Recent progress in the field of quantum magnets has shown that a wide variety of novel spin model can be generated with exploiting the optical lattices. Among them the spin cluster interaction gets some attention. In this work, we have considered a spin-1∕2 XXZ Heisenberg chain with added cluster interaction. In the absence of the cluster interaction, the XXZ chain is solvable by Bethe ansatz. Introducing a small defect in the middle of chain drives the XXZ model to the chaotic phase not for strong anisotropic parameter. Using the numerical full diagonalization method, we have shown that in presence of a cluster interaction, how quantum chaos may develop even for strong Ising interaction region. In addition, the effect of the cluster length and strength on the crossover from the integrable to the chaotic regime is also addressed.
Recent progress in the field of quantum magnets has shown that a wide variety of novel spin model can be generated with exploiting the optical lattices. Among them the spin cluster interaction gets some attention. In this work, we have considered a spin-1∕2 XXZ Heisenberg chain with added cluster interaction. In the absence of the cluster interaction, the XXZ chain is solvable by Bethe ansatz. Introducing a small defect in the middle of chain drives the XXZ model to the chaotic phase not for strong anisotropic parameter. Using the numerical full diagonalization method, we have shown that in presence of a cluster interaction, how quantum chaos may develop even for strong Ising interaction region. In addition, the effect of the cluster length and strength on the crossover from the integrable to the chaotic regime is also addressed.
Using one-dimensional spin-1/2 systems as prototypes of quantum many-body systems, we study the emergence of quantum chaos. The main purpose of this work is to answer the following question: how the spin-orbit interaction, as a pure quantum interaction, may lead to the onset of quantum chaos? We consider the three integrable spin-1/2 systems: the Ising, the XX, and the XXZ limits and analyze whether quantum chaos develops or not after the addition of the Dzyaloshinskii-Moriya interaction. We find that depending on the strength of the anisotropy parameter, the answer is positive for the XXZ and Ising models, whereas no such evidence is observed for the XX model. We also discuss the relationship between quantum chaos and thermalization.
We study the chaotic properties of a large-spin XXZ chain with onsite disorder and a small number of excitations above the fully polarized state. We show that while the classical limit, which is reached for large spins, is chaotic, enlarging the spin suppresses quantum chaos features. We examine ways to facilitate chaos by introducing additional terms to the Hamiltonian. Interestingly, perturbations that are diagonal in the basis of product states in the z z -direction do not lead to significant enhancement of chaos, while off-diagonal perturbations restore chaoticity for large spins, so that only three excitations are required to achieve strong level repulsion and ergodic eigenstates.
Abstract We have considered the spin-1/2 XXZ chain model which is exactly solvable with the Bethe anzats method. It is shown adding a single defect away from edges is sufficient to see a chaos signature in the model. In this work, we have studied the time behavior of the Loschmidt-echo (LE) by considering domain-wall and Neel states as two product initial states and also a Bell state as an entangled initial state. We have used the numerical full diagonalization method to have full spectrum of the model. It is shown that very short times behavior of the LE is independent from defect and initial state, and the decay is quadratic. Moreover, results show that in the chaotic regime and for intermediate times, the exponential decay behavior of the LE shows some changes by removing defect from the chain in the domain-wall and Bell states, while no changes is seen in the Neel state. To get some intuition of these changes, we have addressed the relation between the dynamics of the LE and the local density of states features.
Spectral properties of XXZ spin chains are studied intensively. The integrable XXZ spin chain is one of the most important quantum integrable systems, and its Hamiltonian is given by H = J L ∑ j=1 (S j S x j+1 + S y j S y j+1 +∆S z jS z j+1). (1) Here, L is the number of sites; S j = 1 2 σ j and σ α j (α = x, y, z) is the Pauli matrix on the jth site. Equation (1) corresponds to the Heisenberg spin chain at ∆ = 1, to the XY model at ∆ = 0, and to the Ising spin chain when ∆ → ∞. In this thesis, we study on XXZ spin chains with next-nearest-neighbor (NNN) coupling and/or with a magnetic field as well as Eq. (1). First, we discuss level crossings in the energy spectra of the integrable XXZ spin chain. Some of the level crossings are counterexamples of a theorem by von Neumann and Wigner: two energy levels with the same symmetry rarely cross each other in the flow of the spectrum. Multiple degeneracies appear at the particular crossing points where the symmetry of the system is enhanced in the spectral flow. The eigenvalues and the eigenvectors are obtained in an analytical form for 3-site and 4-site chains. The spectral flow of a 6-site chain is shown by numerical calculation. Not only level crossings but also “level bifurcations” are found in the flow of eigenvalues for the inhomogeneous transfer matrix of the XXZ spin chain. The “level bifurcations” are branches such as bifurcation or multiple branches in the flow of eigenvalues and not observed for the homogeneous transfer matrix. The observations are obtained by numerical calculation. We also confirm the appearance of some of the branches through the Bethe ansatz method. The behavior of some branching points are analyzed in the flow of eigenvalues for a particular inhomogeneous case. Level statistics is discussed for the XXZ spin chains with NNN coupling or with a random magnetic field. Evaluating the level-spacing distribution, the spectral rigidity and the number variance, we confirm the following conjecture: the level-spacing distribution should be described by the Poisson distribution for an integrable model and by the Wigner distribution for a nonintegrable model. The XXZ spin chains with the NNN coupling (J2) or with a random magnetic field (h) are nonintegrable, so that mostly the level statistics shows Wigner behavior. However, the level statistics even for a nonintegrable model sometimes shows Poisson-like behavior because of the finite-size effects. For the NNN coupled XXZ spin chain, the behavior of level statistics comes close to the Poissonian around J2 = 0 and/or ∆ = 1. As h/J becomes large, Poisson-like behavior appears for the level statistics of the XXZ spin chain under random magnetic field. When h/J 0, however, Poisson-like behavior appears for small L. In addition to the Poisson-like behavior, unexpected non-Wigner behavior of level statistics can appear due to mixed symmetry. When we discuss level statistics, we have to consider the whole symmetry of a system and perform desymmetrization. The NNN coupled XXZ chains are invariant under spin rotation around the z axis, translation, reflection. However, we have to consider spin reversal symmetry only when S z tot = 0. If spin reversal symmetry is not considered for S z tot = 0, the behavior of level statistics is described by almost the numerical average of Poisson and Wigner behavior. To observe the dynamical properties of the energy spectra, we investigate energy diffusion for NNN coupled XXZ spin chains in the presence of a periodically oscillating magnetic field. Diffusion coefficients (D) are found to obey the power law with respect to both the field strength (B0) and driving frequency (ω): D ∝ (B0ω) . We observe that β = 2 for “linear response regime” and β = 1 for “non-perturbative regime”. In the linear response regime, the diffusion coefficient can be calculated using the Kubo formula. On the other hand, B0ω is so large in the non-perturbative regime that the perturbation theory fails. The diffusion coefficient is smaller than that predicted by the Kubo formula and the perturbation theory fails in the non-perturbative regime. The ranges of the linear response and the non-perturbative regimes depend on J2. When J2 is small, we observe large-amplitude oscillations, which is attributed to a non-diffusive and ballistic nature of the underlying energy diffusion. In this way, we investigate the energy spectra of finite quantum XXZ spin chains in several different aspects. We hope that the results of the thesis will contribute to the development of the physics of low-dimensional systems and mesoscopic systems.
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.
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.