Abstract
The existence or not of Landau poles is one of the oldest open questions in non-asymptotic 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 $\phi^4$ theory. We find that the irrelevant but symmetry-allowed couplings, such as the $\phi^6$ potential, can significantly change the fate of the Landau pole in the $\phi^4$ theory. We obtain the coupled beta functions of a $\phi^4 + \phi^6$ potential at both small and large orders. Already from the 1-loop calculation, the Landau pole is replaced by an ultraviolet fixed point. A Lipatov analysis at large orders reveals that the inclusion of a $\phi^6$ term also has important repercussions for the high-order expansion of the beta functions. We also investigate the role of the Landau pole in 2+1 dimensional Dirac fermions with Coulomb interactions, e.g., graphene. Both the weak-coupling perturbation theory up to 2 loops and a low-order large-N calculation show the absence of a Landau pole. Furthermore, we calculate the asymptotic expansion coefficients of the beta function. We find that the asymptotic coefficient is bounded by that of a $\phi^4$ theory, so graphene is free from Landau poles if the $\phi^4$ 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.
Highlights
The concept of Landau poles is a long-standing issue in quantum field theory that raises questions about fundamental aspects of the renormalization group (RG), in particular the asymptotic behavior of renormalized couplings [3,12,13]
The most recent Quantum electrodynamics (QED) calculation up to tenth order in the fine structure coupling constant involves the accurate determination of 389 different high-dimensional integrals contributed by 6354 Feynman vertex diagrams, with a resultant electron anomalous magnetic moment agreeing quantitatively with experiments up to ten significant digits [1]
Quite apart from the fact that such large energy scales are experimentally unattainable, new physics, outside the scope of QED, comes in at high energy scales, and the predictions of QED for the Landau pole become academic since QED itself is no longer a correct description of nature at such high energies. This is the context of the current theoretical work, where we investigate the condensed matter analogs of the Landau pole in the theories of critical quantum magnets and the physics of graphene
Summary
Quantum electrodynamics (QED) is the most successful theory in physics. The most recent QED calculation up to tenth order in the fine structure coupling constant (i.e., the fifth order in the QED perturbation theory) involves the accurate determination of 389 different high-dimensional integrals contributed by 6354 Feynman vertex diagrams, with a resultant electron anomalous magnetic moment agreeing quantitatively with experiments up to ten significant digits [1]. Our goal is to study the presence or absence of Landau poles in these two concrete condensed matter physics examples to motivate further theoretical work that could shed light on the fundamental issue of triviality in the quantum field theories which are not asymptotically free. Another important objective of our work is to motivate experimental work in condensed matter systems to directly probe the existence or not of Landau poles in these two systems, which are described by continuum field theories containing Landau poles in the leading-order perturbative analysis. The question of the existence or not of Landau poles is of sufficient fundamental significance that anything we can learn from condensed matter systems about the possible presence or absence of Landau poles would be valuable for future progress in the subject
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