Realizing topological surface states in a lower-dimensional flat band

The anomalous surface states of symmetry protected topological (SPT) phases are usually thought to be only possible in conjunction with the higher-dimensional topological bulk. However, it has recently been realized that a class of anomalous SPT surface states can be realized in the same dimension if symmetries are allowed to act in a nonlocal fashion. An example is the particle-hole symmetric half-filled Landau level, which effectively realizes the anomalous surface state of a $3d$ chiral topological insulator (class AIII). A dual description in terms of Dirac composite fermions has also been discussed. Here we explore generalizations of these constructions to multicomponent quantum Hall states. Our results include a duality mapping of the bilayer case to composite bosons with Kramers degeneracy and the possibility of a particle-hole symmetric integer quantum Hall state when the number of components is a multiple of eight. Next, we make a further extension by half-filling other classes of topological bands and imposing particle-hole symmetry. When applied to time-reversal invariant topological insulators, we realize a different chiral class (CII) topological surface state. Notably, half-filling a $3d$ TI band allows for the realization of the surface of the otherwise inaccessible $4d$ topological insulator, which supports an anomalous $3d$ Dirac cone. Surface topological orders equivalent to the $3d$ Dirac cone (from the global anomaly standpoint) are constructed and connections to Witten's SU(2) anomaly are made. These observations may also be useful for numerical simulations of topological surface states and of Dirac fermions without fermion doubling.

Figure 1

Half-filled flat band/TI surface state correspondence. $N\times$ half-filled Landau levels of $2d$ electrons, corresponds to the zeroth Landau levels of the $N$-Dirac cone surface state of a $3d$ chiral (class AIII) topological insulator.

Figure 2

Disconnecting the $3d$ class CII surface and bulk states. Schematic depiction of the surface projected density of states. Gray shaded regions denote bulk states, blue lines are surface states, and vertical dashed lines mark the surface Brillouin zone boundaries. Starting from a surface termination where the surface states run continuously across the bulk band gap (a), one may add a perturbation to break off a surface band that is completely isolated within the band gap (b), and then consider the narrow bandwidth limit where this isolated band becomes flat, and corresponds to the purely $2d$ flat class AII band (c).