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

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## Abstract

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## 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

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**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.

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**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