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  • 6 - The Lieb-Schultz-Mattis Theorem

  • Christopher Mudry, Paul Scherrer Institute, Villigen, Switzerland, Claudio Chamon, Boston University
  • Book:
    Fractionalization of Particles in Physics
    Published online:
    09 January 2025
    Print publication:
    23 January 2025, pp 419-515

  • Summary

    Chapter 6 ties invertible topological phases to extensions of the original Lieb–Schultz–Mattis theorem. A review is made of the original Lieb–Schultz–Mattis theorem and how it has been refined under the assumption that a continuous symmetry holds. Two extensions of the Lieb–Schultz–Mattis theorem are given that apply to the Majorana chains from Chapter 5 when protected by discrete symmetries. To this end, it is necessary to introduce the notion of projective representations of symmetries and their classifications in terms of the second cohomology group. A precise definition is given of fermionic invertible topological phases and how they can be classified by the second cohomology group in one-dimensional space. Stacking rules of fermionic invertible topological phases in one-dimensional space are explained and shown to deliver the degeneracies of the boundary states that are protected by the symmetries.