Abstract
We study Weyl-loop semi-metals with short range interactions, focusing on the possible interaction driven instabilities. We introduce an ϵ expansion regularization scheme by means of which the possible instabilities may be investigated in an unbiased manner through a controlled weak coupling renormalization group (RG) calculation. The problem has enough structure that a ‘functional’ RG calculation (necessary for an extended Fermi surface) can be carried out analytically. The leading instabilities are identified, and when there are competing degenerate instabilities a Landau–Ginzburg calculation is performed to determine the most likely phase. In the particle-particle channel, the leading instability is found to be to a fully gapped chiral superconducting phase which spontaneously breaks time reversal symmetry, in agreement with general symmetry arguments suggesting that Weyl loops should provide natural platforms for such exotic forms of superconductivity. In the particle hole channel, there are two potential instabilities—to a gapless Pomeranchuk phase which spontaneously breaks rotation symmetry, or to a fully gapped insulating phase which spontaneously breaks mirror symmetry. The dominant instability in the particle hole channel depends on the specific values of microscopic interaction parameters.
Highlights
The most generic metallic states occur in systems that host Fermi surfaces whose dimension is one less than the dimension of the system
Since short range interactions are expected to be strongly irrelevant in the presence of linear band-touching, we develop a convenient generalization of the model in terms of the degree of band-curvature, which allows us to access interaction driven instabilities within the regime of applicability of a weak coupling renormalization group (RG)
The rotational symmetry of the Weyl loop further endows the problem with enough structure that the functional RG analysis necessary for an extended Fermi surface can be carried out analytically
Summary
The most generic metallic states occur in systems that host Fermi surfaces whose dimension is one less than the dimension of the system. In this paper we investigate the effect of short-range interactions on a Weyl-loop semi-metal, and identify the symmetry broken states that are probable at finite interactions. We derive an effective theory that is appropriate for understanding the universal low energy properties of the Weyl-loop semi-metal in the presence of short range interactions. Since the latter two symmetry transformations are locally defined on the loop, they lead to distinct emergent U (1)¥ symmetries which we will distinguish as pseudospin-U (1)¥ and charge-U (1)¥, respectively While the former corresponds to the conservation of qcomponent of total angular momentum, the latter originates from particle number conservation at each θ. We use the polar coordinates introduced in section 2.1.1 to generalize the dispersion of fermions as eh (k, j) = kh (cos j s1 + sin j s2) for any real number h > 0.
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