  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.
