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arXiv:0707.1331 · Physical Review E

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

Winton Brown, Léa Ferreira dos Santos, David J. Starling, Lorenza Viola

Published Feb 7, 2008• 69 citations
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Abstract

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.

DOI
10.1103/physreve.77.021106

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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.
We investigate disordered oneand 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 spinand 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.
We consider a one-dimensional (1D) quantum many-body system and investigate how the interplay between interaction and on-site disorder affects spatial localization and quantum correlations. The hopping amplitude is kept constant. To measure localization, we use the number of principal components (NPC), which quantifies the spreading of the system eigenstates over vectors of a given basis set. Quantum correlations are determined by a global entanglement measure Q , which quantifies the degree of entanglement of multipartite pure states. Our studies apply analogously to a 1D system of interacting spinless fermions, hard-core bosons, or yet to an XXZ Heisenberg spin-1/2 chain. Disorder is characterized by both uncorrelated and long-range correlated random on-site energies. Dilute and half-filled chains are analyzed. In half-filled clean chains, delocalization is maximum when the particles do not interact, whereas multi-partite entanglement is largest when they do. In the presence of uncorrelated disorder, NPC and Q show a nontrivial behavior with interaction, peaking in the chaotic region. The inclusion of correlated disorder may further extend two-particle states, but the effect decreases with the number of particles and the strength of their interactions. In half-filled chains with large interaction, correlated disorder may even enhance localization.
We investigate multipartite entanglement dynamics in disordered spin-1∕21∕21∕2 lattice models exhibiting a transition from integrability to quantum chaos. Borrowing from the generalized entanglement framework, we construct measures for correlations relative to arbitrary local and bilocal spin observables, and show how they naturally signal the crossover between distinct dynamical regimes. Analytical estimates are obtained in the short- and long-time limits. Our results are in qualitative agreement with predictions from the random matrix theory and are relevant to both condensed-matter physics and to the stability of quantum information in disordered quantum computing hardware.
We uncover signatures of quantum chaos in the many-body dynamics of a Bose-Einstein condensate-based quantum ratchet in a toroidal trap. We propose measures including entanglement, condensate depletion, and spreading over a fixed basis in many-body Hilbert space, which quantitatively identify the region in which quantum chaotic many-body dynamics occurs, where random matrix theory is limited or inaccessible. With these tools, we show that many-body quantum chaos is neither highly entangled nor delocalized in the Hilbert space, contrary to conventionally expected signatures of quantum chaos.
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Understanding the emergence of chaos in many-body quantum systems away from semi-classical limits, particularly in spatially local interacting spin Hamiltonians, has been a long-standing problem. In these intrinsically quantum regimes, quantum chaos has been primarily understood through the correspondence between the eigensystem statistics of midspectrum eigenstates and the universal statistics described by random matrix theory (RMT). However, this correspondence no longer holds for finite-temperature eigenstates. Here we show that the statistical properties of finite-temperature eigenstates of quantum chaotic Hamiltonians can be accurately described by pure random states constrained by a local charge, with the average charge density of the constrained random state ensemble playing the same role as the average energy density of the eigenstates. By properly normalizing the energy density using a single Hamiltonian-dependent parameter that quantifies the typical energy per degree of freedom, we find excellent agreement between the entanglement entropy statistics of eigenstates and that of constrained random states. Interestingly, in small pockets of Hamiltonian parameter phase space which we previously identified as `maximally chaotic' [PRX 14, 031014 (2024)], we find excellent agreement not only at the level of the first moment, including O(1) corrections, but also at the level of statistical fluctuations. These results show that notions of maximal chaos -- in terms of how much randomness eigenstates contain -- can still be defined at finite temperature in physical Hamiltonian models away from semi-classical and large-NNN limits.