Global approximations to correlation functions of strongly interacting quantum field theories

TL;DR

This paper introduces a two-point Padé interpolation method to construct global approximations to correlation functions in strongly interacting quantum field theories by combining weak- and strong-coupling expansions. It applies the method to lattice φ^4 theory and the 2D Hubbard model, showing uniform/global convergence of the interpolants and providing a heuristic explanation based on analytic function theory. The approach reduces the required perturbative order and complements existing nonperturbative techniques, offering a scalable pathway for more accurate simulations of strongly correlated systems. The work suggests a versatile framework that could be extended with higher-order expansions and integrated with diagrammatic quantum Monte Carlo methods.

Abstract

We introduce a method for constructing global approximations to correlation functions of strongly interacting quantum field theories, starting from perturbative results. The key idea is to employ interpolation method, such as the two-point Padé expansion, to interpolate the weak and strong coupling expansions of correlation function. We benchmark this many-body interpolation approach on two prototypical models: the lattice $φ^4$ field theory and the 2D Hubbard model. For the $φ^4$ theory, the resulting two point Padé approximants exhibit uniform and global convergence to the exact correlation function. For the Hubbard model, we show that even at second order, the Padé appproximant already provides reasonable characterization of the Matsubara Green's function for a wide range of parameters. Finally, we offer a heuristic explanation for these convergence properties based on analytic function theory.

Figures (4)

• Figure 1: (Left) Two-point Padé approximations of $G(\tilde{g})$ for the $0d$ - $\phi^4$ model, compared with 5-th order WCE Eq. (\ref{['0d_wce']}) and 5-th order SCE Eq. (\ref{['0d_sce']}) expansion results. (Right) Approximation error of two-point Padé expansions relative to the exact result Eq. (\ref{['od_G_analytic']}).
• Figure 2: (First row) Correlation function $G_{00}(\tilde{g})$ for the 1D lattice $\phi^4$ model with $N_s = 4, 64$, computed using various Padé-$w_r$-$s_{2N+1-r}$ schemes and compared with Langevin Monte Carlo simulations (numerical accuracy $\sim 10^{-3}$). (Second row) Approximation error of the Padé schemes. To better illustrate the numerical convergence, the error curves for different Padé approximates are plotted relative to the highest-order result that we could obtain: Padé-$w_6$-$s_3$.
• Figure 3: Top-left: Padé approximation error of the Matsubara Green's function $\frac{1}{N}\sum_{n=1}^{N}\bigl|G_{00}(i\omega_n)-G^{\mathrm{P}}_{00}(i\omega_n)\bigr|$ for the Hubbard dimer, using the analytic $G_{00}(i\omega_n)$ as the reference. Other panels: Approximation results for 2D Hubbard model; solid lines show AFQMC reference and dashed lines show the Padé-$w_2$-$s_1$ approximation results. All simulations are at $t=1.0,\beta=20$. The Padé-Taylor expansion interpolates between two points: $U=0$ and $U=10$.
• Figure 4: Schematic illustration of the convergence mechanism via analytic continuation. If $f(z)$ is pole-free within a $\delta$-tube enclosing $\{0\} \cup \mathbb{R}^{+}$, it can be obtained by successive AC from the germ at $z=0$ to the whole tube, thereby recovering $f(x)$ for $x \ge 0$. This extrapolation procedure may converge slowly for large $x$ (e.g. using one-point Padé expansion); however, knowing an additional germ at $0<z_1\leq +\infty$ allows interpolative AC between the two germs that may converge rapidly to the original function (e.g. using two-point Padé expansion). Red asterisks (*) mark the singularities of $f(z)$, and $r$ denotes the Taylor-series convergence radius at $z=0$. Note that at two ends, the series convergence regimes may be much larger and the displayed analytic germs only show their overlapping with the $\delta$-tube.