Mesoscopic transport signatures of disorder-induced non-Hermitian phases

We investigate the impact on basic quantum transport properties of disorder-induced exceptional points (EPs) that emerge in a disorder-averaged Green's function description of two-dimensional Dirac semimetals with spin- or orbital-dependent potential scattering. Remarkably, we find that EPs may promote the nearly vanishing conductance of a finite sample at the Dirac point to a sizable value that increases with disorder strength. This striking behavior exhibits a strong directional anisotropy that is closely related to the Fermi arcs connecting the EPs. We corroborate our results by numerically exact simulations, thus revealing the fingerprints of characteristic non-Hermitian spectral features also on the localization properties of the considered systems. Finally, several candidates for the experimental verification of our theoretical analysis are discussed, including disordered electronic square-net materials and cold atoms in spin-dependent optical lattices.

Figure 1

(a) Illustration of a disordered two-dimensional (2D) material, which hosts a semimetallic Dirac dispersion in the clean limit, e.g., realized as a square-net material [47, 48] with positive and negative ions adsorbed to the surface. (b) For orbital-selective disorder, the Dirac cones may split into topologically stable exceptional points exhibiting a characteristic square-root-like dispersion (absolute value of complex energy is shown). (c) For a sample of mesoscopic size, the vanishing zero-energy transmission seen in the clean regime (left panel) may be promoted to a finite value (right panel) that increases with disorder strength.

Figure 2

Band structure of the free Hamiltonian $H_0$ [see Eq. (2)] exhibiting four Dirac cones with the parameters set to $t_{\mathrm{x}}=0.5, t_{\mathrm{z}}=0.5$.

Figure 3

(a) Numerically exact spectrum of $H_e\left(k\right)$ in the exceptional phase. Nodal points of the clean are split into EPs connected by Fermi arcs (marked by red lines). Lines of vanishing imaginary part (marked in green) entail propagating quasiparticles with infinite lifetime. The group velocity along the green lines is parallel to the $k_{\mathrm{x}}$ axis, which leads to a directional dependence of transport signatures. Disorder parameters are $\alpha =1.5, s_0=1.0, \gamma =1, \varphi =0$. (b) Cut along the line where $k_{\mathrm{x}}=\frac{\pi }{2}$, showing the profile of the two Fermi arcs in the $(+,+)$ and $(+,-)$ quadrants of Fig. 3. Note that the imaginary Fermi bubbles touch the real axis. Disorder parameters are $\alpha =1.5, s_0=1.0, \varphi =0$. Different colors correspond to $\gamma =1$ (blue), $\gamma =0.9$ (orange), and $\gamma =0.8$ (red), respectively, where for $\gamma <1$ the disorder potential is no longer exactly restricted to one orbital.

Figure 4

(a) Energy-dependent two-terminal conductance in $x$ direction for the disorder-free system of size $N_{\mathrm{x}}=100, N_{\mathrm{y}}=200$. (b) Energy-dependent two-terminal conductance in $x$ direction for various disorder parameters at system size $N_{\mathrm{x}}=100, N_{\mathrm{y}}=200$. Orange line: Exceptional phase with disorder parameters $s_0=1, \gamma =0, \varphi =0, \alpha =1.5$. Blue line: Exceptional phase with disorder parameters $s_0=1, \gamma =0, \varphi =\frac{\pi }{2}, \alpha =1.5$. Green line: Conventional phase with disorder parameters $s_0=1, \gamma =0, \alpha =1.5$. (c) Normalized DOS and average IPR at each energy in the conventional phase. Disorder parameters are $s_0=1, \gamma =0, \alpha =1.5$. Data from exact diagonalization of a system with $N_{\mathrm{x}}=50, N_{\mathrm{y}}=50$ sites. (d) Normalized DOS and average IPR at each energy in the exceptional phase. Disorder parameters are $s_0=1, \gamma =1, \varphi =0, \alpha =1.5$. Data from exact diagonalization of a system with $N_{\mathrm{x}}=50, N_{\mathrm{y}}=50$. All disordered data averaged over ten independently drawn realizations.

Figure 5

Conductance in $x$ direction at zero energy in the exceptional phase as a function of $\varphi$ along with a fit of $T\left(\varphi \right)=\frac{e^2}{h}exp[\tilde{A}-\tilde{\tau }{\left(N_{\mathrm{x}}\sqrt{1+{tan}^2\left(\varphi \right)}\right)}^p]$, where $p=\frac{1}{5}$, with the parameters $\tilde{A}=4.67, \tilde{\tau }=1.08$. Remaining disorder parameters are $s_0=1, \gamma =1, \alpha =1.5$. System size $N_{\mathrm{x}}=100, N_{\mathrm{y}}=200$. Insets show the decay of the conductance at zero energy for increasing system size along with a fit of $T\left(l\right)=\frac{e^2}{h}exp[A-\tau {\left(N_{\mathrm{x}}\right)}^p]$. Remaining disorder parameters are $s_0=1, \gamma =1, \varphi =0, \alpha =1.5$. System $N_{\mathrm{y}}=200$. (a) $p=1$ and $A=1.92, \tau =0.0012$. (b) $p=\frac{1}{2}$ and $A=2.61, \tau =0.068$. (c) $p=\frac{1}{5}$ and $A=4.63, \tau =1.06$.

Figure 6

Conductance in $x$ direction at zero energy in the exceptional phase as a function of disorder strength $\alpha$. Disorder parameters are $s_0=1, \varphi =0$ for all curves, while we vary $\gamma =1$, 0.9, and 0.8 to study deviations from perfectly orbital-selective disorder ($\gamma =1$). System size is $N_{\mathrm{x}}=100, N_{\mathrm{y}}=200$. For comparison, the length of the Fermi arc is shown for $s_0=1, \varphi =0, \gamma =1$.

Figure 7

Scatter plot of the microscopic Hamiltonian eigenstates over energy and IPR for a randomly generated system of $60\times 60$ sites. The point size and color indicate the overlap ${\left|\langle \mathit{k}|\text{ES}\rangle \right|}^2$ of the eigenstate $|\text{ES}\rangle$ with the eigenstate $|\mathit{k}\rangle$ of the effective Hamiltonian $H_{\mathrm{e}}$ at $k_{\mathrm{x}}=\pi /2, k_{\mathrm{y}}=\pi /2$ with the smaller imaginary part, i.e., lower damping. (a) Result for a system in the conventional phase with parameters $s_0=1, \gamma =0, \alpha =1.5$. (b) Result for a system in the exceptional phase with parameters $s_0=1, \gamma =1, \varphi =0, \alpha =1.5$. Note that the plot range only covers the energies for which transport occurs and also cuts off some of the more strongly localized states with very high IPRs.

Figure 8

The correlation function $g(r,\phi =0)$ [see Eq. (5)] for system size $N_{\mathrm{x}}=5000, N_{\mathrm{y}}=200$ along with a fit of the form $g(r,\phi =0)=\lambda r^p$. (a) Result for the conventional phase with parameters $s_0=1, \gamma =0, \alpha =1.5$. Fit parameters are $\lambda =0.5, p=0.73$, where only the data up to the saturation resulting from the limited machine precision was used. (b) Result for the exceptional phase with parameters $s_0=1, \gamma =1, \varphi =0, \alpha =1.5$. Fit parameters are $\lambda =6.82, p=0.07$.

Figure 9

A wave front (in blue) propagates through the sample at an angle $\varphi$.