The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.
We report our results in investigating edge effects of open antiferromagnetic Heisenberg spin chains with spin magnitudes S=1/2,1,3/2,2 using the density-matrix renormalization-group method initiated by White. For integer spin chains, we find that edge states with spin magnitude
Sedge=S/2 exist, in agreement with the valence-bond-solid model picture. For half-integer spin chains, we find that no edge states exist for the S=1/2 spin chain, but an edge state exists in the S=3/2 spin chain with a
Sedge=1/2, in agreement with a previous conjecture by Ng. The properties of the edge state in the S=3/2 chain is studied in detail. Strong finite-size effects associated with spin dimerization in half-integer spin chains are also discussed.
We demonstrate the existence of nonlocal topological (string) order in half-integer-spin antiferromagnetic Heisenberg chains on macroscopic scale on the basis of analytical scaling analysis and density matrix renormalization group calculations. Strong numerical evidence leads to a conjecture that chains with S = (2m-1)/2 and m (m = integers) belong to the same topological class defined by the topological angle theta/pi = 1/m that plays a role similar to the fictitious gauge field in the fractional quantum Hall effect.
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 present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be
e0\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from
S=23 to
S=211 using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of
J1J2 containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and
22S+1 valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with
S and converge to a single point at
J1J2=41 in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for
S=21, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of
J1J2 values.