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The resistance of quantum entanglement to temperature in the KugelKhomskii model
by V. E. Valiulin, A. V. Mikheyenkov, N. M. Chtchelkatchev, K. I. Kugel
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Submission summary
Authors (as registered SciPost users):  Valerii Valiulin 
Submission information  

Preprint Link:  https://arxiv.org/abs/2203.08254v2 (pdf) 
Date accepted:  20230123 
Date submitted:  20220628 14:45 
Submitted by:  Valiulin, Valerii 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The KugelKhomskii model with entangled spin and orbital degrees of freedom is a good testing ground for many important features in quantum information processing, such as robust gaps in the entanglement spectra. Here, we demonstrate that the entanglement can be also robust under effect of temperature within a wide range of parameters. It is shown, in particular, that the temperature dependence of entanglement often exhibits a nonmonotonic behavior. Namely, there turn out to be ranges of the model parameters, where entanglement is absent at zero temperature, but then, with an increase in temperature, it appears, passes through a maximum, and again vanishes.
Author comments upon resubmission
Hereafter is the list of our corrections
The referee 1 wrote: 1. Nonmonotonicity of nearestneighbor 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. Our reply The temperature effects on the entanglement (including nonmonotonicity) have been studied mainly for two or three qubits (spins etc.), where T is the bath temperature [17]. In some fewparticle works, it is noted that the cause of the nonmonotonicity is the following. The ground state is nonentangled, though the excited ones are entangled. There are numerous works on infinite Hubbard and spin systems, mainly chains, see e.g. [826]). Note that only few of them deal with temperature effects. As far as we know, the analysis of the temperature entanglement and its nonmonotonicity in the spinpseudospin model is still absent. We are grateful to the reviewer for the useful remark, and in the revised manuscript (Version 2), we added references to some important works in this area.
The referee wrote: 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 spin1/2 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? Our reply The spinpseudospin model considered in this paper known since the 1970s, has get a second wind after the experimental discovery of orbital waves ([27, 28]) and is now widely used. Thus, reviews of only recent works in this and closely related areas ([29, 30]) contain several hundred of references. Moreover, the model is being studied in application not only to transition metal compounds, but also to other systems with two degrees of freedom, which can be described by spin (pseudospin) operators. Such a situation is realized, for example, in some systems of ultracold atoms in optical lattices. The version of the model, for which both the spin and the pseudospin are equal to 1/2, is studied much more often than others, both due of its relative simplicity and because the field of its experimental implementation is wider (in the future, we intend to investigate entanglement in other versions of the model). Attention to the entanglement in this model is related primarily to its physical nature it is clear that the analysis of manyparticle quantum entanglement between nonidentical degrees of freedom (spin and orbital in the canonical implementation) is of considerable interest. A relatively outdated review of the subject ([31]) the later ones are unknown to us also includes more than 150 references. We believe that these circumstances explain the impossibility of an exhaustive bibliography within the framework of a regular article. We have tried to provide references to the most important works for the issue under study, to the most recent ones and to the key reviews. We also have not met any works on the temperature behavior of the entanglement in the spinpseudospin model, however, we have added several references to the works where temperature effects are studied in the purely spin models. As for nonmonotonicity, we are not aware of works in the model under study where it would have been discovered (there are only indirect indications of the effect in the work of the same authors). Moreover, we point out a range of parameters where the entanglement, which is absent at zero temperature, arises and passes through a maximum at T >0. It seems that such a phenomenon should not be overlooked. We have added several extensions to the revised manuscript (Version 2) explaining the importance of the model under study with the corresponding references. As far as the entanglement in manybody systems is concerned, it is usually important for finding out the range of existence for quantum phase transitions and revealing the areas in the phase diagram exhibiting the enhanced quantum fluctuations. In the spinpseudospin models under study, it is especially important since it highlights the ranges, where the entangled spinorbital excitations play a crucial role in the thermodynamics of the system. In such a case, the nonmonotonocity, especially, the emergence of it only at infinite temperature can reveal important specific features of the thermal characteristics of the system. We have added a more detailed description of the subject to the revised manuscript (Version 2) in the Conclusions section.
Reply to the referee 2 The referee wrote: 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. Our reply The temperature effects on the entanglement (including nonmonotonicity) have been studied mainly for two or three qubits (spins etc.) where T is the bath temperature [17]. In some fewparticle works it is noted that the cause of the nonmonotonicity is the following. The ground state is nonentangled, though the excited ones are entangled. Of course, the model under study can in some sense be related to spin chains and to the Hubbard model. Both have an extensive bibliography dealing with the study of entanglement (including its temperature evolution, see e.g. [826]). Note, that only few of them deal with temperature effects. Among these works, using different measures of entanglement, there are both numerical and analytical ones, and even an experiment. However, the present spinpseudospin model is distinguished from the Heisenberg spin chain (in this case, the ladder) by a fundamentally different type of intersubsystem interaction and the independent tuning of exchange parameters in the subsystems. As for the relationship to the Hubbard model the symmetric spinpseudospin model under study here, with certain restrictions on the coefficients, can be obtained from the twoband Hubbard model and is very far from its standard oneband version. That is why, we initially had not found it possible to mention entanglement in the Hubbard and purely spin models in the bibliography. We are grateful to the reviewer for the useful remark, and in the revised manuscript (Version 2), we added references to some important works in this area. Then, as far as we know, the analysis of the temperature entanglement and its nonmonotonicity in the spinpseudospin model is still absent. The spinpseudospin model considered in this paper known since the 1970s, has get a second wind after the experimental discovery of orbital waves ([27, 28]) and is now widely used. Thus, reviews of only recent works in this and closely related areas ([29, 30]) contain several hundred of references. Moreover, the model is being studied in application not only to transition metal compounds, but also to other systems with two degrees of freedom, which can be described by spin (pseudospin) operators. Such a situation is realized, for example, in some systems of ultracold atoms in optical lattices. The version of the model, for which both the spin and the pseudospin are equal to 1/2, is studied much more often than others, both due of its relative simplicity and because the field of its experimental implementation is wider (in the future, we intend to investigate entanglement in other versions of the model).
The referee wrote: Another major aw in this manuscript, in my opinion, is the lack of details on how the results were obtained. For example, the Hamiltonian given by Eqs.(14) 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 righthand side of Eq.(5) are normalized, then the total density operator on the lefthand side is not normalized. Our reply We in fact had limited ourselves to the onedimensional case. For larger dimensions, even for a smallscale cluster, the required calculational resources are currently unavailable. Therefore, most of the work in this area is focused on the onedimensional case. We have added a more detailed description of the calculation scheme to the revised manuscript (Version 2) after Eq.(6). As regards the comment on the normalization of the right and lefthand sides of Eq.(5), we are grateful to the Referee for a useful remark. We corrected the typo in the formula, of course, in the actual calculation, the normalization is taken into account.
[1] H. B. Fei, B. M. Jost, S. Popescu, B. E. A. Saleh, and M. C. Teich, EntanglementInduced TwoPhoton Transparency, Phys. Rev. Lett. 78, 1679 (1997), publisher: American Physical Society. [2] I. Sinaysky, F. Petruccione, and D. Burgarth, Dynamics of nonequilibrium thermal entanglement, Phys. Rev. A 78, 062301 (2008). [3] J. Dajka, M. Mierzejewski, and J. Luczka, NonMarkovian entanglement evolution of two uncoupled qubits, Phys. Rev. A 77, 042316 (2008). [4] M. Urbaniak, S. B. Tooski, A. Ramsak, and B. R. Bul ka, Thermal entanglement in a triple quantum dot system, Eur. Phys. J. B 86, 505 (2013). [5] Z. Wang, W. Wu, and J. Wang, Steadystate entanglement and coherence of two coupled qubits in equilibrium and nonequilibrium environments, Phys. Rev. A 99, 042320 (2019). [6] T. Seidelmann, F. Ungar, A. Barth, A. Vagov, V. Axt, M. Cygorek, and T. Kuhn, PhononInduced Enhancement of Photon Entanglement in Quantum DotCavity Systems, Phys. Rev. Lett. 123, 137401 (2019). [7] A. Ghannadan and J. Strecka, MagneticFieldOrientation Dependent Thermal Entanglement of a Spin1 Heisenberg Dimer: The Case Study of Dinuclear Nickel Complex with an Uniaxial SingleIon Anisotropy, Molecules 26, 3420 (2021). [8] V. E. Korepin, Universality of Entropy Scaling in One Dimensional Gapless Models, Phys. Rev. Lett. 92, 096402 (2004). [9] S. J. Gu, S. S. Deng, Y. Q. Li, and H. Q. Lin, Entanglement and quantum phase transition in the extended Hubbard model, Phys. Rev. Lett. 93, 086402 (2004). [10] B.Q. Jin and V. E. Korepin, Localizable entanglement in antiferromagnetic spin chains, Phys. Rev. A 69, 062314 (2004). [11] Y. Xu, H. Katsura, T. Hirano, and V. E. Korepin, Entanglement and Density Matrix of a Block of Spins in AKLT Model, J. Stat. Phys. 133, 347 (2008). [12] A. R. Its and V. E. Korepin, The FisherHartwig Formula and Entanglement Entropy, J. Stat. Phys. 137, 1014 (2009). [13] C. C. Chang, R. R. P. Singh, and R. T. Scalettar, Entanglement properties of the antiferromagneticsinglet transition in the Hubbard model on bilayer square lattices, Phys. Rev. B 90, 155113 (2014), publisher: American Physical Society. [14] F. Iemini, T. O. Maciel, and R. O. Vianna, Entanglement of indistinguishable particles as a probe for quantum phase transitions in the extended Hubbard model, Phys. Rev. B 92, 075423 (2015). [15] O. Vafek, N. Regnault, and B. A. Bernevig, Entanglement of Exact Excited Eigenstates of the Hubbard Model in Arbitrary Dimension, SciPost Phys. 3, 043 (2017). [16] F. Parisen Toldin and F. F. Assaad, Entanglement hamiltonian of interacting fermionic models, Phys. Rev. Lett. 121, 200602 (2018). [17] V. K. Vimal and V. Subrahmanyam, Quantum correlations and entanglement in a Kitaevtype spin chain, Phys. Rev. A 98, 052303 (2018). [18] F. Sugino and V. Korepin, Renyi entropy of highly entangled spin chains, Int. J. Mod. Phys. B 32, 1850306 (2018). [19] C. Walsh, P. Semon, D. Poulin, G. Sordi, and A.M. S. Tremblay, Local entanglement entropy and mutual information across the mott transition in the twodimensional Hubbard model, Phys. Rev. Lett. 122, 067203 (2019). [20] J. Spalding, S.W. Tsai, and D. K. Campbell, Critical entanglement for the half lled extended Hubbard model, Phys. Rev. B 99, 195445 (2019). [21] I. Kleftogiannis, I. Amanatidis, and V. Popkov, Exact results for the entanglement in 1D Hubbard models with spatial constraints, J. Stat. Mech: Theory Exp. 2019, 063102 (2019). [22] P. Padmanabhan, F. Sugino, and V. Korepin, Quantum phase transitions and localization in semigroup Fredkin spin chain, Quantum Inf. Process. 18, 69 (2019). [23] G. Mathew, S. L. L. Silva, A. Jain, A. Mohan, D. T. Adroja, V. G. Sakai, C. V. Tomy, A. Banerjee, R. Goreti, A. V. N., R. Singh, and D. JaiswalNagar, Experimental realization of multipartite entanglement via quantum Fisher information in a uniform antiferromagnetic quantum spin chain, Phys. Rev. Res. 2, 043329 (2020). [24] S. Fraenkel and M. Goldstein, Symmetry resolved entanglement: exact results in 1D and beyond, J. Stat. Mech: Theory Exp. 2020, 033106 (2020). [25] C. Kokail, B. Sundar, T. V. Zache, A. Elben, B. Vermersch, M. Dalmonte, R. van Bijnen, and P. Zoller, Quantum variational learning of the entanglement hamiltonian, Phys. Rev. Lett. 127, 170501 (2021). [26] D. L. B. Ferreira, T. O. Maciel, R. O. Vianna, and F. Iemini, Quantum correlations, entanglement spectrum, and coherence of the twoparticle reduced density matrix in the extended Hubbard model, Phys. Rev. B 105, 115145 (2022). [27] E. Saitoh, S. Okamoto, K. T. Takahashi, K. Tobe, K. Yamamoto, T. Kimura, S. Ishihara, S. Maekawa, and Y. Tokura, Observation of orbital waves as elementary excitations in a solid, Nature 410, 180 (2001). [28] J. Schlappa, K. Wohlfeld, K. J. Zhou, M. Mourigal, M. W. Haverkort, V. N. Strocov, L. Hozoi, C. Monney, S. Nishimoto, S. Singh, A. Revcolevschi, J.S. Caux, L. Patthey, H. M. Rønnow, J. van den Brink, and T. Schmitt, Spin orbital separation in the quasionedimensional Mott insulator Sr2CuO3, Nature 485, 82 (2012). [29] Z. Nussinov and J. van den Brink, Compass models: Theory and physical motivations, Rev. Mod. Phys. 87, 1 (2015). [30] D. I. Khomskii and S. V. Streltsov, Orbital E ects in Solids: Basics, Recent Progress, and Opportunities, Chem. Rev. 121, 2992 (2021). [31] A. Oles, Fingerprints of spinorbital entanglement in transition metal oxides, J. Phys. Condens. Matter 24, 313201 (2012).
List of changes
• We have expanded the Abstract section by significantly increasing the review of the entangled states (in particular for Hubbard model), and other cuttingedge research on temperature entanglement.
• We have added more detailed explanation of the importance of the model in hand (in the Conclusions section).
• In the Methods section, we have noticeably extended the description of our computational procedure, detailing each step and highlighting some of the key features.
• We have extended the section Conclusions. Here we explain in more detail the importance of the obtained results on the thermodynamics of entangled states in spinpseudospin systems.
• We have corrected some misprints.
Published as SciPost Phys. Core 6, 025 (2023)
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 20221125 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.08254v2, delivered 20221125, doi: 10.21468/SciPost.Report.6192
Report
In the new version of the manuscript, the authors have taken into consideration the suggestions of both referees and have implemented a few changes in the text. The most noticeable modification is an increased number of the referenced works, which is accompanied by an increased discussion on how the model considered in the paper is different from others in the literature in the context of study of temperature effect on entanglement. Additionally, the authors have included some more details about their calculations in the new version of their manuscript.
Taking into account the implemented changes and the reply, unfortunately, I still cannot recommend the current work for publication. My main concern regarding originality of the results is still present. Perhaps, I should have stressed more in my first report on the importance of the comparison of the results of the manuscript with the results of the cited works. This comparison should not only concern the type of model considered, but also the effects that are observed there, i.e., the results themselves. For example, is it the case that nonmonotonicity of entanglement occurs exclusively in the Kugel–Khomskii model, or, perhaps, this effect is much more profound than in other systems? In a way, in their reply the authors have even confirmed that very similar problems have been studied before, and that similar effects have already been reported. For example, the authors write: "The temperature effects on the entanglement (including nonmonotonicity) have been studied mainly for two or three qubits." The authors' reply to this criticism is: "As far as we know, the analysis of the temperature entanglement and its nonmonotonicity in the spinpseudospin model is still absent". However, it is not made clear from the text or the reply how this choice of the model affects the results, i.e., the dependence of entanglement on temperature. Is the considered model the most sensitive one or, contrary, the most robust to temperature fluctuations? Without such questions being clearly answered in the manuscript, it is impossible to judge about the impact of the current work, and hence to recommend the manuscript for publication.
Report
I have read the authors' response to the comments of both the referees. As far as I know, the dependence of the nonmonotonicity of entanglement wrt temperature was not checked before. It would still be good to know why such dependence is of interest, and this is not answered clearly in the paper. However, I think that the study could lead to such answers and hence I recommend its publication in SciPost.