Topological charge distributions of an interacting two-spin system

Quantum systems are often described by parameter-dependent Hamiltonians. Points in parameter space where two levels are degenerate can carry a topological charge. Here we theoretically study an interacting two-spin system where the degeneracy points form a nodal loop or a nodal surface in the magnetic parameter space, similarly to such structures discovered in the band structure of topological semimetals. Key results of our work are that (1) we determine the topological charge distribution along these degeneracy geometries and (2) we show that these nonpointlike degeneracy patterns can be obtained not only by fine-tuning but they can be stabilized by spatial symmetries. Since simple spin systems such as the one studied here are ubiquitous in condensed-matter setups, we expect that our findings, and the physical consequences of these nontrivial degeneracy geometries, are testable in experiments with quantum dots, molecular magnets, and adatoms on metallic surfaces.

Figure 1

Interacting two-spin system and its nonpointlike magnetic degeneracy patterns. (a) Two localized spinful electrons in a double quantum dot, in the presence of spin-orbit coupling, interacting with each other via exchange interaction ($J, \widehat{\mathit{R}}$), and an external magnetic (Zeeman) field $\mathit{B}$ via $g$ tensors ${\widehat{\mathit{g}}}_{\text{L}}$ and ${\widehat{\mathit{g}}}_{\text{R}}$. Figure shows one potential realization [7] defined in a nanowire (brown) by electrostatic gates (yellow). (b) Nonpointlike magnetic degeneracy patterns of the two-spin system. (II) Two degeneracy points with unit ($+1$) topological charge each (red) and a neutral degeneracy ellipse. (IV) Degeneracy ellipsoid surface carrying a total topological charge of $+2$. (V) Degeneracy ellipse carrying a total topological charge of $+2$. Classes (II), (IV), (V) were defined in Table I of Ref. [11].

Figure 2

Inferring the linear charge density of a charged loop from the electric field it creates. (a) Charged loop (blue) and a torus (colored) enclosing the loop, with meridian radius $r$. (b) The torus is parametrized by the map ${\mathit{p}}_r:[0,2\pi R)\times [0,2\pi )\to {\mathbb{R}}^3$, using the longitude path length $s$ and the meridian angle $\theta$. The electric flux density on a torus reveals the linear charge density of the loop as meridian radius of the torus shrinks to zero, $r\to 0$, see Eq. (7).

Figure 3

Neutral degeneracy ellipse has vanishing linear topological charge distribution. (a) Berry flux density ${\mathcal{B}}_n$ (temperature map) on the surface of a torus surrounding a neutral degeneracy circle (black). Meridian radius, $r/R=0.2$ (b) Two-dimensional Berry curvature ${\mathcal{B}}_{\text{2D}}(s,\vartheta )$ [Eq. (12)] on the preimage of the torus, i.e., as a function of longitude path length $s$ and meridian angle $\vartheta$. (c) Apparent linear topological charge density ${\tilde{\nu }}_r\left(s\right)$ [Eq. (13)] as function of longitude path length $s$ and meridian radius $r$. As $r\to 0$, this function converges to the constant zero function (white), showing that the linear topological charge density vanishes, $\nu \left(s\right)=0$. (d) A benchmark for the numerical integration: Numerically evaluated ground-state Chern number $\mathcal{Q}$ on the torus, as function of the meridian radius $r$. Numerical error grows slightly as $r\to 0$, due to the divergence of the Berry curvature in the vicinity of the degeneracy circle, but it remains well below ${10}^{-5}$ even for the smallest $r$ values considered.

Figure 4

Charged degeneracy ellipse has two charged points. (a) Berry flux density ${\mathcal{B}}_n$ (temperature map) on the surface of a torus surrounding a charged degeneracy circle (black). Meridian radius, $r/R=0.2$ (b) Two-dimensional Berry curvature ${\mathcal{B}}_{\text{2D}}(s,\vartheta )$ [Eq. (12)] on the preimage of the torus, i.e., as a function of longitude path length $s$ and meridian angle $\vartheta$. (c) Apparent linear topological charge density ${\tilde{\nu }}_r\left(s\right)$ [Eq. (13)], as function of longitude path length $s$ and meridian radius $r$. As $r\to 0$, this function converges to the constant zero function (white), showing that the linear topological charge density vanishes, $\nu \left(s\right)=0$. (d) A benchmark for the numerical integration: Numerically evaluated ground-state Chern number on the torus, as function of the meridian radius $r$. Numerical error grows slightly as $r\to 0$, but it remains well below ${10}^{-3}$ even for the smallest $r$ values considered.

Figure 5

Topological surface charge distribution on a degeneracy ellipsoid. See Sec. 4 for parameter values. This topological charge distribution is the same as the electric charge distribution on the surface of a charged conducting ellipsoid.

Figure 6

Electric flux created by a charged wire (red) through a disk (blue). Appendix pp1-s1 proves that the flux converges to zero as the disk radius $r$ approaches zero.