Electronic topology with bound defect charges promotes intermediate hexatic phase in two-dimensional melting

Disclinations play an amusing role in topological electronic states because of the ability to trap charges residing in the cores of these defects. According to the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory of two-dimensional (2D) melting, unbinding the bound pairs of disclinations (i.e., dislocations) induces the structural transition from the hexatic to liquid phase, corresponding to the destruction of quasilong range order. In this work, we elucidate the interplay between topological electronic states and the KTHNY structural transition. We verify the existence of electronic topology-induced charges that are trapped at disclinations during the 2D melting of hard disk particles. Resorting to the Ewald technique, we calculate the real-space distribution of the electrostatic potential generated by nonuniform Coulomb gas composed of these trapped charges. We found that the nonzero gradient of the electrostatic potential gives rise to additional contributions to the hydrostatic pressure. Based on the linear elastic theory, we further show that the electronic topology-induced trapped charges add extra contributions to the core energy of dislocations, increase the energy barrier of dislocation unbinding, and eventually decrease the lower critical boundary of the hexatic phase in the phase diagram. Our finding not only gives a special perspective for studying amorphous topological insulators but proposes a new impacting mechanism in the deviation of the phase boundary of structural transitions.

Figure 1

The electric charges are trapped in the topological vortices during 2D melting. (a) Schematic display of melting of hard disks and the bond-orientational correlation functions $g_6\left(r\right)$. The slope of the black dashed line in (a) is $-1/4$, which equals the minimum possible slope for a hexatic phase. (b) The tight-binding model containing 768 melting disks is based on Delaunay triangulation (denoted by thin red lines) with the periodic boundary condition (PBC). The polygons marked by black lines are the Voronoi cells where atomic sites are located in the center of each cell and the intersite hoppings are represented by the edges of Delaunary triangles (i.e., the thin red lines). (c) The average of electric charges according to different types of topological defects in the hexatic lattice with $\rho =0.7$. Over 20 realizations are considered in the calculation. The horizontal black dashed line stands for zero value, while the vertical one is the location of the topological transition of the amorphous topological model. (d) The real-space distribution of electric charges and the color bar represents the amount of charges. The fixed parameters used for calculations in (c) and (d) are ${\omega }_s=0.18$, ${\omega }_p=-0.65$, $T_{ss\sigma }=-0.04$, $T_{sp\sigma }=0.09$, $T_{pp\sigma }=0.18$, and $T_{pp\pi }=0.05$ eV.

Figure 2

The hydrostatics of the network stem from the trapped electrons on the site. (a) Sketch of the net charges in the disordered triangular lattice. (b) The electrostatic pressure $P_e$ and potential $V$ as the functions of particle density $\rho$, where 20 realizations are considered. The dashed vertical cyan line denotes the critical density $\rho =0.65$. ${\lambda }_{so}=1.1$ and 1.3 eV are assigned topological and trivial phases of the amorphous tight-binding model defined as Eq. (1), respectively. (c) Phase diagram of electrostatic potential $V$. The horizontal dotted blue and vertical dashed black lines indicate the structural and topological transitions, respectively.

Figure 3

Summaries about charged topological defects and their impact on structural transitions, originating from nontrivial topological bands. (a) The number density of two types of disclinations as the functions of $\rho$, and the inset shows their average of electric charges compulsively localized in the defect core. (b) The electrostatic interaction $E_{\mathrm{int}}$ and self-energies $E_{\mathrm{self}}$ of the entire system, with their corresponding values depending on the particle densities. (c) The core energy of dislocation ${\mathcal{E}}_c$ as the functions of $\rho$. The red and green dashed lines represent the ${\mathcal{E}}_c$ whether considering electrostatic force, respectively. (d) Phase diagram of ${\mathcal{E}}_c$ in the (${\lambda }_{so},\rho$) plane. The meaning of the horizontal dotted blue and vertical dashed black lines is the same as that in Fig. 2.