Eigenstate Thermalization in a Locally Perturbed Integrable System.

Eigenstate thermalization is widely accepted as the mechanism behind thermalization in generic isolated quantum systems. Using the example of a single magnetic defect embedded in the integrable spin-1/2 XXZ chain, we show that locally perturbing an integrable system can give rise to eigenstate thermalization. Unique to such setups is the fact that thermodynamic and transport properties of the unperturbed integrable chain emerge in properties of the eigenstates of the perturbed (nonintegrable) one. Specifically, we show that the diagonal matrix elements of observables in the perturbed eigenstates follow the microcanonical predictions for the integrable model, and that the ballistic character of spin transport in the integrable model is manifest in the behavior of the off-diagonal matrix elements of the current operator in the perturbed eigenstates.

Density Functional Theory Prediction of the Electrocatalytic Mechanism of Proton Reduction by a Dicobalt Tetrakis(Schiff Base) Macrocycle.

A dicobalt tetrakis(Schiff base) macrocycle has recently been reported to electrochemically catalyze the reduction of H+ to H2 in an acetonitrile solution. Density functional theory (DFT) calculations using the ωB97X-D functional are shown to produce structural and thermodynamic results in good agreement with the experimental data. A mechanistic model based on thermodynamics is developed that incorporates electrochemical and magnetic details of the complex, accounting for electron-spin reorganization of the metal center after redox steps. The model is validated through a comparison of the predicted electrochemical potentials with the irreversible cyclic voltammogram of [Co2LAc]+, which shows redox-coupled spin-crossover (RCSCO) behavior for the CoII/III transitions. Using our model, we predict the thermodynamically favored mechanism of H2 evolution by [Co2L]2+ to be one of heterolytic proton attack on a [CoII2L(µ-H)]+ species. Understanding the electronic details and thermodynamically preferred mechanism of this catalyst will aid in improving its efficiency and the future design of bimetallic Co-based H+ electrocatalysts. Also, this work will assist in the future DFT modeling of bimetallic RCSCO complexes.

Eigenstate thermalization for observables that break Hamiltonian symmetries and its counterpart in interacting integrable systems.

We study the off-diagonal matrix elements of observables that break the translational symmetry of a spin-chain Hamiltonian, and as such connect energy eigenstates from different total quasimomentum sectors. We consider quantum-chaotic and interacting integrable points of the Hamiltonian, and focus on average energies at the center of the spectrum. In the quantum-chaotic model, we find that there is eigenstate thermalization; specifically, the matrix elements are Gaussian distributed with a variance that is a smooth function of ω=E_{α}-E_{ß} (E_{α} are the eigenenergies) and scales as 1/D (D is the Hilbert space dimension). In the interacting integrable model, we find that the matrix elements exhibit a skewed log-normal-like distribution and have a variance that is also a smooth function of ω that scales as 1/D. We study in detail the low-frequency behavior of the variance of the matrix elements to unveil the regimes in which it exhibits diffusive or ballistic scaling. We show that in the quantum-chaotic model the behavior of the variance is qualitatively similar for matrix elements that connect eigenstates from the same versus different quasimomentum sectors. We also show that this is not the case in the interacting integrable model for observables whose translationally invariant counterpart does not break integrability if added as a perturbation to the Hamiltonian.

Entanglement and matrix elements of observables in interacting integrable systems.

We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω=E_{α}-E_{ß} ({E_{α}} are the eigenenergies) proportional to 1/D, where D is the Hilbert space dimension.