Parafermion braiding in fractional quantum Hall edge states with a finite chemical potential

Parafermions are non-Abelian anyons which generalize Majorana fermions and hold great promise for topological quantum computation. We study the braiding of ${\mathbb{Z}}_{2n}$ parafermions which have been predicted to emerge as localized zero modes in fractional quantum Hall systems at filling factor $\nu =1/n$ ($n$ odd). Using a combination of bosonization and refermionization, we calculate the energy splitting as a function of distance and chemical potential for a pair of parafermions separated by a gapped region. Braiding of parafermions in quantum Hall edge states can be implemented by repeated fusion and nucleation of parafermion pairs. We simulate the conventional braiding protocol of parafermions numerically, taking into account the finite separation and finite chemical potential. We show that a nonzero chemical potential poses challenges for the adiabaticity of the braiding process because it leads to accidental crossings in the spectrum. To remedy this, we propose an improved braiding protocol which avoids those degeneracies.

Figure 1

Braiding protocol in 1D that exchanges parafermionic zero modes at times $0<{\tau }_1<{\tau }_2<T$. The black lines serve as graphical aid and represent which parafermions are approaching each other. They do not represent an experimental setup. The empty circles represent zero-energy modes and the full circles represent finite energy modes. Initially ($\tau =0$), ${\chi }_{2,3}$ are nucleated at a finite energy, whereas ${\chi }_{1,4}$ are zero-energy states. At $\tau ={\tau }_1$, ${\chi }_{1,2}$ are then fused together such that the zero-energy state of ${\chi }_1$ gets transferred unto ${\chi }_3$. At $\tau ={\tau }_2$, we fuse ${\chi }_{2,4}$ such that the state of ${\chi }_4$ gets transferred unto ${\chi }_1$. Finally, at $\tau =T$, we fuse ${\chi }_{2,3}$ back into the vacuum and return to the initial configuration, albeit with the zero-energy states of ${\chi }_{1,4}$ exchanged.

Figure 2

Two FQH edge states with opposite spins, gapped out by either superconducting pairing or backscattering, can host ${\mathbb{Z}}_{2n}$ parafermions at the interfaces. If backscattering ($g$) dominates, the fields ${\varphi }_{1,2}$ are pinned inside the backscattering region, whereas for dominant superconducting pairing ($\mathrm{\Delta }$), the field ${\theta }_1$ is pinned.

Figure 3

Two FQH edges with opposite spin polarizations, gapped by either a SC or backscattering region, hosting ${\mathbb{Z}}_{2n}$ parafermion at the interfaces between the SC and the backscattering region. Such opposite spin polarization can be realised using 2DEG with opposite $g$ factors [27]. For strong SC and backscattering interaction, the fields ${\theta }_{1,2}$ and ${\varphi }_1$ are pinned in their respective regions.

Figure 4

Two loops ${\mathrm{\Gamma }}_I$ and ${\mathrm{\Gamma }}_{\mathrm{II}}$ in the configuration space spanned by the coupling constants $|t_{23}|$, $|t_{12}|$ and $|t_{24}|$. The loop ${\mathrm{\Gamma }}_I$ passes through the points $|t_{23}|=1$, $|t_{12}|=1$, $|t_{24}|=1$, and $|t_{23}|=1$ at the times $\tau =0<{\tau }_1<{\tau }_2<T,$ respectively. In the loop ${\mathrm{\Gamma }}_{\mathrm{II}}$, we add the intermediate stages where $|t_{23}|=|t_{12}|=1$, $|t_{12}|=|t_{12}|=1$, and $|t_{24}|=|t_{23}|=1$ at the times ${\tau }_{1/2}<{\tau }_{3/2}<{\tau }_{5/2}$.

Figure 5

Comparison of the two braiding protocols under consideration. In the simultaneous exchange protocol (top), at most two parafermions are strongly coupled at any given time. For the sequential exchange protocol (bottom), in contrast, up to three parafermions are strongly coupled.

Figure 6

(Top) Spectrum of the braiding Hamiltonian (31) for $T=50$ using the simultaneous exchange protocol without chemical potential ($\mu =0$). (Bottom) Overlap $O_0\left(T\right)$ for initial state parity $q=0$ as a function of braiding time $T$. The green dots correspond to averaging over eight different braiding times. The orange curve connects the dots to indicate the general trend of the overlap as a function of braiding time.

Figure 7

(Top) Spectrum of the braiding Hamiltonian (31) for $T=50$ using the simultaneous exchange protocol with nonzero chemical potential $\mu =2$. Exponentially small avoided crossings occur between the energy levels. (Bottom) Overlap $O_0\left(T\right)$ for initial state parity $q=0$ as a function of total braiding time $T$. Averaging over 20 braiding times is depicted by the green dots, the orange curve indicates the trend. The chemical potential induces many avoided level crossings with exponentially small gaps, so the adiabatic limit is not reached within the simulated time.

Figure 8

The sequential exchange braiding protocol we propose, for the times $0<{\tau }_{1/2}<{\tau }_1<{\tau }_{3/2}<{\tau }_2<{\tau }_{5/2}<T$. Empty circles represent zero energy modes and full circles represent finite energy modes. The initial pair ${\chi }_{2,3}$ stays coupled as one brings ${\chi }_1$ close to ${\chi }_2$. This assures that the gap between the ground-state manifold and the first excited state stays large. When ${\chi }_{1,2}$ are maximally coupled, one can then decouple ${\chi }_{2,3}$. One then continue the steps until ${\chi }_{2,3}$ are the only coupled parafermionic pairs.

Figure 9

(Top) Spectrum of the braiding Hamiltonian (31) for $T=50$ without chemical potential ($\mu =0$) using the sequential braiding scheme. (Bottom) Overlap $O_0\left(T\right)$. We see that adiabaticity is reached much faster compared to the simultaneous exchange protocol.

Figure 10

(Top) Spectrum of the braiding Hamiltonian (31) for $T=50$ with maximum length $L=10$ using the sequential braiding scheme with $\mu =2$. (Bottom) Overlap $O_0\left(T\right)$ for every time step and running average over ten time steps. The adiabatic limit is reached for much smaller braiding times compared to the simultaneous exchange protocol.

Figure 11

(Top, left) Spectrum of the Hamiltonian (31) for a total braiding time $T=50$ using the simultaneous exchange protocol. (Top, right and bottom) The probabilities $V_0^j\left(\tau \right)$ to end up in the different states for different total braiding times $T=50$, 100, and 550, using the parameters $\mu =0$ and global phase ${\phi }_0=0.4\pi$. The red part indicates the probability to stay in the ground state, other colors represent the excited states.

Figure 12

Spectrum and overlaps for finite chemical potential, with global phase ${\phi }_0=0.1\pi$. (Top) Chemical potential $\mu =0.8$ and simultaneous exchange protocol. (Bottom) Chemical potential $\mu =3$ and sequential exchange protocol. In the simultaneous exchange protocol, multiple nonadiabatic transitions occur and destroy the overlap even for relatively low chemical potential and long braiding times. A much higher overlap can be obtained for shorter braiding times and higher chemical potential using the improved sequential exchange protocol.

Figure 13

The Berry phase difference for ${\mathbb{Z}}_4$ parafermions for the simultaneous exchange protocol (left, and for the simultaneous exchange protocol with inverted path in parameter space (right). The phase differences are plotted in units of $2\pi$. (Left) The blue line at $1/2$ indicates that the Berry phase ${\beta }_2\to \pi$, whereas the purple line at $3/8$ shows that the Berry phase ${\beta }_{1,3}\to 3\pi /4$. (Right) For the inverted path, ${\beta }_2\to \pi$, whereas ${\beta }_{1,3}\to 5\pi /4=-3\pi /4\left(\text{mod}2\pi \right)$. These result are in accordance with the analytical Berry phase in Eq. (B4).

Figure 14

(Top) Berry phase differences as a function of the global phase for ${\mathbb{Z}}_4$ parafermions using the simultaneous exchange protocol, from numerical simulation (left) and using Eq. (B4) (right). (Bottom) The Berry phase differences converge to the expected values also using the sequential exchange braiding protocol.

Figure 15

(Left) Energy spectrum of the individual Hamiltonians $H_{12}$ and $H_{23}$, for $\mu =2$. Crossings occur in the spectra due to oscillations. (Right) Energy gap between the ground state and the first excited state of $H_{12}$, $H_{23}$ and $H_{12}+H_{23}$. In $H_{12}+H_{23}$, $H_{12}$ lifts the degeneracy of the crossings of $H_{23}$ and vice versa. The gap sizes are of the order $e^{-L_{ij}\left(\tau \right)}$.