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The ground-state phase diagram of Heisenberg spin-1/2 system on a two-leg ladder with rung alternation is studied by combining analytical approaches with numerical simulations. For the case of ferromagnetic leg exchanges a unique ferrimagnetic ground state emerges, whereas for the case of antiferromagnetic leg exchanges several different ground states are stabilized depending on the ratio between exchanges along legs and rungs. For the more general case of a honeycomb-ladder model for the case of ferromagnetic leg exchanges besides usual rung-singlet and saturated ferromagnetic states we obtain a ferrimagnetic Luttinger liquid phase with both linear and quadratic low energy dispersions and ground state magnetization continuously changing with system parameters. For the case of antiferromagnetic exchanges along legs, different dimerized states including states with additional topological order are suggested to be realized.

In a comment on arXiv:1006.5070v2, Drechsler et al. claim that the frustrated ferromagnetic spin-1/2 chain LiCuVO4 should be described by a strong rather than weak ferromagnetic nearest-neighbor interaction, in contradiction with their previous work. Their comment is based on DMRG and ED calculations of the magnetization curve and the magnetic excitations. We show that their parameters are at odds with the magnetic susceptibility and the magnetic excitation spectrum, once intensities are taken into account, and that the magnetization curve cannot discriminate between largely different parameter sets within experimental uncertainties. We further show that their new exact diagonalization results support the validity of the RPA-approach, and strongly reinforce our conclusion on the existence of a four-spinon continuum in LiCuVO4, see Enderle et al., Phys. Rev. Lett. 104 (2010) 237207., Comment: 1 page

In a comment on arXiv:1006.5070v1, Drechsler et al. present new band-structure calculations suggesting that the frustrated ferromagnetic spin-1/2 chain LiCuVO4 should be described by a strong rather than weak ferromagnetic nearest-neighbor interaction, in contradiction with their previous calculations. In our reply, we show that their new results are at odds with the observed magnetic structure, that their analysis of the static susceptibility neglects important contributions, and that their criticism of the spin-wave analysis of the bound-state dispersion is unfounded. We further show that their new exact diagonalization results reinforce our conclusion on the existence of a four-spinon continuum in LiCuVO4, see Enderle et al., Phys. Rev. Lett. 104 (2010) 237207., Comment: 2 pages, 1 figure

The effect of thermal fluctuations on the dynamics of a gapped quantum magnet is studied using inelastic neutron scattering on copper nitrate, a model material for the one-dimensional (1D) bond alternating Heisenberg chain, combined with theoretical and numerical analysis. We observe and interpret the thermally induced central peak due to intraband scattering as well as the thermal development of an asymmetric continuum of scattering. We relate this asymmetric line broadening to hard core constraints and quasi-particle interactions. Our findings are a counter example to recent assertions of universality of line broadening in 1D systems and are to be considered as a new paradigm of behaviour, applicable to a broad range of quantum systems., Comment: 5 pages 4 figures

We investigate the phase diagram of TmB4, an Ising magnet on a frustrated Shastry-Sutherland lattice by neutron diffraction and magnetization experiments. At low temperature we find Neel order at low field, ferrimagnetic order at high field and an intermediate phase with magnetization plateaus at fractional values M/Msat = 1/7, 1/8, 1/9 ... and spatial stripe structures. Using an effective S = 1/2 model and its equivalent two-dimensional (2D) fermion gas we suggest that the magnetic properties of TmB4 are related to the fractional quantum Hall effect of a 2D electron gas., Comment: 4 pages, 3 figures

We present results on the dynamics of the distorted diamond chain, S=1/2 dimers alternating with single spins 1/2 and exchange couplings $J_1$ and $J_3$ in between. The dynamics in the spin fluid (SF) and tetramer-dimer (TD) phases is investigated numerically by exact diagonalisation for up to 24 spins. Representative excitation spectra are presented, both for zero magnetic field and in the 1/3 plateau phase and the relevant parameters are determined across the phase diagram. The behavior across the SF-TD phase transition line is discussed for the specific heat and for excitation spectra. The relevance of the distorted diamond chain model for the material Cu$_3$(CO$_3$)$_2$(OH)$_2$ (azurite) is discussed with particular emphasis on inelastic neutron scattering experiments, a recent suggestion of one possibly ferromagnetic coupling constant is not confirmed., Comment: 24 pages, 8 figures

Magnetic field effects on the one-dimensional frustrated ferromagnetic chain are studied by means of effective field theory approaches in combination with numerical calculations utilizing Lanczos diagonalization and the density matrix renormalization group method. The nature of the ground state is shown to change from a spin-density-wave region to a nematic-like one upon approaching the saturation magnetization. The excitation spectrum is analyzed and the behavior of the single spin-flip excitation gap is studied in detail, including the emergent finite-size corrections., Comment: 6 pages

We present results for the dynamic structure factor of the S=1/2 bond alternating Heisenberg chain over a large range of frequencies and temperatures. Data are obtained from a numerical evaluation of thermal averages based on the calculation of all eigenvalues and eigenfunctions for chains of up to 20 spins. Interpretation is guided by the exact temperature dependence in the noninteracting dimer limit which remains qualitatively valid up to an interdimer exchange $\lambda \approx 0.5$. The temperature induced central peak around zero frequency is clearly identified and aspects of the crossover to spin diffusion in its variation from low to high temperatures are discussed. The one-magnon peak acquires an asymmetric shape with increasing temperature. The two-magnon peak is dominated by the S=1 bound state which remains well defined up to temperatures of the order of J. The variation with temperature and wavevector of the integrated intensity for one and two magnon scattering and of the central peak are discussed., Comment: 8 pages, 8 figures

This paper has been withdrawn., Comment: This paper has been withdrawn because it is an excerpt

We investigate analytically and numerically the dynamics of domain walls in a spin chain with ferromagnetic Ising interaction and subject to an external magnetic field perpendicular to the easy magnetization axis (transverse field Ising model). The analytical results obtained within the continuum approximation and numerical simulations performed for discrete classical model are used to analyze the quantum properties of domain walls using the semiclassical approximation. We show that the domain wall spectrum shows a band structure consisting of 2$S$ non-intersecting zones., Comment: 15 pages, 9 figures

We study S=1/2 dimers which are weakly coupled by three-dimensional antiferromagnetic interactions J'. It is commonly known that doping with non-magnetic impurities immediately induces a long-range Neel order. We show that application of an external magnetic field H may drive the system back to a disordered phase. We discuss the zero temperature phase diagram in the (H,J') plane and we propose suggestions for experiments., Comment: 4 Revtex pages, 2 figures

We study the phase diagram of spin ladders with ferromagnetic legs under the influence of a symmetry breaking magnetic field in the weak coupling effective field theory by bosonization. For antiferromagnetic interleg coupling we identify two phase transitions introduced by the external magnetic field. In order to establish the universality of the phases we supplement the bosonization approach by results from a strong coupling (rung) expansion and from spin wave analysis., Comment: 7 revtex pages, 6 figures

We discuss the transition strength between the disordered ground state and the basic low-lying triplet excitation for interacting dimer materials by presenting theoretical calculations and series expansions as well as inelastic neutron scattering results for the material KCuCl_3. We describe in detail the features resulting from the presence of two differently oriented dimers per unit cell and show how energies and spectral weights of the resulting two modes are related to each other. We present results from the perturbation expansion in the interdimer interaction strength and thus demonstrate that the wave vector dependence of the simple dimer approximation is modified in higher orders. Explicit results are given in 10th order for dimers coupled in 1D, and in 2nd order for dimers coupled in 3D with application to KCuCl_3 and TlCuCl_3., Comment: 17 pages, 6 figures, part 2 is based on cond-mat/0211335

We report the results of elastic and inelastic neutron scattering experiments on the Haldane gap quantum antiferromagnet Ni(C5D14N2)2N3(PF6) performed at mK temperatures in a wide range of magnetic field applied parallel to the S = 1 spin chains. Even though this geometry is closest to an ideal axially symmetric configuration, the Haldane gap closes at the critical field Hc~4T, but reopens again at higher fields. The field dependence of the two lowest magnon modes is experimentally studied and the results are compared with the predictions of several theoretical models. We conclude that of several existing theories, only the recently proposed model [Zheludev et al., cond-mat/0301424 ] is able to reproduce all the features observed experimentally for different field orientations., Comment: 11 pages 8 figures submitted to Phys. Rev. B

We have performed electron spin resonance measurements over a wide frequency and magnetic field range on a single crystal of the S=1 quasi-one-dimensional Heisenberg antiferromagnet $\rm Ni(C_5H_{14}N_2)_2N_3(PF_6)$. We observed gapped excitation branches above the critical field ($H_{\rm c}$) where the Haldane gap closes. These branches are analyzed by a phenomenological field-theory using complex-field $\phi^{4}$ model. A satisfactory agreement between experiment and theory is obtained., Comment: 4 revtex pages, 3 eps figures included

We discuss the transition strength between the disordered ground state and the basic low-lying triplet excitation for the Heisenberg chain with alternating exchange based on series expansions in the interdimer interaction up to tenth order. We demonstrate that the wave vector dependence of the simple dimer approximation is strongly modified in higher orders., Comment: 9 pages, 3 figures, corrected typos in equations (11)-(13)

The low-temperature properties of the spin S=1/2 ladder with anisotropic ferromagnetic legs are studied using the continuum limit bosonization approach. The weak-coupling ground state phase diagram of the model is obtained for a wide range of coupling constants and several unconventional gapless ''spin-liquid'' phases are shown to exist for ferromagnetic coupling. The behavior of the ladder system in the vicinity of the ferromagnetic instability point is discussed in detail., Comment: 11 pages, 4 figures

We investigate low-lying singlet and triplet excitations and the magnetization process of quasi-1D spin systems composed of tetrahedral spin clusters. For a class of such models, we found various exact low-lying excitations; some of them are responsible for the first-order transition between two different ground states formed by local singlets. Moreover, we find that there are two different kinds of magnetization plateaus which are separated by a first-order transition., Comment: To appear in Phys.Rev.B (Issue 01 August 2002). A short comment is added

The isotropic S=1/2 antiferromagnetic spin ladder with additional four-spin ring exchange is studied perturbatively in the strong coupling regime with the help of cluster expansion technique, and by means of bosonization in the weak coupling limit. It is found that a sufficiently large strength of ring exchange leads to a second-order phase transition, and the shape of the boundary in the vicinity of the known exact transition point is obtained. The critical exponent for the gap is found to be $\eta\simeq1$, in agreement both with exact results available for the dimer line and with the bosonization analysis. The phase emerging for high values of the ring exchange is argued to be gapped and spontaneously dimerized. The results for the transition line from strong coupling and from weak coupling match with each other naturally., Comment: 8 pages, 4 figures, some minor changes in text and references

We study the dynamical structure factor of gapped one-dimensional spin systems in the critical phase in high magnetic field. It is shown that the presence of a ``condensate'' in the ground state in the high-field phase leads to interesting signatures in the response functions., Comment: uses ptptex.sty (included), 10 pages, 3 figs, to appear in Prog. Theor. Phys. Suppl. (Proc. of the 16th Nishinomiya Yukawa Memorial Symposium)

TlCuCl$_3$ has a singlet ground state with the excitation gap $\Delta =0.65$ meV. The magnetic excitations in TlCuCl$_3$ have been investigated by means of neutron inelastic scattering experiments. The constant-$\it{\pmb Q}$ energy scan profiles were collected in the $a^*-c^*$ plane. A well-defined single magnetic excitation mode was observed. The dispersion relations along four different directions were determined. The lowest excitation occurs at ${\it{\pmb Q}}=(h, 0, l)$ with integer $h$ and odd $l$, as observed in KCuCl$_3$. A cluster series expansion up to the sixth order was applied to analyze the dispersion relations, and the individual exchange interactions were evaluated. It was demonstrated that TlCuCl$_3$ is a strongly coupled spin-dimer system., Comment: 20 pages, 6 figures, 12 eps files, revtex style

We have studied dimerized spin systems by realizing the cluster expansion to high order. We have extended our previous dimer expansion for one-dimensional systems to cover weakly interacting chains for a quantitative description of three dimensional materials like PHCC and KCuCl_3. By comparison with recent inelastic neutron scattering data we are able to determine the exchange energies between individual spins. We have further investigated the incommensurate region of zigzag chains with isotropic exchange coupling constants near the disorder-line where the dispersion curve exhibits a minimum at a finite wavevector. Our approach clearly shows the gradual transition between the minimum of the dispersion at wavevector 0 and wavevector Pi within this region. The extent of the incommensurate regime is given analytically in an expansion in the coupling constants., Comment: 3 pages, 3 figures; contribution to ICNS2001; uses svjour.clo, svglobal.clo (included)

Motivated by recent experiments on Ni(C_{2}H_{8}N_{2})_{2}Ni(CN)_{4} (commonly known as NENC), we study the electron spin resonance in the critical high-field phase of the antiferromagnetic S=1 chain with strong planar anisotropy and show that the ESR spectra exhibit several peculiarities in the critical phase. Possible relevance of those results for other gapped spin systems is discussed., Comment: 8 revtex pages, 1 eps figure included

We study, both numerically and variationally, the interplay between different types of elementary excitations in the model of a spin chain with anisotropic spin-orbit coupling, in the vicinity of the "dimer line" with an exactly known dimerized ground state. Our variational treatment is found to be in a qualitative agreement with the exact diagonalization results. Soliton pairs are shown to be the lowest excitations only in a very narrow region of the phase diagram near the dimer line, and the phase transitions are always governed by magnon-type excitations which can be viewed as soliton-antisoliton bound states. It is shown that when the anisotropy exceeds certain critical value, a new phase boundary appears. In the doped model on the dimer line, the exact elementary charge excitation is shown to be a hole bound to a soliton. Bound states of those "charged solitons" are studied; exact solutions for N-hole bound states are presented., Comment: 11 pages revtex, 6 figures

We investigate the elementary excitations of quasi one-dimensional S=1/2 systems built up from zig-zag chains with general isotropic exchange constants, using exact (Lanczos) diagonalization for 24 spins and series expansions starting from the decoupled dimer limit. For the ideal one-dimensional zig-zag chain we discuss the systematic variation of the basic (magnon) triplet excitation with general exchange parameters and in particular the presence of practically flat dispersions in certain regions of phase space. We extend the dimer expansion in order to include the effects of 3D interactions on the spectra of weakly interacting zig-zag chains. In an application to KCuCl_3 we show that this approach allows to determine the exchange interactions between individual pairs of spins from the spectra as determined in recent neutron scattering experiments., Comment: 8 pages, 9 figures; some changes, figure added; final version

Specific heat measurements down to 0.5 K have been performed on a single crystal sample of a spin-ladder like compound Cu$_{2}$(C$_{5}$H$_{12}$N$_{2}$)$_{2}$Cl$_{4}$ under magnetic fields up to 12 T. The temperature dependence of the observed data in a magnetic field below 6 T is well reproduced by numerical results calculated for the S=1/2 two-leg ladder with $J_{\rm{rung}}$/$J_{\rm{leg}}$=5. In the gapless region above 7 T ($H_{\rm{c1}}$), the agreement between experiment and calculation is good above about 2 K and a sharp and a round peak were observed below 2 K in a magnetic field around 10 T, but the numerical data show only a round peak, the magnitude of which is smaller than that of the observed one. The origin of the sharp peak and the difference between the experimental and numerical round peak are discussed., Comment: 14 pages, 11 figures, Submitted to PRB

A model of two-leg spin-S ladder with two additional frustrating diagonal exchange couplings J_{D}, J_{D}' is studied within the framework of the nonlinear sigma model approach. The phase diagram has a rich structure and contains 2S gapless phase boundaries which split off the boundary to the fully saturated ferromagnetic phase when J_{D} and J_{D}' become different. For the S=1/2 case, the phase boundaries are identified as separating two topologically distinct Haldane-type phases discussed recently by Kim et al. (cond-mat/9910023)., Comment: revtex 4 pages, figures embedded (psfig)

We investigate the influence of biquadratic exchange interactions on the low-lying excitations of a S=1/2-ladder using perturbation theory, numerical diagonalization of finite systems and exact results for ladders with matrix product ground states. We consider in particular the combination of biquadratic exchange interactions corresponding to ring exchange on the basic ladder plaquette. We find that a moderate amount of ring exchange reduces the spin gap substantially and makes equal bilinear exchange on legs and rungs consistent with experimentally observed spectra., Comment: RevTeX, 17 pages, 6 embedded figures, uses epsfig

We propose an effective theory for the critical phase of a quantum ferrimagnetic chain with alternating spins 1 and 1/2 in an external magnetic field. With the help of the matrix product variational approach, the system is mapped to a spin-1/2 XXZ chain in an (effective) magnetic field; as a byproduct, we obtain an excellent description of the optical magnon branch in the gapped phase. Recent finite-temperature DMRG results for the low-temperature part of the specific heat are well described by the present approach, and the ``pop-up'' peaks, developing near the critical field values and in the middle of the critical phase, are identified with the contributions from two different spinon bands of the effective spin-1/2 chain. The effect should be as well observable in other spin-gap systems in an external field, particularly in spin ladders., Comment: 4 RevTeX pages, 2 ps figures included

We investigate the specific heat and magnetisation of a ferrimagnet with gS=1 and S=1/2 spins in a finite magnetic field using the transfer matrix DMRG down to T=0.025J. Ferromagnetic gapless and antiferromagnetic gapped excitations for H=0 lead to rich thermodynamics for H > 0. While the specific heat is characterized by a generic double peak structure, magnetisation reveals two critical fields, Hc1=1.76(1) and Hc2=3.00(1) with square-root behaviour in the T=0 magnetisation. Simple analytical arguments allow to understand these experimentally accessible findings., Comment: 5 pages, 7 eps figures, uses RevTeX, submitted to PRB

We study two-leg S=1/2 ladders with general isotropic exchange interactions between spins on neighboring rungs, whose ground state can be found exactly in a form of finitely correlated (matrix product) wave function. Two families of models admitting an exact solution are found: one yields translationally invariant ground states and the other describes spontaneously dimerized models with twofold degenerate ground state. Several known models with exact ground states can be obtained as particular cases from the general solution of the first family, which includes also a set of models with only bilinear interactions. Those two families of models have nonzero intersection, which enables us to determine exactly the phase boundary of the second-order transition into the dimerized phase and to study the properties of this transition. The structure of elementary excitations in the dimerized phase is discussed on the basis of a variational ansatz. For a particular class of models, we present exact wave functions of the elementary excitations becoming gapless at second-order transition lines. We also propose a generalization of the Bose-Gayen ladder model which has a rich phase diagram with all phase boundaries being exact., Comment: 13 pages revtex, 5 figures embedded

Ground-state and thermodynamic properties of the one-dimensional Heisenberg antiferromagnet in which two S=1/2 and two S=1 spins are arranged alternatively are studied by a quantum Monte Carlo method and by analytical estimates. It is found that the ground state of the system undergoes a second-order phase transition which accompanies the vanishing of the energy gap., Comment: 6 pages; 3 JPG figures; plain TEX; to be published in J. Magn. Magn. Mater. (Proc. Int. Conf. Magnetism, Cairns, 1997)

We present a family of spin ladder models which admit exact solution for the ground state and exhibit non-Haldane spin liquid properties as predicted recently by Nersesyan and Tsvelik [Phys. Rev. Lett. v.78, 3939 (1997)], and study their excitation spectrum using a simple variational ansatz. The elementary excitation is neither a magnon nor a spinon, but a pair of propagating triplet or singlet solitons connecting two spontaneously dimerized ground states. Second-order phase transitions separate this phase from the Haldane phase and the rung-dimer phase., Comment: 5 pages RevTeX, final published version

We study the "mixed spin" isotropic ladder system having S=1 spins on one leg and S=1/2 spins on the other, with general-type exchange interactions between spins on neighboring rungs. A set of model Hamiltonians with exact ground states in the form of a certain matrix product wave function is obtained. We show that sufficiently strong frustration can lead to exotic singlet ground states with infinite (exponential) degeneracy. We also list a couple of rather simple models with nontrivial ground states, including a model with only bilinear exchange., Comment: 8 pages, 2 figures, RevTeX, uses psfig.sty

We numerically investigate elementary excitations of the Heisenberg alternating-spin chains with two kinds of spins 1 and 1/2 antiferromagnetically coupled to each other. Employing a recently developed efficient Monte Carlo technique as well as an exact diagonalization method, we verify the spin-wave argument that the model exhibits two distinct excitations from the ground state which are gapless and gapped. The gapless branch shows a quadratic dispersion in the small-momentum region, which is of ferromagnetic type. With the intention of elucidating the physical mechanism of both excitations, we make a perturbation approach from the decoupled-dimer limit. The gapless branch is directly related to spin 1's, while the gapped branch originates from cooperation of the two kinds of spins., Comment: 7 pages, 7 Postscript figures, RevTex

We use the variational matrix-product ansatz to study elementary excitations in the S=1/2 ladder with additional diagonal coupling, equivalent to a single S=1/2 chain with alternating exchange and next-nearest neighbor interaction. In absence of alternation the elementary excitation consists of two free S=1/2 particles ("spinons") which are solitons in the dimer order. When the nearest-neighbor exchange alternates, the "spinons" are confined into one S=1 excitation being a soliton in the generalized string order. Variational results are found to be in a qualitative agreement with the exact diagonalization data for 24 spins. We argue that such an approach gives a reasonably good description in a wide range of the model parameters., Comment: RevTeX, 13 pages, 11 embedded figures, uses psfig and multicol

We investigate the isotropic two-leg S=1/2 ladder with general bilinear and biquadratic exchange interactions between spins on neighboring rungs, and determine the Hamiltonians which have a matrix product wavefunction as exact ground state. We demonstrate that a smooth change of parameters leads one from the S=1/2 dimer and Majumdar-Ghosh chains to the S=1 chain with biquadratic exchange. This proves that these model systems are in the same phase. We also present a new set of models of frustrated S=1/2 spin chains (including only bilinear NN and NNN interactions) whose ground states can be found exactly., Comment: 4 pages, RevTeX, uses psfig.sty, submitted to Phys. Rev. Lett

We study the low-temperature properties of S=1 and 1/2 alternating spin chains with antiferromagnetic nearest-neighbor exchange couplings using analytical techniques as well as a quantum Monte Carlo method. The spin-wave approach predicts two different low-lying excitations, which are gapped and gapless, respectively. The structure of low-lying levels is also discussed by perturbation theory in the strength of the Ising anisotropy. These analytical findings are compared with the results of quantum Monte Carlo calculations and it turns out that spin-wave theory well describes the present system. We conclude that the quantum ferrimagnetic chain exhibits both ferromagnetic and antiferromagnetic aspects., Comment: 13 pages, RevTeX, six figures, submitted to J. Phys. Cond. Mat

We propose a new version of the matrix product (MP) states approach to the description of quantum spin chains, which allows one to construct MP states with certain total spin and its z-projection. We show that previously known MP wavefunctions for integer-spin antiferromagnetic chains and ladders correspond to some particular cases of our general ansatz. Our method allows to describe systems with spontaneously broken rotational symmetry, like quantum ferrimagnetic chains whose ground state has nonzero total spin. We apply this approach to describe the ground state properties of the isotropic ferrimagnetic Heisenberg chain with alternating spins 1 and 1/2 and compare our variational results with the high-precision numerical data obtained by means of the quantum Monte Carlo (QMC) method. For both the ground state energy and the correlation functions we obtain very good agreement between the variational results and the QMC data., Comment: 4 pages, RevTeX, uses psfig.sty, submitted to Phys. Rev. B

We investigate antiferromagnetic spin ladders with nonmagnetic impurities by variational and numerical (Lanczos and DMRG) methods. The interaction between the two unpaired spins opposite to the impurities is described by an effective exchange interaction $J_{eff}$, the magnitude of which depends on the impurity distance. The magnitude of $J_{eff}$ is different for unpaired spins at the edges of an open ladder and in the bulk. This difference is related to the different distribution of the unpaired spin into the bulk of the ladder. The numerical results are interpreted using matrix product states. Using the DMRG we calculate the spectrum of low-lying energy levels for up to 6 impurities and find that these spectra can be reproduced assuming pair interactions with an accuracy of better than 10\%. We discuss the filling of the ladder gap with impurity states and argue that in the thermodynamic limit the spin ladder with a finite concentration of impurities always shows a Curie susceptibility at low temperatures., Comment: 21 pages, RevTex, 5 figures, 3 tables, submitted to Phys. Rev. B

We investigate the phase diagram of antiferromagnetic spin ladders with additional exchange interactions on diagonal bonds by variational and numerical methods. These generalized spin ladders interpolate smoothly between the $S=1/2$ chain with competing nn and nnn interactions, the $S=1/2$ chain with alternating exchange and the antiferromagnetic $S=1$ chain. The Majumdar-Ghosh ground states are formulated as matrix product states and are shown to exhibit the same type of hidden order as the af $S=1$ chain. Generalized matrix product states are used for a variational calculation of the ground state energy and the spin and string correlation functions. Numerical (Lanczos) calculations of the energies of the ground state and of the low-lying excited states are performed, and compare reasonably with the variational approach. Our results support the hypothesis that the dimer and Majumdar-Ghosh points are in the same phase as the af $S=1$ chain., Comment: 23 pages, REVTEX, 7 figures