RG Flow from $φ^4$ Theory to the 2D Ising Model

TL;DR

The paper develops and applies conformal truncation to the $1+1$D $\phi^4$ theory, starting from the UV free scalar CFT and using a Casimir-basis truncation in lightcone quantization to access the full RG flow toward the IR Ising fixed point. It provides nonperturbative spectral densities for local operators and tracks the Zamolodchikov $C$-function along the flow, demonstrating IR universality by matching Ising-model predictions for $\varepsilon$ and $\sigma$ and verifying the IR behavior of the stress-tensor sector. The critical coupling is extracted via finite-$\Delta_{\max}$ extrapolation, yielding $\frac{\bar{\lambda}_*}{4\pi}=1.84\pm 0.03$, with results consistent with, yet distinct from, prior lightcone methods due to quantization schemes. The study showcases the method’s capability to compute real-time, infinite-volume observables nonperturbatively and highlights UV-correction effects that influence IR observables like the $C$-function, outlining directions for extensions to other CFT deformations and higher dimensions.

Abstract

We study 1+1 dimensional $φ^4$ 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.

Figures (15)

• Figure 1: Integrated spectral densities for $\phi^2$ (upper left), $\phi^3$ (upper right), $\phi^4$ (lower left), and $\phi^5$ (lower right) in massive free field theory ($\bar{\lambda}=0$), both the raw value (main plot) and normalized by the theoretical prediction (inset). The conformal truncation results (blue dots) for each plot are computed using the $\Delta_{\max}$ shown, with the corresponding number of $n$-particle basis states, and compared to the theoretical prediction (black curve).
• Figure 2: The two lowest mass eigenvalues in the odd sector and the lowest eigenvalue in the even sector as a function of $\bar{\lambda}$ for $\Delta_{\max}=34$ (12,310 basis states).
• Figure 3: Two examples of the dependence of $\mu_{1,\textrm{odd}}^2$ (green), $\mu_{1,\textrm{even}}^2$ (blue), and $\mu_{2,\textrm{odd}}^2$ (red) on $\Delta_{\max}$, at fixed $\frac{\bar{\lambda}}{4\pi}=0.55$ (left) and $\frac{\bar{\lambda}}{4\pi}=1.75$ (right). The solid lines show the best fit for each $\mu_i^2(\Delta_{\max})$ to the functional form in eq. \ref{['eq:fit']}, with the resulting powers $n=2.0$ (left) and $n=1.0$ (right). The $y$-intercept for each fit provides the extrapolated value of $\mu_i^2$ for $\Delta_{\max}\rightarrow\infty$, and the error is estimated by varying the slope by $15\%$ about the mean of the data points.
• Figure 4: The two lowest mass eigenvalues in the odd sector and the lowest eigenvalue in the even sector as a function of $\bar{\lambda}$ in the extrapolated limit $\Delta_{\max}\rightarrow\infty$.
• Figure 5: The ratio of two lowest mass eigenvalues in the odd sector and the lowest eigenvalue in the even sector to the mass gap as a function of $\mu^2_{1,\textrm{odd}}$ in the extrapolated limit $\Delta_{\max}\rightarrow\infty$.