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 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.
The recently proposed classification of integrability-breaking perturbations according to their strength is studied in the context of quantum field theories. Using random matrix methods to diagnose the resulting quantum chaotic behaviour, we investigate the \phi^4 ϕ 4 and \phi^6 ϕ 6 interactions of a massive scalar, by considering the crossover between Poissonian and Wigner-Dyson distributions in systems truncated to a finite-dimensional Hilbert space. We find that a naive extension of the scaling of crossover coupling with the volume observed in spin chains does not give satisfactory results for quantum field theory. Instead, we demonstrate that considering the scaling of the crossover coupling with the number of particles yields robust signatures, and is able to distinguish between the strengths of integrability breaking in the \phi^4 ϕ 4 and \phi^6 ϕ 6 quantum field theories.
The level-spacing distribution of a spin-21XXZ chain is numerically studied under random magnetic field. We show explicitly how the level statistics depends on the lattice size L, the anisotropy parameter $\ensuremath{\Delta},$ and the mean amplitude of the random magnetic field h. In the energy spectrum, quantum integrability competes with nonintegrability derived from the randomness, where the XXZ interaction is modified by the parameter $\ensuremath{\Delta}.$ When $\ensuremath{\Delta}\ensuremath{\ne}0,$ the level-spacing distribution mostly shows Wigner-like behavior, while when $\ensuremath{\Delta}=0,$ Poisson-like behavior appears although the system is nonintegrable due to randomness. Poisson-like behavior also appears for $\ensuremath{\Delta}\ensuremath{\ne}0$ in the large h limit. Furthermore, the level-spacing distribution depends on the lattice size L, particularly when the random field is weak.
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.
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.
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.
The many-body localization transition (MBLT) between ergodic and many-body localized phases in disordered interacting systems is a subject of much recent interest. The statistics of eigenenergies is known to be a powerful probe of crossovers between ergodic and integrable systems in simpler examples of quantum chaos. We consider the evolution of the spectral statistics across the MBLT, starting with mapping to a Brownian motion process that analytically relates the spectral properties to the statistics of matrix elements. We demonstrate that the flow from Wigner-Dyson to Poisson statistics is a two-stage process. First, a fractal enhancement of matrix elements upon approaching the MBLT from the delocalized side produces an effective power-law interaction between energy levels, and leads to a plasma model for level statistics. At the second stage, the gas of eigenvalues has local interactions and the level statistics belongs to a semi-Poisson universality class. We verify our findings numerically on the XXZ spin chain. We provide a microscopic understanding of the level statistics across the MBLT and discuss implications for the transition that are strong constraints on possible theories.