Chapter 7 opens with the description of superconductivity in terms of Bogoliubov–de-Gennes Hamiltonians. The 10-fold way in terms of the Altland–Zirnbauer symmetry classes is applied to random matrix theory and two disordered quantum wires. The chapter closes with the 10-fold way for the gapped phases of quantum wires.

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.

Chapter 3 is devoted to fractionalization in polyacetylene. Topological defects (solitons) in the dimerization of polyacetylene are introduced and shown to bind electronic zero modes. The fractional charge of these zero modes is calculated by different means: (1) The Schrieffer counting formula(2) Scattering theory(3) Supersymmetry(4) Gradient expansion of the current(5) Bosonization.

The effects of temperature on the fractional charge and the robustness of the zero modes to interactions in polyacetylene are studied.

Chapter 2 is a gentle introduction to the many-body physics of polyacetylene. Band theory for electrons hopping along a one-dimensional lattice is explained. The continuum limit is taken. An example of a quantum critical point with its emergent symmetries is given. The Su–Schrieffer–Heeger (SSH) model for polyacetylene is defined and solved at the mean-field level.

Chapter 5 introduces a family of exactly soluble spin-1/2 lattice Hamiltonians: The spin-1/2 cluster “c” chains. Each member of this family is gapped and nondegenerate when periodic boundary conditions hold. The nondegeneracy of the ground state is lifted for all members of this family except for one member, owing to the presence of zero modes bound to the boundaries when open boundary conditions hold. The notion of symmetry fractionalization is thereby introduced. This family of exactly soluble Hamiltonians is mapped of a family of exactly soluble Majorana lattice Hamiltonians, one of which is an example of a Kitaev chain, though the Jordan–Wigner transformation. The stability of the degeneracy of the zero modes to integrability-breaking but symmetry-preserving interactions is derived through the explicit construction of the stacking rules.

Chapter 4 aims at establishing that the fractional charge calculated in Chapter 3 is sharp. To this end, the calculation of the mean value and second cumulant of the electronic charge localized in one of two wells of a double-well potential in quantum mechanics is contrasted to that of the mean value and second cumulant of the fractional charge localized around a soliton in a dimerization profile of polyacetylene support a pair of soliton and anti-soliton defects far apart from each other.

Chapter 8 extends the 10-fold way of gapped phases from one to any dimension of space. This is done by presenting the homotopy groups of the classifying spaces of normalized Dirac masses. There follows two applications. First, there is the interplay between Anderson localization and the topology of classifying spaces for disordered quantum wires. Second, it is possible to derive the breakdown of the 10-fold way due to short-range interactions in any dimension. The chapter closes with the relationship between invertible topological phases and invertible topological field theories.

Chapter 1 describes how the book is organized.