Nested-sphere description of the $N$-level Chern number and the generalized Bloch hypersphere

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Nested-sphere description of the $N$ -level Chern number and the generalized Bloch hypersphere

Cameron J. D. Kemp, Nigel R. Cooper, and F. Nur Ünal

Phys. Rev. Research 4, 023120 – Published 13 May 2022

Abstract

The geometric interpretation of (pseudo)spin- $1 / 2$ systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high-energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings, or have involved restrictions to sub manifolds within the full Hilbert space. We here employ a coherence vector description to theoretically characterize a general $N$ -level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying $SU (N)$ algebra and construct physically intuitive geometric pictures for topological concepts. Focusing on two spatial dimensions, we reveal a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising $N - 1$ two-spheres. We demonstrate that for the $N$ -level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up another avenue for the interpretation of the topological classification and the dynamical illustration of multilevel systems, which in turn is anticipated to help in the design of new experimental probes.

Received 14 October 2021
Revised 18 November 2021
Accepted 7 April 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.023120

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Topological phases of matter

Chern insulators Ultracold gases

Tomography

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Cameron J. D. Kemp ¹, Nigel R. Cooper^2,3, and F. Nur Ünal ^2,*

¹Robinson College, University of Cambridge, Grange Road, Cambridge CB3 9AN, United Kingdom
²TCM Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
³Department of Physics and Astronomy, University of Florence, Via G. Sansone 1, 50019 Sesto Fiorentino, Italy

^*fnu20@cam.ac.uk

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Issue

Vol. 4, Iss. 2 — May - July 2022

Subject Areas

Atomic and Molecular Physics

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Images

Figure 1
Illustration of the nested two-spheres structure (a) in three-level systems. The exterior sphere (red) is a unit two-sphere spanned by the angles $(θ, ϕ)$ defined through Eq. (5), which corresponds to mixing between the ground state and the rest of the bands via the last (diagonal) Gell-Mann matrix ( ${\hat{λ}}_{8}$ in this case) upon expressing the wave function in an eigenstate basis as detailed in the text. The poles of the interior unit two-sphere (blue), associated with the mixing in the interior two-level via ${\hat{λ}}_{3}$ , requires $θ = 0$ on the exterior one [Eq. (7)], and hence the name “nested.” (b) Generalization to $N$ levels harboring $N - 1$ diagonal Gell-Mann matrices ( ${\hat{λ}}_{k^{2} - 1}$ for $2 \leq k \leq N$ ). The $(k^{2} - 1)$ -th Gell-Mann matrix consists of $k - 1$ ones in its diagonal and one more nonzero term necessary to render it traceless. Similar to the diagonal Pauli matrix ${\hat{σ}}_{z} \to {\hat{λ}}_{3}$ , these diagonal Gell-Mann matrices can be associated with the $\hat{z}$ axis of a two-sphere. When we start from the south pole of the exterior sphere, $n_{ext}$ vector must be inverted somewhere in the BZ and reach the north pole for a nontrivial Chern number.
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Figure 2
Distributions on the exterior (top panel) and interior (bottom) two-spheres for the ground state of the Harper-Hofstadter model (10) with reference to the eigenstates at $Q = Γ$ , as a function of crystal momentum $q$ with lattice constant set to one. (a) Starting from the exterior south pole ( $θ_{Q} = π$ ), the state vector reaches the exterior north pole ( $θ_{Q_{⊥}} = 0$ ) once within the BZ, a prerequisite for nontrivial $C$ , (c) while simultaneously the interior vector is found to be on its north pole $S_{3} (Q_{⊥}) = + 1$ . The total winding of the respective azimuthal angles $Δ (ϕ + φ) = 2 π$ at this band inversion point gives $C = 1$ .
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Figure 3
Similar to Fig. 2 but for $Q = (π, 0)$ . (a) The vector $n_{ext} (q)$ gets inverted on the exterior sphere again once, visiting both poles. (c) $S_{3} (q)$ can reach one of the interior sphere poles at this point, and here $S_{3} (Q_{⊥}) = - 1$ is at the south pole. [(b), (d)] The exterior sphere azimuthal angle $ϕ$ winds by $π$ around this singularity (black arrow), while the interior winding is $- π$ but around the south pole, adding up to $C = 1$ via Eq. (9).
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Figure 4
Polar and azimuthal angles on the exterior sphere in the Harper-Hofstadter model (10). (a) The ground-state wave function becomes orthogonal three times with respect to the eigenstate at $Q = (π, π / 3)$ ; at $Γ$ and two M points where $θ (q)$ vanishes. (b) Adding the winding of $| Δ ϕ | = 2 π$ around these singularities, one of which has opposite chirality, we find $C = 1$ . Here, also the entire contribution to the Chern number comes from the exterior sphere, where the interior sphere azimuthal angle is found to be constant and $| S_{3} (q) | < 1$ everywhere.
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