Introduction to Lightcone Conformal Truncation: QFT Dynamics from CFT Data

TL;DR

The paper introduces Lightcone Conformal Truncation (LCT) as a Hamiltonian truncation framework for studying dynamical quantities in QFTs with UV CFT fixed points, focusing on 2D theories. It develops and contrasts three practical methods to construct the truncated Hilbert space and compute matrix elements: Fock Space, Wick Contraction, and Radial Quantization, with key innovations such as CFT data-based basis construction and SUSY-inspired chiral-symmetry-preserving counterterms. The work provides detailed scaffolding and code to study models including 2D φ^4 theory, Yukawa theory, and 2D QCD at both large and finite Nc, demonstrating how to extract spectra, spectral densities, and Zamolodchikov C-functions, and revealing how truncation defines emergent UV/IR scales and affects convergence. By combining analytic and numerical techniques, the authors offer a practical pipeline for deriving QFT observables from CFT data in an infinite-volume setting, complemented by a public Mathematica package and tutorials. Overall, the framework offers a concrete, scalable route to nonperturbative real-time dynamics in 2D QFTs and provides insights into phase structure, resonance behavior, and gauge dynamics within a unified LCT approach.

Abstract

We both review and augment the lightcone conformal truncation (LCT) method. LCT is a Hamiltonian truncation method for calculating dynamical quantities in QFT in infinite volume. This document is a self-contained, pedagogical introduction and "how-to" manual for LCT. We focus on 2D QFTs which have UV descriptions as free CFTs containing scalars, fermions, and gauge fields, providing a rich starting arena for LCT applications. Along our way, we develop several new techniques and innovations that greatly enhance the efficiency and applicability of LCT. These include the development of CFT radial quantization methods for computing Hamiltonian matrix elements and a new SUSY-inspired way of avoiding state-dependent counterterms and maintaining chiral symmetry. We walk readers through the construction of their own basic LCT code, sufficient for small truncation cutoffs. We also provide a more sophisticated and comprehensive set of Mathematica packages and demonstrations that can be used to study a variety of 2D models. We guide the reader through these packages with several examples and illustrate how to obtain QFT observables, such as spectral densities and the Zamolodchikov $C$-function. Specific models considered are finite $N_c$ QCD, scalar $φ^4$ theory, and Yukawa theory.

Figures (26)

• Figure 1: Ground state energy $E_0$ and residual $\Delta E_0 \equiv E_0 - E_0^{(n_{\rm max}=\infty)}$ as a function of truncation level $n_{\rm max}$ for the anharmonic oscillator at coupling $g=3$. $E_0^{(n_{\rm max}=\infty)}$ is shown in dashed, black. The residuals decay exponentially as a function of $n_{\rm max}$
• Figure 2: Cartoon of Space of CFTs and the RG flows between them. Not drawn to scale. One of the goals of conformal truncation is to turn this cartoon into a sharp computational tool.
• Figure 3: Mass term Hamiltonian matrix elements involving $\partial\phi$ and $(\partial\phi)^3$. The matrix element corresponding to the middle diagram, which involves the creation of particles from the vacuum, vanishes in lightcone quantization.
• Figure 4: Quartic interaction Hamiltonian matrix elements involving $\partial\phi$ and $(\partial\phi)^3$. The $1 \rightarrow 1$ matrix element has been removed by normal-ordering the $\phi^4$ interaction, and the $1 \rightarrow 5$ matrix element, which involves the creation of particles from the vacuum, vanishes in lightcone quantization.
• Figure 5: Example of a potential $V(\phi)$ with $(a)$ first-order or $(b)$ second-order phase transition as couplings vary in $\phi^6$ theory. In $(a)$, the lowest eigenvalue of the LC Hamiltonian jumps discontinuously from a positive to negative value, indicating the presence of a new global minimum, while in $(b)$, the lowest eigenvalue smoothly crosses zero.