# RG Flow from $\phi^4$ Theory to the 2D Ising Model

## TL;DR

This paper employs conformal truncation to analyze the 1+1D $^4$ theory, successfully capturing the RG flow to the 2D Ising model and computing spectral densities across the entire coupling range.

## Contribution

It introduces a conformal truncation approach to study the RG flow from free field theory to the 2D Ising model, providing non-perturbative spectral densities at strong coupling.

## Key findings

- Spectral densities match 2D Ising model near criticality.
- Method computes the Zamolodchikov C-function along the RG flow.
- Numerical diagonalization captures IR dynamics at strong coupling.

## Abstract

We study 1+1 dimensional $\phi^4$ theory using the recently proposed method of conformal truncation. Starting in the UV CFT of free field theory, we construct a complete basis of states with definite conformal Casimir, $\mathcal{C}$. We use these states to express the Hamiltonian of the full interacting theory in lightcone quantization. After truncating to states with $\mathcal{C} \leq \mathcal{C}_{\max}$, we numerically diagonalize the Hamiltonian at strong coupling and study the resulting IR dynamics. We compute non-perturbative spectral densities of several local operators, which are equivalent to real-time, infinite-volume correlation functions. These spectral densities, which include the Zamolodchikov $C$-function along the full RG flow, are calculable at any value of the coupling. Near criticality, our numerical results reproduce correlation functions in the 2D Ising model.