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arXiv:cond-mat/9704072 · Physical review. B, Condensed matter

Spin chains with a periodic array of impurities

Takahiro Fukui, Norio Kawakami

Published Jun 1, 1997• 49 citations
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Abstract

We investigate a spin-chain model composed of a periodic array of two kinds of spins ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$, 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.

DOI
10.1103/physrevb.55.r14709

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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 investigate low-energy properties of the alternating-spin-chain model composed of spins s1{s}_{1}s1​ and s2{s}_{2}s2​ with a singlet ground state. After examining the spin-wave spectrum in detail, we map low-energy spin excitations to the O(3) nonlinear \ensuremath{\sigma} model in order to take into account quantum fluctuations. Analyzing the topological term in the resulting \ensuremath{\sigma} model, we discuss how the massless or massive excitations are developed, especially according to the topological nature of the alternating spin system.
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 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.
Two quantum spin chains problems are discussed. Firstly, using the exact form factors of the 0(3) nonlinear a model the three-magnon contribution to the spin correlation function of a spin-1 chain is calculated. A very broad peak with a small amplitude is predicted. The integrated intensity of this contribution relative to the single magnon peak is of the order 2%. Experimentally, such a contribution would be difficult, if not impossible, to observe. In the second problem we consider the problem of doping spin-1 magnetic materials with nonmagnetic impurities. In the case where interchain coupling is weak, the nonmagnetic impurities effectively cut decoupled spin chains into finite lengths. We impose free boundary conditions at the chain ends and apply bosonization, mapping the problem onto a torus. Using this formalism we are able to numerically calculate the alternating susceptibility for any reasonable length chain at any temperature. The Neel temperature is then calculated for a three dimensional square lattice as a function of average chain length.