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arXiv:quant-ph/0611248 · Physical Review A

Entanglement, avoided crossings, and quantum chaos in an Ising model with a tilted magnetic field

J. Karthik, Auditya Sharma, Arul Lakshminarayan

Published Feb 6, 2007• 90 citations
arXivHTMLPDF

Abstract

We study a one-dimensional Ising model with a magnetic field and show that tilting the field induces a transition to quantum chaos. We explore the stationary states of this Hamiltonian to show the intimate connection between entanglement and avoided crossings. In general, entanglement gets exchanged between the states undergoing an avoided crossing with an overall enhancement of multipartite entanglement at the closest point of approach, simultaneously accompanied by diminishing two-body entanglement as measured by concurrence. We find that both for stationary as well as nonstationary states, nonintegrability leads to a destruction of two-body correlations and distributes entanglement more globally.

DOI
10.1103/physreva.75.022304

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We study a one-dimensional Ising model with a magnetic field and show that tilting the field induces a transition to quantum chaos. We explore the stationary states of this Hamiltonian to show the intimate connection between entanglement and avoided crossings. In general, entanglement gets exchanged between the states undergoing an avoided crossing with an overall enhancement of multipartite entanglement at the closest point of approach, simultaneously accompanied by diminishing two-body entanglement as measured by concurrence. We find that both for stationary as well as nonstationary states, nonintegrability leads to a destruction of two-body correlations and distributes entanglement more globally.
We study the entanglement properties in a one-dimensional Ising chain with a tilted magnetic field that is capable of showing both integrable and nonintegrable behaviors. Here the pairwise entanglement is characterized by concurrence and the multipartite entanglement is characterized by the Q measure. According to the entanglement properties of the ground state in the Ising mode, which have tilt angle, we can find that the Q measure decreases with the increasing of the strength of external field. And the phase transition property of the system is changed with the increase of tilt angle for the external magnetic field. We also consider the evolution of entanglement in this model, and find that the nonintegrability can suppress the pairwise entanglement but promotes the multipartite entanglement with the integrable system.
Abstract We study the ground and first excited state of the finite one dimensional Majumdar-Ghosh model with quenched disorder about the avoided crossings. We find a relation between the shift of the first avoided crossing and average value of random numbers. For low disorder regime, the two-qubit, odd and even partition entanglement gets exchanged at avoided crossings for low spin, and former two for any value of spin. This effect has a deep relation with the dimerization nature of these states. We develop an expression sans entanglement measures to detect and study these crossings.
Controlling entanglement and coherence is central to quantum information, yet the two resources often exhibit antagonistic trends and are difficult to optimize within a single platform. Here we show that chaos enables switchable eigenstate resources: avoided crossings in soft- versus strong- chaos windows selectively realize an entanglement-peak mode or a coherence-peak mode within the same system. Crucially, this chaos-controlled inversion is not tied to a particular notion of subsystems, appearing both in single-wave settings and in genuine many-body settings. From the quantum-chaos perspective, conventional diagnostics based on avoided-crossing phenomenology and eigenmode delocalization are insufficient; eigenfunction entanglement and basis coherence provide the missing discriminants. Using two wave-chaotic billiards and a tilted-field Ising chain, we track the information-theoretic response of eigenstates across localized hybridization windows. Even when avoided-crossing phenomenology and delocalization are comparable, the entanglement and coherence responses invert between soft- and strong-chaos regimes. In the Ising chain, a single microscopic knob, the global field tilt, toggles between the two operating modes and reveals a trade-off in which off-diagonal correlations grow as diagonal populations dip. Our diagnostics require only reduced states (or their spectra) and are compatible with mode imaging in wave-chaos resonators and randomized measurements in programmable spin simulators.
We investigate the entanglement arising naturally in a one-dimensional Ising chain in a magnetic field in an arbitrary direction. We find that for different temperatures, different orientations of the magnetic field give maximum entanglement. In the high-temperature limit, this optimal orientation corresponds to the magnetic field being perpendicular to the Ising orientation (z(z(z direction). In the low-temperature limit, we find that varying the angle of the magnetic field very slightly from the z direction leads to a rapid rise in entanglement. We also find that the orientation of the magnetic field for maximum entanglement varies with the field amplitude. Furthermore, we have derived a simple rule for the mixing of concurrences (a measure of entanglement) due to the mixing of pure states satisfying certain conditions.
Entanglement sharing among sites of one-particle states is considered using the measure of concurrence. These are the simplest in a hierarchy of number-specific states of many qubits and correspond to ``one-magnon'' states of spins. We study the effects of onsite potentials that are both integrable and nonintegrable. In the integrable case, we point to a metal-insulator transition that reflects on the way entanglement is shared. In the nonintegrable case, the average entanglement content increases and saturates along with a transition to classical chaos. Such quantum chaotic states are shown to have universal concurrence distributions that are modified Bessel functions derivable within random matrix theory. Time-reversal breaking and time-evolving states are shown to possess significantly higher entanglement sharing capacity than eigenstates of time-reversal symmetric systems. We use the ordinary Harper and the kicked Harper Hamiltonians as model systems.
We study the relation between entanglement and quantum chaos in one- and two-dimensional spin-1∕21∕21∕2 lattice models, which exhibit mixing of the noninteracting eigenfunctions and transition from integrability to quantum chaos. Contrary to what occurs in a quantum phase transition, the onset of quantum chaos is not a property of the ground state but takes place for any typical many-spin quantum state. We study bipartite and pairwise entanglement measures---namely, the reduced von Neumann entropy and the concurrence---and discuss quantum entanglement sharing. Our results suggest that the behavior of the entanglement is related to the mixing of the eigenfunctions rather than to the transition to chaos.