Dynamics of one-dimensional quantum spin systems with competing interactions at zero temperature

Abstract

This dissertation reports three projects on the zero-temperature dynamics of some prominent one-dimensional (1D) quantum spin systems. A common feature of all three model systems is the presence of competing interactions. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis. The first model investigated is the 1D s = 1 XXZ model with an additional uniaxial single-site term. The results for the in-plane and out-of-plane spin dynamic structure factors $S\\sb{\\mu\\mu}(q,\\omega),\\ \\mu = x,\\ z$ are found to bear the characteristic signatures of several phase transitions between the Haldane phase, the Neel phase, the singlet phase, and two different critical phases. The second model investigated is the 1D s = 1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling. The focus here is on the dynamic structure factors $S\\sb{zz}(q,\\omega)$ and $S\\sb{DD}(q,\\omega$), which describe (for q = $\\pi$) the fluctuations of the Neel and dimer order parameters, respectively. By means of a weak-coupling continued-fraction analysis, the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distributions of S$\\sb{zz}(\\pi,\\omega$) and $S\\sb{DD}(\\pi,\\omega$) in the weak-coupling regime of the spin-fluid phase are calculated. For some parameter values, a discrete branch of excitations above the spinon continuum is found. The discrete states contribute to $S\\sb{zz}(q,\\omega$) but not to $S\\sb{DD}(q,\\omega$). A strong-coupling continued-fraction analysis is employed for the study of the dynamically relevant dispersions and spectral-weight distributions of $S\\sb{zz}(q,\\omega$) and $S\\sb{DD}(q,\\omega$) in the strong-coupling regime of the spin-fluid phase and in the dimer or Neel ordered phases. Special attention is given to the case where the pure dimer ground state is realized exactly. The third model investigated is the 1D s = 1 XXX model with bilinear and biquadratic exchange. Here the study is focused on the special coupling ratio where the valence-bond-solid ground state is realized. The excitation spectra which are dynamically relevant for the dynamic spill and dimer structure factors, $S\\sb{zz}(q,\\omega$) and $S\\sb{DD}(q,\\omega$), are investigated via two different types of continued-fraction analysis. The spectral weight in $S\\sb{ZZ}(q,\\omega$) is found to be carried predominantly by a single branch of quasi-particles and a continuum of two-particle scattering states thereof, whereas the spectral weight in $S\\sb{DD}(q,\\omega$) is dominated by the continuum alone, which shrinks to a single mode at $q = \\pi$.