Citation: Hastings, M. B. "Topology and phases in fermionic systems." Journal of Statistical Mechanics: Theory and Experiment 2008.01 (2008): L01001.
Web: https://arxiv.org/abs/0710.3324
Tags: Fermionic-order, Invertible-phases, Lieb-Robinson
This paper demonstrates a lovely little result - every free fermion phase is invertible, and its inverse is given by its complex conjugate. The proof of this fact is completely explicit. He takes an arbitrary free-fermion Hamiltonian, and then adds a the complex conjugate on top as a second layer. He then constructs an explicit path in parameter space between the stacked Hamiltonian and a product state. He then shows that in a very simple way that the Hamiltonians along the whole path are gapped.
Hastings also makes a very nice connection between being in the trivial phase and having exponentially localized Wannier functions. Wannier functions are a distinguished choice of basis for the occupied states of the free-fermion system. That is, we say that a system has exponentially localized Wannier functions if one can find a basis for the space of occupied states, such that each basis vector is exponentially localized in real space. For time-reversal symmetric functions, it was recently shown (after being long conjectured) that time-reversal invariant topological insulators always have exponentially localized Wannier functions:
> Fiorenza, Domenico, Domenico Monaco, and Gianluca Panati. "Construction of real-valued localized composite Wannier functions for insulators." Annales Henri Poincaré. Vol. 17. No. 1. Cham: Springer International Publishing, 2016.
If a state is in the trivial phase, then you can deform to a product state. The product state has Wannier functions which are localized to points. Doing quasi-adiabatic flow on these Wannier functions, one obtains exponentially localized Wannier functions in the original state. So, trivial topological insulator implies exponentially localized Wannier functions. In particular, Hastings has shown that if you double a free fermion system then it will have exponentially localized Wannier functions.