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

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.

Quantum chimera states

“…This ambiguous behaviour is certainly the manifestation of the nature of the chimera system which is a hybrid of a both chaotic and regular system. For the case with a LSD being a Poisson distribution, the survival 3 Note that | ↓ ... ↓ is an eigenstate like for the totally chaotic models [8,9,10,11]. But for these cases | ↓ ... ↓↑ presents a survival probability with a chaotic behaviour.…”

“…But in contrast with the classical case where the entropy measures the disorder, in this quantum context the entropy measures the entanglement. In comparison, the computation of the same quantities for different models of chaotic or random spin chains or glasses [8,9,10,11,26] shows eigenstates with a large entanglement which is uniform on the chain (or with small variations between nearest neighbour spins). These models do not involve states with both some spins highly entangled and the other ones totally not entangled as in figure 2.…”

“…The quantum states like figures 2 and 3 can be considered as quantum chimera states. Note that the present model like the chaotic or the random models [8,9,10,11,26] presents also totally regular (non entangled) states (as classical chimera states coexist with fully synchronized states). figure) where the Bloch sphere of each spin is represented by azimuthal projection centered on the north pole (θ = 0) the limit circle being the south pole (θ = π 2 ); and up populations, coherences and linear entropies (bottom figure) of the spins of the chain in the eigenstate |χ = 0 with N = 12, M = 3, ω 1 = ... = ω N = 1.…”

“…Quantum chaos is an ambiguous concept since in classical dynamics the chaos is strongly linked to the non-linear effects whereas the quantum dynamics is fundamentally a linear theory. A commonly used criterion of quantum chaos for spin systems is the level spacing distribution (LSD) of the spectrum [8,9,10,11]. A regular system presents a LSD as Dirac peaks, a (pseudo)-random system presents a LSD as a Poisson distribution (characterizing the disorder of the energy levels without correlation) and a chaotic system presents a LSD as a Wigner-Dyson distribution (characterizing the disorder of the energy levels with correlations).…”

“…In this paper we show that a simple quantum system, a closed chain of spins, offers quantum analogues of the chimera states if the couplings between the spins have the same structure that the couplings of the classical models. It is well known that spin chains can exhibit kinds of quantum disorder and of quantum chaos [8,9,10,11], and that quantum synchronization is related to the entanglement [12,13,14,15]. To involve a kind of chimera states, our model consists of a non-hermitian spin chain [16,17,19,22] which can be assimilated to a spin chain in contact with an environment.…”