Proposal for a topological spin Chern pump

The quantum Hall (QH) effect discovered in 1980 (1) is the first example of topological state in the field of condensed matter physics. Since then, there has been continuously strong interest in topological phenomena of condensed matter systems. Laughlin (2) interpreted the integer QH effec as a quantum charge pump. In- creasing the magnetic flux by a single flux quantum that threads a looped QH ribbon constitutes a cycle of the pump due to gauge invariance, transferring an integer- quantized amount of charge from one edge of the rib- bon to the other. Thouless, Kohmoto, Nightingale, and Nijs (3) showed that the QH state can be classified by a topological invariant, the Chern number. Thouless and Niu (4, 5) also established a general relation between the Chern number and the charge pumpedduring a period of slow variation of potential in the Schrodinger equation. Recently, an important discovery was the topological insulator, (6-9) a new quantum stateof matterexisting in nature. Different from the QH systems, the topolog- ical insulators preserve the time-reversal (TR) symme- try. Two-dimensional topological insulators, also called the quantum spin Hall (QSH) systems, have a bulk band gap and a pair of gapless helical edge states traversing the bulk gap. When electron spin is conserved, the topo- logical properties of the QSH systems can be easily un- derstood, as a QSH system can be viewed as two inde- pendentQH systems without Landau levels. (10) When the spin conservation is destroyed, unconventional topo- logical invariants are needed to classify the QSH systems. The Z2 index (11) and the spin Chern numbers (12-14) have been proposed to describe the QSH systems. While the two different invariants are found to be equivalent to each other for TR-invariant systems, (13, 14) they lead to controversial predictions when the TR symmetry is bro- ken. The definition of the Z2 index explicitly relies on the presence of TR symmetry, suggesting that the QSH state turns into a trivial insulator once the TR symme- try is broken. However, calculations (15) based upon the spin Chern numbers showed that the nontrivial topolog- ical properties of the QSH systems remain intact when the TR symmetry is broken, as long as the band gap and spin spectrum gap stay open. The nonzero spin Chern numbers guarantee that the edge states must appear on the sample boundary, (16) which could be either gaped or gapless, depending on symmetries or spatial distributions of the edge states. (17) This prediction was supported by the recent experimental observation of the QSH effect in InAs/GaSb bilayers under broken TR symmetry. (18) Spin pumpspromise broad applications in spintronics,