# Quantum (spin/thermal) Hall v.s. (Spin/Thermal) Quantum Hall effects

Are the following concepts defined correctly, as I understand:

1. Quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductance $\sigma_{xy}=J_x/E_y$ undergoes quantum Hall transitions to take on the quantized values $\sigma_{xy}=\text{quantized} \cdot e^2/\hbar$.

2. Spin quantum Hall effect is characterized by the spin current induced in the transverse direction respect to a spatially varying Zeeman field ($B$). See for example Ref Phys. Rev. B 60, 4245 1999

3. Quantum spin Hall effect describes the transverse spin current induced by an electrical voltage. It is also an example of 2-spatial dimensional topological insulator or symmetry protected topological order. (Protected by charge conservation symmetry and spin-$S_{z}$ conservation symmetry but NO time reversal symmetry)

4. Topological insulator: Some people distinguish topological insulator (charge conservation symmetry and time reversal symmetry) from Quantum spin Hall effect (charge conservation symmetry and spin-$S_{z}$ conservation symmetry but NO time reversal symmetry). But some people mix Topological insulator with Quantum spin Hall effect.

And what are the differences between

v.s.

How are they defined and contrast to each other?

Are there also standard conventions of

1. quantized (spin/thermal) Hall

and

1. (Spin/Thermal) quantized Hall effects

as well occasionally seen in the literature? What conceptual and physical phenomenon do they characterize?