Observation of the possible chiral edge mode in Bi1−xSbx

After the classification of topological states ofmatter has been clarified for non-interacting electron systems, the theoretical connection between gapless boundarymodes and nontrivial bulk topological structures, and their evolutions as a function of dimensions are nowwell understood.However, such dimensional hierarchy has not beenwell established experimentally although some indirect evidences were reported, for example, such as the half-quantizedHall conductance via quantumHall effect and extrapolation in the quantum-oscillationmeasurement. In this paper, we report the appearance of the possible chiral edgemode from the surface state of topological insulators undermagnetic fields, confirming the dimensional hierarchy in three-dimensional topological insulators. Applying laser pulses to the surface state of Bi1−xSbx, wefind that the sign of voltage relaxation in one edge becomes opposite to that in the other edge onlywhenmagnetic fields are applied to the topological insulating phase.We show that this sign difference originates from the chirality of edge states, based on coupled time-dependent Poisson andBoltzmann equations. Chiral edge states are boundary states emerging in two-dimensional (2D) electron systems and reflect its bulk topology. These exotic states have been observed in quantumHall systems [1–4], graphene [5, 6], HgTe [7, 8], surface of three-dimensional (3D) topological insulators (TI) [9–11], and quantum anomalousHall systems [12, 13]. In fact, prerequisite for observing the edge states has been thought to be to apply high perpendicular magnetic fieldB so thatwell-separated Landau levels are established.When the Fermi energyEF is placed in between the bulk Landau gap, as a consequence of band bending, there are states crossing it at the edges. In contrast to usual quantumHall systems, 2DDirac electron systems exhibit half-integer quantumHall effect, as first observed in graphene [5, 6] and later inHeTe [7, 8]. The Landau levels of 2DDirac fermions have  ( ) ( ∣ ∣) / N v eB N sign 2 , F 1 2 where sign is the sign function,N is the Landau index, vF is the Fermi velocity, e is the electron charge,  is Planck constant divided by 2π. This is in contrast to usual Landau levels ωc(N+1/2) and leads to half-quantizedHall conductance σxy=g(N+1/2)e /,where g is the number of degenerate species of Dirac fermions. The half-quantizedHall conductance is expected to occur in 3DTI. In this case, interestingly, a hierarchy structure is observable, starting from the bulk 3DTI to one-dimensional (1D) edge state through topological 2D surface state. Topological 2D surface state is a boundary state of the bulk 3DTI and 1D edge state occurs as a boundary state of the topological 2D surface state when gap opens at theDirac node and EF of the 2D surface state is placed at the node. In general, in the topological state ofmatter [14–18], there exist structures called bulk-edge correspondence between gapless boundarymodes and nontrivial bulk topological structures, and their evolutions as a function of dimensions. This dimensional hierarchy shows how gapless boundary states evolve as dimensional reduction is performed. Theoretically, the existence of gapless surface states in 3D topological insulators is attributed to the topological   · E B termwith an angle coefficient θ=π [19–23]. For the 2D surface state, this termdevelops OPEN ACCESS