RG studies of scalar-field models of long-range interactions

TL;DR

This paper uses the functional renormalization group to study nonlocal scalar field theories with a $φ\Box^{-1}φ$ interaction and its generalization to $φ\Box^{σ/2}φ$, focusing on the infrared fixed-point structure and phase behavior. Through the Local Potential Approximation (LPA) and its refinement LPA', along with spike analyses and two-field local reformulations, it identifies the nonlocal Gaussian fixed point as IR-stable and tracks how fixed points and critical exponents depend on the nonlocal parameters and dimensionality. It demonstrates that nonlocality can shift fixed-point locations without altering universal exponents in certain regimes, while in others it induces new nonlocal fixed points and symmetry-breaking effects, and it connects these findings to Lifshitz criticality via an effective dimension mapping. The results illuminate how long-range interactions and nonlocal operators influence IR physics, offering a framework applicable to nonlocal gravity-inspired models and long-range condensed matter systems. Overall, the work provides a coherent FRG-based picture of how nonlocal terms shape fixed points, phase structure, and universality across a range of σ and d, with potential extensions to finite temperature and O(N) generalizations.

Abstract

In this work we studies the long-range interactions in non-gravitational field theories and their behaviour in the deep infrared. To model such effects, we consider a nonlocal scalar theory obtained by adding a $φ\Box^{-1}φ$ term to the local action. Using the functional renormalisation group, we analyse its infrared fixed-point structure. Within the LPA, we show that nonlocality modifies phase-transition patterns and can induce symmetry breaking. Extending the LPA beyond polynomial truncations, we examine the convexity property of the effective potential as $k\rightarrow 0$ and find that the flow becomes singular for $λ^{2}>0$ before reaching the deep infrared. In the LPA$'$ framework, we find that the infrared-stable fixed point is the nonlocal Gaussian fixed point. We then generalise the model to $φ\Box^{σ/2}φ$ and analyse how the infrared properties depend on $σ$. With appropriate scaling choices, we show that the infrared behaviour remains unchanged up to $σ=d/2$ and follows Sak's prediction up to $σ=2$. Finally, we study higher-derivative cases within the LPA, focusing on $σ=4$, which corresponds to isotropic Lifshitz criticality, and obtain results consistent with earlier work.

Figures (10)

• Figure 1: Red solid line shows the modified cutoff propagator using Eq. \ref{['regulator']}, Dashed green line shows the modified propagator with standard Litim cutoff and black dotted line show the original propagator.
• Figure 2: (a) Spike plot for different values of $\tilde{\lambda}^{2}$ in $d=3$. The black line corresponds to the local theory $\tilde{\lambda}^{2}=0$ and as we increase the $\tilde{\lambda}^{2}$ the spike shift towards left. The plot corresponds to $\tilde{\lambda}^{2}=0$ to $\tilde{\lambda}^{2}=1$ at an interval of $0.2$. In the inset we plot the variation $s_{0}$ corresponds to WF fixed point as a function of non-local coupling ($\tilde{\lambda}^{2}$). We also shows the best line fit for the data point, which explicitly shows the linear behavior. (b) Spike analysis and the linear shift of non Gaussian fixed points in $d=2.6$
• Figure 3: RG-flow diagram with parameters $g_{2}$ and $g_{4}$ for $N=2$ truncation and $d=3$. Separatrix of different phases for different values of $\lambda^{2}$ are shown in different colors. Red dashed lines shows the separatrix for $\lambda^{2}=0$, the region flowing towards large positive $g_{2}$ corresponds to the symmetric phase and the other phase corresponds to the broken symmetry phase. Green and blue dashed lines shows the separatrix for non zero values of $\lambda^{2}$. The red dot corresponds to the Gaussian and WF FP with $\lambda^{2}=0$ and the blue dot corresponds to WF FP with $\lambda^{2}=0.2$. Point "P" represents a symmetric phase for $\lambda^{2}=0$ and broken phase for $\lambda^{2}\geq 0.1$
• Figure 4: The generalized "mass" $m^2 = u"(x, t)$ obtained from the integration of the flow in the limit $t \rightarrow \infty$ for $r = -0.3$ and $g = 0.01$. The solid blue line corresponds to $\lambda^2 = 0$, the orange line to $\lambda^2 = -0.01$, and the red line to $\lambda^2 = -0.04$.
• Figure 5: Spike analysis plot for Eq. \ref{['uflowwithlog']}. Here $s=u"\left(\phi=0\right)$ and $\chi_{s}$ represents the value of $\chi$ where the solution diverges. For the plot we have further scaled the field value as $\tilde{\chi}=10^{-2}\chi$. The plot remains the same for $d=2.2,2.4,2.6,2.8,3,4$.