Topological Amorphous Metals

We study amorphous systems with completely random sites and find that, through constructing and exploring a concrete model Hamiltonian, such a system can host an exotic phase of topological amorphous metal in three dimensions. In contrast to the traditional Weyl semimetals, topological amorphous metals break translational symmetry, and thus they cannot be characterized by the first Chern number defined based on the momentum space band structures. Instead, their topological properties will manifest in the Bott index and the Hall conductivity as well as the surface states. By studying the energy band and quantum transport properties, we find that topological amorphous metals exhibit a diffusive metal behavior. We further introduce a practical experimental proposal with electric circuits where the predicted phenomena can be observed using state-of-the-art technologies. Our results open the door to exploring topological gapless phenomena in amorphous systems.

Figure 1

Schematic of a 3D random site configuration with the allowed hopping inside the light red sphere for a typical site at the center.

Figure 2

(a) The Bott index and the Hall conductivity in units of $e^2/(2h)$ as a function of $m_z$ for distinct system sizes. The black line denotes the Bott index for a cubic lattice configuration. Three different phases are identified: amorphous Anderson insulator (AAI), topological amorphous metal (TAM), and amorphous insulator (AI). (b) Schematic of a four terminal setup used to compute the Hall conductivity, where we consider the cubic geometry for all leads (see the dotted part for $V_2=V$). (c),(d) The local density of states for $m_z=2$ and $m_z=6$, respectively.

Figure 3

(a) The gap versus $m_z$ for different system sizes calculated via the Lanczos method, with the inset plotting the same thing in the logarithmic scale. The arrows show the universal dips. (b) The DOS at zero energy $\rho (0)$ as a function of $m_z$ (with the logarithmic scale figure shown in the inset) calculated by the KPM for $L=55$ and $N_c=2^{11}$. (c) The DOS $\rho (E)$ versus $E$ for various $m_z$ in different phases for $L=55$ and $N_c=2^{11}$. (Inset) Plots of $\rho (E)$ versus $E$ when $m_z=6$ for $L=55$, $N_c=2^{11}$ (red line), $L=60$, $N_c=2^{11}$ (green line), and $L=60$, $N_c=2^{13}$ (blue line). (d) The level-spacing ratio (LSR) $r(E=0)$ (left vertical axis) and inverse participation ratio (IPR) $I(E=0)$ (right vertical axis) for $L=24$ for the states around zero energy (only the states with negative energy are considered) computed via the Lanczos algorithm. In all the above subfigures, samples in a cubic box are considered.

Figure 4

(a) Conductivity ${\sigma }_{zz}$ in units of $e^2/h$ and (b) Fano factor versus $m_z$ for $L=25$ in a cubic box. (Inset) Plots of the conductivity in the logarithmic scale, showing its steep drops across the phase transitions. The dashed lines correspond to $F=1/3$ and $F=1/3+1/(6\ln 2)$, respectively.