A hypersphere-like non-Abelian Yang monopole and its topological characterization

Abstract

Synthetic monopoles, which correspond to degeneracies of Hamiltonians, play a central role in understanding exotic topological phenomena. Dissipation-induced non-Herminicity (NH), extending the eigenspectra of Hamiltonians from the real to complex domain, largely enriches the topological physics associated with synthetic monopoles. We here investigate exceptional points (EPs) in a four-dimensional NH system, finding a hypersphere-like non-Abelian Yang monopole in a five-dimensional parameter space, formed by EP2 pairs. Such an exotic structure enables the NH Yang monopole to exhibit a unique topological transition, which is inaccessible with the point-like counterpart. We characterize such a topological phenomenon with the second Chern number.

Figures (4)

• Figure 1: Conceptual schematic and energy spectra. (a) Schematic of the EHS in 5D parameter manifold, depicted as a hypersphere (red ring). Projections onto any 3D subspace manifested as spherical shells (red surfaces). Real (b) and imaginary (c) parts of the spectra with respect to $q_{j,k}\in{\{q_1,q_2,q_3,q_5\}}$.
• Figure 2: The second Chern number ($C_2$) versus the radius R of 5D parameter manifold. $C_2$ reaches 1 when the parameter manifold (projected onto blue sphere) encloses the EHS (projected onto red ring), and changes to 0 when unenclosed, with a sharp transition occurring at the critical boundary of $R=1$.
• Figure 3: The characterization of the Wilson loop and of its relations. (a) Parameter manifold (projected onto yellow sphere) enclosing the EHS (projected onto orange sphere). The traversing path $\mathcal{L}$ to acquire (b) the Wilson loop $W_\mathcal{L}$ at distinct phase angle $\theta_2$. (c) Trajectory of loop $\mathcal{L}$ when the parameter manifold is within the EHS and (d) its corresponding $W_\mathcal{L}$ versus $\theta_2$. Expectation values of the three Pauli operators $\langle \sigma_j\rangle$ (left) and of their square summation $\langle\sigma^2\rangle$ (right) under the system evolution with respect to angle $\phi_1$ (e) for the traversing path $\mathcal{L}$ with $\theta_2=\pi/4$. (f) Minimal values of $W_\mathcal{L}$ versus the radius $R$ of the parameter manifold. (g) Real and (h) imaginary parts of the spectra on the Riemann surface for path $\mathcal{L}$ encircling the ring which is the projection of the EHS in $\{q_{1},q_2,q_4\}$ subspace. (i) Both real and imaginary parts of the spectra, where the EP is the projection of the EHS in $q_{1,2}$ axis, and (j) the corresponding $W_\mathcal{L}$ as function of $\Delta/R$, respectively, when path $\mathcal{L}$ encircling ($\Delta/R<2$) and non-encircling ($\Delta/R>2$) the EHS. The shadow region marks the critical boundary at $\Delta/R = 2$ where the transition happens.
• Figure 4: Schematic of the experimental protocol. Schematic of the driving strategy with circular coupling: $\{|fg0\rangle\xleftrightarrow{\Lambda_1} |1_+\rangle\xleftrightarrow{\Lambda_2}|gf0\rangle\xleftrightarrow{\Lambda_3}|1_-\rangle\xleftrightarrow{\Lambda_4}|fg0\rangle\}$, where $\Lambda_j$ ($j=1,2,3,4$) are the effective coupling strengths between states, $\Xi$ describes the detuning, and $\kappa$ is the single-photon loss rate of the resonator mode.