The repartition of the separation between energy levels of various isotropic S=1/2 antiferromagnetic chains is studied numerically with the aim of investigating the transition from integrable to noninte- grable systems. We begin by displaying the level separation distribution of the integrable Bethe chain. Then two nonintegrable systems, two coupled chains, and a next-nearest-neighbor coupled chain, are studied as a function of the coupling. We examine how the level spacing evolves from the Poisson distribution to the Gaussian-orthogonal-ensemble distribution. Finally we consider the Haldane-Shastry 1/${\mathit{r}}^{2}$ model. A number of conclusions regarding the behavior and relevance of the level-spacing distribution in these spin systems is drawn.
The repartition of the separation between energy levels of various isotropic S=1/2 antiferromagnetic chains is studied numerically with the aim of investigating the transition from integrable to noninte- grable systems. We begin by displaying the level separation distribution of the integrable Bethe chain. Then two nonintegrable systems, two coupled chains, and a next-nearest-neighbor coupled chain, are studied as a function of the coupling. We examine how the level spacing evolves from the Poisson distribution to the Gaussian-orthogonal-ensemble distribution. Finally we consider the Haldane-Shastry 1/r2 model. A number of conclusions regarding the behavior and relevance of the level-spacing distribution in these spin systems is drawn.
The level-spacing distributions of XXZ spin chains with next-nearest-neighbor couplings are studied under periodic boundary conditions. We confirm that integrable XXZ spin chains mostly have the Poisson distribution as expected. On the contrary, the level-spacing distributions of next-nearest-neighbor coupled XXZ chains are given by non-Wigner distributions. It is against the expectations, since the models are nonintegrable.
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.
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 derive an exact expression for the partition function of the su(m) Haldane-Shastry spin chain, which we use to study the density of levels and the distribution of the spacing between consecutive levels. Our computations show that when the number of sites N is large enough, the level density is Gaussian to a very high degree of approximation. More surprisingly, we also find that the nearest-neighbor spacing distribution is not Poissonian, so that this model departs from the typical behavior for an integrable system. We show that the cumulative spacing distribution of the model can be well approximated by a simple functional law involving only three parameters.
Level statistics is discussed for XXZ spin chains with discrete symmetries for some values of the next-nearest-neighbor (NNN) coupling parameter. We show how the level statistics of the finite-size systems depends on the NNN coupling and the XXZ anisotropy, which should reflect competition among quantum chaos, integrability and finite-size effects. Here discrete symmetries play a central role in our analysis. Evaluating the level-spacing distribution, the spectral rigidity and the number variance, we confirm the correspondence between non-integrability and Wigner behavior in the spectrum. We also show that non-Wigner behavior appears due to mixed symmetries and finite-size effects in some nonintegrable cases.
Abstract The full spectrum and integrability of unitary equivalent models are the same. A standard diagnostic tool for integrability is level spacing statistics, which requires separating the full spectrum into sectors according to the symmetry. When two unitary equivalent models have different symmetries, it is interesting to see how their level spacing statistics exhibit consistent conclusions. In this work, we examine the level spacing statistics of two unitary equivalent models with distinct local symmetries. The first model is the spin-1 XXZ chain H , and the second model is obtained by Kennedy–Tasaki transformation U KT H U KT . We found that the level spacings of model H follow the statistics of the Gaussian orthogonal ensemble (GOE). However, model U KT H U KT only displays GOE statistics in some sectors after a ‘hidden nonlocal symmetry’ is resolved. In addition, we study how the bond-centered inversion transforms under U KT . A mapping relation between the levels in the sector { Z , X , I } of model H and the sector { Z , X , I ′ } of model U KT H U KT is found, where Z , X , I correspond to the quantum numbers of π rotation around z , x axes, and bond-centered inversion.