Mixed quantum spin chains with periodicity are transformed into a nonlinear σ model (NLSM). The NLSM deduces a general formula which determines whether the system has a spin gap or not. The formula is applied to some concrete systems and their ground-state phase diagrams are derived. The odd period case is generally discussed.
We study a period-4 antiferromagnetic mixed quantum spin chain consisting of three kinds of spins. When the ground state is singlet, the spin magnitudes in a unit cell are arrayed as (s-t, s, s+t, s) with integer or half-odd integer s and t (0 <= t < s). The spin Hamiltonian is mapped onto a nonlinear sigma model (NLSM) in a previously developed method. The resultant NLSM includes only two independent parameters originating from four exchange constants for fixed s and t. The topological angle in the NLSM determines the gapless phase boundaries between disordered phases in the parameter space. The phase diagrams for various s and t shows rich structures. We systematically explain the phases in the singlet-cluster-solid picture.
We study mixed quantum spin chains consisting of two kinds of spins with magnitudes sa and sb. The spins are arrayed as ${s}_{a}\ensuremath{-}{s}_{a}\ensuremath{-}{s}_{b}\ensuremath{-}{s}_{b}$ in a unit cell and the exchange couplings are accordingly periodic with period 4. The spin Hamiltonian is mapped onto a nonlinear $\ensuremath{\sigma}$ model based on the general formula for periodic inhomogeneous spin chains. The gapless condition given by the nonlinear $\ensuremath{\sigma}$ model determines boundaries between disordered phases in the space of the exchange parameters. The phase diagram has a rich phase structure characterized by the values of sa and sb. We explain all phases in the singlet-cluster-solid picture which is an extension of the valence-bond-solid picture.
We derive a nonlinear $\ensuremath{\sigma}$ model (NLSM) that represents antiferromagnetic Heisenberg spin chains with inhomogeneous spin magnitudes and inhomogeneous nearest-neighbor exchange constants arrayed in finite periods. The only restriction is that the average spin magnitude on a sublattice is the same as that on another sublattice. The NLSM yields the gapless condition in terms of the spin magnitudes and exchange constants. We apply this condition to several cases, including systems with impurities. The result shows that the spin gap persists, despite impurities, in many cases.
We study a quantum spin system on a honeycomb lattice, when it includes frustration and distortion in antiferromagnetic (AF) exchange interactions. We transform the spin system onto a nonlinear $\ensuremath{\sigma}$ model (NLSM), preserving the original spin degrees of freedom. Assisted by a renormalization-group argument, the NLSM provides a ground-state phase diagram consisting of an ordered AF phase and a disordered spin-gap phase. The spin-gap phase extends from a strong frustration regime to a strong distortion regime, showing that the disordered ground states are essentially the same in both regimes. In the spin-1/2 case, the spin-gap phase for spin system on a honeycomb lattice is larger than that for the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model on a square lattice.
We study a frustrated mixed spin chain with side chains, where the spin species and the exchange interactions are spatially varied. A nonlinear $\ensuremath{\sigma}$ model method is formulated for this model, and a phase diagram with two disordered spin-gap phases is obtained for typical cases. Among them, we examine the case with a main chain, which consists of an alternating array of spin-1 and spin-21 sites, and side chains, each of which consists of a single spin-21 site, in great detail. Based on numerical, perturbational, and variational approaches, we propose a singlet cluster solid picture for each phase, where the ground state is expressed as a tensor product of local singlet states.
The well known Haldane map from spin chains into the O(3) nonlinear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, in exactly the same way as it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the parameter, which multiplies the topological term, is equal to , where S is the spin of the ladder. Hence even ladders should have a dynamically generated spin gap, while odd ladders with half-integer spin should stay gapless and physically equivalent to a perturbed Wess - Zumino - Witten model in the infrared regime. We also derive some consequences from the dependence of the sigma model coupling constant on the ladder Heisenberg couplings constants.