Statistical and dynamical properties in the energy spectra of XXZ spin chains

N Kitanine, Jean Michel Maillet, N. A. Slavnov +1 more

Kazue Kudo

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.

Jonathan P. Keating, Noah Linden, H. J. Wells

Ding-Zu Wang, Hao Zhu, Cui, Jian +5 more

The eigenstate thermalization hypothesis (ETH) is a successful theory that establishes the criteria for ergodicity and thermalization in isolated quantum many-body systems. In this work, we investigate the thermalization properties of spin-$ 1/2 $ XXZ chain with linearly-inhomogeneous interactions. We demonstrate that introduction of the inhomogeneous interactions leads to an onset of quantum chaos and thermalization, which, however, becomes inhibited for sufficiently strong inhomogeneity. To exhibit ETH, and to display its breakdown upon varying the strength of interactions, we probe statistics of energy levels and properties of matrix elements of local observables in eigenstates of the inhomogeneous XXZ spin chain. Moreover, we investigate the dynamics of the entanglement entropy and the survival probability which further evidence the thermalization and its breakdown in the considered model. We outline a way to experimentally realize the XXZ chain with linearly-inhomogeneous interactions in systems of ultracold atoms. Our results highlight a mechanism of emergence of ETH due to insertion of inhomogeneities in an otherwise integrable system and illustrate the arrest of quantum dynamics in presence of strong interactions.

Joachim Stolze, Taras Verkholyak, Oleg Derzhko +1 more

Ding-Zu Wang, Hao Zhu, Jian Cui +5 more

The eigenstate thermalization hypothesis (ETH) is a successful theory that establishes the criteria for ergodicity and thermalization in isolated quantum many-body systems. In this work, we investigate the thermalization properties of a spin-$1/2$ XXZ chain with linearly inhomogeneous interactions. We demonstrate that introduction of the inhomogeneous interactions leads to an onset of quantum chaos and thermalization, which, however, becomes inhibited for sufficiently strong inhomogeneity. To exhibit ETH, and to display its breakdown upon varying the strength of interactions, we probe statistics of energy levels and properties of matrix elements of local observables in eigenstates of the inhomogeneous XXZ spin chain. Moreover, we investigate the dynamics of the entanglement entropy and the survival probability which further evidence the thermalization and its breakdown in the considered model. We outline a way to experimentally realize the XXZ chain with linearly inhomogeneous interactions in systems of ultracold atoms. Our results highlight a mechanism of emergence of ETH due to insertion of inhomogeneities in an otherwise integrable system and illustrate the arrest of quantum dynamics in the presence of strong interactions.