Field-theory approach to flat polymerized membranes

TL;DR

This paper reviews the field-theoretic renormalization-group approach to understanding the critical properties of flat polymerized membranes, including detailed three-loop calculations and analysis of critical exponents.

Contribution

It provides a comprehensive three-loop renormalization-group analysis of flat polymerized membranes, including derivations of critical exponents and comparison with other methods.

Findings

Calculated critical exponents for flat membranes

Analyzed the flow diagram for the critical flat phase

Presented four-loop results and perturbative series structure

Abstract

We review the field-theoretic renormalization-group approach to critical properties of flat polymerized membranes. We start with a presentation of the flexural effective model that is entirely expressed in terms of a transverse (flexural) field with non-local interactions. We then provide a detailed account of the full three-loop computations of the renormalization-group functions of the model within the dimensional regularization scheme. The latter allows us to consider the general case of a $d$-dimensional membrane embedded in $D$-dimensional space. Focusing on the critical flat phase of two-dimensional membranes $(d = 2)$ in three-dimensional space $(D = 3)$, we analyse the corresponding flow diagram and present the derivation of the anomalous stiffness. The latter controls all the other critical exponents of the theory such as the roughness exponent and the scaling of the elastic constants. State-of-the-art four-loop results as well as discussions on the structure of the perturbative series and comparison with other approaches are also provided.