Violation of Bohigas-Giannoni-Schmit conjecture using an integrable many-body Floquet system

Earlier studies have given enough evidence in support of the BGS conjecture, with few exceptions violating it. Here, we provide one more counterexample using a many-body system popularly known as the model of quantum kicked top consisting of $N$ qubits with all-to-all interaction and kicking strength $k=N\pi/2$. We show that it is quantum integrable even though the corresponding semiclassical phase-space is chaotic, thus violating the BGS conjecture. We solve the cases of $N=5$ to $11$ qubits analytically, finding its eigensystem, the dynamics of the entanglement, and the unitary evolution operator. For the general case of $N>11$ qubits, we provide numerical evidence of integrability using degenerate spectrum, and the exact periodic nature of the time-evolved unitary evolution operator and the entanglement dynamics.Comment: 4.5 pages (two-column) + 25 pages (one-column) + 3 figures; Comments are welcom

Entanglement transitions in random definite particle states

Entanglement within qubits are studied for the subspace of definite particle states or definite number of up spins. A transition from an algebraic decay of entanglement within two qubits with the total number $N$ of qubits, to an exponential one when the number of particles is increased from two to three is studied in detail. In particular the probability that the concurrence is non-zero is calculated using statistical methods and shown to agree with numerical simulations. Further entanglement within a block of $m$ qubits is studied using the log-negativity measure which indicates that a transition from algebraic to exponential decay occurs when the number of particles exceeds $m$. Several algebraic exponents for the decay of the log-negativity are analytically calculated. The transition is shown to be possibly connected with the changes in the density of states of the reduced density matrix, which has a divergence at the zero eigenvalue when the entanglement decays algebraically.Comment: Substantially added content (now 24 pages, 5 figures) with a discussion of the possible mechanism for the transition. One additional author in this version that is accepted for publication in Phys. Rev.

Entanglement between two subsystems, the Wigner semicircle and extreme value statistics

The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose, $\rho_{12}^{T_2}$. The density of states of $\rho_{12}^{T_2}$ is close to the semicircle law when both subsystems have dimensions which are not too small and are of the same order. A simple random matrix model for the partial transpose is found to capture the entanglement properties well, including a transition across a critical dimension. Log-negativity is used to quantify entanglement between subsystems and analytic formulas for this are derived based on the simple model. The skewness of the eigenvalue density of $\rho_{12}^{T_2}$ is derived analytically, using the average of the third moment over the ensemble of random pure states. The third moment after partial transpose is also shown to be related to a generalization of the Kempe invariant. The smallest eigenvalue after partial transpose is found to follow the extreme value statistics of random matrices, namely the Tracy-Widom distribution. This distribution, with relevant parameters obtained from the model, is found to be useful in calculating the fraction of entangled states at critical dimensions. These results are tested in a quantum dynamical system of three coupled standard maps, where one finds that if the parameters represent a strongly chaotic system, the results are close to those of random states, although there are some systematic deviations at critical dimensions.Comment: Substantially improved version (now 43 pages, 10 figures) that is accepted for publication in Phys. Rev.

Universal scaling of higher-order spacing ratios in Gaussian random matrices

Higher-order spacing ratios in Gaussian ensembles are investigated analytically. A universal scaling relation, known from earlier numerical studies, of the higher-order spacing ratios is proved in the asymptotic limits.Comment: 8 pages. Comments are welcom