Topology and Phases in Fermionic Systems

Abstract

There can exist topological obstructions to continuously deforming a gapped Hamiltonian for free fermions into a trivial form without closing the gap. These topological obstructions are closely related to obstructions to the existence of exponentially localized Wannier functions. We show that by taking two copies of a gapped, free fermionic system with complex conjugate Hamiltonians, it is always possible to overcome these obstructions. This allows us to write the ground state in matrix product form using Grassman-valued bond variables, and show insensitivity of the ground state density matrix to boundary conditions.

4 pages, see also arxiv:0710.3299

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• Kitaev chains with long-range pairing
• Disordered Topological Insulators via $C^{\ast }$-Algebras
• Self-Organized Error Correction in Random Unitary Circuits with Measurement
• Quantum Adiabatic Brachistochrone
• Topological Insulators and C^*-Algebras: Theory and Numerical Practice
• Renormalization group constructions of topological quantum liquids and beyond
• Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Making Almost Commuting Matrices Commute
• Local markers for crystalline topology
• Construction of real-valued localized composite Wannier functions for insulators
• Hall conductance and the statistics of flux insertions in gapped interacting lattice systems
• Search for localized Wannier functions of topological band structures via compressed sensing
• Almost commuting self-adjoint matrices --- the real and self-dual cases
• Almost commuting unitary matrices related to time reversal