# Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model and Basic Questions

## 3. Methods

#### 3.1. Entanglement Entropy and Correlation Functions

#### 3.2. Effective Hamiltonian in the Large $\lambda $ Limit

## 4. Results and Discussion

#### 4.1. Gradual Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling

#### 4.2. Entanglement Evolution for the Quantum Phases along the $\alpha +\beta =0$ Line

#### 4.3. Entanglement Evolution for the Quantum Phases along the $\alpha +\beta =-1$ Line

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gotfryd, D.; Pärschke, E.M.; Chaloupka, J.; Oleś, A.M.; Wohlfeld, K. How spin-orbital entanglement depends on the spin-orbit coupling in a Mott insulator. Phys. Rev. Res.
**2020**, 2, 013353. [Google Scholar] [CrossRef] [Green Version] - Bentsson, I.; Zyczkowski, K. Geometry of Quantum States: An Introduction to Quantum Entanglement; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Oleś, A.M.; Horsch, P.; Feiner, L.F.; Khaliullin, G. Spin-Orbital Entanglement and Violation of the Goodenough-Kanamori Rules. Phys. Rev. Lett.
**2006**, 96, 147205. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kugel, K.I.; Khomskii, D.I. Jahn-Teller Effect and Magnetism: Transition Metal Compounds. Usp. Fiz. Nauk
**1982**, 25, 621–664. [Google Scholar] [CrossRef] - Feiner, L.F.; Oleś, A.M.; Zaanen, J. Quantum Melting of Magnetic Order due to Orbital Fluctuations. Phys. Rev. Lett.
**1997**, 78, 2799–2802. [Google Scholar] [CrossRef] [Green Version] - Chen, Y.; Wang, Z.D.; Li, Y.Q.; Zhang, F.C. Spin-orbital entanglement and quantum phase transitions in a spin-orbital chain with SU(2)×SU(2) symmetry. Phys. Rev. B
**2007**, 75, 195113. [Google Scholar] [CrossRef] [Green Version] - Brzezicki, W.; Dziarmaga, J.; Oleś, A.M. Noncollinear Magnetic Order Stabilized by Entangled Spin-Orbital Fluctuations. Phys. Rev. Lett.
**2012**, 109, 237201. [Google Scholar] [CrossRef] [Green Version] - You, W.-L.; Horsch, P.; Oleś, A.M. Entanglement driven Phase Transitions in Spin-Orbital Models. New J. Phys.
**2015**, 17, 083009. [Google Scholar] [CrossRef] - Snamina, M.; Oleś, A.M. Magnon Dressing by Orbital Excitations in Ferromagnetic Planes of K
_{2}CuF_{4}and LaMnO_{3}. New J. Phys.**2019**, 21, 023018. [Google Scholar] [CrossRef] [Green Version] - Jackeli, G.; Khaliullin, G. Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models. Phys. Rev. Lett.
**2009**, 102, 017205. [Google Scholar] [CrossRef] [Green Version] - Khaliullin, G.; Maekawa, S. Orbital Liquid in Three-Dimensional Mott Insulator: LaTiO
_{3}. Phys. Rev. Lett.**2000**, 85, 3950–3953. [Google Scholar] [CrossRef] [Green Version] - Normand, B.; Oleś, A.M. Frustration and Entanglement in the t
_{2g}Spin–Orbital Model on a Triangular Lattice: Valence–Bond and Generalized Liquid States. Phys. Rev. B**2008**, 78, 094427. [Google Scholar] [CrossRef] [Green Version] - Normand, B. Multicolored quantum dimer models, resonating valence-bond states, color visons, and the triangular-lattice t
_{2g}spin-orbital system. Phys. Rev. B**2011**, 83, 064413. [Google Scholar] [CrossRef] [Green Version] - Chaloupka, J.; Oleś, A.M. Spin-Orbital Resonating Valence-Bond Liquid on a Triangular Lattice: Evidence from Finite Cluster Diagonalization. Phys. Rev. B
**2011**, 83, 094406. [Google Scholar] [CrossRef] [Green Version] - Brzezicki, W.; Dziarmaga, J.; Oleś, A.M. Topological Order in an Entangled SU(2)⊗XY Spin-Orbital Ring. Phys. Rev. Lett.
**2014**, 112, 117204. [Google Scholar] [CrossRef] [PubMed] - Chaloupka, J.; Khaliullin, G. Orbital Order and Possible Superconductivity in LaNiO
_{3}/LaMO_{3}Superlattices. Phys. Rev. Lett.**2008**, 100, 016404. [Google Scholar] [CrossRef] [Green Version] - Miyasaka, S.; Okimoto, Y.; Tokura, Y. Anisotropy of Mott-Hubbard Gap Transitions due to Spin and Orbital Ordering in LaVO
_{3}and YVO_{3}. J. Phys. Soc. Jpn.**2002**, 71, 2086–2089. [Google Scholar] [CrossRef] [Green Version] - Khaliullin, G.; Horsch, P.; Oleś, A.M. Theory of Optical Spectral Weights in Mott Insulators with Orbital Degrees of Freedom. Phys. Rev. B
**2004**, 70, 195103. [Google Scholar] [CrossRef] [Green Version] - Kovaleva, N.N.; Oleś, A.M.; Balbashov, A.M.; Maljuk, A.; Argyriou, D.N.; Khaliullin, G.; Keimer, B. Low-energy Mott-Hubbard excitations in LaMnO
_{3}probed by optical ellipsometry. Phys. Rev. B**2010**, 81, 235130. [Google Scholar] [CrossRef] [Green Version] - Laurita, N.J.; Deisenhofer, J.; Pan, L.-D.; Morris, C.M.; Schmidt, M.; Johnsson, M.; Tsurkan, V.; Loidl, A.; Armitage, N.P. Singlet-Triplet Excitations and Long-Range Entanglement in the Spin-Orbital Liquid Candidate FeSc
_{2}S_{4}. Phys. Rev. Lett.**2015**, 114, 207201. [Google Scholar] [CrossRef] [Green Version] - Man, H.Y.; Halim, M.; Sawa, H.; Hagiwara, M.; Wakabayashi, Y.; Nakatsuji, S. Spin-orbital entangled liquid state in the copper oxide Ba
_{3}CuSb_{2}O_{9}. J. Phys. Condens. Matter**2018**, 30, 443002. [Google Scholar] [CrossRef] [Green Version] - Becker, M.; Hermanns, M.; Bauer, B.; Garst, M.; Trebst, S. Spin-orbit physics of j = 1/2 Mott insulators on the triangular lattice. Phys. Rev. B
**2015**, 91, 155133. [Google Scholar] [CrossRef] [Green Version] - Hermanns, M.; Kimchi, I.; Knolle, J. Physics of the Kitaev Model: Fractionalization, Dynamic Correlations, and Material Connections. Ann. Rev. Cond. Mat. Phys.
**2018**, 9, 17–33. [Google Scholar] [CrossRef] [Green Version] - Rusnačko, J.; Gotfryd, D.; Chaloupka, J. Kitaev-like Honeycomb Magnets: Global Phase Behavior and Emerging Effective Models. Phys. Rev. B
**2019**, 99, 064425. [Google Scholar] [CrossRef] [Green Version] - Natori, W.M.H.; Knolle, J. Dynamics of a Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of Fermionic Magnons. Phys. Rev. Lett.
**2020**, 125, 067201. [Google Scholar] [CrossRef] - Takagi, H.; Takayama, T.; Jackeli, G.; Khaliullin, G.; Nagler, S.E. Concept and realization of Kitaev quantum spin liquids. Nat. Rev. Phys.
**2019**, 1, 264–280. [Google Scholar] [CrossRef] - Ulrich, C.; Khaliullin, G.; Sirker, J.; Reehuis, M.; Ohl, M.; Miyasaka, S.; Tokura, Y.; Keimer, B. Magnetic Neutron Scattering Study of YVO
_{3}: Evidence for an Orbital Peierls State. Phys. Rev. Lett.**2003**, 91, 257202. [Google Scholar] [CrossRef] [Green Version] - Sirker, J.; Herzog, A.; Oleś, A.M.; Horsch, P. Thermally Activated Peierls Dimerization in Ferromagnetic Spin Chains. Phys. Rev. Lett.
**2008**, 101, 157204. [Google Scholar] [CrossRef] [Green Version] - Horsch, P.; Oleś, A.M.; Feiner, L.F.; Khaliullin, G. Evolution of Spin-Orbital-Lattice Coupling in the RVO
_{3}Perovskites. Phys. Rev. Lett.**2008**, 100, 167205. [Google Scholar] [CrossRef] [Green Version] - Fujioka, J.; Yasue, T.; Miyasaka, S.; Yamasaki, Y.; Arima, T.; Sagayama, H.; Inami, T.; Ishii, K.; Tokura, Y. Critical competition between two distinct orbital-spin ordered states in perovskite vanadates. Phys. Rev. B
**2010**, 82, 144425. [Google Scholar] [CrossRef] - Yan, J.-Q.; Tian, W.; Cao, H.B.; Chi, S.; Ye, F.; Llobet, A.; Puretzky, A.; Chen, Q.; Ma, J.; Ren, Y.; et al. Lattice distortion in the spin-orbital entangled state in RVO
_{3}perovskites. Phys. Rev. B**2019**, 100, 184423. [Google Scholar] [CrossRef] [Green Version] - Chen, C.C.; van Veenendaal, M.; Devereaux, T.P.; Wohlfeld, K. Fractionalization, entanglement, and separation: Understanding the collective excitations in a spin-orbital chain. Phys. Rev. B
**2015**, 91, 165102. [Google Scholar] [CrossRef] [Green Version] - Ronquillo, D.C.; Vengal, A.; Trivedi, N. Signatures of magnetic-field-driven quantum phase transitions in the entanglement entropy and spin dynamics of the Kitaev honeycomb model. Phys. Rev. B
**2019**, 99, 140413. [Google Scholar] [CrossRef] [Green Version] - Daghofer, M.; Oleś, A.M.; von der Linden, W. Orbital Polarons versus Itinerant e
_{g}Electrons in Doped Manganites. Phys. Rev. B**2004**, 70, 184430. [Google Scholar] [CrossRef] [Green Version] - Avella, A.; Oleś, A.M.; Horsch, P. Defect-Induced Orbital Polarization and Collapse of Orbital Order in Doped Vanadium Perovskites. Phys. Rev. Lett.
**2019**, 122, 127206. [Google Scholar] [CrossRef] [Green Version] - Yaji, K.; Kuroda, K.; Toyohisa, S.; Harasawa, A.; Ishida, Y.; Watanabe, S.; Chen, C.T.; Kobayashi, K.; Komori, F.; Shin, S. Spin-dependent quantum interference in photoemission process from spin-orbit coupled states. Nat. Commun.
**2017**, 8, 14588. [Google Scholar] [CrossRef] [Green Version] - Oleś, A.M. Fingerprints of Spin-Orbital Entanglement in Transition Metal Oxides. J. Phys. Condens. Matter
**2012**, 24, 313201. [Google Scholar] [CrossRef] - Brzezicki, W. Spin, orbital and topological order in models of strongly correlated electrons. J. Phys. Condens Matter
**2020**, 32, 023001. [Google Scholar] [CrossRef] [Green Version] - Witczak-Krempa, W.; Chen, G.; Kim, Y.B.; Balents, L. Correlated Quantum Phenomena in the Strong Spin-Orbit Regime. Ann. Rev. Cond. Mat. Phys.
**2014**, 5, 58–82. [Google Scholar] [CrossRef] [Green Version] - Bianconi, A. Superstripes and complexity in High-Temperature Superconductors. J. Supercond. Nov. Magn.
**2014**, 27, 909–912. [Google Scholar] [CrossRef] - Keimer, B.; Kivelson, S.A.; Norman, M.R.; Uchida, S.; Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature
**2015**, 518, 179–186. [Google Scholar] [CrossRef] - Campi, G.; Bianconi, A. High-Temperature Superconductivity in a Hyperbolic Geometry of Complex Matter from Nanoscale to Mesoscopic Scale. J. Supercond. Nov. Magn.
**2016**, 29, 627–631. [Google Scholar] [CrossRef] - Bussmann-Holder, A.; Kohler, J.; Simon, A.; Whangbo, M.-H.; Bianconi, A.; Perali, A. The Road Map toward Room-Temperature Superconductivity: Manipulating Different Pairing Channels in Systems Composed of Multiple Electronic Components. Condens. Matter
**2017**, 2, 24. [Google Scholar] [CrossRef] - Bianconi, A. Lifshitz Transitions in Multi-band Hubbard Models for Topological Superconductivity in Complex Quantum Matter. J. Supercond. Novel Magn.
**2018**, 31, 603–610. [Google Scholar] [CrossRef] [Green Version] - Khomskii, D.I. Transition Metal Compounds; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Pati, S.K.; Singh, R.R.P.; Khomskii, D.I. Alternating Spin and Orbital Dimerization and Spin-Gap Formation in Coupled Spin-Orbital Systems. Phys. Rev. Lett.
**1998**, 81, 5406–5409. [Google Scholar] [CrossRef] [Green Version] - Itoi, C.; Qin, S.; Affleck, I. Phase diagram of a one-dimensional spin-orbital model. Phys. Rev. B
**2000**, 61, 6747. [Google Scholar] [CrossRef] [Green Version] - Lundgren, R.; Chua, V.; Fiete, G.A. Entanglement entropy and spectra of the one-dimensional Kugel–Khomskii model. Phys. Rev. B
**2012**, 86, 224422. [Google Scholar] [CrossRef] [Green Version] - Kugel, K.I.; Khomskii, D.I.; Sboychakov, A.O.; Streltsov, S.V. Spin-orbital interaction for face-sharing octahedra: Realization of a highly symmetric SU(4) model. Phys. Rev. B
**2015**, 91, 155125. [Google Scholar] [CrossRef] [Green Version] - Valiulin, V.E.; Mikheyenkov, A.V.; Kugel, K.I.; Barabanov, A.F. Thermodynamics of Symmetric Spin-Orbital Model: One- and Two-Dimensional Cases. JETP Lett.
**2019**, 109, 546–551. [Google Scholar] [CrossRef] [Green Version] - Horsch, P.; Khaliullin, G.; Oleś, A.M. Dimerization versus Orbital Moment Ordering in a Mott Insulator YVO
_{3}. Phys. Rev. Lett.**2003**, 91, 257203. [Google Scholar] [CrossRef] [Green Version] - Solovyev, V.I. Spin-orbital superexchange physics emerging from interacting oxygen molecules in KO
_{2}. New J. Phys.**2008**, 10, 013035. [Google Scholar] [CrossRef] - Veenstra, C.N.; Zhu, Z.-H.; Raichle, M.; Ludbrook, B.M.; Nicolaou, A.; Slomski, B.; Landolt, G.; Kittaka, S.; Maeno, Y.; Dil, J.H.; et al. Spin-Orbital Entanglement and the Breakdown of Singlets and Triplets in Sr
_{2}RuO_{4}Revealed by Spin- and Angle-Resolved Photoemission Spectroscopy. Phys. Rev. Lett.**2014**, 112, 127002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bahr, S.; Alfonsov, A.; Jackeli, G.; Khaliullin, G.; Matsumoto, A.; Takayama, T.; Takagi, H.; Büchner, B.; Kataev, V. Low-energy magnetic excitations in the spin-orbital Mott insulator Sr
_{2}IrO_{4}. Phys. Rev. B**2014**, 89, 180401(R). [Google Scholar] [CrossRef] [Green Version] - Zhang, G.; Gorelov, E.; Sarvestani, E.; Pavarini, E. Fermi Surface of Sr
_{2}RuO_{4}: Spin-Orbit and Anisotropic Coulomb Interaction Effects. Phys. Rev. Lett.**2016**, 116, 106402. [Google Scholar] [CrossRef] [Green Version] - Oleś, A.M.; Wohlfeld, K.; Khaliullin, G. Orbital Symmetry and Orbital Excitations in High-T
_{c}Superconductors. Condens. Matter**2019**, 4, 46. [Google Scholar] [CrossRef] [Green Version] - Khaliullin, G. Excitonic Magnetism in Van Vleck-type d
^{4}Mott Insulators. Phys. Rev. Lett.**2013**, 111, 197201. [Google Scholar] [CrossRef] [Green Version] - Souliou, S.-M.; Chaloupka, J.; Khaliullin, G.; Ryu, G.; Jain, A.; Kim, B.J.; Le Tacon, M.; Keimer, B. Raman scattering from Higgs mode oscillations in the two-dimensional antiferromagnet Ca
_{2}RuO_{4}. Phys. Rev. Lett.**2017**, 119, 067201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Svoboda, C.; Randeria, M.; Trivedi, N. Effective magnetic interactions in spin-orbit coupled d
^{4}Mott insulators. Phys. Rev. B**2017**, 95, 014409. [Google Scholar] [CrossRef] [Green Version] - Li, F.-Y.; Chan, G. Spin-orbital entanglement in d
^{8}Mott insulators: Possible excitonic magnetism in diamond lattice antiferromagnets. Phys. Rev. B**2019**, 100, 045103. [Google Scholar] [CrossRef] [Green Version] - Chaloupka, J.; Khaliullin, G. Highly frustrated magnetism in relativistic d
^{4}Mott insulators: Bosonic analog of the Kitaev honeycomb model. Phys. Rev. B**2019**, 100, 224413. [Google Scholar] [CrossRef] [Green Version] - Pärschke, E.M.; Wohlfeld, K.; Foyevtsova, K.; van den Brink, J. Correlation induced electron-hole asymmetry in quasi- two-dimensional iridates. Nat. Commun.
**2017**, 8, 686. [Google Scholar] [CrossRef] [PubMed] - Kim, C.H.; Baidya, S.; Cho, H.; Gapontsev, V.V.; Streltsov, S.V.; Khomskii, D.I.; Park, J.-G.; Go, A.; Jin, H. Theoretical evidence of spin-orbital-entangled Jeff=12 state in the 3d transition metal oxide CuAl
_{2}O_{4}. Phys. Rev. B**2019**, 100, 161104. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematic phase diagram of the spin-orbital model (2) in the $\{\alpha ,\beta \}$ plane inspired by figure 1 in [48] and figure 7 in [1]. The model has four distinct phases: product disentangled phases FM⊗AO, AF⊗FO, FM⊗FO, and the phase around the SU(4) point with finite spin-orbital entanglement on superexchange bonds.

**Figure 2.**Evolution of the entanglement entropy ${\mathcal{S}}_{\mathrm{vN}}$ (4) with increasing $\lambda $ for the Hamiltonian (1) calculated for a ring of $L=4$ sites. Panels present slices of the $\{\alpha ,\beta \}$ parameter space for a fixed value of $\lambda /J$. (

**a**–

**c**) reproduce surprisingly well the result presented in figure 1 of Ref. [1] for $L=12$. (

**d**–

**i**) reveal intermediate stages between the (

**b**,

**c**). The three white dashed lines represent (from left to right) hidden FM Heisenberg, XY, and AF Heisenberg models emerging in the $\lambda \to \infty $ limit at $\alpha +\beta =-1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}2$, $\alpha +\beta =0$, and $\alpha +\beta =1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}2$ (white dashed lines from bottom to top). For longer chains, e.g., $L=12$, the “quantum stripe” (red region near the $\alpha +\beta =0$ line) in (

**c**) fits between hidden FM Heisenberg and AF Heisenberg lines. Red dashed lines in the panels indicate the cross section along which the $\{\alpha ,\lambda \}$ maps are calculated in Figure 3 and Figure 4.

**Figure 3.**Evolution of the von Neumann spin-orbital entanglement entropy and the three spin-orbital correlation functions in the ground state of Hamiltonian (1) with $\alpha +\beta =0$ for $\alpha \in [-1.0,1.0]$, calculated with ED for the ring of $L=4$ sites for logarithmically increasing spin-orbit coupling $\lambda $: (

**a**) the von Neumann spin-orbital entanglement entropy ${\mathcal{S}}_{\mathrm{vN}}$ (4); (

**b**) the intersite spin-orbital correlation function ${\mathcal{C}}_{\mathrm{SO}}$ (6); (

**c**) the on-site spin-orbit correlation function ${\mathcal{O}}_{\mathrm{SO}}$ (7); (

**f**) the spin correlation function $\mathcal{S}$ (8). (

**d**,

**e**) show the von Neumann spin-orbital entanglement entropy ${\mathcal{S}}_{\mathrm{vN}}$ (4) obtained for increasing $\lambda $ at $\left|\alpha \right|=1\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}2$ [cut A in ((

**a**)] and fitted with a logistic function (black thin line), and at $\alpha =0$ [cut B in (

**a**)], respectively.

**Figure 4.**Absence of large spin-orbital entanglement along the red $\alpha +\beta =-1$ line in Figure 2 for the 1D ring of $L=4$ sites. Top panels show the maps for the $(\alpha ,\lambda )$-plane: (

**a**) entanglement entropy ${\mathcal{S}}_{\mathrm{vN}}$ (4); (

**b**) spin-orbital ${\mathcal{C}}_{\mathrm{SO}}$ correlation (6), (

**c**) on-site ${\mathcal{O}}_{\mathrm{SO}}$ correlation (7), and (

**f**) spin-only correlation $\mathcal{S}$ (8). (

**d,e**) present full energy spectrum along cross section C (D), with the ground state energy marked in blue; (

**g**) the energy differences $\Delta E={E}_{i}-{E}_{0}$ between the first four excited states and the ground state around the sharp transition from non-degenerate perturbed AF⊗FO (FM⊗AO) level into 2-fold degenerate Ising FM ground state; (

**h**) presents the splitting of 25-fold degenerate FM⊗FO ground state at infinitesimal $\lambda >0$—it gives the 2-fold degenerate Ising FM ground state. In (

**i**), we contrast the entanglement entropy ${\mathcal{S}}_{\mathrm{vN}}$ (4) along line C and line A in Figure 3.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gotfryd, D.; Pärschke, E.; Wohlfeld, K.; Oleś, A.M.
Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling. *Condens. Matter* **2020**, *5*, 53.
https://doi.org/10.3390/condmat5030053

**AMA Style**

Gotfryd D, Pärschke E, Wohlfeld K, Oleś AM.
Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling. *Condensed Matter*. 2020; 5(3):53.
https://doi.org/10.3390/condmat5030053

**Chicago/Turabian Style**

Gotfryd, Dorota, Ekaterina Pärschke, Krzysztof Wohlfeld, and Andrzej M. Oleś.
2020. "Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling" *Condensed Matter* 5, no. 3: 53.
https://doi.org/10.3390/condmat5030053