Quantum spin Hall state and 2+1D topological insulator

October 3, 2020

What is quantum spin Hall effect? According to Bernevig-Zhang and Kane-Mele, quantum spin Hall effect refers to quantized transverse Sz-spin current induced by force acting on electric charges (i.e. a quantized mixed-electro-spin Hall conductance). The 2+1D topological insulator is defined by Kane-Mele

According to the above two definitions, quantum spin Hall state and 2+1D topological insulator are not the same. They even have different symmetries: quantum spin Hall state have Uup(1) × Udown(1) symmetry, while topological insulator have G--(U,T) = U(1) ⋊ Z4T/Z2 symmetry. In fact, quantum spin Hall state and topological insulator, having different symmetries, are different symmetry protected trivial (SPT) states for fermions. The non-interacting fermionic SPT orders were classified by Ryu-Schnyder-Furusaki-Ludwig and Kiteav

Even though topological insulator arises from the studies of quantum spin Hall effect, it is incorrect to think topological insulator to be due to quantum spin Hall effect. In particular, Kane-Mele pointed out that that even without quantum spin Hall effect, an insulator can still be non-trivial. This led to the notion of topological insulator. This is a very surprising discovery which started the very active field of topological insulator.

The quantum spin Hall effect is very simple and not surprising. It just comes from the spin-up and spin-down electrons with opposite integer quantum Hall effect. When the Uup(1) × Udown(1) symmetry is broken down to U(1) symmetry (ie when Sz is no longer conserved), the quantum spin Hall state will become a trivial insulator. However, before 2005, there were a lot of research efforts, trying to define quantum spin Hall conductance even when Sz is not conserved. If those efforts were used in the right direction, the topological insulator could be discovered earlier. There are many papers refer to 2+1D topological insulator as quantum spin Hall state, which is misleading.

Kane-Mele's paper has a title “Z_2 Topological Order and the Quantum Spin Hall Effect”. Despite the term “Topological Order” in the title, we now know that the topological insulator is a short-range entangled SPT state. It does not has the topological order previously defined by Wen which involves the long-range entanglement of Chen-Gu-Wen. Although the title contains "Quantum Spin Hall Effect", it was used in the negative sense, i.e. the state studied in the paper is non-trivial even without quantum spin Hall effect. So both the terms in the title actually should be regarded in the negative sense: the non-trivial state studied in the paper has no topological order nor quantum spin Hall effect. The paper discovered a new state of matter: topological insulator, a fermionic SPT state protected by G--(U,T) = U(1) ⋊ Z4T/Z2 symmetry.

It is known that topological orders have non-trivial boundary state (many gapless). Such a holographic boundary-bulk relation becomes a powerful way to study topological orders. Similar to topological orders, topological insulators also have non-trivial boundary (many gapless). This similar holographic boundary-bulk relation also becomes a powerful way to study topological insulators and other more general SPT states.

For details, see Zoo of quantum-topological phases of matter.

Two kinds fo topology