At large values of the anisotropy Δ, the open-boundary Heisenberg spin- chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this ‘edge-locking’ phenomenon in the entire Δ > 1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors. Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large Δ there are two eigenstates with imaginary Bethe momenta, related to localization at the two edges. For any finite chain size, one of the two solutions become real as Δ is lowered below a certain value. For two particles, a richer scenario is observed, with eigenstates having the possibility of both particles locked on the same or different edge, one locked and the other free, or both free either as single magnons or as bound composites corresponding to ‘string’ solutions. For finite chains, some of the edge-locked spins get delocalized at certain values of Δ (‘exceptional points’), corresponding to imaginary solutions becoming real. We characterize these phenomena thoroughly by providing analytic expansions of the Bethe momenta for large chains, large anisotropy Δ, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (Δ = 1) and edge-locking becomes stable in the whole Δ > 1 region.
At large anisotropies, the spectrum of the Heisenberg XXZ spin chain separates into ``bands'' with energies largely determined by the number of domain walls. The band structure is richer with open-boundary conditions: there are more bands and the bands develop intricate fine structures. We characterize and explain these structures and substructures in the open-boundary chain. The fine structures are explained using degenerate perturbation theory. We also present some dynamical consequences of these subband structures, through explicit time evolution of the wave function from initial states motivated by the fine-structure analysis.
We investigate an open XYZ spin- chain driven out of equilibrium by boundary reservoirs targeting different spin orientations, aligned along the principal axes of anisotropy. We show that by tuning local magnetic fields, applied to spins at sites near the boundaries, one can change any nonequilibrium steady state to a fully uncorrelated Gibbsian state at infinite temperature. This phenomenon occurs for strong boundary coupling and on a critical manifold in the space of the fields amplitudes. The structure of this manifold depends on the anisotropy degree of the model and on the parity of the chain size.
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
SG when $S\ensuremath{\ge}1$. The ground state of open chains of
N spins has
SG=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.
We investigate an open XYZ spin-1/2 chain driven out of equilibrium by boundary reservoirs targeting different spin orientations, aligned along the principal axes of anisotropy. We show that by tuning local magnetic fields, applied to spins at sites near the boundaries, one can change any nonequilibrium steady state to a fully uncorrelated Gibbsian state at infinite temperature. This phenomenon occurs for strong boundary coupling and on a critical manifold in the space of the fields amplitudes. The structure of this manifold depends on the anisotropy degree of the model and on the parity of the chain size.