Using the density-matrix renormalization-group (DMRG) technique, we carry out a large scale numerical calculation for the
S=2 antiferromagnetic Heisenberg (AFH) chain. We obtain the magnetization curve of
S=2 chain near the critical field. The corrections to the magnetization due to the magnon-magnon interaction are the same order for
S=2 and
S=1 chains. By comparing the experimental magnetization with our calculation, we find that the gap obtained experimentally is about
0.08J which is close to the accurate DMRG estimation for the
S=2 AFH chain. Numerical errors in our large scale calculation are also discussed.
We report on density-matrix renormalization-group and analytical work on
S=1 antiferromagnetic Heisenberg spin chains. We study the finite-size behavior within the framework of the nonlinear $\ensuremath{\sigma}$ model. We study the effect of magnon-magnon interactions on the finite-size spectrum and on the magnetization curve close to the critical magnetic field, determine the magnon scattering length, and compare it to the prediction from the nonlinear $\ensuremath{\sigma}$ model.
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 zero-temperature magnetization curve $(M\ensuremath{-}H$ curve) of the one-dimensional quantum antiferromagnet of spin one. The Hamiltonian H we consider is of the bilinear-biquadratic form: $H={\ensuremath{\sum}}_{i}f({s}_{i}\ensuremath{\cdot}{s}_{i+1})$ (+Zeeman term) where
si is the spin operator at site i and $f(X)=X+\ensuremath{\beta}{X}^{2}$ with $0<~\ensuremath{\beta}<1.$ We focus on validity of the $\ensuremath{\delta}$-function Bose-gas picture near the two critical fields: upper-critical field
Hs above which the magnetization saturates and the lower-critical field
Hc associated with the Haldane gap. As for the behavior near
Hs, we take ``low-energy effective S matrix'' approach, where the correct effective Bose-gas coupling constant c is extracted from the two down-spin S matrix in its low-energy limit. We find that the resulting value of c differs from the spin-wave value. We draw the $M\ensuremath{-}H$ curve by using the resultant Bose gas, and compare it with numerical calculation where the product-wave-function renormalization-group (PWFRG) method, a variant of White's density-matrix renormalization group method, is employed. Excellent agreement is seen between the PWFRG calculation and the correctly mapped Bose-gas calculation. We also test the validity of the Bose-gas picture near the lower-critical field
Hc. Comparing the PWFRG-calculated $M\ensuremath{-}H$ curves with the Bose-gas prediction, we find that there are two distinct regions, I and II, of $\ensuremath{\beta}$ separated by a critical value ${\ensuremath{\beta}}_{c}(\ensuremath{\approx}0.41).$ In region I, $0<\ensuremath{\beta}<{\ensuremath{\beta}}_{c},$ the effective Bose coupling c is positive but rather small. The small value of c makes the ``critical region'' of the square-root behavior $M\ensuremath{\sim}\sqrt{H\ensuremath{-}{H}_{c}}$ very narrow. Further, we find that in the $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\beta}}{\ensuremath{\beta}}_{c}\ensuremath{-}0,$ the square-root behavior transmutes to a different one, $M\ensuremath{\sim}(H\ensuremath{-}{H}_{c}{)}^{\ensuremath{\theta}}$ with $\ensuremath{\theta}\ensuremath{\approx}1/4.$ In region II, ${\ensuremath{\beta}}_{c}<\ensuremath{\beta}<1,$ the square-root behavior is more pronounced as compared with region I, but the effective coupling c becomes negative.
This thesis examines the Heisenberg antiferromagnetic spin chain in one dimension (1D) with a crystal field splitting term and applied magnetic field term. We use theoretical techniques from quantum field theory and conformal field theory (CFT) to make predictions about the excitation spectrum for our model. We then use Density Matrix Renormalization Group (DMRG) numerical techniques to simulate our spin chain and extract the energy spectrum as we vary our crystal field splitting and magnetic field terms. These results are compared and we examine where theoretical calculations accurately describe our system. This work is motivated by recent experimental work done on SrNi₂Vi₂O₈ by Bera et al. [1] which is a quasi-1D material with weakly coupled spin chains in the bulk. These 1D chains are expected to be described by the Hamiltonian we study in this thesis, and we neglect interchain coupling. We first consider our system where the crystal field splitting term is set to zero, which can be described theoretically using a mapping to the non linear sigma model (NLSM). Near the critical field, it undergoes a Bose condensation transition whose excitation spectrum can be mapped to non-interacting fermions in 1D. We then consider large negative crystal field splitting, and find that near small applied magnetic field we can describe some excited states using Landau-Ginsburg theory. Near critical field, we show that the transition is in the Ising universality, and use results from CFT to predict the spectrum for finite size systems. This allows us to make predictions about where the transition field would be for very large or infinite system size. Finally, we examine our crystal field splitting tuned to the value obtained in Ref. 1, which is a small, negative value. We observe qualitative elements in this spectrum from the spectra obtained at zero and large negative crystal field splitting.
We study the 1D quantum Heisenberg chain with randomly ferromagnetic or antiferromagnetic couplings [a model previously studied by approximate strong-disorder renormalization group (RG)]. We find that, at least for sufficiently large spin S , the ground state has “spin glass” order. The spin waves on top of this state have the dynamical exponent z = 3 / 2 , intermediate between the values z = 1 of the antiferromagnet and z = 2 of the ferromagnet. Density matrix renormalization group (DMRG) simulations are in good agreement with the analytical results for spins S = 1 and S = 3 / 2 . The case S = 1 / 2 shows large finite size effects: We suggest that this case is also ordered, but with a small ordered moment.