# Are there any gapped systems that aren't invertible?

Assume the following definitions:

A gapped phase of matter is a collection of (quantum-mechanical) systems with a unique ground state and an energy gap to all excitations in the limit of infinite volume. Any two systems in this collection can be deformed into each other without closing the gap.

An invertible system is a gapped system $$A$$ such that there exists a system $$A^{-1}$$ that can be stacked with $$A$$ to give a system in the 'trivial' gapped phase.

So there seem to be gapped systems that are not invertible. Could you suggest an example?

• When 0 lies in the relevant gap, then yes, your hamiltonian is necessarily invertible. – Max Lein Mar 18 at 2:34