quantum field theories. TSMs, in their various flavors, involve a division of a computational Hilbert space, H, into two parts, one part, H1 that is “kept” for the numerical computations, and one part, H2, that is discarded or “truncated.” Even though H2 is discarded, truncated spectrum methodologies will often try to incorporate the effects of H2 in some effective way. In these terms, we propose to keep the dimension of H1 small. We pair this choice of H1 with a Krylov subspace iterative approach able to take into account the effects of H2. This iterative approach can be taken to arbitrarily high order and so offers the ability to compute quantities to arbitrary precision. In many cases it also offers the advantage of not needing an explicit UV cutoff. To compute the matrix elements that arise in the Krylov iterations, we employ a Feynman diagrammatic representation that is then evaluated with Monte Carlo techniques. Each order of the Krylov iteration is variational and is guaranteed to improve upon the previous iteration. The first Krylov iteration is akin to the next-to-leading order approach of Elias-Miró [NLO renormalization in the Hamiltonian truncation, ]. To demonstrate this approach, we focus on the (1+1d)-dimensional ϕ4 model and compute the bulk energy and mass gaps in both the Z2-broken and unbroken sectors. We estimate the critical ϕ4 coupling in the broken phase to be gc=0.2645±0.002. Published by the American Physical Society 2024