Interferometric Braiding of Anyons in Chern Insulators

TL;DR

The paper presents impurity-based interferometry to directly measure the geometric phases associated with anyons in fractional Chern insulators. By binding spin-1/2 impurities to quasiholes and using Ramsey interferometry augmented by spin-echo sequences, it separates Aharonov-Bohm and exchange contributions, enabling direct observation of braiding statistics. Numerical simulations in non-interacting Chern insulators quantify the system size required to faithfully extract these phases and illustrate how edge effects can obscure signals, while also outlining realistic cold-atom and solid-state implementations. The work sets the stage for experimental anyon braiding studies and provides a framework for extending to non-Abelian statistics via Wilson loop matrices, with potential impact on topological quantum control.

Abstract

Coherent control and braiding of anyons remain central challenges in realizing topologically protected quantum operations. We propose a Ramsey interferometry protocol to directly access the geometric phases associated with anyons in fractional Chern insulators. Our approach employs impurities with individually addressable internal states that bind to the anyons, allowing their adiabatic motion and exchange under full spatial control. By combining Ramsey and spin-echo sequences using one and two impurities, the protocol gives independent access to the Aharonov-Bohm and exchange contributions to the total geometric phase, thereby providing an unambiguous probe of anyonic statistics. Our scheme can potentially be implemented in cold-atom quantum simulators as well as in van der Waals heterostructures. Complementary finite-size simulations in non-interacting Chern insulators quantify the system sizes required to faithfully extract geometric phases, highlighting the role of edge effects. Our results establish impurity-based interferometry as a feasible route toward direct anyon braiding experiments in quantum simulators and lay the groundwork for future explorations of non-Abelian braiding and topological quantum control.

Figures (8)

• Figure 1: Interferometric sequences allowing for measurements of (a) the Aharonov-Bohm phase and (b) the exchange phase using impurities and state-dependent paths. Due to the combined Ramsey and spin echo sequence, the dynamical phase at the end of the sequence vanishes, resulting only in the measurement of the geometric contribution. Notably, for this sequence to work it is sufficient to bind $\ket{\uparrow}$ to the quasihole, whereas $\ket{\downarrow}$ does not need to move and can be completely decoupled from the many-body system.
• Figure 2: Aharonov-Bohm phase obtained from Eq. \ref{['Eq:GeometricPhaseDiscretizedPath']} for a single particle encircling a flux $\delta\alpha$ through the central four plaquettes along a circular path of radius $R/a$ for $V_{\rm pin}/J=-1.5$ (full symbols) and $-5.0$ (open symbols), respectively. For $R/a\!=\!1$ the path does not encircle the local flux completely, resulting in substantial deviations from the expected behavior. For radii $R/a \!\gtrsim\! 2$ the entire flux is encircled, so that we expect a scaling $\varphi_{\rm AB} \!\propto\! 2\pi \times 4\delta\alpha \times q^{\star}$ according to Eq. \ref{['eq:ABPhase:singleParticle']}, with $q^{\star}$ the charge of the pinned object. The solid black line indicates this scaling for $q^{\star}\!=\!1$. We find that for a sufficiently strong pinning potential the numerically extracted Aharonov-Bohm matches this prediction as well as the expected $q^{\star}\!=\!1$ (see inset). Here, $q^{\star}$ is obtained using a fit of $\varphi_{\rm AB}^{(\rm pred)}(\delta\alpha)$ to the numerical data.
• Figure 3: (a) Aharonov-Bohm phase for a single particle in a magnetic field of flux $\alpha\!=\!0.2$ per plaquette, encircling an additional flux $\delta\Phi$ through the central four plaquettes along a circular path of radius $R/a$. Already for a relatively weak pinning potential ($V_{\rm pin}/J\!=\!-1,~ \sigma/a\!=\!1$) and $R/a\!\gtrsim\! 3$ the numerically extracted Aharonov-Bohm phase (symbols) matches the prediction (lines). The solid lines indicate the expected scaling for the respective $\delta\Phi$, whereas the gray shaded area indicates a band of width $2\pi$ around the expected value for $\delta\alpha\!=\!0$. (b) Pinned charge $q^{\star}$ as extracted from the Aharonov-Bohm (circles) and full geometric (squares) phases. For $R/a\gtrsim3$ the pinned charge agrees well with the expected $q^{\star}\!=\!1$. Here, $q^{\star}$ is obtained using a fit of $\varphi_{\rm AB/geo}^{(\rm pred)}(\delta\Phi)$ to the numerical data, respectively. Note, that measurements of $\varphi_{\rm AB}$ involve only one particle, whereas measurements of the full geometric phase $\varphi_{\rm geo}$ require to manipulate two particles. (c) Extracted exchange phase for two fermions. For $3 \!\lesssim\! R/a \!\lesssim\! 4.5$, the phase is consistent with the fermionic $\varphi_{\rm exc}\!=\!\pi$, whereas for smaller radii the effect of not fully encircling the additional flux $\delta\Phi$ is visible for $\delta\Phi \!\neq\! 0$. For larger radii, edge effects become substantial and lead to deviations from the expected exchange phase. The insets show the different paths taken by the pinning potentials to determine the Aharonov-Bohm (left) and full geometric (right) phases, respectively.
• Figure 4: Density profiles for a system of $N\!=\!35$ particles in a magnetic field of flux $\alpha\!=\!0.2$ per plaquette subject to no (left), one (middle) and two (right) pinning potentials ($V_{\rm pin}/J \!=\! 1.5,~ \sigma/a\!=\!1.0$). The ground state without any pinning potentials realizes a Chern insulator in the lowest Hofstadter band.
• Figure 5: (a) Aharonov-Bohm phase for a single pinning potential ($V_{\rm pin}/J\!=\!1.5,~ \sigma/a\!=\!1$) moved along a circular path of radius $R/a$ in a system of $N\!=\!35$ particles on $15\times15$ sites with a flux $\alpha\!=\!0.2$ per plaquette, realizing a Chern insulator in the lowest Hofstadter band. The solid lines indicate the expected scaling for the respective $\delta\alpha$, whereas the gray shaded area indicates a band of width $2\pi$ around the expected value for $\delta\Phi\!=\!0$. (b) Pinned charge $q^{\star}$ as extracted from the Aharonov-Bohm (circles) and full geometric (squares) phases. For $3 \!\lesssim\! R/a \!\lesssim\! 4$ the pinned charge agrees well with the expected $q^{\star}\!=\!-1$. Here, $q^{\star}$ is obtained using a fit of $\varphi_{\rm AB/geo}^{(\rm pred)}(\delta\Phi)$ to the numerical data, respectively. (c) Extracted exchange phase for two pinning potentials. For $3 \!\lesssim\! R/a \!\lesssim\! 4$, the phase is consistent with the expected $\varphi_{\rm exc}\!=\!\pi$, whereas for smaller radii the effect of not fully encircling the additional flux $\delta\Phi$ is visible for $\delta\Phi \!\neq\! 0$. For larger radii, edge effects become substantial and lead to deviations from the expected exchange phase.