The Loschmidt Index

In this work the authors study ground state properties of noninteracting topological systems and their implications on nonequilibrium real-time dynamics in the context of dynamical quantum phase transitions. More specifically, they investigate overlaps of ground state wave functions of a parameter-dependent Hamiltonian. They show that such overlaps can exhibit nodes which are topological and which can be used to classify both the system's equilibrium and nonequilibrium properties. In particular, they introduce a so-called Loschmidt index, which is the number of nodes in these wavefunction overlaps. Importantly, their analysis not only applies to the mostly studied two-band models but can be equally applied to multi-band systems. Finally, the authors apply their theory to a paradigmatic model system, the so-called one-dimensional XY model and they discuss thoroughly the connection to other previously introduced topological classifications in this context.

The manuscript is well-structured and well written and is accessible also to non-expert readers. The results are sound and the analysis thorough. The main topic covered in this work is also timely. Overall, I consider this a nice piece of work and I am therefore in favor of recommending it for publication in SciPost.

I could, however, imagine that the present manuscript could profit significantly from also applying their theory to a regime, which could particularly highlight the strength of the introduced Loschmidt index. In the presentation the authors already consider the aforementioned XY model, which, however, is relatively simple and whose properties have been already very well described by other topological quantum numbers introduced recently. In particular, the XY model realizes a two-band model. The challenging regime is the multi-band case and I would consider the present manuscript an even stronger contribution when their theory would have been applied to such a more difficult case.

While I wouldn't consider this point as a mandatory addition, I feel that the manuscript could profit significantly from extending the discussion to the multi-band case.

In this paper, the authors introduce a new topological metric, the Loschmidt index, which they show can be used to count zeros of a generic real-valued function f. They specifically focus on the case where f is the Loschmidt echo, whose zeros play a defining role in dynamical phase transitions. They demonstrate that this works both equilibrium and non-equilibrium phase transitions in the XY model.
This idea is new and relevant, and therefore I recommend publication in SciPost Physics. The paper is well-written and I don’t have major notes regarding the text. My main comment to the authors is that it would be useful to more clearly address how the method is expected to be useful. Since the topological invariant is constructed my mapping f to a d+1 dimensional function, one of whose parameters is f itself, it seems like one could simply count the zeros of f by putting the function on a grid and looking for zero crossings. This would be much easier than numerically calculating the higher-dimensional winding number integral. My question to the authors is what are the conditions under which the topological integral would be more useful than simple zero counting?