Marginal Fermi liquid versus excitonic instability in three-dimensional Dirac semimetals

We study the different phases in the Quantum Electrodynamics of three-dimensional Dirac semimetals depending on the number $N$ of Dirac fermions, using renormalization group methods and the self-consistent resolution of the Schwinger-Dyson equation. We find that, for $N<4$, a phase with dynamical generation of mass prevails at sufficiently strong coupling, sharing the same physics of the excitonic instability in two-dimensional Dirac semimetals. For $N\ge 4$, we show that the phase diagram has instead a line of critical points characterized by the suppression of the quasiparticle weight at low energies, making the system fall into the class of marginal Fermi liquids. Such a boundary marks the transition to a kind of strange metal which can still be defined in terms of electron quasiparticles, but with parameters that have large imaginary parts implying an increasing deviation from the conventional Fermi liquid picture.

Figure 1

(a) Phase boundaries given in terms of the bare interaction strength $\lambda$ [defined in Eq. (14)], marking the transition to a strongly renormalized Fermi liquid (SRFL) at large $N$ and to a phase with chiral symmetry breaking (CSB) for small $N$. (b) Plot of the anomalous scaling dimension ${\gamma }_d\left(g\right)$ (full line) and the rate of variation $\beta \left(g\right)$ of the Fermi velocity with respect to energy (dashed line), both multiplied by $N$.

Figure 2

Plot of the factors $z_{\psi }(0,i\omega )$ and $z_m(\mathbf{k},0)/z_{\psi }(\mathbf{k},0)$ (expressed in eV) for $N=4$ and values of the bare interaction strength $\lambda =42.4,31.6,19.4$, and $8.1$ (from top to bottom, with the two lowest curves in (b) collapsed down to the horizontal axis). The inset shows the plot of $z_v(\mathbf{k},0)/z_{\psi }(\mathbf{k},0)$ for the same sequence of couplings, from bottom to top.

Figure 3

Plot of the real and imaginary parts of the renormalization factors $z_{\psi }(0,i\omega )$ and $z_v(\mathbf{k},0)$ for $N=8$ and values of the bare interaction strength $\lambda =21.08,21.13$, and $21.77$ [curves from top to bottom in both sides of (a), from lower to higher absolute value in both sides of (b)].