Three-loop order approach to flat polymerized membranes

TL;DR

This paper derives three-loop renormalization group equations for the flat phase of polymerized membranes, providing refined theoretical predictions for critical exponents that align well with nonperturbative and numerical results.

Contribution

It extends previous one- and two-loop calculations to three-loop order, offering more accurate analytical insights into the critical behavior of polymerized membranes.

Findings

Computed the field anomalous dimension η at three-loop order.

Found η=0.8872 at the stable fixed point in D=2, consistent with known results.

Demonstrated close agreement with nonperturbative and numerical approaches.

Abstract

We derive the three-loop order renormalization group equations that describe the flat phase of polymerized membranes within the modified minimal subtraction scheme, following the pioneering one-loop order computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)] and the recent two-loop order one of Coquand, Mouhanna and Teber [Phys. Rev. E 101, 062104 (2020)]. We analyze the fixed points of these equations and compute the associated field anomalous dimension $\eta$ at three-loop order. Our results display a marked proximity with those obtained using nonperturbative techniques and reexpanded in powers of $\epsilon=4-D$. Moreover, the three-loop order value that we get for $\eta$ at the stable fixed point, $\eta=0.8872$, in $D=2$, is compatible with known theoretical results and within the range of accepted numerical values.