Effective Hamiltonians and Counterterms for Hamiltonian Truncation

Abstract

We outline a procedure for applying Hamiltonian Truncation to Quantum Field Theories (QFTs) that have UV divergences. To do this, we derive a novel representation of an Effective Hamiltonian which makes manifest some of its important properties (e.g. the non-perturbative matching of the spectra between the UV theory and the theory described by the Effective Hamiltonian). We check the consistency of our procedure using Conformal Perturbation Theory. Finally we comment on how the Effective Hamiltonian, which incorporates non-local interactions, describes a local QFT.

Hamiltonian truncation crafted for UV-divergent QFTs

“…Considerable progress has been made in reducing the cutoff dependence of HT calculations through the use of Effective Hamiltonians [3][4][5][6][7][8][9][10][11][12][13]. In these approaches, extra terms are added to the Hamiltonian that act only on low energy states.…”

“…In Ref. [12], a general procedure was outlined for determining the required nonlocal counterterms in QFTs with UV divergences, which made use of an Effective Hamiltonian. In this procedure, the QFT was first regulated using a local short distance cutoff regulator and renormalized using local counterterms.…”

“…In this paper, we revisit the procedure of [12] and apply it in numerical studies of three strongly-coupled two-dimensional QFTs as examples. Our goals are firstly to provide a demonstration of the procedure, secondly to investigate its performance and efficiency against wellstudied benchmark QFTs, and then to boldly go and examine the phase structure of a new QFT, that has not been studied using HT before.…”

“…We begin in section 2 by introducing our notation and reviewing both the TCSA and the procedure from [12]. In section 3 we present the three QFTs we will study, and discuss the TCSA in the context of minimal models.…”

“…Increasing ∆ T is computationally costly and therefore it is worth finding strategies to improve convergence without necessarily increasing ∆ T . One such strategy is to use an Effective Hamiltonian [3][4][5][6][7][8][9][10][11][12][13]. The intuitive idea is to add EFT operators to (3), which act on the truncated Hilbert space, and take into account the effect of integrating out the higher energy eigenstates with ∆ i > ∆ T .…”