Doubly excited ferromagnetic spin chain as a pair of coupled kicked rotors

We show that the dynamics of a doubly excited Heisenberg spin chain, subject to short pulses from a parabolic magnetic field may be analyzed as a pair of quantum kicked rotors. By focusing on the two-magnon dynamics in the kicked $XXZ$ model we investigate how the anisotropy parameter—which controls the strength of the magnon-magnon interaction—changes the nature of the coupling between the two “image” coupled kicked rotors. We investigate quantum state transfer possibilities and show that one may control whether the spin excitations are transmitted together, or separate from each other.

Figure 1

Showing the production of “bound state” accelerator modes (AM2) which move slowly and fast “scattering state” AM. We plot the time evolution of on-site magnetization $⟨P_n^{\downarrow }⟩$ for two initially neighboring spin flips $|\psi \left(0\right)⟩=|400,401⟩$ on spin chains with Ising terms of strength: $\Delta =0$ (left), $\Delta =1$ (center), and $\Delta =2$ (right). When $\Delta =0$, only the AM are present, in this case the dynamics maps to two independent QKRs. With increasing $\Delta$ the AM2 become dominant. We have chosen here, the parameters $K=13$, $B_Q=0.1$, $n_0=400$, and $T=1$ for chains of 800 spins, which are known to produce Gaussian excitations in a singly-excited chain.

Figure 2

Spin-spin correlations corresponding to Fig. 1 at $t=T$: (left) $\Delta =0$, (center) $\Delta =1$, and (right) $\Delta =2$. The two-site correlation function $⟨P_{n_1}^{\downarrow }{P}_{n_2}^{\downarrow }⟩$, equal to the probability of finding one flip on site $n_1$ and the other on $n_2$, is shown. At $\Delta =0$ the spins are anticorrelated in contrast to Fig. 1 which suggests the dynamics of uncoupled particles. $\Delta =1$ and 2 have an anticorrelated component (flips separate) as well as an additional component where the flips travel together.

Figure 3

Decay of QKR-like behaviour of neighbouring spin flips on the $\Delta =2$ kicked Heisenberg chain. At large anisotropy $\left(\Delta ⪢1\right)$, the matrix element in Eq. (22) implies a correspondence between the dynamics on the nearest-neighbor subspace and that of a QKR with stochasticity parameter $K_b=JT{B}_Q/\Delta$ and effective Planck’s constant ${\tau }_b=2B_Q$. Here we show how this correspondence holds when $\Delta =2$, for the initial state $|100,101⟩$ on a chain of 200 spins. We plot the fidelity $F=\left|⟨\psi \left(nT\right)|{\psi }_{\Delta ⪢1}\left(nT\right)⟩\right|$ of evolution on the chain calculated numerically $|\psi \left(nT\right)⟩={\left[U\left(T\right)\right]}^n|100,101⟩$ with the state evolved by the matrix element in Eq. (22), $|{\psi }_{\Delta ⪢1}\left(t=nT\right)⟩={\left[U_H^{\Delta ⪢1}\left(T\right)\right]}^n|100,101⟩$. We see that even at this relatively small anisotropy, QKR-like behavior is seen for short times. However, this correspondence decays more rapidly for larger $B_Q\approx 1$. This may be due to increased coupling between the scattering and bound states due to the kicking field.

Figure 4

Influence of the ${\sigma }_i^z{\sigma }_{i+1}^z$ coupling on the growth of the “center of mass” second moments for two flips initialized 10 sites apart and parameters $K=5$ and $B_Q=1$ on a chain of 200 spins.