Topological materials with extensive flat-band surface states

Abstract

Materials that have zero-energy flat band states on the surface may show surface superconductivity. Here we report a theoretical observation that a Hamiltonian describing a thin slab of topological nodal line semimetal, has zero energy eigenstate spanning the entire surface of the Brillouin zone under certain conditions, namely (i) the hopping amplitude of fermions in the direction of thickness is more than that in other directions (ii) the onsite energy should be less than some limiting value determined by the hopping probability. Our claim is substantiated by analytic and numerical approach. We also report new phase transitions in a region of parameter space and indicate that the Hamiltonian can also be realised by stacked layers described by a suitable Hamiltonian.

Figures (14)

• Figure 1: The phase diagram (at zero temperature) in $t-J$ plane. There are six phases, irrespective of whether we have open or closed boundary in the Z direction. Fig. 1a is for periodic boundary in X and Y direction and open boundary in Z direction. We show here the regions on the toroidal $k_x-k_y$ surface at which zero-energy states, localized at surfaces $n_z=1$ and $n_z=L_z$, appear. Fig. 1b is for fully periodic boundaries. We show here the nodal lines (where the energy is zero) in the $k_x-k_y-k_z$ Brillouin zone. In both the figures, the nature of non-analyticity at the phase boundary is indicated by colour, dashed magenta corresponds to divergence of $\partial^3 E_0/\partial t^3$ and red, to a discontinuous change in the same quantity. (For computing the location of nodal lines we used $N=400^3$. For computing the surface states, we used $L_z=200$) Phase I: Topological insulator with zero-energy surface state ranging all over the $k_x-k_y$ plane. Phase II: Nodal line semimetal phase, with one nodal line, which touches $k_x-k_z$ and $k_y-k_z$ boundary surfaces. The zero-energy states form a multiply connected region, but the non-zero energy states form a simply connected region (a patch). Phase III: Another nodal line phase, but with two nodal lines, which touch all the surfaces. Both the zero-energy and non-zero-energy states form multiply connected regions. Phase IV: Two nodal lines in this phase touch only the $k_x-k_y$ boundary surfaces, but not the $k_x-k_z$ and $k_y-k_z$ surfaces. The non-zero-energy states form a multiply connected region, but the zero-energy states form two patches. Phase V: This phase contains only one nodal loop, which does not touch any boundary surface of the BZ. The non-zero-energy states form a multiply connected region, but the zero-energy states form one patch. Phase VI: This is a normal insulator phase without any node and zero-energy state on surface. The equations of critical lines separating a) II and V, III and IV is $J=1$, b) II and III, V and IV is $J-2t=-1$ c) I and II is $J+2t=1$ d) V and VI is $J-2t=1$
• Figure 2: Derivatives of ground state energy calculated from Eq. (\ref{['E_0(J,t)']}) for $t=1$ and $N=300^3$.
• Figure 3: The probability distribution of all the $E=0$ states vs $n_z$ in phase III(left) and phase V(right).
• Figure 4: Regions (coloured) for which the winding number is 1 on the $k_x$-$k_y$ plane with fully periodic boundaries. Note that for $J<1$, these states form a multiply connected region, and for $J>1$ they form one or more simply connected region(s). Also, these regions are the same as the regions for zero energy surface states in Fig. 1a under periodic boundary condition in XY plane.
• Figure 5: Dispersion relation for Phase I : The dispersion relations are calculated for $J=0.2$ and $t=0.2$ in (a),(b) and (c) . We have also added the plot of the dispersion relation over the whole $k_x-k_y$ plane in (d).