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

TL;DR

The paper develops a nonperturbative Hamiltonian-truncation approach, Lightcone Conformal Truncation (LCT), to study (2+1)d $\phi^4$ theory with UV divergences. It introduces a practical state-dependent counterterm prescription at $O(g^2)$ in lightcone quantization to cure UV sensitivities, enabling reliable strong-coupling and near-critical results. The authors compute the spectrum, Lorentz-invariant two-point functions, and Källén-Lehmann densities, finding mass-gap closure at a scheme-dependent critical coupling and IR universality consistent with the 3d Ising CFT, including vanishing $\rho_{T^\mu_{\ \mu}}(\mu)$ near criticality. Although limited by truncation, the work demonstrates IR-dominated convergence and provides a framework for applying LCT to higher-dimensional QFTs and critical phenomena.

Abstract

We use Lightcone Conformal Truncation (LCT) -- a version of Hamiltonian truncation -- to study the nonperturbative, real-time dynamics of $φ^4$-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 $φ^4$-theory in lightcone quantization. We then use LCT with this counterterm prescription to study $φ^4$-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.

Figures (24)

• Figure 1: (a) Sunset diagram; (b) Sunset diagram with spectators.
• Figure 2: Schematic representation of the integrated spectral density (see \ref{['eq:ISDFormula']}) of a generic operator ${\cal O}$ in the undeformed CFT, with our choice of $\mu$ discretization \ref{['eq:gDef']} (blue line), compared to the exact expression (black dashed line). There are $\mathfrak{i}_{\mathrm{max}}$ bins, each with constant support on the intervals $[\mu_{\mathfrak{i}-1}^2,\mu_\mathfrak{i}^2]$. The largest value $\mu_{\mathfrak{i}_{\mathrm{max}}}^2$ sets the UV cutoff, and the lowest value $\mu_1^2$ sets the IR resolution (see \ref{['eq:LUVIR']}). The parameter $r$ controls the relative width of successive bins, such that $r=1$ corresponds to uniform bins, and $r < 1$ has smaller bins in the IR.
• Figure 3: Integrated spectral densities of $\phi^2$ (top left), $\phi^3$ (top right), $\phi^4$ (bottom left), and $\phi^5$ (bottom right) in free massive field theory ($g=0$). Each plot shows the LCT data (blue), computed using $\Delta_{\max}=16$, $\mathfrak{i}_{\max}=65$, $\frac{\Lambda_{\mathrm{IR}}}{m}=0.5$, and $r=0.8$ (corresponding to $\frac{\Lambda_{\mathrm{UV}}}{m} = 1411$), along with the known analytical result (black line). The insets show the ratio of the data to the analytical result.
• Figure 4: The free massive theory ($\bar{g}=0$) two-point functions $\left\langle \phi^n(x)\phi^n(0) \right\rangle$ for $\phi^2$ (top left), $\phi^3$ (top right), $\phi^4$ (bottom left), and $\phi^5$ (bottom right). The LCT data (blue) was obtained with the same parameters as figure \ref{['fig:FreeSpectralDensities']}, and is compared to the theoretical prediction (black line). The insets show the ratio of the data to the analytical result. To guide the reader, red dashed lines and text show the scale at which the correlator deviates from the theory prediction by 20 percent.
• Figure 5: Perturbative corrections to the 1-particle mass at $O(g^2)$, given by eq. \ref{['eq:1pMassShift']}, and $O(g^3)$, eq. \ref{['eq:1pShiftg3']}, in LC quantization. Both corrections only involve three-particle intermediate states.