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arXiv:cond-mat/9804055 · Physical Review Letters

Nonlinear σ Model for Inhomogeneous Spin Chains

Ken'ichi Takano

Published Jun 21, 1999• 23 citations
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Abstract

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.

DOI
10.1103/physrevlett.82.5124

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We derive a nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ 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 mixed quantum spin chains consisting of two kinds of spins with magnitudes sa{s}_{a}sa​ and sb.{s}_{b}.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{s}_{a}sa​ and sb.{s}_{b}.sb​. We explain all phases in the singlet-cluster-solid picture which is an extension of the valence-bond-solid picture.
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.
The antiferromagnetic Heisenberg model on a chain with nearest- and next-nearest-neighbor couplings is mapped onto the SO(3) nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 branch at all orders in 1/N. We argue that this feature should be present in the spin-1 Heisenberg model itself. Exact diagonalizations are used to support this claim. The inapplicability of this model to half-integer spin chains is discussed.
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.
We investigate a spin-chain model composed of a periodic array of two kinds of spins S1{\mathrm{S}}_{1}S1​ and S2{\mathrm{S}}_{2}S2​, which allows us to study the spin chains with impurities, as well as the alternating spin chains, in a unified fashion. By using the Lieb-Shultz-Mattis theorem, we first study the model rigorously, and then by mapping it to the nonlinear \ensuremath{\sigma} model, we extensively investigate low-energy properties with particular emphasis on the competition between the massive and massless phases.