Fig. 3: Filling anomaly and density of states.
From: Two-dimensional higher-order topology in monolayer graphdiyne
a A schematic figure displaying the energy spectrum of a finite-size MGD composed of n × n unit cells, where the total number of states is N = 72n2. Here, a thick gray strip near the Fermi level EF indicates the non-bonding states arising from carbon atoms along the boundary. The upper panel corresponds to the case at half-filling where an odd number of holes is below the gap, which demonstrates the higher-order band topology with w2 = 1. The lower panel shows the case when two hydrogen atoms are added to the finite-size MGD. Here, the total number of states \(N^{\prime}\) increases by two (\(N^{\prime} =N+2\)). The hybridization between two hydrogen states and two non-bonding states generates two bonding and two anti-bonding states, so that the number of states below the gap changes from \(\frac{N}{2}+4n-1\) to \(\frac{N^{\prime} }{2}+4n-2\). Then, w2 also changes from 1 to 0. Thus, to maintain the value of w2, the number of hydrogen atoms attached for passivation should be an integer multiple of four. b Density of states (DOS) of a finite-size MGD without hydrogen passivation. Here, the carbon atom at each corner has a non-bonding state. There are (4n − 1) holes below the gap at half-filling. The green color indicate the gapped region. c DOS of a finite-size MGD with hydrogen passivation where (8n − 4) hydrogen atoms are attached along the boundary. Here, only two carbon atoms at My-invariant corners have non-bonding states. There is a single hole below the gap at half-filling. Since both systems in b and c have an odd number of holes, both are HOTIs with w2 = 1.
