Disordered Haldane-Shastry model

The Haldane-Shastry model is one of the most studied interacting spin systems. The Yangian symmetry makes it exactly solvable, and the model has semionic excitations. We introduce disorder into the Haldane-Shastry model by allowing the spins to sit at random positions on the unit circle and study the properties of the eigenstates. At weak disorder, the spectrum is similar to the spectrum of the clean Haldane-Shastry model. At strong disorder, the long-range interactions in the model do not decay as a simple power law. The eigenstates in the middle of the spectrum follow a volume law, but the coefficient is small, and the entropy is hence much less than for an ergodic system. In addition, the energy level spacing statistics is neither Poissonian nor of the Wigner-Dyson type. The behavior at strong disorder hence serves as an example of a nonergodic phase, which is not of the many-body localized kind, in a model with long-range interactions and SU(2) symmetry.

Figure 1

Left: Uniform lattice on a circle with $N=6$ spins. Right: Disordered lattice with $N=6$ spins. When we divide the system into two parts to compute the entanglement entropy, we choose part $A$ of the system to be the $N/2$ spins with the smallest angles ${\varphi }_j$ and part $B$ to be the remaining spins.

Figure 2

Top left: Uniform lattice with $N=20$ spins. Top right: Behavior of the coupling strengths $|h_{ij}|$ in the Hamiltonian on the clean lattice. Bottom left: One realization of the disordered lattice with $N=20$ spins for maximum disorder strength $\delta =N$. Bottom right: Behavior of the coupling strengths for the disordered lattice shown on the left. In both the clean and disordered cases, we fix the $i\mathrm{th}$ spin to be the one with the smallest angle ${\varphi }_i$ and plot $|h_{ij}|$ as a function of ${\varphi }_j-{\varphi }_i$. Note that for the disordered case, the coupling strength can be quite large between spins that are far from each other.

Figure 3

Top: Entanglement entropy $S$ of half of the chain for the state closest to the middle of the spectrum of the Hamiltonian (5) averaged over ${10}^4$ disorder realizations as a function of the disorder strength $\delta$. The different curves are for different system sizes $N$ as indicated. The entropy $[Nln(2)-1]/2$ for an ergodic system is shown with horizontal dashed lines. Bottom: Variance ${\sigma }^2$ of the entanglement entropy computed from the same set of data. It is seen that the transition happens around $\delta =1$.

Figure 4

Scaling of the disorder-averaged half chain entanglement entropy $\langle S\rangle$ with the system size $N$ for weak and strong disorder. The slope of a linear fit for the points computed at weak disorder is around 0.3, while for those computed at maximum disorder it is around 0.06. The dashed blue line shows the entropy $[Nln(2)-1]/2$ of an ergodic system for comparison. The results shown are averaged over ${10}^4$ disorder realizations.

Figure 5

The energy level spacing distribution of the disordered Haldane-Shastry model computed for 16 spins for one disorder realization at maximal disorder strength. It follows neither the Poisson distribution, nor the Gaussian orthogonal ensemble.

Figure 6

The adjacent gap ratio $r$ for the spectrum of the Hamiltonian (5) as a function of the system size $N$ at maximal disorder strength. The results shown are averaged over ${10}^4$ disorder realizations. The curve neither approaches the Poisson value, nor the value for the Gaussian orthogonal ensemble.

Figure 7

Energy spectrum of the Haldane-Shastry model at no and weak disorder computed for 12 spins. The spectrum at weak disorder closely resembles that of the clean Haldane-Shastry model.