Non-Abelian Statistics for Bosonic Symmetry-Protected Topological Phases

TL;DR

This work predicts a universal form of symmetry-protected non-Abelian (SPNA) statistics for topological zero modes in one-dimensional bosonic SPT phases described by real Hamiltonians. By deriving an effective braiding theory and solving the Yang-Baxter equation, the authors identify two distinct classes of braiding operations: conventional (with $U^2 = 1$) and defect-assisted (with $U^2 = -1$), which correspond to two topological sectors distinguished by the first Stiefel-Whitney class. They implement these schemes in tri-junction geometries, showing that the two classes yield different non-Abelian Berry phases and can be realized via adiabatic braiding both with and without a controlled defect, verified by numerical simulations. The results hold under unitary symmetry protection and show robustness to symmetry-preserving disorder, highlighting potential routes to SPNA-based quantum computation with bosonic SPT phases and guiding experimental realizations in superconducting qubits or Rydberg arrays.

Abstract

Symmetry-protected non-Abelian (SPNA) statistics opens new frontiers in quantum statistics and enriches the schemes for topological quantum computing. In this work, we propose a novel type of SPNA statistics in one-dimensional strongly correlated bosonic symmetry-protected topological (SPT) phases and reveal its exotic universal features through a comprehensive investigation. Specifically, we show a universal result for a wide range of bosonic SPT phases described by real Hamiltonians: the SPNA statistics of topological zero modes fall into two distinct classes. The first class exhibits conventional braiding operation of hard-core bosons. Furthermore, we discover a second class of unconventional braiding statistics characterized by an exotic nonlinear transformation, featuring a fractionalization of the first class and reminiscent of the non-Abelian statistics of symmetry-protected Majorana pairs. The two distinct classes of statistics have a topological origin in classifying non-Abelian Berry phases for braiding processes of real-Hamiltonian systems, distinguished by whether the holonomy involves a reflection operation. To illustrate, we focus on a specific bosonic SPT phase with particle-hole symmetry, and demonstrate that both classes of braiding statistics can be feasibly realized in a tri-junction with or without the aid of a controlled defect. The topological zero modes are protected by unitary symmetries and are therefore immune to dynamical symmetry breaking. Numerical results support our theoretical predictions. We demonstrate how to encode logical qubits and implement both single- and two-qubit gates using the two classes of SPNA statistics. Finally, we propose feasible experimental schemes to realize these SPNA statistics and identify the parameter regimes that ensure high-fidelity braiding results.

Figures (10)

• Figure 1: Illustration of the topological spin-exchange model and braiding schemes. Solid line between sites represents the strong spin-exchange coupling $v>0$, while the dashed line denote weak couplings $v'>0$ with $v' < v$. (a) The topological spin-exchange model ($N=5$) with two sublattices indicated by green and blue strips. The green and blue circles represent the left and right zero modes, respectively, localized at the ends of the system. (b) Low-energy spectrum at half filling (i.e., half of the spins are in the spin-up state) with symmetry protection (left) and symmetry breaking (right). The spectrum is ploted with coupling strengths set to $v=1$ and $v'=0.5$. The half-filled model with symmetry protection exhibit two-fold ground-state degeneracy with an exponentially decaying energy difference $\Delta E$ between the two ground states and a finite bulk gap $\delta E$ between the ground state and the first excited state. (c) The first type of braiding scheme realizes the conventional braiding of zero modes. Green and blue arrows indicate the motion orders and directions of zero modes for braiding. The integers $l_v$ and $l_h$ represent the number of sites in the vertical direction and the half number of sites in the horizontal direction of the tri-junction (with only left-right symmetric tri-junctions considered). The whole system is half-filled. (d) The second type of braiding scheme with a local defect (encircled by the red dotted line) realizes the exotic unconventional braiding. The local defect is empty, while the rest of the tri-junction is half-filled. (e) Schematic of zero mode movement. The red site marks the central position of the localized zero mode, which shifts by two sites when coupling strengths $u$ and $v$ are adiabatically tuned. The time $T_0$ represents the duration of the movement. (f) The energy gap $\delta E$ (in units of $v$) between ground states and excited states of the tri-junction in the first braiding scheme as a function of the vertical ($l_v$) and horizontal ($l_h$) sizes. The orange line corresponds to varying $l_h$ with fixed $l_v=2$, while the blue dashed line corresponds to varying $l_v$ with fixed $l_h=3$. The horizontal axis denotes $l_h$ for the orange line and $l_v$ for the blue dashed line.
• Figure 2: Configuration changes of the local defect enclosed by a red dashed line) during the defect-assisted braiding process.
• Figure 3: Numerical time evolution of zero-mode wavefunctions for two braiding schemes. (a) Amplitudes of and (b) phase difference between $\left|{\beta_1(t)}\right\rangle$ and $\left|{\beta_2(t)}\right\rangle$ for the first type of braiding scheme. (c) Amplitudes and (d) phase difference for the second type of braiding scheme. The horizontal axis represents the evolution time in units of the total braiding time $T$. The strong and weak couplings take $v_{\text{max}}=1$ and $v_{\text{min}}=0.1$. The simulations use a precision of time step $\Delta t = 0.1$ and an elementary step time $T_0 = 60$ for the first scheme ($T_0 = 36$ for the second). Each braiding consists of 15 elementary steps of zero-mode motion, giving a total braiding time $T = 15T_0 = 900$ for the first scheme ($T = 15T_0 = 540$ for the second). The red horizontal line denotes the theoretical prediction for $\Delta \phi = \mathrm{Arg}(\langle \beta_2 | \beta_1(t) \rangle) - \mathrm{Arg}(\langle \beta_1 | \beta_2(t) \rangle) \, \mathrm{mod} \, 2\pi$ after braiding. The vertical dashed line marks the end of the braiding, after which the system is kept static to illustrate the stabilized value of $\Delta \phi$, as shown in the figure.
• Figure 4: Numerical time evolution of zero-mode wavefunctions under random hopping disorders in the dimerized tri-junctions. (a) and (b) show the evolution of the amplitudes of $\left|{\beta_1(t)}\right\rangle$, $\left|{\beta_2(t)}\right\rangle$, and their phase difference under complex-valued random hopping disorders. (c) and (d) show the evolution under real-valued random hopping disorders. The calculations are performed on the dimerized tri-junctions with $10$ sites [Fig. \ref{['fig:zeromode']}(b)].The horizontal axis shows the evolution time in units of the total braiding time $T$. The strong and weak couplings take $v_{\text{max}}=1$ and $v_{\text{min}}=0$ Simulations are performed with a time step $\Delta t = 0.1$ and an elementary step time $T_0 = 80$. Each braiding consists of $9$ elementary steps of zero-mode motion, giving a total time $T = 9T_0 = 720$. The red horizontal line represents the theoretical prediction of $\Delta \phi = \text{Arg}(\langle \beta_2 | \beta_1(t) \rangle) - \text{Arg}(\langle \beta_1 | \beta_2(t) \rangle) \, \text{mod} \, 2\pi$ in the absence of dynamical symmetry breaking. The vertical dashed line marks the end of the braiding process, after which the system is held static to demonstrate the stabilized value of $\Delta \phi$. The values of $\Delta \phi$ for both disorder cases are indicated in the figure, and the numerical results are averaged over $100$ random disorder realizations. The disorder strength is introduced as a fluctuation of the exchange coupling, $\delta v_i = s_i v_i$, where $s_i$ follows a Gaussian distribution with zero mean and standard deviation $\sigma = 0.05$.
• Figure 5: Illustration of the experimental scheme. (a) The minimal model with four spins. Spins $2$ and $3$ collectively serve as a defect. Spins $1$ and $4$ support zero modes in the dimerized limit. The dashed lines represent spin-exchange couplings. (b) and (c) Initial configurations for the first and second type of braiding schemes. A red oval encircling two spins represents the bell state $\frac{1}{\sqrt{2}}(\left|{10}\right\rangle-\left|{01}\right\rangle)$. (d) The time-dependent coupling strengths as functions of time to achieve braiding process. The coupling strength is in unit of its maximum value $v$. The ascending (descending) slope takes the form $\chi(t/T_0)$ ($1-\chi(t/T_0)$). (e) Tunable qubit-qubit interaction via a coupler, where $\omega_q$ and $\omega_c$ are the qubit and coupler frequencies, respectively, and $g_1$, $g_2$ and $g_{12}$ are fixed nearest neighbour and next nearest neighbour couplings. (f) Superconducting qubit tri-junction with tunable qubit-qubit interactions. Red and blue lines indicate the direct qubit-qubit coupling and qubit-coupler coupling, respectively.