Equivalence of higher-dimensional tilting paths

Determine whether all paths for taking the limits of the multiple tilting parameters \(\{\lambda_i\}_{i=1,\dots,d}\) in higher-dimensional tilted stochastic transition matrices are equivalent, and, if they are not, identify the correct path for defining the topological invariant.

Background

The paper proves that, in one-dimensional gapped systems, the authors’ shifted-point-gap approach is equivalent to the alternative tilting approach based on $W^\lambda(k)=W(k-\mathrm{i}\lambda)$. In higher dimensions, however, introducing one tilting parameter for each momentum component produces infinitely many possible paths for taking the limiting values of the parameters.

The unresolved issue is whether these paths yield the same topological classification. If they do not, a principled choice of path is required. This ambiguity also obstructs a unified classification based solely on tilted transition matrices, particularly because the tilting construction has no direct momentum-space parameter in zero dimensions.