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AIP conference proceedings

The spectral weight of low-lying excitations in the one-dimensional spin 1/2 Heisenberg antiferromagnet

J. C. Bonner, Bill Sutherland, Peter M. Richards

Published Jan 1, 1975• 9 citations
Publisher

Abstract

We have examined numerical calculations on a spin 1/2 one‐dimensional (1D) Heisenberg antiferromagnet for N=8 spins, both for the states and their spectral weights. This work is of interest because of the recent neutron scattering experiments in CuCl2⋅2NC5D5 which show the spectral weight to be concentrated at the des Cloizeaux and Pearson (dCP) spin wave frequencies. The states form a two‐parameter continuum between the dCp value E1(k) =π‖J‖ ‖sin k‖ and the upper limit E2(k) =2π‖J‖ ‖ sin(k/2) ‖. Though most of the spectral weight is at the dCP frequency for N=8, about 10% of it is found near E2 for k=π. There is no numerical evidence for the existence of singlet bound states as predicted by Ovchinnikov.

DOI
10.1063/1.30115

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Spin Waves in a One-Dimensional Heisenberg Antiferromagnet

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The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.

Progress of Theoretical Physics•Jan 1, 1973•7 citations

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

Jill C. Bonner

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.

Journal of Applied Physics•Apr 15, 1991•1 citation

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The neutron-scattering intensity and energy eigenvalues of the elementary excitations of the S=1 antiferromagnetic Heisenberg chain with single-ion anisotropy [scrH=J ${\mathit{tsum}}_{\mathit{i}=1}^{\mathit{N}}$ ${\mathbf{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathit{i}+1}$+D tsum(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$${)}^{2}$] are calculated with a numerical diagonalization method and a projector Monte Carlo method. We set D/J=0.2,0.0,-0.2 and N=14,16,18,32. For D\ensuremath{\ne}0, the elementary excitations in the subspace of ${\mathit{S}}^{\mathit{z}}$=\ifmmode\pm\else\textpm\fi{}1 and in the subspace of ${\mathit{S}}^{\mathit{z}}$=0 are different. The ${\mathit{S}}^{\mathit{z}}$=\ifmmode\pm\else\textpm\fi{}1 branch is lower than the ${\mathit{S}}^{\mathit{z}}$=0 branch near Q=\ensuremath{\pi}; however, it becomes slightly higher in other regions. The crossing of levels occurs at Q\ensuremath{\simeq}\ifmmode\pm\else\textpm\fi{}0.7\ensuremath{\pi}. We find that the low-lying excited state has a weight of more than 90% in the dynamical structure factor, except for Q\ensuremath{\simeq}0. Our results explain recent experimental results on Ni(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4}$.

Physical review. B, Condensed matter•Jul 1, 1993•51 citations

Quantum effects in the spin dynamics of the linear Heisenberg antiferromagnet

Gerhard H. Muller, Hans Beck, Jill C. Bonner

We present an approximate analytic expression for the dynamical spin correlation function of the S=1/2 linear Heisenberg antiferromagnet at T=0. The basis for our approach is that in zero field the spectrum is dominated by a double continuum [in (q,ω) -space] of triplet spin wave excitations. The S=1/2 integrated intensity agrees very well with recent neutron scattering results on CPC, unlike the corresponding classical intensity. Moreover, the S=1/2 spectral weight function shows increasing asymmetry as q→π, a quantum effect, observable in more recent neutron scattering data. In non-zero magnetic field, there exist two, partly overlapping, double continua, each giving rise to a peak situated at the lower boundary. The (zz component of) spectral weight function therefore has a double-peaked structure, as observed experimentally. Theory and experiment are in apparent agreement concerning the energy difference between the peaks.

Journal of Applied Physics•Nov 1, 1979•9 citations
We have examined numerical calculations on a spin 1/2 one‐dimensional (1D) Heisenberg antiferromagnet for N=8 spins, both for the states and their spectral weights. This work is of interest because of the recent neutron scattering experiments in CuCl2⋅2NC5D5 which show the spectral weight to be concentrated at the des Cloizeaux and Pearson (dCP) spin wave frequencies. The states form a two‐parameter continuum between the dCp value E1(k) =π‖J‖ ‖sin k‖ and the upper limit E2(k) =2π‖J‖ ‖ sin(k/2) ‖. Though most of the spectral weight is at the dCP frequency for N=8, about 10% of it is found near E2 for k=π. There is no numerical evidence for the existence of singlet bound states as predicted by Ovchinnikov.
We report a detailed neutron-scattering study of the spin dynamics in CuCl2{\mathrm{Cl}}_{2}Cl2​\ifmmode\cdot\else\textperiodcentered\fi{}2N(C5{\mathrm{C}}_{5}C5​D5{\mathrm{D}}_{5}D5​), a physical realization of the one-dimensional S=12S=\frac{1}{2}S=21​ Heisenberg antiferromagnet. At T=1.3T=1.3T=1.3 K well-defined excitations are observed over the whole zone with energies given by $E(q)=\ensuremath{\pi}{J}_{\mathrm{nn}}sin(\mathrm{qc})$, the celebrated des Cloizeaux-Pearson exact solution for the spectrum of first excited states, but with intensities approximately those expected from classical spin-wave theory. At T=8T=8T=8 K the excitations are broad and ill defined.
The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.
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 neutron-scattering intensity and energy eigenvalues of the elementary excitations of the S=1 antiferromagnetic Heisenberg chain with single-ion anisotropy [scrH=J tsumi=1N{\mathit{tsum}}_{\mathit{i}=1}^{\mathit{N}}tsumi=1N​ Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathbf{S}}_{\mathit{i}+1}Si+1​+D tsum(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2] are calculated with a numerical diagonalization method and a projector Monte Carlo method. We set D/J=0.2,0.0,-0.2 and N=14,16,18,32. For D\ensuremath{\ne}0, the elementary excitations in the subspace of Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1 and in the subspace of Sz{\mathit{S}}^{\mathit{z}}Sz=0 are different. The Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1 branch is lower than the Sz{\mathit{S}}^{\mathit{z}}Sz=0 branch near Q=\ensuremath{\pi}; however, it becomes slightly higher in other regions. The crossing of levels occurs at Q\ensuremath{\simeq}\ifmmode\pm\else\textpm\fi{}0.7\ensuremath{\pi}. We find that the low-lying excited state has a weight of more than 90% in the dynamical structure factor, except for Q\ensuremath{\simeq}0. Our results explain recent experimental results on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​.
We present an approximate analytic expression for the dynamical spin correlation function of the S=1/2 linear Heisenberg antiferromagnet at T=0. The basis for our approach is that in zero field the spectrum is dominated by a double continuum [in (q,ω) -space] of triplet spin wave excitations. The S=1/2 integrated intensity agrees very well with recent neutron scattering results on CPC, unlike the corresponding classical intensity. Moreover, the S=1/2 spectral weight function shows increasing asymmetry as q→π, a quantum effect, observable in more recent neutron scattering data. In non-zero magnetic field, there exist two, partly overlapping, double continua, each giving rise to a peak situated at the lower boundary. The (zz component of) spectral weight function therefore has a double-peaked structure, as observed experimentally. Theory and experiment are in apparent agreement concerning the energy difference between the peaks.