Fine structures in the spectrum of the open-boundary Heisenberg chain at large anisotropies

At large anisotropies, the spectrum of the Heisenberg XXZ spin chain separates into “bands” with energies largely determined by the number of domain walls. The band structure is richer with open-boundary conditions: there are more bands and the bands develop intricate fine structures. We characterize and explain these structures and substructures in the open-boundary chain. The fine structures are explained using degenerate perturbation theory. We also present some dynamical consequences of these subband structures, through explicit time evolution of the wave function from initial states motivated by the fine-structure analysis.

Figure 1

Energy and $\langle {\widehat{N}}_{\text{AFM}}\rangle$ (average number of domain walls) of eigenstates, indexed in order of increasing energy, for (a) open-boundary conditions (OBC) and (b) periodic-boundary conditions (PBC). We show the $N_{\uparrow }=3$ sector, for an $L=13$ chain at large anisotropy $\Delta =10$. There are $2N_{\uparrow }=6$ bands in the OBC spectrum and $N_{\uparrow }=3$ bands in the PBC spectrum.

Figure 2

Energy and $\langle {\widehat{N}}_{\text{AFM}}\rangle$ of eigenstates for $N_{\uparrow }=3$, $L=10$ chain with open-boundary conditions for several $\Delta$ values. The bands in energy start appearing at high $\Delta$; at moderate $\Delta$, the band structure is easier to see through the jumps in $\langle {\widehat{N}}_{\text{AFM}}\rangle$.

Figure 3

Blow-ups of third and fourth bands from the top, for an open $L=10$ chain in the $N_{\uparrow }=3$ sector; $\Delta =10$. (a) Third band, containing $4(L-4)=24$ states, consists of a flat patch between two staircases of nearly degenerate pairs. (b) Fourth band has $(L-4)$ steps each containing $(L-4)$ states, thus having a total of ${(L-4)}^2=36$ states.

Figure 4

Third band from the top, $N_{\uparrow }=3$ sector, for open chains of size $L=10$ and 11. Dashed horizontal lines are from first-order degenerate perturbation theory ${\epsilon }_3+\lambda e_r$. (a) Perturbation theory is seen to be quite accurate for $\Delta =100$. (b) At smaller $\Delta$, there are visible deviations from perturbation theory and a dispersion in the central “flat” patch. For the even case $L=10$, one of the $e_r$'s is zero, so that the central patch has length $2(L-4)=12$. For the odd case $L=11$, the central group contains $2(L-5)=12$ states.

Figure 5

Time evolution for chains initiated with configurations $|\alpha \rangle$ and $|\beta \rangle$, as defined in Eq. (6). Time units are $J_x^{-1}$; $\Delta =5$. For initial state $|\alpha \rangle =|\uparrow \downarrow \uparrow \uparrow \downarrow \downarrow \downarrow ...\rangle$, the propagation is fast as a single magnon. The initial state $|\beta \rangle =|\uparrow \downarrow \downarrow \uparrow \uparrow \downarrow \downarrow ...\rangle$ is part of the flat patch of the third band, not connected by first-order hopping processes; dynamics therefore involves slower bimagnon propagation.

Figure 6

Fourth band from the top of the $N_{\uparrow }=4$ spectrum, for open chains of size $L=10$ and 11. Dashed horizontal lines are from first-order degenerate perturbation theory ${\epsilon }_4+\lambda e_r$. Panel (a) has a large anisotropy $\Delta =100$, so that the perturbative calculations are quite accurate. At smaller $\Delta =10$, the essential structure is the same, but each subband now deviates slightly from the first-order prediction, has visible dispersion, and also visible higher-order substructures. One of the groups of $(L-4)$ merges with the “zeros block” at ${\epsilon }_4=(L-9)/4$ for the odd-sized chain (b), but not for the even-site chain (a).