Comparing numerically evaluated excitation gaps of dimerized spin-
21 XXZ chains with the gap formula for the low-energy effective sine-Gordon theory, we determine coefficients
dxy and
dz of bosonized dimerization operators in spin-
21 XXZ chains, which are defined as ${(\ensuremath{-}1)}^{j}({S}_{j}^{x}{S}_{j+1}^{x}+{S}_{j}^{y}{S}_{j+1}^{y})={d}_{xy}\text{ }\text{sin}[\sqrt{4\ensuremath{\pi}}\ensuremath{\phi}(x)]+\ensuremath{\cdots}$ and ${(\ensuremath{-}1)}^{j}{S}_{j}^{z}{S}_{j+1}^{z}={d}_{z}\text{ }\text{sin}[\sqrt{4\ensuremath{\pi}}\ensuremath{\phi}(x)]+\ensuremath{\cdots}$. We also calculate the coefficients of both spin and dimer operators for the spin-
21 Heisenberg antiferromagnetic chain with a nearest-neighbor coupling
J and a next-nearest-neighbor coupling
J2=0.2411J. As applications of these coefficients, we present ground-state phase diagrams of dimerized spin chains in a magnetic field and antiferromagnetic spin ladders with a four-spin interaction. The optical conductivity and electric polarization of one-dimensional Mott insulators with Peierls instability are also evaluated quantitatively.
We study the ground-state phase diagram of a spin-
21 frustrated XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung and diagonal interactions, ${J}_{\ensuremath{\perp}}$ and ${J}_{\ifmmode\times\else\texttimes\fi{}}$. Previous studies on the isotropic model have revealed that a fluctuation-induced effective dimer attraction between the legs stabilizes the columnar dimer (CD) phase in the highly frustrated regime ${J}_{\ensuremath{\perp}}\ensuremath{\approx}2{J}_{\ifmmode\times\else\texttimes\fi{}}$, especially for ferromagnetic ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}<0$. By means of effective field theory and numerical analyses, we extend this analysis to the XXZ model, and we obtain a rich phase diagram. The diagram includes four gapped featureless phases with no symmetry breaking: the rung singlet (RS) and Haldane phases as well as their twisted variants, the RS* and Haldane* phases, which are all distinct in the presence of certain symmetries. Significantly, the Haldane-CD transition point in the isotropic model turns out to be a crossing point of two transition lines in the XXZ model, and the stripe N\'eel and RS* phases appear between these lines. This indicates a nontrivial interplay between the effective dimer attraction and the exchange anisotropy. In the easy-plane regime, the four featureless phases and two critical phases are found to compete in a complex manner depending on the signs of ${J}_{\ensuremath{\perp},\ifmmode\times\else\texttimes\fi{}}$.
We investigate the phase diagram of antiferromagnetic spin ladders with additional exchange interactions on diagonal bonds by variational and numerical methods. These generalized spin ladders interpolate smoothly between the [Formula: see text] chain with competing nn and nnn interactions, the [Formula: see text] chain with alternating exchange and the antiferromagnetic (AF) S = 1 chain. The Majumdar - Ghosh ground states are formulated as matrix product states and are shown to exhibit the same type of hidden order as the AF S = 1 chain. Generalized matrix product states are used for a variational calculation of the ground state energy and the spin and string correlation functions. Numerical (Lanczos) calculations of the energies of the ground state and of the low-lying excited states are performed, and compare reasonably with the variational approach. Our results support the hypothesis that the dimer and Majumdar - Ghosh points are in the same phase as the AF S = 1 chain.