Interplay of Coulomb interactions and disorder in three-dimensional quadratic band crossings without time-reversal symmetry and with unequal masses for conduction and valence bands

Coulomb interactions famously drive three-dimensional quadratic band crossing semimetals into a non-Fermi liquid phase of matter. In a previous work [Nandkishore and Parameswaran, Phys. Rev. B 95, 205106 (2017)], the effect of disorder on this non-Fermi liquid phase was investigated, assuming that the band structure was isotropic, assuming that the conduction and valence bands had the same band mass, and assuming that the disorder preserved exact time-reversal symmetry and statistical isotropy. It was shown that the non-Fermi liquid fixed point is unstable to disorder and that a runaway flow to strong disorder occurs. In this paper, we extend that analysis by relaxing the assumption of time-reversal symmetry and allowing the electron and hole masses to differ (but continuing to assume isotropy of the low energy band structure). We first incorporate time-reversal symmetry breaking disorder and demonstrate that there do not appear any new fixed points. Moreover, while the system continues to flow to strong disorder, time-reversal-symmetry-breaking disorder grows asymptotically more slowly than time-reversal-symmetry-preserving disorder, which we therefore expect should dominate the strong-coupling phase. We then allow for unequal electron and hole masses. We show that whereas asymmetry in the two masses is irrelevant in the clean system, it is relevant in the presence of disorder, such that the ‘effective masses’ of the conduction and valence bands should become sharply distinct in the low-energy limit. We calculate the RG flow equations for the disordered interacting system with unequal band masses and demonstrate that the problem exhibits a runaway flow to strong disorder. Along the runaway flow, time-reversal-symmetry-preserving disorder grows asymptotically more rapidly than both time-reversal-symmetry-breaking disorder and the Coulomb interaction.

Figure 1

These two diagrams determine the $O\left(\varepsilon \right)$ correction to the Green's function with each solid line representing the bare Green's function. A dashed line may represent either a disorder or the Coulomb interaction. If the dashed line represents disorder, then it connects two fermion lines at the same point in real space, but they may have different time and replica indices. For disorder interaction, the diagram (b) is then proportional to the number of replica flavors $n$ and vanishes upon taking the replica limit $n\to 0$. If the dashed line represents the Coulomb interaction, then it connects two fermions with the same time index and same replica index but with different spatial positions.

Figure 2

These diagrams determine the $O\left(\varepsilon \right)$ correction to the four-fermion vertices (either disorder or interaction) and are denoted by ZS, BCS, ${\mathrm{ZS}}^{\prime }$, and VC, respectively, following the convention of Ref. [12]. Each solid line represents the bare Green function. A dashed line may represent either a disorder or the Coulomb interaction. Note that unlike the ZS, BCS, and ${\mathrm{ZS}}^{\prime }$ diagrams, the VC diagrams are generically not symmetric under interchange of indices, i.e., ${\mathrm{\Gamma }}_{\alpha \beta }^{\text{VC}}\ne {\mathrm{\Gamma }}_{\beta \alpha }^{\text{VC}}$.

Figure 3

Quadratic band touching with $m/m^{\prime }=0.75$.

Figure 4

Behavior of the scalar-vector disorder ratio as a function of $r_m$, at the stable fixed point of the ratio space.