Nonperturbative dynamics of (2+1)d $\phi^4$-theory from Hamiltonian truncation

TL;DR

This paper employs Lightcone Conformal Truncation to nonperturbatively analyze the real-time dynamics of (2+1)d $^4$-theory, addressing UV divergences with a novel state-dependent renormalization scheme and exploring critical phenomena.

Contribution

It introduces a state-dependent counterterm prescription for Hamiltonian truncation in (2+1)d $^4$-theory and applies it to study the spectrum and correlation functions near criticality.

Findings

Demonstrates the closing of the mass gap at a scheme-dependent critical coupling.

Computes Lorentz-invariant two-point functions at strong coupling.

Shows IR universality and vanishing stress tensor trace near the critical point.

Abstract

We use Lightcone Conformal Truncation (LCT) -- a version of Hamiltonian truncation -- to study the nonperturbative, real-time dynamics of $\phi^4$-theory in 2+1 dimensions. This theory has UV divergences that need to be regulated. We review how, in a Hamiltonian framework with a total energy cutoff, renormalization is necessarily \emph{state-dependent}, and UV sensitivity cannot be canceled with standard local operator counterterms. To overcome this problem, we present a prescription for constructing the appropriate state-dependent counterterms for (2+1)d $\phi^4$-theory in lightcone quantization. We then use LCT with this counterterm prescription to study $\phi^4$-theory, focusing on the $\mathbb{Z}_2$ symmetry-preserving phase. Specifically, we compute the spectrum as a function of the coupling and demonstrate the closing of the mass gap at a (scheme-dependent) critical coupling. We also compute Lorentz-invariant two-point functions, both at generic strong coupling and near the critical point, where we demonstrate IR universality and the vanishing of the trace of the stress tensor.