Callan-Symanzik-Lifshitz approach to generic competing systems

TL;DR

This paper develops a renormalization approach for analyzing critical behaviors in systems with competing interactions, providing exact calculations of critical exponents and establishing the method's consistency with existing formalisms.

Contribution

It introduces the Callan-Symanzik-Lifshitz method for arbitrary competing systems, proving renormalizability and computing critical exponents up to two-loop order.

Findings

Existence of Callan-Symanzik-Lifshitz equations at critical dimension

Multiplicative renormalizability of vertex functions

Exact calculation of critical exponents for isotropic behaviors

Abstract

We present the Callan-Symanzik-Lifshitz method to approaching the critical behaviors of systems with arbitrary competing interactions. Every distinct competition subspace in the anisotropic cases define an independent set of renormalized vertex parts via normalization conditions with nonvanishing distinct masses at zero external momenta. Otherwise, only one mass scale is required in the isotropic behaviors. At the critical dimension, we prove: i) the existence of the Callan-Symanzik-Lifshitz equations and ii) the multiplicative renormalizability of the vertex functions using the inductive method. Away from the critical dimension, we utilize the orthogonal approximation to compute higher loop Feynman integrals, anisotropic as well as isotropic, necessary to get the exponents $\eta_{n}$ and $\nu_{n}$ at least up to two-loop level. Moreover, we calculate the latter exactly for isotropic behaviors at the same perturbative order. Similarly to the computation in the massless formalism, the orthogonal approximation is found to be exact at one-loop order. The outcome for all critical exponents matches exactly with those computed using the zero mass field-theoretic description renormalized at nonvanishing external momenta.