Boundary-induced spin density waves in linear Heisenberg antiferromagnetic spin chains with $\mathbf{S \ge 1}$

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 in the thermodynamic limit, but not the same spin $S_G$ when $S \ge 1$. 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 (DMRG) calculations with different algorithms for even and odd N are presented up to N = 500 for the energy and spin densities $\rho(r,N)$ of edge states in HAFs with $S = 1$, 3/2 and 2. The edge states are boundary-induced spin density waves (BI-SDWs) with $\rho(r,N)\propto(-1)^{r-1}$ for $r=1,2,\ldots N$. The SDWs are in phase when N is odd, out of phase when N is even, and have finite excitation energy $\Gamma(N)$ 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 $5 \xi$ spins by two parameters, the correlation length $\xi$ and the SDW amplitude, with $\xi = 6.048$ 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 $|\Gamma(N)|$ and weakens the size dependence. The nonlinear sigma model (NL$\sigma$M) has been applied the HAFs, primarily to $S = 1$ with even N, to discuss spin densities and exchange between localized states at the ends as $\Gamma(N) \propto (-1)^N \exp(-N/\xi)$...