Strong-coupling expansion and two-point Padé approximation for lattice $φ^4$ field theory

Abstract

Reliable approximations for correlation functions at intermediate and strong coupling remain hard to obtain for general quantum field theories. Perturbative expansions are often asymptotic or have a finite radius of convergence, which limits their applicability beyond weak coupling. Here we combine weak- and strong-coupling expansions and propose to use two-point Padé schemes to construct approximants. For lattice $φ^4$ theory, we show that this two-point interpolation strategy yields accurate global approximations to the two-point correlation function across broad coupling regimes and compares favorably with standard one-point resummation methods. We also provide heuristic explanations for the observed convergence behavior and discuss the practical range of validity of the approach.

Figures (5)

• Figure 1: Different approximations of $G(\tilde{g})$ for the zero-dimensional $\phi^4$ field theory, where $\tilde{g} = \sqrt{g}$ is used. All the subscripts correspond to the order of the approximation. The second panel shows the approximation error relative to the exact result \ref{['od_G_analytic']}. Note that the error plots for the WCE and SCE expansions are shown on a log-log scale, which clearly illustrates the divergence of the SCE at small values of $g$.
• Figure 2: Convergence rate comparison between SCE-1Padé, Borel-Padé, and 2Padé expansion for zero-dimensional $\phi^4$ field theory at selected interaction strength $\tilde{g}=\sqrt{g}$.
• Figure 3: (Left) Different Padé approximation for $G_{00}(\tilde{g})$ of the 1D lattice $\phi^4$ field and (Right) their corresponding approximation error $e_{00}(\tilde{g})=|G^{MC}_{00}(\tilde{g})-G^{app}_{00}(\tilde{g})|$ . All the subscripts correspond to the order of the approximation. Note that using only WCE or SCE up to the third order, the highest order 1Padé or Borel-Padé expansion one could get is $[1/2]$, which is what we displayed here. On the contrary, 2Padé combines WCE and SCE hence we can get expansion up to the order $[3/4]$. The simulated $\phi^4$ field has lattice site $N_s=16$ with $t=\mu=1.0$. The Langevin MC results are based on $10^8$ samples, yielding a numerical accuracy of approximately $O(10^{-4})$. Due to this limited accuracy, the error curves for 2Padé is less accurate. By the analysis of our work zhu2025global, the numerical accuracy of 2Padé$_{[3/4]}$ is expected to be slightly smaller than $10^{-3}$ for all $\tilde{g}$.
• Figure 4: Top row: Domain coloring plots of $G(g)$ and $\mathcal{G}(s)$ in the complex plane. The grayness indicates the modulus of the function, while the hue represents the complex argument $\theta$. The branch point at $g = 0$ is fully resolved in the $s$-plane, where $\mathcal{G}(s)$ is analytic at $s = 0$. Bottom row: Plots of $G(g \pm i\epsilon)$ and $\mathcal{G}(s \pm i\epsilon)$ with a small imaginary shift $\epsilon = 10^{-4}$. The branch-point singularity of $G(g)$ at $g = 0$ is clearly visible, in contrast to the regular behavior of $\mathcal{G}(s)$. We emphasize that the plot of $\mathcal{G}(s)$ outside the moon-shaped convergence disk is inaccurate, as it is generated from the truncated 50th-order power series expansion \ref{['0d_sce_remove']}. As we discuss below, the correct continuation of $\mathcal{G}(s)$ beyond this disk is well captured by Padé approximation (see Fig. \ref{['fig:domain_coloring_pade']}).
• Figure 5: (Top Left) Comparison between the analytic continuation (AC) of the exact solution $G(g)$ given in Eq. \ref{['od_G_analytic']} and that of its 4th-order 2Padé approximation $G_{\text{Pad\'e}}(g)$ in the complex plane. The red contour lines represent the modulus $|G(g)|$, and the black contour lines show $|G_{\text{Pad\'e}}(g)|$. The near-perfect overlap in the right half-plane ($\text{Re}[g] > 0$) demonstrates good agreement between the two. (Bottom Left) Plot of the approximation error $G_{\text{Pad\'e}}(g) - G(g)$, highlighting the accuracy of the 2Padé approximation for $\text{Re}[g] \geq 0$. (Top Right) The same comparison between $\mathcal{G}(s)$ and that of its 5th-order SCE-1Padé approximation $\mathcal{G}_{\text{Pad\'e}}(s)$ in the complex plane. (Bottom Right) Plot of the approximation error $\mathcal{G}_{\text{Pad\'e}}(s) - \mathcal{G}(s)$, highlighting the accuracy of the SCE-1Padé approximation for $\text{Re}[s] \geq 0$.