We analyze the onset of incommensurabilities around the VBS point of the
S=1 bilinear-biquadratic model. We propose a simple effective field theory which is capable of reproducing all known properties of the commensurate-incommensurate transition at the disorder point ${\ensuremath{\theta}}_{\mathrm{VBS}}.$ Moreover, the theory predicts another special point ${\ensuremath{\theta}}_{\mathrm{disp}},$ distinct from the VBS point, where the Haldane gap behaves singularly. The ground state energy density is an analytic function of the model parameters everywhere, thus we do not have phase transitions in the conventional sense.
Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model are investigated using an infinite matrix product state algorithm. The bipartite entanglement entropy can detect a transition point between the two phases. In both phases, the entanglement spectrum shows double degeneracy. We calculate the nonlocal order parameter of the bond-centered inversion in both phases, which rapidly approaches a saturation value of −1 as the segment length increases. The nonlocal order parameter of the bond-centered inversion with a saturation value −1 and the nonzero value string order indicate that the Haldane phase is a symmetry-protected topological phase. To distinguish the commensurate and incommensurate Haldane phases, the transversal spin correlation and corresponding momentum distribution of the structure factor are analyzed. As a result, the transversal spin correlations exhibit different decay forms in both phases.
We systematically study the effects of higher-order interactions on the
S=23,1 orthogonal dimer chains using exact diagonalization and density matrix renormalization group. Due to frustration and higher spin, there are rich quantum phases, including three Haldane phases, two gapless phases, and several magnetically ordered phases. To characterize these phases and their phase transitions, we study various physical quantities such as energy gap, energy-level crossing, fidelity susceptibility, spin correlation, entanglement spectrum, and central charge. According to our calculations, the biquadratic term can enhance the Haldane phase regions. In particular, we numerically identify that a Haldane phase in
S=23 case is adiabatically connected to the exact AKLT (Affleck, Kennedy, Lieb, and Tasaki) point when adding a bicubic term. Our study on the orthogonal dimer model, which is a 1D version of the Shastry-Sutherland model, provides insights into understanding the possible
S=23,1 Haldane phases in quasi-1D and 2D frustrated magnetic materials.
The
S=21 chain-boundary excitations occurring in the Haldane phase of
S=1 antiferromagnetic spin chains are investigated. The bilinear-biquadratic Hamiltonian is used to study these excitations as a function of the strength of the biquadratic term, \ensuremath{\beta}, between $\ensuremath{-}1<~\ensuremath{\beta}<~1.$ At the Affleck-Kennedy-Lieb-Tasaki point, $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3},$ we show explicitly that these excitations are localized at the boundaries of the chain on a length scale equal to the correlation length $\ensuremath{\xi}=1/\mathrm{ln}3$, and that the on-site magnetization for the first site is
〈S1z〉=32. Applying the density matrix renormalization group we show that the chain-boundary excitations remain localized at the boundaries for $\ensuremath{-}1<\ensuremath{\beta}<1.$ As the two critical points \ensuremath{\beta}=\ifmmode\pm\else\textpm\fi{}1 are approached the size of the
S=21 objects diverges and their amplitude appears to vanish.
The excitation spectra of the
T=0 dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the center fluctuation operator in the one-dimensional
S=1 Heisenberg model with isotropic bilinear $(J\mathrm{cos}\ensuremath{\theta})$ and biquadratic $(J\mathrm{sin}\ensuremath{\theta})$ exchange are investigated via the recursion method for systems with up to
N=18 sites over the predicted range, $\ensuremath{-}\ensuremath{\pi}/4<\ensuremath{\theta}\ensuremath{\lesssim}\ensuremath{\pi}/4,$ of the topologically ordered Haldane phase. The four static and dynamic structure factors probe the ordering tendencies in the various coupling regimes and the elementary and composite excitations which dominate the
T=0 dynamics. At $\ensuremath{\theta}=\mathrm{arctan}\frac{1}{3}$ (valence-bond solid point), the dynamically relevant spectra in the invariant subspaces with total spin
ST=0,1,2 are dominated by a branch of magnon states
(ST=1), by continua of two-magnon scattering states
(ST=0,1,2), and by discrete branches of two-magnon bound states with positive interaction energy
(ST=0,2). The dimer and trimer spectra at $q=\ensuremath{\pi}$ are found to consist of single modes with
N-independent excitation energies ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{D}/|{e}_{0}|=5$ and ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{T}/|{e}_{0}|=6,$ where
e0=E0/N is the ground-state energy per site. The basic structure of the dynamically relevant excitation spectrum remains the same over a substantial parameter range within the Haldane phase. At the transition to the dimerized phase $(\ensuremath{\theta}=\ensuremath{-}\ensuremath{\pi}/4),$ the two-magnon excitations turn into two-spinon excitations.
We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field
h. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor $S(q,\ensuremath{\omega})$ as a function of
h. At
h=0, the envelope of the lowest-energy excitations goes soft at two points, ${q}_{1}=2\ensuremath{\pi}/3$ and ${q}_{2}=4\ensuremath{\pi}/3$, dubbed the phase A. With increasing field, the spectral peaks at each of the gapless points bifurcate, making in total four soft modes, and combine to form a new set of excitations that soften at a single point $q=\ensuremath{\pi}$ at ${h}_{c1}\ensuremath{\approx}0.94$. Beyond
hc1 the system enters another gapless B phase until the transition at
hc2=4 to the fully polarized phase. We compare the ULS model results with those for the Affleck-Kennedy-Lieb-Tasaki model as a representative of the gapped Haldane phase. We explain the mechanism of the gapless to gapless transition in the ULS model using its conserved charges and a spinon band picture. We also discuss the universality of central charges of the BLBQ family of models subjected to a magnetic field.