Boundary-induced spin-density waves in linear Heisenberg antiferromagnetic spin chains with S≥1

Abstract

Linear Heisenberg antiferromagnets (HAFs) are chains of spin-$S$ sites with isotropic exchange $J$ between neighbors. Open and periodic boundary conditions return the same ground-state energy per site in the thermodynamic limit, but not the same spin ${S}_{G}$ when $S\ensuremath{\ge}1$. The ground state of open chains of $N$ spins has ${S}_{G}=0$ or $S$, respectively, for even or odd $N$. Density-matrix renormalization-group calculations with different algorithms for even and odd $N$ are presented up to $N=500$ for the energy and spin densities $\ensuremath{\rho}(r,N)$ of edge states in HAFs with $S=1$, 3/2, and 2. The edge states are boundary-induced spin density waves (BI-SDWs) with $\ensuremath{\rho}(r,N)\ensuremath{\propto}{(\ensuremath{-}1)}^{r\ensuremath{-}1}$ for $r=1,2,...,N$. The SDWs are in phase when $N$ is odd, are out of phase when $N$ is even, and have finite excitation energy $\mathrm{\ensuremath{\Gamma}}(N)$ that decreases exponentially with $N$ for integer $S$ and faster than $1/N$ for half integer $S$. The spin densities and excitation energy are quantitatively modeled for integer $S$ chains longer than $5\ensuremath{\xi}$ spins by two parameters, the correlation length $\ensuremath{\xi}$ and the SDW amplitude, with $\ensuremath{\xi}=6.048$ for $S=1$ and 49.0 for $S=2$. The BI-SDWs of $S=3/2$ chains are not localized and are qualitatively different for even and odd $N$. Exchange between the ends for odd $N$ is mediated by a delocalized effective spin in the middle that increases $|\mathrm{\ensuremath{\Gamma}}(N)|$ and weakens the size dependence. The nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$) has been applied to the HAFs, primarily to $S=1$ with even $N$, to discuss spin densities and exchange between localized states at the ends as $\mathrm{\ensuremath{\Gamma}}(N)\ensuremath{\propto}{(\ensuremath{-}1)}^{N}exp(\ensuremath{-}N/\ensuremath{\xi})$. $S=1$ chains with odd $N$ are fully consistent with the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$; $S=2$ chains have two gaps $\mathrm{\ensuremath{\Gamma}}(N)$ with the same $\ensuremath{\xi}$ as predicted whose ratio is 3.45 rather than 3; the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is more approximate for $S=3/2$ chains with even $N$ and is modified for exchange between ends for odd $N$.