Many-body effects and ultraviolet renormalization in three-dimensional Dirac materials

Abstract

We develop a theory for electron-electron interaction-induced many-body effects in threedimensional Weyl or Dirac semimetals, including interaction corrections to the polarizability, electron self-energy, and vertex function, up to second order in the effective fine structure constant of the Dirac material. These results are used to derive the higher-order ultraviolet renormalization of the Fermi velocity, effective coupling, and quasiparticle residue, revealing that the corrections to the renormalization group flows of both the velocity and coupling counteract the leading-order tendencies of velocity enhancement and coupling suppression at low energies. This in turn leads to the emergence of a critical coupling above which the interaction strength grows with decreasing energy scale. In addition, we identify a range of coupling strengths below the critical point in which the Fermi velocity varies non-monotonically as the low-energy, non-interacting fixed point is approached. Furthermore, we find that while the higher-order correction to the flow of the coupling is generally small compared to the leading order, the corresponding correction to the velocity flow carries an additional factor of the Dirac cone flavor number (the multiplicity of electron species, e.g. ground-state valley degeneracy arising from the band structure) relative to the leading-order result. Thus, for materials with a larger multiplicity, the regime of velocity non-monotonicity is reached for modest values of the coupling strength. This is in stark contrast to an approach based on a large-N expansion or the random phase approximation (RPA), where higher-order corrections are strongly suppressed for larger values of the Dirac cone multiplicity. This suggests that perturbation theory in the coupling constant (i.e., the loop expansion) and the RPA/large-N expansion are complementary in the sense that they are applicable in different parameter regimes of the theory. We show how our results for the ultraviolet renormalization of quasiparticle properties can be tested experimentally through measurements of quantities such as the optical conductivity or dielectric function (with carrier density or temperature acting as the scale being varied to induce the running coupling). Although experiments typically access the finite-density regime, we show that our zero-density results still capture clear many-body signatures that should be visible at higher temperatures even in real systems with disorder and finite doping.

Excitonic pairing and insulating transition in two-dimensional semi-Dirac semimetals

“…Generating 2D semi-Dirac fermions by merging pairs of Dirac points was predicted to take place in deformed graphene [102][103][104][105][106], pressured organic compound α-(BEDT-TTF) 2 I 3 [104,[106][107][108], few-layer black phosphorus subject to pressure or perpendicular electric field [109,110] or doping [111], and some sorts of artificial optical lattices [112,113]. Experimentally, the merging of distinct Dirac points and the appearance of semi-Dirac fermions were recently observed in ultracold Fermi gas of 40 K atoms in honeycomb lattice [114], and microwave cavities with graphene-like structure [115]. Kim et al [116] realized semi-Dirac semimetals in few-layer black phosphorus at critical surface doping with potassium.…”

“…The role of Coulomb interaction depends crucially on the fermion dispersion and the dimension. Extensive renormalization group (RG) analysis [53] have revealed that Coulomb interaction is marginally irrelevant in 2D Dirac semimetal [5,36,37], 3D Dirac/Weyl semimetal [38][39][40][41], and also 3D double Weyl semimetal [49,50]. The Fermi liquid (FL) theory is valid in 2D Dirac semimetal [5].…”

“…The effects of long-range Coulomb interaction have been studied in various types of semimetals [5,[34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. The role of Coulomb interaction depends crucially on the fermion dispersion and the dimension.…”

Temperature-dependent many-body effects in Dirac-Weyl materials: Interacting compressibility and quasiparticle velocity

“…Using the renormalization group (RG) theory it is possible to express the interacting compressibility at one density (i.e., one k F value) simply in terms of that at another density, thus completely eliminating the unknown ultraviolet scale associated with k c as has been extensively discussed recently in Ref. 60 . Note that the interaction coupling constant α = e 2 /(κv F ) is the ratio of the interaction to kinetic energy (the effective fine-structure constant), which is density independent for Dirac materials.…”

“…60 . A practical way to compare theory with experiment would be to eliminate k c and express the experimental compressibility at one density with that at another density.…”