Second Chern Number and Non-Abelian Berry Phase in Topological Superconducting Systems

Topology ultimately unveils the roots of the perfect quantization observed in complex systems. The two-dimensional quantum Hall effect is the celebrated archetype. Remarkably, topology can manifest itself even in higher-dimensional spaces in which control parameters play the role of extra, synthetic dimensions. However, so far, a very limited number of implementations of higher-dimensional topological systems have been proposed, a notable example being the so-called four-dimensional quantum Hall effect. Here we show that mesoscopic superconducting systems can implement higher-dimensional topology and represent a formidable platform to study a quantum system with a purely nontrivial second Chern number. We demonstrate that the integrated absorption intensity in designed microwave spectroscopy is quantized and the integer is directly related to the second Chern number. Finally, we show that these systems also admit a non-Abelian Berry phase. Hence, they also realize an enlightening paradigm of topological non-Abelian systems in higher dimensions.

Popular Summary

Many physicists have started to apply concepts derived from topology—a branch of mathematics dealing with the shapes of objects and their arrangement in space—to the study of physical problems. Topology nowadays plays a central role in many research areas ranging from quantum to classical physics. Interestingly, topology is not restricted to low-dimensional systems, but it can also emerge in higher-dimensional spaces in which control parameters play the role of extra synthetic dimensions. Unfortunately, current theories in the area of condensed matter rarely enable for implementations of higher-dimensional topological systems that often require hardly tunable exotic materials and by design are restricted to three spatial dimensions. Therefore, only a small number of such implementations have been experimentally realized, thus making the progress in this field rather slow. Hence, there is an urgent need for engineered and scalable realizations of higher-dimensional topological quantum systems.

In this work, we theoretically demonstrate that engineered superconducting systems can implement higher-dimensional topological systems and therefore represent a formidable platform to explore the intriguing physics of quantum systems with nontrivial four-dimensional topology. We show that the nontrivial topology is experimentally accessible in the microwave response of the system and that a non-Abelian Berry phase can be generated and measured through absorption in this topologically exotic state of matter.

Our proposal sets the stage for the future experimental explorations of higher-dimensional topological phases in superconducting quantum systems that are in principle scalable. Such quantum systems constitute possible building blocks for holonomic quantum hardware with potentially superior stability.

Figure 1

(a) Double-quantum-dot system with mediated interactions $F(\mathit{\lambda })$, $K(\mathit{\lambda })$, and $Z(\mathit{\lambda })$ via four parameters $\mathit{\lambda }=({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4)$. (b) For ${ϵ}_L\ne {ϵ}_R$, there are two pairs of states in the double-dot system. At the symmetric point for ${ϵ}_L={ϵ}_R$, there is only one pair of twofold degenerate states. (c) Sketch of an energy crossing of the two twofold degenerate bands appearing at the crossing point $B_{z,c}$ for $\mathit{\lambda }={\mathit{\lambda }}_c$, which leads to a change of the second Chern number.

Figure 2

(a) Two dots (red) coupled to two SC leads with $w_0\ll v$. (b) The effective hopping between the dots is mediated by two crossed Andreev reflection processes in which a Cooper pair is combined (left lead) and one Cooper pair is split (right lead). (c) This mechanism leads to an effective tunneling $K\propto v^2{w}_0{e}^{i({\varphi }_1-{\varphi }_2)}$ depending on the two phases of the leads. (d) One dot (red) coupled to two SC leads that are themselves connected via spin-orbit coupling $\mathbf{w}$. (e) The spin-flip of an electron on the dot is mediated by two crossed Andreev reflection processes in which a Cooper pair is combined (right lead) and one Cooper pair is split (left lead). For this mechanism, a Rashba-like spin-orbit interaction between the superconductors is needed. (f) The process results in an effective phase-dependent spin-flip term $F\propto v^2(w_x+i{w}_y)\sin ({\varphi }_1-{\varphi }_2)$ on the dot. (g) One dot (red) coupled to two SC leads that are themselves coupled via spin-orbit interaction $\mathbf{w}$. (h) The mechanism is mediated by two crossed Andreev reflection processes in which a Cooper pair is combined (right lead) and one Cooper pair is split (left lead). (i) The process results in an effective Zeeman term $Z\propto v^2{w}_z\sin ({\varphi }_1-{\varphi }_2)$ on the dot.

Figure 3

(a) Full system of Example A with an environmental network of nine SC leads coupled via normal hoppings ($w_0,v_0,v$) or via spin-orbit couplings (${\mathbf{w}}_j$). Each lead has a SC phase ${\varphi }_j$ and the dots have a magnetic field $B_z$ in opposing directions. Note that the phases of each lead can be controlled by connecting neighboring leads via SC loops. Since the fluxes through these loops can be independently controlled, the phases are effectively independent of its neighbors and can be tuned to the proposed setup [30, 31, 32, 33, 34, 35, 48, 49]. (b) Full system of Example B with an environmental network of four SC leads with normal hopping ($v,w_0$) and spin-dependent hopping ($\sigma w$), as well as four SC leads with a local exchange field $h$ with magnetization direction ${\theta }_j$ (with respect to the direction of a magnetic field $B_z$ on the dots) coupled via normal hopping ($\tilde{v},{\tilde{w}}_0$). Each lead has a SC phase ${\varphi }_j$ and the dots have a magnetic field $B_z$ in opposing directions.

Figure 4

(a) Second Chern number numerically evaluated with Monte Carlo integration for $N={10}^9$ points in $\mathit{\lambda }$ space. In Example A, we set $v_0={ϵ}_T$, ${\mathbf{w}}_1/w_0=(-0.5,0,0.5)$, ${\mathbf{w}}_2/w_0=(0,0.5,1)$, and ${\mathbf{w}}_3/w_0=(0.5,0,0.5)$. In Example B, we set ${ϵ}_T=4{\widetilde{ϵ}}_T/3={ϵ}_h$ and $h=\mathrm{\Delta }/2$. (b) Energy bands with a crossing point for Example B at $B_z\approx 3.14{ϵ}_T$, which changes the second Chern number from $-1$ to $0$. In both cases ${ϵ}_L={ϵ}_R=0$.

Figure 5

(a) Protocol for qubit manipulations for any initial state [here $|E_{-}^{(1)}⟩$] via the non-Abelian Berry phase in the case of Example A. The rotation from $|E_{-}^{(1)}⟩$ to $|E_{-}^{(2)}⟩$ (black) for the adiabatic change ${\varphi }_1,{\varphi }_2:0\to 2\pi$ is depicted on a torus. A second rotation around the $z$ axis of the Bloch sphere (gray) can be achieved via the adiabatic change ${\varphi }_1:0\to 2\pi$ and ${\varphi }_2=0$. ${\varphi }_3={\varphi }_4=0$ during the whole process. (b) Bloch sphere of the degenerate subspace $E_{-}$ with the possible rotations in (a) (top-left corner). Numerically determined angles of the Berry rotations $\theta$ (by changing adiabatically ${\varphi }_1,{\varphi }_2:0\to 2\pi$) and $\phi$ (by changing adiabatically ${\varphi }_1:0\to 2\pi$) as a function of the magnetic field $B_z$ in the case of Example A for $v_0={ϵ}_T$, ${\mathbf{w}}_1/w_0=(-0.5,0,0.5)$, ${\mathbf{w}}_2/w_0=(0,0.5,1)$, ${\mathbf{w}}_3/w_0=(0.5,0,0.5)$, and ${ϵ}_R={ϵ}_L=0$.

Figure 6

(a),(b) Microwave spectroscopy protocol for the diagonal elements of the non-Abelian Berry curvature in the degenerate subspace. A small time-dependent modulation is applied to two parameters ${\lambda }_j(t)=2A\cos (\omega t)/\hslash \omega$ and ${\lambda }_k(t)=2A\cos (\omega t\pm \pi /2)/\hslash \omega$, $j\ne k$, with $A/\hslash \omega \ll 1$, leading to a transition from the initial state $|I_j⟩$ to an unknown final state $|F⟩$. From the averaged response over the excited states, the diagonal part of the Berry curvature can be measured. Similarly, the diagonal Berry curvature with respect to the rotated states $|I_3⟩$ (c) and $|I_4⟩$ (d) can be measured. From these, the off-diagonal part of the non-Abelian Berry curvature with respect to the initial basis can be constructed by means of Eqs. (11a).

Figure 7

(a) Second Chern number for Example A for ${\mathbf{w}}_1/w_0=(-0.5,0,0.5)$, ${\mathbf{w}}_2/w_0=(0,0.5,1)$, and ${\mathbf{w}}_3/w_0=(0.5,0,0.5)$ as a function of the direct coupling between the dots $v_0$ for different magnetic fields $B_z$. (b) Second Chern number for Example A as a function of the magnetic field $B_z$ for different couplings $v_0$. All results are evaluated numerically with Monte Carlo integration for a maximum of $N={10}^8$ integration points.

Figure 8

(a) Second Chern number for Example A for ${\mathbf{w}}_1/w_0\propto (1,0,0)$, ${\mathbf{w}}_2/w_0\propto (0,1,0)$, and ${\mathbf{w}}_3/w_0\propto (0,0,1)$ as a function of the magnetic field $B_z$ for different absolute magnitudes of the SOC with $v_0={ϵ}_T$. (b) Second Chern number for Example A as a function of the magnetic field $B_z$ with $v_0=0.1{ϵ}_T$, ${\mathbf{w}}_1/w_0=(0.5,0.5g,0.5g)$, ${\mathbf{w}}_2/w_0=(0.5g,0.5,0.5g)$, and ${\mathbf{w}}_3/w_0=(0.5g,0.5g,0.5)$, such that, for $g=0$, the SOCs are pairwise orthogonal and, for $g=1$, they are equal. All results are evaluated numerically with Monte Carlo integration for a maximum of $N={10}^8$ integration points.

Figure 9

Second Chern number for Example B for $h/\mathrm{\Delta }=0.5$ as functions of (a) ${\widetilde{ϵ}}_T/{ϵ}_T$ with ${ϵ}_h={ϵ}_T$ and (b) ${ϵ}_h/{ϵ}_T$ with ${ϵ}_T=4{\widetilde{ϵ}}_T/3$ for different magnetic fields $B_z$. All results are evaluated numerically with Monte Carlo integration for a maximum of $N={10}^8$ integration points.

Figure 10

Second Chern number in Example B for ${ϵ}_T={ϵ}_h=4{\widetilde{ϵ}}_T/3$ as a function of the ratio $h/\mathrm{\Delta }$ for different magnetic fields $B_z$. All results are evaluated numerically with Monte Carlo integration for a maximum of $N={10}^8$ integration points.

Figure 11

Bloch sphere of the degenerate subspace $E_{-}$ with the possible rotations around the polar angle $\theta$ and the azimuthal angle $\phi$ for Example B (top-right corner). Numerically determined angles of the Berry rotations $\phi$ (by changing adiabatically ${\varphi }_2:0\to 2\pi$) and $\theta$ (by changing adiabatically ${\theta }_1,{\theta }_2:0\to 2\pi$) as a function of the magnetic field $B_z$ for ${ϵ}_T={ϵ}_h=4{\widetilde{ϵ}}_T/3$ and $h/\mathrm{\Delta }=0.5$.