Some aspects of the quantum Hall effects

The quantum Hall effects have introduced many important developments to physics, ranging from advances in metrology to the incorporation of topology in the understanding of phases of matter. Some general aspects of the integer quantum Hall effect (IQHE) and the fractional quantum Hall effect (FQHE) will be discussed in detail. The existence of frustration-free parent Hamiltonians in the FQH context has not been realized as a consequence of the infinite-bond-dimension matrix-product-state (MPS) representation of FQHS. We expose the direct link between a MPS (which is a tensor network) structure and the existence of frustration-free parent Hamiltonians for Laughlin and Moore-Read FQH states and their zero energy bosonic and/or Majorana excitations.