Topological Quantum Synchronization of Fractionalized Spins

The gapped symmetric phase of the Affleck-Kennedy-Lieb-Tasaki model exhibits fractionalized spins at the ends of an open chain. We show that breaking SU(2) symmetry and applying a global spin-lowering dissipator achieves synchronization of these fractionalized spins. Additional local dissipators ensure convergence to the ground state manifold. In order to understand which aspects of this synchronization are robust within the entire Haldane-gap phase, we reduce the biquadratic term, which eliminates the need for an external field but destabilizes synchronization. Within the ground state subspace, stability is regained using only the global lowering dissipator. These results demonstrate that fractionalized degrees of freedom can be synchronized in extended systems with a significant degree of robustness arising from topological protection. A direct consequence is that permutation symmetries are not required for the dynamics to be synchronized, representing a clear advantage of topological synchronization compared to synchronization induced by permutation symmetries.

Figure 1

Evolution of the local transverse spin $⟨S_j^{\mathrm{x}}⟩$ of the synchronized AKLT model for an open chain of length $N=6$ (sites $j=1$, 2, 3 in solid lines, sites $j=4$, 5, 6 in dashed lines). (a) Starting from a random pure state, i.e., a vector of dimension $3^6$ with random complex amplitudes (we also investigated random mixed states with equivalent results), the two halves of the chain are perfectly antisynchronized with each other after a transient time because the dynamical symmetry operator $A=|G_{1,-1}⟩⟨G_{0,0}|$ is antisymmetric upon inversion of the chain. The (anti)synchronized amplitudes after the transient time decay exponentially into the bulk. (b) Same plot as in (a) but focusing on the early time dynamics: The random initial conditions result in transient random spin dynamics. (c) The balanced superposition of $|G_{0,0}⟩$ and $|G_{1,-1}⟩$ as initial state is immune to dissipation and maximizes the observed (anti)synchronization amplitudes. The oscillation frequency is $\omega =B/N$. Parameters: $B=0.2$, $\gamma =\kappa =0.2$.

Figure 2

(a) Eigenvalues of the Liouvillian superoperator $\mathcal{L}$ close to the imaginary axis for an open chain of length $N=6$ and different values of $ϵ$, considering both global and local dissipators. The purely imaginary eigenvalues for $ϵ=0$ move away from the imaginary axis as the biquadratic term is decreased. Simultaneously, the oscillation frequency increases resulting in fast but damped (anti)synchronization. Parameters: $B=0.2$, $\kappa =\gamma =0.2$. (b),(c) Stable synchronization may be recovered for finite values of $ϵ$ within the ground state subspace for $B=0$ if one considers only the global dissipator $L_{\mathrm{G}}$ (i.e., $\kappa =0$). Sites $j=1$, 2, 3 are in solid lines, sites $j=4$, 5, 6 in dashed lines. The oscillation frequency in panel (c) with $ϵ=1/6$ is larger compared to panel (b) with $ϵ=0.1$ because of the increased gap above the threefold degenerate ground state. The initial state is the infinite temperature state within the ground state manifold.