Bott Periodicity for \({\mathbb{Z}_2}\) Symmetric Ground States of Gapped Fre

详细信息    查看全文

• 作者：R. Kennedy ; M. R. Zirnbauer
• 刊名：Communications in Mathematical Physics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：342
• 期：3
• 页码：909-963
• 全文大小：1,310 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mathematical and Computational Physics
  Quantum Physics
  Quantum Computing, Information and Physics
  Complexity
  Statistical Physics
  Relativity and Cosmology
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0916

文摘

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.