Landau poles in condensed matter systems

Abstract

The existence or not of Landau poles is one of the oldest open questions in nonasymptotic quantum field theories. We investigate the Landau pole issue in two condensed matter systems whose long-wavelength physics is described by appropriate quantum field theories: the critical quantum magnet and Dirac fermions in graphene with long-range Coulomb interactions. The critical quantum magnet provides a classic example of a quantum phase transition, and it is well described by the φ4 theory. We find that the irrelevant but symmetry-allowed couplings, such as the φ6 potential, can significantly change the fate of the Landau pole in the emergent φ4 theory. We obtain the coupled β functions of a φ4+φ6 potential at both small and large orders. Already from the one-loop calculation, the Landau pole is replaced by an ultraviolet fixed point. A Lipatov analysis at large orders reveals that the inclusion of a φ6 term also has important repercussions for the high-order expansion of the β functions. We also investigate the role of the Landau pole in a very different system: Dirac fermions in 2+1 dimensions with long-range Coulomb interactions, e.g., graphene. Both the weak-coupling perturbation theory up to two loops and a low-order large-N calculation show the absence of a Landau pole. Furthermore, we calculate the asymptotic expansion coefficients of the β function. We find that the asymptotic coefficient is bounded by that of a pure bosonic φ4 theory, and consequently graphene is free from Landau poles if the pure φ4 theory does not manifest a Landau pole. We briefly discuss possible experiments that could potentially probe the existence of a Landau pole in these systems. Studying Landau poles in suitable condensed matter systems is of considerable fundamental importance since the relevant Landau pole energy scales in particle physics, whether it is quantum electrodynamics or Higgs physics, are completely unattainable.