In this thesis we present an approach, similar to random matrix ensembles, in order to study the integrable-chaotic transition in the Heisenberg spin model. We consider three ways to break the integrability: presence on an external field on a single spin, coupling of an external random field with each spin in the chain and next nearest neighbor interaction between spins. We propose a transition described by a power law in the spectral density, i.e. S(k) ∝ 1/kα, where α = 2 for the integrable case and α = 1 for the chaotic case, with 1 < α < 2 for systems in the crossover regime. The transition is also described by the behavior of the "burstiness" B and the Kullback–Leibler divergence DLK(PW−D(s)|Pdata(s)), where PW−D(s) and Pdata(s) are the Wigner-Dyson and the system’s spacing distribution respectively. The B coefficient is associated to a sequence of events in the system. The Kullback–Leibler divergence provides information on how two distributions differ from each other. From analyzing the behavior of these three quantities, we obtain a universal description of integrable-chaotic transition in the spin chains.
In this thesis we present an approach, similar to random matrix ensembles, in order to study the integrable-chaotic transition in the Heisenberg spin model. We consider three ways to break the integrability: presence on an external field on a single spin, coupling of an external random field with each spin in the chain and next nearest neighbor interaction between spins. We propose a transition described by a power law in the spectral density, i.e. S(k) ∝ 1/kα, where α = 2 for the integrable case and α = 1 for the chaotic case, with 1 < α < 2 for systems in the crossover regime. The transition is also described by the behavior of the "burstiness" B and the Kullback–Leibler divergence DLK(PW−D(s)|Pdata(s)), where PW−D(s) and Pdata(s) are the Wigner-Dyson and the system’s spacing distribution respectively. The B coefficient is associated to a sequence of events in the system. The Kullback–Leibler divergence provides information on how two distributions differ from each other. From analyzing the behavior of these three quantities, we obtain a universal description of integrable-chaotic transition in the spin chains.
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 study numerically the evolution of energy-level statistics as an integrability-breaking term is added to the XXZ Hamiltonian. For finite-length chains, physical properties exhibit a crossover from behavior corresponding to the Poisson level statistics characteristic of integrable models to behavior corresponding to the Wigner-Dyson statistics characteristic of the random-matrix theory used to describe chaotic systems. Different measures of the level statistics are observed to follow different crossover patterns. The range of numerically accessible system sizes is too small to establish with certainty the scaling with system size, but the evidence suggests that in a thermodynamically large system an infinitesimal integrability breaking would lead to Wigner-Dyson behavior.
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.
The statistics of gaps between quantum energy levels is a hallmark criterion in quantum chaos and quantum integrability studies. The relevant distributions corresponding to exactly integrable vs. fully chaotic systems are universal and described by the Poisson vs. Wigner-Dyson curves. In the transitional regime between integrability and chaos, the distributions are much less universal and have not been understood quantitatively until now. We point out that the relevant statistics that controls these distributions is that of the matrix elements of the nonintegrable perturbation Hamiltonian in the energy eigenbasis of the unperturbed integrable system. With this insight, we formulate a simple random matrix ensemble that correctly reproduces the level spacing distributions in a variety of test systems. For the distribution of matrix elements appearing in our construction, we furthermore discover surprising universal features: across a variety of physical systems with diverse degrees of freedom, these distributions are dominated by simple power laws.
We study the spectral properties of and spectral crossovers between different random matrix ensembles [Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE)] in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term, competing with the usual isotropic and time-reversal invariant Heisenberg term. We have investigated these crossovers in the context of the level-spacing distribution and the level-spacing ratio distribution. We use random matrix theory (RMT) analytical results to fit the observed Poissonian-to-GOE and GOE-to-GUE crossovers, and examine the relationship between the RMT crossover parameter $\ensuremath{\lambda}$ and scaled physical parameters of the spin-chain systems in terms of a scaling exponent. We find that the crossover behavior exhibits universality, in the sense that, it becomes independent of lattice size in the large Hamiltonian matrix dimension limit. Moreover, the scaling exponent obtained from such a finite size scaling analysis, seems to be quite robust and independent of the type of crossover considered or the specific spectral correlation measure used.
We present a systematic investigation of the short-range spectral fluctuation properties of three non-Hermitian spin-chain Hamiltonians using complex spacing ratios (CSRs). Specifically, we focus on the non-Hermitian variants of the standard one-dimensional anisotropic XY model having intrinsic rotation-time (RT) symmetry that has been explored analytically by Zhang and Song [Phys. Rev. A 87, 012114 (2013)1050-294710.1103/PhysRevA.87.012114]. The corresponding Hermitian counterpart is also exactly solvable and has been widely employed as a toy model in several condensed matter physics problems. We show that the presence of a random field along the x direction together with the one along the z direction facilitates integrability and RT-symmetry breaking, leading to the emergence of quantum chaotic behavior. This is evidenced by a spectral crossover closely resembling the transition from Poissonian to Ginibre unitary ensemble (GinUE) statistics of random matrix theory. Additionally, we consider two phenomenological random matrix models in this paper to examine 1D Poisson to GinUE and 2D Poisson to GinUE crossovers and the associated signatures in CSRs. Here 1D and 2D Poisson correspond to real and complex uncorrelated levels, respectively. These crossovers reasonably capture spectral fluctuations observed in the spin-chain systems within a certain range of parameters.