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

TL;DR

This paper investigates boundary-induced spin density waves in linear Heisenberg antiferromagnetic chains with spins greater than or equal to 1, using DMRG calculations to analyze edge states, their phase relations, and decay properties.

Contribution

It provides a detailed quantitative analysis of boundary-induced spin density waves in HAFs with various spins, including modeling of spin densities and excitation energies for long chains.

Findings

Edge states are boundary-induced spin density waves with alternating phase.

Excitation energy decreases exponentially with chain length for integer spins.

Spin densities and energies are modeled using correlation length and SDW amplitude.

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 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 $\Gamma(N) \propto (-1)^N \exp(-N/\xi)$...