Interaction distance in the extended XXZ model

We employ the interaction distance to characterise the physics of a one-dimensional extended XXZ spin model, whose phase diagram consists of both integrable and non-integrable regimes, with various types of ordering, e.g., a gapless Luttinger liquid and gapped crystalline phases. We numerically demonstrate that the interaction distance successfully reveals the known behaviour of the model in its integrable regime. As an additional diagnostic tool, we introduce the notion of "integrability distance" and particularise it to the XXZ model in order to quantity how far the ground state of the extended XXZ model is from being integrable. This distance provides insight into the properties of the gapless Luttinger liquid phase in the presence of next-nearest neighbour spin interactions which break integrability.