Self-similar spectral structures and edge-locking hierarchy in open-boundary spin chains

For an anisotropic Heisenberg ($\mathit{XXZ}$) spin chain, we show that an open boundary induces a series of approximately self-similar features at different energy scales, high up in the eigenvalue spectrum. We present a nonequilibrium phenomenon related to this fractal structure, involving states in which a connected block near the edge is polarized oppositely to the rest of the chain. We show that such oppositely polarized blocks can be “locked” to the edge of the spin chain and that there is a hierarchy of edge-locking effects at various orders of the anisotropy. The phenomenon enables dramatic control of quantum-state transmission and magnetization control.

Figure 1

A few $\uparrow$ spins at the left edge of an almost polarized spin chain. The leftmost ten spins are shown; the remaining spins to the right are all $\downarrow$’s. The configurations on the left (a–f) are edge locked, while those on the right (g–l) are not.

Figure 2

Dynamics of open $\mathit{XXZ}$ chain ($\Delta =4$), initiated with $N_{\uparrow }=2$ or $N_{\uparrow }=3$ oppositely polarized spins near the left edge. Initial configurations are shown on top of each panel. In the edge-locked cases (a, c, d), local spin values $\langle S_j^z\rangle$ do not vary much from their initial values.

Figure 3

Energy spectra for (a) periodic $\mathit{XXZ}$ chain and (b) open $\mathit{XXZ}$ chain. In each case, $L=13$, $N_{\uparrow }=3$, and $\Delta =10$. Compared to the periodic chain, the open-chain spectrum has extra features, some of which are highlighted in insets. With $N_{\uparrow }=3$, there are two classes of edge-locked states, the two states of the top band (main plot) and two of the four states separating out from the next band (upper inset). The lower insets show additional spectral structures induced by the edge.

Figure 4

Left: hierarchy of subband splittings seen by zooming in successively. Here $N_{\uparrow }=8$ ($L=20$, $\Delta =10$), so the first four levels of the hierarchy are present. For large $N_{\uparrow }$ and $L$, the splitting structures are self-similar. Right: Energy splittings scale as ${\delta }_1~{\Delta }^0$, ${\delta }_2~{\Delta }^{-2}$, ${\delta }_3~{\Delta }^{-4}$, and ${\delta }_4~{\Delta }^{-6}$.