Dual energy-differentiated topological transition in artificial red phosphorus chains

Abstract

We investigate the spectral and transport properties of an atomic chain of red phosphorus. We reveal the separation of flat-band states from the rest of the system and calculate its energy bands analytically. The topological properties of the system are established through the evaluation of the Berry (Zak) phase of the energy bands, revealing nontrivial topology. The Berry phase depends on the relative strength of the hopping parameters and exhibits dual energy-dependent topological phase transitions. Remarkably, the emergence of inert band edges provides a direct spectral signature of these transitions, acting as energy-resolved indicators of the redistribution of topological charge between bands. The existence of the associated edge states is proved numerically for finite lattices. The theoretical predictions, particularly the band structure and the existence of edge states, are further confirmed by numerical simulations of red phosphorus through a phononic lattice in the form of a highly structured aluminum plate.

Figures (10)

• Figure 1: (a) Pictorial representation of the P$_4$ molecule also known as white phosphorus. (b) Red phosphorus lattice formed by a series of open-bond P$_4$ molecules, where $t_1$ and $t_2$ are the intra and inter hopping parameters, respectively.
• Figure 2: (a) Energy spectrum of the finite red phosphorus chain as a function of $\omega = t_1/t_2$, we fix $t_2 = 1$. The blue bands correspond to bulk states, while the red ones correspond to edge states. (b)-(d) Squared modulus of the wave function for the topological edge (red) and bulk states (blue) as a function of the site number. In (d) is shown the wave function for the unique blue flat band in (a) for non-dispersive bulk states.
• Figure 3: Auxiliary atomic chain described by the Hamiltonian (\ref{['H3']}). Flat-band states are located on the atoms of the $D$-chain.
• Figure 4: Energy bands $E_{1,2,3}/t_2$ in dependence on $\Theta=k a$ for five different values of $\omega=t_1/t_2$. We fixed $a=1$ and $t_2=1$.
• Figure 5: Energy bands in dependence on $\omega=t_1/t_2$. The gap between the lower and middle band gets closed for $\omega=1$, whereas the gap between the second and the third band closes for $\omega=1/2$. The edges of the energy gap are solutions of (\ref{['ebe1']}) and (\ref{['ebe2']}).