The W-state signature conjecture for Lifshitz transitions

Let $L$ be the system size, let $X_i$ denote the local operator creating the one-excitation component at site $i$, and let $|\alpha\rangle$ be a single-site product state. A zero-mode $|W_1\rangle$-like state has the form

$$\frac{1}{\sqrt{L}}\sum_i e^{i \frac{2\pi}{L}m x_i}X_i|\alpha\rangle^{\otimes L},$$

where $m\in\mathbb{Z}$, and the Hamiltonian is translation invariant.

The W-state signature conjecture. If such a translation-invariant Hamiltonian is tuned so that the zero-mode $|W_1\rangle$-like state and the product state $|\alpha\rangle^{\otimes L}$ become ground states, then the system is at a Lifshitz transition.

This proposed signature covers simple Lifshitz transitions, such as transitions between an insulator and a metal, and is consistent with the occurrence of quadratic or higher dispersion at the critical point. It does not cover cases in which a metal or semimetal changes its Fermi-surface shape through a Van Hove singularity or Dirac lines; the authors leave a precise formulation for such cases open.