The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to
5/2. For open chains with spin magnitudes
S=5/2 to
S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes
S> 1.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.
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.
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.
The structure of edge states in general finite antiferromagnetic quantum spin chains with arbitrary spin value S is discussed within the framework of the nonlinear-sigma model (NL\ensuremath{\sigma}M) plus a topological \ensuremath{\theta} term. Based on a large-N theory of SU(N) quantum antiferromagnets and strong-coupling expansion, we argue that edge states with fractionalized spin quantum number S' exist in all spin chains with S\ensuremath{\ge}1, with S'=S/2 for integer spin chains and S'=(S-1/2)/2 for half-integer spin chains.
Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.