Surface quantum critical phenomena in disordered Dirac semimetals

We study a non-Anderson disorder driven quantum phase transition in a semi-infinite Dirac semimetal with a flat boundary. The conformally invariant boundary conditions, which include those that are time-reversal invariant, lead to nodal-like surface states on the boundary. In this case the boundary becomes metallic at a critical disorder that is weaker than that for the semimetal-diffusive metal transition in the bulk. The latter transition takes place in the presence of a metallic surface; in the language of surface critical phenomena this corresponds to the so-called extraordinary transition. The lines of the surface and the extraordinary transitions meet at the special transition point. To elucidate universal properties at different transitions on the phase diagram, we employ renormalization group methods and compute the corresponding surface critical exponents using $\varepsilon$ expansion.

Figure 1

Phase diagram in the $(\tilde{\mathrm{\Delta }},\theta )$ plane of a semi-infinite Dirac semimetal, where $\tilde{\mathrm{\Delta }}$ is the dimensionless strength of disorder and $\theta$ is the angle parametrizing the time reversal boundary conditions (see Sec. 2). The surface transition line and the extraordinary transition line are given by ${\tilde{\mathrm{\Delta }}}_S={cos}^2\theta$ and ${\tilde{\mathrm{\Delta }}}_c=1$, respectively. For $\tilde{\mathrm{\Delta }}<{\tilde{\mathrm{\Delta }}}_S$ both the surface and the bulk are in the semimetal phase, for ${\tilde{\mathrm{\Delta }}}_S<\tilde{\mathrm{\Delta }}<{\tilde{\mathrm{\Delta }}}_c$ metallic eigenstates populate the surface, and for $\tilde{\mathrm{\Delta }}>{\tilde{\mathrm{\Delta }}}_c$ the bulk becomes a diffusive metal as well. The lines of surface and extraordinary transitions meet at the multicritical point of the special transition.

Figure 2

Dimensionless LDOS as a function of $\tilde{z}=\frac{\epsilon z}{\hslash v_F}$ and $\theta$ for $\epsilon >0$. (Left panel) $f_{\theta }^B\left(\tilde{z}\right)$ is the contribution from the extended states which exhibit the Friedel oscillations decaying for $\tilde{z}\to \infty$ so that $f_{\theta }^B\left(\infty \right)=1$. (Right panel) $f_{\theta }^S\left(\tilde{z}\right)$ is the contribution from the surface states localized at $z=0$ whose energy is positive for $\theta <0$ and negative for $\theta >0$.

Figure 3

Spectrum of the Dirac Hamiltonian (1) in a slab geometry for symmetric BCs with $\theta =-\frac{\pi }{4}$. In the limit of $kL\gg 1$ the states with energies $|{\epsilon }_k|/k>1$ (blue lines) form the bulk Dirac bands, while the two states (red lines) whose energy asymptotically approach ${\epsilon }_k/k=1/\sqrt{2}$ (dashed black line) are localized at the boundaries.

Figure 4

Red dashed line: the lowest energy surface state in a slab geometry for symmetric BCs with $\theta =-\frac{\pi }{4}$ and $kL=8$. Blue solid line: the surface state in a semi-infinite geometry given by Eqs. (11) and (15) for the same parameters.

Figure 5

Diagrams contributing to the one- and two-particle vertex functions to two-loop order.