We investigate the transport of energy, magnetization, etc. in several finite one-dimensional (1D) quantum systems only by solving the corresponding time-dependent Schrödinger equation. We explicitly renounce any other transport analysis technique. Varying model parameters we find a sharp transition from non-normal to normal transport and a transition from integrability to chaos, i.e ., from Poissonian to Wigner-like level statistics. These transitions always appear in conjunction with each other. We investigate some rather abstract “design models” and a (locally perturbed) Heisenberg spin chain.
We investigate the transport of energy, magnetization, etc. in several finite one-dimensional (1D) quantum systems only by solving the corresponding time-dependent Schrödinger equation. We explicitly renounce any other transport analysis technique. Varying model parameters we find a sharp transition from non-normal to normal transport and a transition from integrability to chaos, i.e ., from Poissonian to Wigner-like level statistics. These transitions always appear in conjunction with each other. We investigate some rather abstract “design models” and a (locally perturbed) Heisenberg spin chain.
We analyze transport of local magnetization and develop schemes to control transport behavior in finite spin-1/2 Heisenberg chains and spin-1/2 Heisenberg two-leg ladders at zero temperature. By adjusting parameters in the Hamiltonians, these quantum systems may show both integrable and chaotic limits. We provide examples of chaotic systems leading to diffusive and to ballistic transport. In addition, methods of coherent quantum control to induce a transition from diffusive to ballistic transport are proposed.
We study transport of local magnetization in spin-1/2 Heisenberg chains in integrable and chaotic regimes at zero temperature. A clean Heisenberg chain with only nearest-neighbor interactions is integrable, whereas the addition of frustrating next-nearest neighbor interactions or yet of on-site disorder may lead to the onset of chaos. For a particular initial state far from equilibrium, the dynamics of local magnetization in the chaotic and non-chaotic clean chains is surprisingly similar, but differ significantly from the disordered system. We also discuss how quantum control methods consisting of sequences of magnetic pulses may be used to remove the effects of particular terms of the Hamiltonian, such as defects or unwanted interactions, and therefore induce a desired transport behavior.
We study transport of local magnetization in a Heisenberg spin-1/2 chain at zero temperature. The system is initially prepared in a highly excited pure state far from equilibrium and its evolution is analyzed via exact diagonalization. Integrable and nonintegrable regimes are obtained by adjusting the parameters of the Hamiltonian, which allows for the comparison of transport behaviors in both limits. In the presence of nearest-neighbor interactions only, the transport behavior in the integrable clean system contrasts with the chaotic chain with on-site defects, oscillations in the first suggesting ballistic transport, and a fast decay in the latter indicating diffusive transport. The results for a nonintegrable system with frustration are less conclusive, similarities with the integrable chain being verified. We also show how methods of quantum control may be applied to chaotic systems to induce a desired transport behavior, such as that of an integrable system.
In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.
In this Chapter, we present recent theoretical developments on the finite temperature transport of one dimensional electronic and magnetic quantum systems as described by a variety of prototype models. In particular, we discuss the unconventional transport and dynamic - spin, electrical, thermal - properties implied by the integrability of models as the spin-1/2 Heisenberg chain or Hubbard. Furthermore, we address the implication of these developments to experimental studies and theoretical descriptions by low energy effective theories.
We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.