Stability of k-local phases of matter

It is well known that Hamiltonians associated to topological phases of matter on Euclidean geometries are stable against local noise, meaning that local noise can not close their gap or lift their ground state degeneracy in the thermodynamic limit. In this work, we relax the assumption about the existence of an underlying Euclidean geometry and ask whether k-local Hamiltonians associated to (not necessarily geometric) qLDPC error correcting codes are robust against local noise, where locality of noise is now defined with respect to the interaction graph. We find that if there exists constants $varepsilon_1,varepsilon_2>0$ such that the size of balls of radius $r$ on the interaction graph is upper bounded by $O(exp(r^{1-varepsilon_1}))$ and balls of radius $O(log(N)^{1+varepsilon_2})$ are locally correctable, then the associated Hamiltonian is stable against local noise. As a non-trivial example, we show that the semi-Hyperbolic surface code Hamiltonian has a finite perturbation strength threshold.