# Rearrangement of Energy Levels between Energy Super-Bands Characterized by Second Chern Class

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Construction

#### 2.1. Quantum Model Hamiltonian

#### 2.2. Generating Functions for Numbers of States

## 3. Local Spin-Oscillator Approximation and Large-Spin Systems

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Correlation diagram connecting two uncoupled limits (left and right ends) and coupled limit (middle) for the spectrum of quantum Hamiltonian (1). Eigenstates that remain in the same energy band are called “bulk”, and represented by blue or green color depending on the principal contribution to the eigenfunction being by the $|{S}_{1}=1/2,{N}_{1}\rangle $ or $|{S}_{1}=-1/2,{N}_{1}\rangle $, respectively. Eigenstates that change the energy band when control parameter is varied are called “edge”, and are shown in red. Calculation of the first Chern number for energy bands of model Hamiltonian (1) is given in [15].

**Figure 2.**Schematic representation of $(P,\alpha )$ space near one degeneracy point of eigenvalues of semi-quantum Hamiltonian corresponding to isolated value of control parameter $\alpha ={\alpha}_{\mathrm{deg}}=0$. Yellow sphere surrounds isolated degeneracy point and allows to calculate the topological invariant for ${\Delta}_{k}\left({\alpha}_{\mathrm{deg}}\right)$ bundle associated with degeneracy point $({P}_{\mathrm{deg}},{\alpha}_{\mathrm{deg}}=0)$ (red point in the center) and related to the number of redistributed energy levels for corresponding quantum problem. $(\mathbf{p},\mathbf{q})$ subspaces for fixed $\alpha \ne 0$ are the base spaces for ${\Lambda}_{k}\left(\alpha \right)$ bundles.

**Figure 3.**Schematic representation of band structure inversion for Hamiltonian (1) with $S=1/2$ realized in two steps with intermediate formation of coupled basis with $J=N\pm 1/2$.

**Figure 4.**Schematic representation of band structure inversion for Hamiltonian (1) with S = 3/2 realized in two steps with intermediate formation of coupled basis with $J=N\pm 3/2,N\pm 1/2$.

**Figure 5.**Schematic correlation diagram for the crossover of the superband energy level structure of the spin-$\frac{3}{2}$ system with Hamiltonian (10) as function of a single control parameter. The bulk states (blue color) belong to the same superband for all values of the control parameter; the edge states (two shades of red) change the superbands under the control parameter variation. The value of ${j}_{1}$ is displayed for each edge state.

**Figure 6.**Possible geometric representation of the number-of-state functions for the trivial (uncoupled) superband over ${S}^{2}\times {S}^{2}$ slow space. Edge states are situated near the four vertices of the trapezoid. See Section 3 for the detailed local representation in the regions surrounded by the red dash line.

**Figure 7.**Spectrum of Hamiltonian (10) computed numerically as function of control parameter $\alpha $ for $X=Y=5$, $S=3/2$, ${d}_{1q}=1/5,{d}_{2q}=2/5$. The choice of ${c}_{1q},{c}_{2q}$ coefficients influences internal structure of superbands and is not important for analysis of the redistribution. Four edge states are shown by red color. 480 bulk states are marked by three different colors depending on ${j}_{1}$ value. Eigenvalues of the semi-quantum version of (10) are represented by shaded areas, and empty circles mark semi-quantum energies for critical orbits (18a) and (18b). The values of the control parameter $\alpha $, corresponding to the formation of conical degeneracy points of the semi-quantum eigenvalues, and to the redistribution of quantum energy levels between superbands, are precised in the text following. Equation (18).

**Figure 8.**Schematic representation of the redistribution of surplus states between the superbands during the crossover of the band structure for the local model with Hamiltonian (19) and $S=7/2$ in case of resonance $1:1:2$. For each superband, surplus states are shown by red filled circles and arranged by the value of ${j}_{1}$. The whole set of surplus and bulk states for the $S=7/2$ system is represented in Figure 9, left.

**Figure 9.**Representation of basis sets for quaternionic high-spin local models. Surplus states for $1:1:2$ Dirac oscillator (

**left subfigure**) are shown by fill red circles. For $1:1:(-2)$ model (

**right subfigure**) the holes are shown by empty red circles. The representation for $|{S}_{1}|$-superband is valid for any $S\ge |{S}_{1}|$. The vertical axis for each superband can be labeled as $({I}_{Y}+1/2)\phantom{\rule{4pt}{0ex}}\mathrm{sign}(-{S}_{1})$.

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Sadovskii, D.; Zhilinskii, B.
Rearrangement of Energy Levels between Energy Super-Bands Characterized by Second Chern Class. *Symmetry* **2022**, *14*, 183.
https://doi.org/10.3390/sym14020183

**AMA Style**

Sadovskii D, Zhilinskii B.
Rearrangement of Energy Levels between Energy Super-Bands Characterized by Second Chern Class. *Symmetry*. 2022; 14(2):183.
https://doi.org/10.3390/sym14020183

**Chicago/Turabian Style**

Sadovskii, Dmitrii, and Boris Zhilinskii.
2022. "Rearrangement of Energy Levels between Energy Super-Bands Characterized by Second Chern Class" *Symmetry* 14, no. 2: 183.
https://doi.org/10.3390/sym14020183