Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

We investigate disordered one- and two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin- and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.

Figure 1

(Color online) Case $J/\delta \epsilon$, $S_z=0$ subspace, $L=12$. Top panels: Level statistics indicator $\eta$ vs $J/\delta \epsilon$. The dashed line indicates in each case the transition region, where the $\left(S,R\right)$ subspaces are not fully distinct. The arrow indicates the actual limiting value of the LSI as $J/\delta \epsilon \to \infty$. Bottom panels: Number of principal components $\xi$ vs $J/\delta \epsilon$. Left panels: 1D chain. Right panels: 2D $3\times 4$ lattice. Averages over 20 random realizations.

Figure 2

(Color online) Case $\delta J/J$, $\left(S_z,S\right)=\left(0,1\right)$ subspace, $L=12$. Top panels: Level statistics indicator $\eta$ vs $\delta J/J$. Dashed lines indicate in each case the region where the transition to $R$-symmetry sectors occurs. Dotted line: Gaussian disorder with $\sigma =\delta J/4$. The arrow indicates the actual limiting value of $\eta$ as $\delta J/J\to 0$. Bottom panels: Number of principal components $\xi$ vs $\delta J/J$. Left panels: 1D chain. Right panels: $3\times 4$ lattice. Averages over 50 random realizations.

Figure 3

(Color online) Case $\delta J/\delta \epsilon$. $S_z=0$ subspace, $L=12$. Top panels: Level statistics indicator $\eta$ vs $\delta J/\delta \epsilon$. Dashed line indicates the region where the transition to $S^2$-symmetry sectors occurs. Arrow indicates the actual limiting value of $\eta$ as $\delta J/\delta \epsilon \to \infty$. Bottom panels: Number of principal components $\xi$ vs $\delta J/\delta \epsilon$. Left panels: 1D chain. Right panels: $3\times 4$ lattice. Averages over 50 random realizations.

Figure 4

(Color online) Case $\delta J/\delta \epsilon$. Number of principal components vs energy eigenvalue for $\delta J/\delta \epsilon =3$. Eigenstates in the $S_z=0$ subspace for a $3\times 4$ qubit lattice are considered. The dashed horizontal line shows the GOE prediction ${\xi }_{\text{GOE}}=\left(N_0+2\right)/3\sim 308$. Averages over 20 random realizations.

Figure 5

(Color online) Choices of partitions for evaluating $n$–local purities in the $3\times 4$ lattice.

Figure 6

(Color online) Case $J/\delta \epsilon$. Average nearest-neighbor concurrence and $n$–local purity averaged over all eigenvectors of the $S_z=0$ subspace vs $\delta J/\delta \epsilon$. Lattice size $L=12$. Solid black line: concurrence. Left panel: 1D chain. Right panel: $3\times 4$ lattice. Averages over 20 random realizations.

Figure 7

(Color online) Case $\delta J/J$. Average nearest-neighbor concurrence and $n$–local purity averaged over all eigenvectors of the $\left(S_z,S\right)=\left(0,1\right)$ subspace vs $\delta J/J$. Lattice size $L=12$. Solid black line: concurrence. Left panel: 1D chain. Right panel: $3\times 4$ lattice. Averages over 20 random realizations.

Figure 8

(Color online) Case $\delta J/\delta \epsilon$, $S_z=0$ subspace, $3\times 4$ 2D spin lattice, $\delta J/\delta \epsilon =2$. Left panel: $P_2$ vs energy eigenvalue. Right panel: $P_1$ vs ${\xi }_c$. A single disorder realization is considered.

Figure 9

(Color online) Multipartite entanglement vs delocalization in a clean Heisenberg Hamiltonian: $P_6$ vs ${\xi }_c$ in the $\left(S_z,S,R\right)=\left(0,1,+1\right)$ symmetry subspace. Lattice size: $L=12$. Left panel: 1D chain. Right panel: $3\times 4$ 2D lattice.

Figure 10

(Color online) Generalized fermionic entanglement vs delocalization in a clean Heisenberg Hamiltonian: $P_{u\left(L\right)}$ vs ${\xi }_c$ in the $\left(S_z,S,R\right)=\left(0,1,+1\right)$ subspace. Lattice size: $L=12$. Left panel: 1D chain. Right panel: $3\times 4$ 2D lattice.