Abstract
We compute the large $N$ critical exponents $\ensuremath{\eta}$, ${\ensuremath{\eta}}_{\ensuremath{\phi}}$ and $1/\ensuremath{\nu}$ in $d$ dimensions in the chiral Heisenberg Gross-Neveu model to several orders in powers of $1/N$. For instance, the large $N$ conformal bootstrap method is used to determine $\ensuremath{\eta}$ at $O(1/{N}^{3})$ while the other exponents are computed to $O(1/{N}^{2})$. Estimates of the exponents for a phase transition in graphene are given which are shown to be commensurate with other approaches. In particular the behavior of the exponents in $2<d<4$ is in qualitative agreement with a functional renormalization group analysis. The $\ensuremath{\epsilon}$ expansion of each exponent near four dimensions is in exact agreement with recent four loop perturbation theory.
Highlights
One of the more remarkable fundamental quantum field theories is the Gross-Neveu or Ashkin-Teller model, [1,2]
Put another way prior to the discovery of the W and Z vector bosons of the Standard Model the physics of the weak interactions was described by an effective field theory involving 4-point fermion interactions
The general Gross-Neveu class of quantum field theories, several of which were introduced in [1], have enjoyed a renaissance over recent years. This is in the main due to the fact that certain phase transitions in graphene can be described by specific universality classes based on the Gross-Neveu model [1] where the classes derive from the underlying symmetry of the transition
Summary
One of the more remarkable fundamental quantum field theories is the Gross-Neveu or Ashkin-Teller model, [1,2]. Given the current sparsity of theoretical critical exponent estimates for the chiral Heisenberg Gross-Neveu model, it is crucial to bring the analysis up to the same level of precision as that of the Ising Gross-Neveu universality class. For the most part we will analyze the basic 2-point functions of the chiral Heisenberg Gross-Neveu theory by solving the skeleton Schwinger-Dyson equations algebraically in the limit as one approaches the d-dimensional Wilson-Fisher fixed point While this will determine η, ηφ and 1=ν to Oð1=N2Þ, the original method given in [32] pointed the way to determining η at Oð1=N3Þ.
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