Exploring Hamiltonian Truncation in $\bf{d=2+1}$

TL;DR

This work extends Hamiltonian Truncation to $d=2+1$ QFTs by developing a regulator-based framework for UV-divergent perturbations, exemplified through $\phi^4$ in $d=2+1$. It identifies nontrivial issues with naive $E_T$-regulated perturbation theory, notably missing-state effects that spoil cancellations of vacuum bubbles, and proposes two patch-counterterm schemes to restore finiteness as $E_T\to\infty$. The authors validate the approach by analyzing a solvable $\phi^2_3$ test and then applying the full $\phi^4_3$ theory, showing good perturbative agreement at weak coupling and robust strong-coupling results that pass a nontrivial self-duality crosscheck. The results demonstrate the practicality of HT in higher dimensions and UV-divergent settings, with implications for studying strongly coupled QFTs in physically relevant regimes. The study also outlines potential refinements, such as a momentum-cutoff strategy, and points to future work in large-volume behavior and phase transitions.

Abstract

We initiate the application of Hamiltonian Truncation methods to solve strongly coupled QFTs in $d=2+1$. By analysing perturbation theory with a Hamiltonian Truncation regulator, we pinpoint the challenges of such an approach and propose a way that these can be addressed. This enables us to formulate Hamiltonian Truncation theory for $φ^4$ in $d=2+1$, and to study its spectrum at weak and strong coupling. The results obtained agree well with the predictions of a weak/strong self-duality possessed by the theory. The $φ^4$ interaction is a strongly relevant UV divergent perturbation, and represents a case study of a more general scenario. Thus, the approach developed should be applicable to many other QFTs of interest.

Figures (9)

• Figure 1: The size of the Fock space basis as a function of $E_T$ for $m=1$ and different box sizes $L$. For $L=4$ the contributions from basis elements with different occupation numbers $N$ is shown.
• Figure 2: The vacuum energy density [left] and mass gap [right] of the theory defined by Eq. \ref{['free1']} deformed by Eq. \ref{['eq:phi2def']} with $g_2=1$ as a function of the cutoff energy $E_T$. The analytic predictions both in the infinite volume limit and at finite volume are also indicated. The results for the vacuum energy are shown with and without including the sub-leading correction (\ref{['phi2sublead']}), which improves the convergence.
• Figure 3: The vacuum energy density [left] and mass gap [right] as a function of $g_2$ obtained from Hamiltonian Truncation calculations after extrapolation to $E_T \rightarrow \infty$. We also show the exact analytic results [for the vacuum energy density with and without including winding corrections, solid and dashed respectively] and the prediction from perturbation theory [labeled PT] at order $g_2^2$ and $g_2^4$.
• Figure 4: The vacuum and low lying energy levels of $\phi^4_3$ obtained from a HT calculation as a function of the cutoff $E_T$, for a theory with $g=18m$ in a box of size $L=4/m$. We show the results with the compete set of counter-terms needed to render the theory finite ["with C.T.s"], and those obtained when no counter-terms are included ["without C.T.s"].
• Figure 5: A comparison between the mass gap $\Delta$ as a function of $g$ calculated from HT calculations and the perturbative prediction. We plot $m \left(\Delta -m \right)/\left(g/24 \right)^2$ so that the perturbative prediction at $O(g^2)$ is a horizontal line and the $O(g^3)$ prediction is the straight line indicated. Details of the extrapolation of the HT data to $E_T \rightarrow \infty$ and the error estimates are given in the main text.