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Bott periodicity for \(\mathbb Z_2\) symmetric ground states of gapped free-fermion systems. (English) Zbl 1346.81159
In the paper, the authors give a homotopy-theoretical classification of symmetry-periodic topological phases of gapped free-fermion systems with some special symmetries.
Authors’ abstract: Building on the symmetry classification of disordered fermions, we give a proof of the proposal by Kitaev, and others, for a “Bott clock” topological classification of free-fermion ground states of gapped systems with symmetries. Our approach differs from previous ones in that (i) we work in the standard framework of Hermitian quantum mechanics over the complex numbers, (ii) we directly formulate a mathematical model for ground states rather than spectrally flattened Hamiltonians, and (iii) we use homotopy-theoretic tools rather than \(K\)-theory. Key to our proof is a natural transformation that squares to the standard Bott map and relates the ground state of a \(d\)-dimensional system in symmetry class \(s\) to the ground state of a \((d+1)\)-dimensional system in symmetry class \(s+1\). This relation gives a new vantage point on topological insulators and superconductors.

MSC:
81V70 Many-body theory; quantum Hall effect
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
14F45 Topological properties in algebraic geometry
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