The resistance of quantum entanglement to temperature in the Kugel--Khomskii model

The manuscript aims at studying resistance of entanglement in the Kugel–Khomskii model to temperature. The Kugel–Khomskii model is a spin–pseudospin system (the latter being e.g., electron's or vacancy's orbital degree of freedom) defined by a specific type of interaction. According to the authors, this type of interaction can occur in lattices of atoms for which the on-site Hubbard interactions are dominating. The authors start by finding eigenstates of the Kugel–Khomskii Hamiltonian numerically for chains of atoms up to 10. The authors then consider thermal states in which the density operator's dependence on temperature is explicit. The main part of the manuscript is dedicated to measuring entanglement in the thermal states of the Kugel–Khomskii model for various parameters of the Hamiltonian and different temperatures. As a main highlight of their work the authors choose non-monotonicity of entanglement with respect to temperature for certain model parameters.

I find the considered problem interesting and motivating. At the same time I have serious doubts that very similar problems have not been studied earlier. For instance, [V.E. Korepin, Phys. Rev. Lett. 92, 096402 (2004)] studied entanglement in the Hubbard model and its dependence on temperature. Unfortunately, the manuscript does not provide an adequate review of the literature and does not state explicitly in which way the findings of the current work are novel. Without such a comparison, in particular a proper review of the previous results on entanglement in the Hubbard model, I cannot recommend this work for publication.

Another major flaw in this manuscript, in my opinion, is the lack of details on how the results were obtained. For example, the Hamiltonian given by Eqs.(1-4) is very general and allows for arbitrary configurations of atoms in a lattice. At the same time, the authors state in the introduction that they study only chains. Most importantly, the authors do not provide description of their mathematical derivations since the whole analysis is numerical. In this case, however, a computer code should be provided. A few formulas included in the manuscript are not enough for evaluation of the results' correctness and in some cases are even confusing. In particular, there is something clearly missing in the description of the density operator in Eq.(5). If the density operators on the right-hand side of Eq.(5) are normalized, then the total density operator on the left-hand side is not normalized.

Overall, despite a potential interest of the SciPost readers to the considered problem, I cannot recommend the manuscript for publication in its current form.

1. Topical area.
2. Clearly written.

1. Nonmonotonicity of nearest-neighbor entanglement wrt temperature was previously known. For small as well for infinite quantum spin systems. It is possible that it was not known for the model considered.

2. These refs are not given, but more importantly, the paper does not bring anything more than the nonmonotonicity mentioned. Like, is it very important that entanglement is nonmonotonic wrt temperature in this particular spin-1/2 model? Is this model used for some task where entanglement is used as a resource and this nonmonotonicity is crucial or could potentially be crucial?

Given the above points, I do not recommend the publication of this paper.