# American Institute of Mathematical Sciences

September  2020, 12(3): 455-505. doi: 10.3934/jgm.2020021

## Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries

 1 Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606, Japan 2 Department of Physics, Université du Littoral——Côte d'Opale, 59140 Dunkerque, France

* Corresponding author: Boris I. Zhilinskií

Dedicated to James Montaldi

Received  September 2019 Revised  February 2020 Published  September 2020 Early access  July 2020

We investigate the elementary rearrangements of energy bands in slow-fast one-parameter families of systems whose fast subsystem possesses a half-integer spin. Beginning with a simple case without any time-reversal symmetries, we analyze and compare increasingly sophisticated model Hamiltonians with these symmetries. The models are inspired by the time-reversal modification of the Berry phase setup which uses a family of quadratic spin-quadrupole Hamiltonians of Mead [Phys. Rev. Lett. 59, 161–164 (1987)] and Avron et al [Commun. Math. Phys. 124(4), 595–627 (1989)]. An explicit correspondence between the typical quantum energy level patterns in the energy band rearrangements of the finite particle systems with compact slow phase space and those of the Dirac oscillator is found in the limit of linearization near the conical degeneracy point of the semi-quantum eigenvalues.

Citation: Toshihiro Iwai, Dmitrií A. Sadovskií, Boris I. Zhilinskií. Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries. Journal of Geometric Mechanics, 2020, 12 (3) : 455-505. doi: 10.3934/jgm.2020021
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##### References:
Spectrum of the Dirac oscillator with $S = \frac12$ as function of formal control parameter $\mu$. The energies of the bulk states (blue and green) and of the edge state (red) are given by (13)
Correlation diagram of the deformed spin-orbit system with spin $S = \frac12$ and $N\gg S$ as function of formal coupling control parameter $\gamma$. Bold solid lines represent the energies of the bulk states (blue and green) and the edge state (red). When the value of $\gamma$ varies, one single state (edge) changes bands, while all other states (bulk) remain within the same band
Spectrum of the $N,S = \frac12$ system (two coupled angular momenta of conserved lengths) with Hamiltonian $\hat{H}_\gamma$ (22) as function of parameter $\gamma$ (scaled magnetic field strength). Green and red solid lines represent the "bulk" states of the lower and upper band (multiplet); blue solid line marks the energy of the single "edge" state redistributed at $\gamma = \frac12$
, the domain of $\gamma$ is extended from $[0,1]$ to $[-1,1]$; the $\gamma>0$ parts of both diagrams are identical. The value $0$ of $\gamma$ corresponds to the uncoupled system (two bands of equal number of states), while the values $-1$ and $+1$ correspond to the coupled spin-orbit system with negative and positive coupling Hamiltonian (2). Bold solid lines represent the energies of the bulk states (blue and green) and the edge states (red). When the value of $\gamma$ varies, two single states (edge) change bands, while all other states (bulk) remain within the same bands">Figure 4.  Correlation diagram of the deformed spin-orbit system with Hamiltonian (29), spin $S = \frac12$, and $N\gg S$ as function of formal coupling control parameter $\gamma$. In comparison to the diagram in fig. 4, the domain of $\gamma$ is extended from $[0,1]$ to $[-1,1]$; the $\gamma>0$ parts of both diagrams are identical. The value $0$ of $\gamma$ corresponds to the uncoupled system (two bands of equal number of states), while the values $-1$ and $+1$ correspond to the coupled spin-orbit system with negative and positive coupling Hamiltonian (2). Bold solid lines represent the energies of the bulk states (blue and green) and the edge states (red). When the value of $\gamma$ varies, two single states (edge) change bands, while all other states (bulk) remain within the same bands
, the $[-\frac12,\frac12]$ part of the domain of $\gamma$ is shrunk to $0$, while the endpoints of the two diagrams are identical. There is no uncoupled system, the values $-1$ and $+1$ of $\alpha$ correspond to the coupled spin-orbit system with isotropic negative and positive coupling Hamiltonian (2). Bold solid lines represent the energies of the bulk states (blue and green) and one Kramers doublet edge state (red)">Figure 5.  Correlation diagram of the spin-orbit system with the conjectured ${\mathcal{T}}$-equivariant deformation, spin $S = \frac12$, and $N\gg S$ as function of formal coupling control parameter $\alpha$. In comparison to the diagram in fig. 2, the $[-\frac12,\frac12]$ part of the domain of $\gamma$ is shrunk to $0$, while the endpoints of the two diagrams are identical. There is no uncoupled system, the values $-1$ and $+1$ of $\alpha$ correspond to the coupled spin-orbit system with isotropic negative and positive coupling Hamiltonian (2). Bold solid lines represent the energies of the bulk states (blue and green) and one Kramers doublet edge state (red)
Spectrum of the $N,S = \frac12$ system (two coupled angular momenta of conserved lengths) with ${\mathcal{T}}$-invariant Hamiltonian (31) as function of scaled spin-orbit coupling parameter $\alpha$. Each solid line represents a Kramers doublet quantum state. Energies of the bulk states (red and green solid lines) are given by (34), while the solid blue line represents the edge state. Dotted lines indicate critical values of the semi-quantum energies (33)
and 6. The bulk state energies (red and green lines) are given by (39)">Figure 7.  Spectrum of the system with Hamiltonian (36b) obtained as linearization of ${\mathcal{T}}$-reversal ${\boldsymbol{{N}}}{\boldsymbol{{S}}}$, $S = \frac12$ Hamiltonian (32) for coupling parameter $\alpha\in[-1,1]$. Compare to fig. 1 and 6. The bulk state energies (red and green lines) are given by (39)
and 4 which can be computed in the standard way for the $\Lambda_{1,2}$ bundles on ${\mathbb{S}}^2_N$ (see sec. 1.2 and 2.1), the values of $c_1$ in this diagram are conjectured so that they agree with the number of states in each band. See text for more detail">Figure 8.  Correlation diagram for the quadratic dynamical spin-quadrupole system in sec. 2.3.2 with the conjectured ${\mathcal{T}}_S\times {\mathcal{T}}_N$-equivariant deformation in sec. 5, spin $S = \frac32$, and $N\gg S$ as function of formal control parameter $\alpha$ of the isotropic quadrupolar coupling term (18). The values $\pm1$ of $\alpha$ correspond to the coupled system with Hamiltonian (18) times $\pm1$. Bold solid lines represent the energies of the bulk states (blue and green) and of the two Kramers doublet edge states (red) exchanged in opposite directions. Unlike the indices $c_1$ in fig. 2 and 4 which can be computed in the standard way for the $\Lambda_{1,2}$ bundles on ${\mathbb{S}}^2_N$ (see sec. 1.2 and 2.1), the values of $c_1$ in this diagram are conjectured so that they agree with the number of states in each band. See text for more detail
Spectrum of the Kramers degenerate ${\mathcal{T}}_N\times {\mathcal{T}}_S$-invariant axially symmetric system of coupled angular momenta ${\boldsymbol{{S}}}$ (fast) and ${\boldsymbol{{N}}}$ (slow) of conserved lengths $S = \frac32$ and $N = 5$ with Hamiltonian (40) as function of the "quaternionic" spin-orbit coupling parameter $\alpha_0$. Solid color lines represent quantum Kramers doublet states. Specifically, the bulk state energies are depicted in blue and green, while red and purple correspond to the two edge state doublets. For the end values $\pm1$ of $\alpha_0$, the levels within each band reassemble visibly into multiplets with conserved length $J$ of the total angular momentum ${\boldsymbol{{N}}}+{\boldsymbol{{S}}}$ and the respective values of $J$ are marked along the left and right vertical axes, cf. sec. 2.3.3. The edge states are distinguished by the conserved value of $J_1$ displayed near the $\alpha_0 = -1$ end of the plot. The boundaries of the lightly shaded semi-quantum energy domains (gray lines) are given by (42) where $N_1$ takes one of the critical values $\{N,0\}$ and with $N$ replaced by $N_{\rm classical} = N+\frac12$
. Bold solid bars represent the bulk states (blue and green) and two Kramers doublet edge states (red). Above the energy-momentum diagram, we show the composition of the two multiplets in each superband of the spectrum of the isotropic term in sec. 2.3.3. Red arrows indicate which of the subbands takes in the particular edge states, see proposition 1">Figure 10.  Joint spectrum of the ${{\varepsilon}}$-term of the quadratic spin-orbit Hamiltonian (40) and $\hat{J}_1$. The values of spin $S = \frac32$ and slow angular momentum $N = 5$ correspond to fig. 9. Bold solid bars represent the bulk states (blue and green) and two Kramers doublet edge states (red). Above the energy-momentum diagram, we show the composition of the two multiplets in each superband of the spectrum of the isotropic term in sec. 2.3.3. Red arrows indicate which of the subbands takes in the particular edge states, see proposition 1
Circles $C_{\rho}$ and $C_r$ of respective radii $\rho$ and $r$ form the boundary of the annulus $W_{\rho,r}$, and $C_{\rho}$ is also the boundary of disk $D_{\rho}$
Base spaces ${\mathbb{R}}^2_\mu$ and ${\mathbb{S}}^2_r$ of the $\Lambda_\pm$ and $\Delta_\pm$ eigenvector bundles, respectively, in the parameter space ${\mathbb{R}}^3$ of the Dirac oscillator (sec. 2.2). The spaces intersect on the ${\mathbb{S}}^1$ circle $C_{\!\rho}$ involved in the Chern index calculus, see sec. A.4 for details, and compare to fig 11
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