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Critical Properties of 1-D Spin 1/2 Antiferromagnetic Heisenberg Model

J. F. Audet, Andreas Fledderjohann, C. Gerhardt, Michael Karbach, H. Kröger, K.-H. Mütter, M. Schmidt

Published Jan 1, 1997• 1 citation
OpenAlex

Abstract

Abstract. We discuss numerical results for the 1 − D spin 1/2 antiferromagnetic Heisenberg model with next-to-nearest neighbour coupling and in the presence of an uniform magnetic field. The model develops zero frequency excitations at field dependent soft mode momenta. We compute critical quantities from finite size dependence of static structure factors. 1

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Critical behavior of the S=1 antiferromagnetic Heisenberg chain (abstract)

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The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.

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Physical review. B, Condensed matter•May 1, 1994•6 citations
Abstract. We discuss numerical results for the 1 − D spin 1/2 antiferromagnetic Heisenberg model with next-to-nearest neighbour coupling and in the presence of an uniform magnetic field. The model develops zero frequency excitations at field dependent soft mode momenta. We compute critical quantities from finite size dependence of static structure factors. 1
In the presence of a uniform field the one-dimensional spin-1/2 antiferromagnetic Heisenberg model develops zero frequency excitations at field-dependent ``soft-mode'' momenta. We determine three types of critical quantities, which we extract from the finite-size dependence of the lowest excitation energies, the singularities in the static structure factors and the infrared singularities in the dynamical structure factors at the soft mode momenta. We also compare our results with the predictions of conformal field theory. \textcopyright{} 1996 The American Physical Society.
On the basis of a numerical computation of the ground states in the sectors with a given total spin S we study the magnetic properties of the one-dimensional frustrated antiferromagnetic Heisenberg model at zero temperature and at the spin-fluid - dimer phase transition. We find that the magnetization curve M(B) has a quartic-root singularity near saturation, . The longitudinal spin - spin, the dimer - dimer and the transverse spin - spin structure factors develop singularities at the field-dependent momenta and , respectively. The type of each of these singularities depends on the frustration parameter and the magnetization M.
We study the one-dimensional spin-1/2 model with nearest and next-to-nearest-neighbor couplings exposed to a homogeneous magnetic field h3{h}_{3}h3​ and a dimer field with period q and strength $\ensuremath{\delta}$. The latter generates a magnetization plateau at $M=(1\ensuremath{-}q/\ensuremath{\pi})/2$, which evolves with strength $\ensuremath{\delta}$ of the perturbation as ${\ensuremath{\delta}}^{\ensuremath{\epsilon}}$, where $\ensuremath{\epsilon}=\ensuremath{\epsilon}{(h}_{3},\ensuremath{\alpha})$ is related to the $\ensuremath{\eta}$ exponent which describes the critical behavior of the dimer structure factor, if the perturbation is switched off $(\ensuremath{\delta}=0)$. We also discuss the appearance of magnetization plateaus in ladder systems with l legs.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
The properties of the one-dimensional antiferromagnet of spin 1/2 in a strong magnetic field h with nearest- and next-nearest-neighbor interactions are studied in terms of the equivalent Bose-gas problem. We find that below the first critical field hcu{\mathit{h}}_{\mathit{c}\mathit{u}}hcu​ the phase with a gap in the elementary excitations spectrum and corresponding plateau in the magnetization curve should exist due to the two-particle attraction in the vicinity of the commensurate wave number p0u{\mathit{p}}_{0\mathit{u}}p0u​=\ensuremath{\pi}. In the interval between the first and second critical fields hcu{\mathit{h}}_{\mathit{c}\mathit{u}}hcu​hhcf{\mathit{h}}_{\mathit{c}\mathit{f}}hcf​, the system behaves as the one-dimensional Bose gas with particle repulsion and the magnetization has a typical square-root dependence on field.