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arXiv:cond-mat/0005174 · Physical review. B, Condensed matter

Line-shape predictions via Bethe ansatz for the one-dimensional spin- 12 Heisenberg antiferromagnet in a magnetic field

Michael Karbach, Gerhard H. Muller

Published Dec 1, 2000• 51 citations
arXivHTMLPDF

Abstract

The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) $s=\frac{1}{2}$ Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization $0<{M}_{z}<N/2,$ the ground state, which contains ${2M}_{z}$ spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ these collective excitations form a continuum in $(q,\ensuremath{\omega})$ space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.

DOI
10.1103/physrevb.62.14871

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•Jan 1, 2015•0 citations
The spin fluctuations parallel to the external magnetic field in the ground state of the one-dimensional (1D) s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnet are dominated by a two-parameter set of collective excitations. In a cyclic chain of N sites and magnetization 0<Mz<N/2,0<{M}_{z}<N/2,0<Mz​<N/2, the ground state, which contains 2Mz{2M}_{z}2Mz​ spinons, is reconfigured as the physical vacuum for a different species of quasiparticles, identifiable in the framework of the coordinate Bethe ansatz by characteristic configurations of Bethe quantum numbers. The dynamically dominant excitations are found to be scattering states of two such quasiparticles. For N\ensuremath→\ensuremath∞,\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},N\ensuremath→​\ensuremath∞, these collective excitations form a continuum in (q,\ensuremathω)(q,\ensuremath{\omega})(q,\ensuremathω) space with an incommensurate soft mode. Their matrix elements in the dynamic spin structure factor Szz(q,\ensuremathω){S}_{\mathrm{zz}}(q,\ensuremath{\omega})Szz​(q,\ensuremathω) are calculated directly from the Bethe wave functions for finite N. The resulting line- shape predictions for N\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}N\ensuremath→​\ensuremath∞ complement the exact results previously derived via algebraic analysis for the exact two-spinon part of Szz(q,\ensuremathω){S}_{\mathrm{zz}}(q,\ensuremath{\omega})Szz​(q,\ensuremathω) in the zero-field limit. They are relevant for neutron-scattering experiments on quasi-1D antiferromagnetic compounds in a strong magnetic field.
The T=0T=0T=0 dynamical properties of the one-dimensional (1D) s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnet in a uniform magnetic field are studied via the Bethe ansatz for cyclic chains of N sites. The ground state at magnetization 0<Mz<N/2,0<{M}_{z}<N/2,0<Mz​<N/2, which can be interpreted as a state with 2Mz{2M}_{z}2Mz​ spinons or as a state of $N/2\ensuremath{-}{M}_{z}$ magnons, is reconfigured here as the vacuum for a different species of quasiparticles, the psinons and antipsinons. We investigate three kinds of quantum fluctuations, namely the spin fluctuations parallel and perpendicular to the direction of the applied magnetic field and the dimer fluctuations. The dynamically dominant excitation spectra are found to be sets of collective excitations composed of two quasiparticles excited from the psinon vacuum in different configurations. The Bethe ansatz provides a framework for (i) the characterization of the new quasiparticles in relation to the more familiar spinons and magnons, (ii) the calculation of spectral boundaries and densities of states for each continuum, (iii) the calculation of transition rates between the ground state and the dynamically dominant collective excitations, (iv) the prediction of line shapes for dynamic structure factors relevant for experiments performed on a variety of quasi-1D antiferromagnetic compounds, including KCuF3,{\mathrm{KCuF}}_{3},KCuF3​, Cu(C4H4N2)(NO3)2,\mathrm{Cu}({\mathrm{C}}_{4}{\mathrm{H}}_{4}{\mathrm{N}}_{2})({\mathrm{NO}}_{3}{)}_{2},Cu(C4​H4​N2​)(NO3​)2​, and CuGeO3.{\mathrm{CuGeO}}_{3}.CuGeO3​.
Having introduced the magnon in part I and the spinon in part II as the relevant quasi-particles for the interpretation of the spectrum of low-lying excitations in the one-dimensional (1D) s=1/2 Heisenberg ferromagnet and antiferromagnet, respectively, we now study the low-lying excitations of the Heisenberg antiferromagnet in a magnetic field and interpret these collective states as composites of quasi-particles from a different species. We employ the Bethe ansatz to calculate matrix elements and show how the results of such a calculation can be used to predict lineshapes for neutron scattering experiments on quasi-1D antiferromagnetic compounds. The paper is designed as a tutorial for beginning graduate students. It includes 11 problems for further study.
The exact two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional s=1/2,s=1/2,s=1/2, XXZ\mathrm{XXZ}XXZ model at T=0T=0T=0 in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center (Q=0).(Q=0).(Q=0). The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.
Spin-dynamical calculations on one-dimensional systems have relied heavily on classical ($s=\ensuremath{\infty}$) theories, despite abundant evidence that quantum effects can be extremely important at low temperatures. We present a new approach to the spin dynamics of the one-dimensional isotropic s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnetic (HB AF) which does not involve the many-body techniques usually employed. It is based on analytic Bethe ansatz calculations of excitation energies and densities of states combined with finite-chain calculations of matrix elements. An important feature of our method is the use of rigorous selection rules and the introduction of new selection rules, which are valid for macroscopic systems in a magnetic field. We show that in zero field the dynamical two-spin correlation function ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ at T=0T=0T=0 is governed by a two-parameter continuum of spin-wave-type excitations. In nonzero field, the longitudinal component ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ and the transverse components ${S}_{\mathrm{xx}}(q,\ensuremath{\omega})\ensuremath{\equiv}{S}_{\mathrm{yy}}(q,\ensuremath{\omega})$ behave quite differently because they are dominated by different continua of excitations. The former is characterized by a lowest excitation branch with a zero-frequency mode moving from the zone boundary ($q=\ensuremath{\pi}$) towards the zone center (q=0q=0q=0) as the field increases, whereas the latter is characterized by a lowest branch with a zero frequency mode moving from $q=0 \mathrm{to} \ensuremath{\pi}$ with increasing field. The first part of our work features an approximate analytic expression for ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ at zero temperature and in zero field. Although our expression is not rigorous, exact sum rules are violated only by a small amount, and good agreement exists with the few known exact results. Our studies are extended to nonzero temperatures by placing major reliance on exact finite-chain calculations. Our work was stimulated by recent neutron scattering experiments and is oriented towards experimental comparisons. Our result for the s=12s=\frac{1}{2}s=21​ integrated intensity is in much better agreement with neutron scattering data on CuCl2{\mathrm{Cl}}_{2}Cl2​\ifmmode\cdot\else\textperiodcentered\fi{}2N(C5{\mathrm{C}}_{5}C5​D5{\mathrm{D}}_{5}D5​) (CPC) than the corresponding semiclassical result. Moreover, the spectral-weight distribution in ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ shows increasing asymmetry as $q\ensuremath{\rightarrow}\ensuremath{\pi}$, a quantum effect, again in agreement with more recent neutron scattering data. The second part of our work deals with the effects of an applied magnetic field. We extend the analytic work of Ishimura and Shiba to obtain expressions for the energies and densities of states of the various excitation continua. It is shown that these continua are expected to give rise to multiple structures in the scattering intensity. Our results appear to be in quantitative agreement with preliminary results of a neutron study in CPC in a field of 70 kOe, revealing anomalous scattering intensity peaks. Our results repeatedly demonstrate the inadequacy of classical spin-wave theory for this problem. They call for additional experimental studies on quasi-one-dimensional antiferromagnets to examine other unusual and interesting phenomena predicted by our approach.
Spectral properties of the spin-1/2 antiferromagnetic Heisenberg chain in a magnetic field are investigated by using exact Bethe-ansatz solutions. We argue that not only quasiparticles called psinon and antipsinon but also a quasiparticle representing a 2-string in the Bethe ansatz plays an important role for dynamical properties in a magnetic field. Combined with psinon and antipsinon, the quasiparticle for a 2-string forms a continuum in the high-energy regime for transverse dynamical structure factor S +− ( κ,ω ). In the zero-field limit, the continuum is located on the mode of the lowest excited states in zero field called the des Cloizeaux-Pearson mode. In a magnetic field, the continuum separates from other low-energy continua, and reduces to the mode of bound states of overturned spins from the fully polarized state near the saturation field. We confirm the relevance through comparisons with available experimental results on the quasi-one-dimensional antiferromagnet CuCl 2 ·2N(C 5 D 5 ).
The research presented here focuses on the theoretical investigation of spinons in quasi one-dimensional quantum spin chains from the perspective of neutron scattering, with emphasis placed on the calculation of dynamic spin structure factors for both unpolarised and polarised incident neutrons. Two anisotropic versions of the Heisenberg spin chain are considered for such an examination: the spin-1/21/21/2 antiferromagnetic XXZXXZXXZ model and the spin-1/21/21/2 ferromagnetic XYZXYZXYZ model.The XYZXYZXYZ model supports spinon scattering (modelled at finite temperature) and is related to the so-called Villain mode. The introduction of a perturbation in the form of an external transverse magnetic field or Dzyaloshinskii-Moriya interaction (DMI) leads to the emergence of incommensurability in the system; such a result is a signature of fractional excitations. The presence of these interactions also give rise to a chiral response associated with the spinons in the polarised term of the scattering cross section. The inelastic scattering cross section is computed for both unpolarised and polarised neutrons.In relation to the antiferromagnetic XXZXXZXXZ model, emphasis is on spinon pair creation (modelled at T=0rmKT = 0 {rm K}T=0rmK). Motivated by the presence of a chiral response in the case of the XYZXYZXYZ model mentioned above, a new technique is developed in order to incorporate DMI and the transverse magnetic field so that one can determine if there is a chiral property associated with the two-spinon system. This technique is numerical in nature and is based on Green function methods. Using the results produced by the Green function approach, the inelastic cross section in the presence of an external transverse magnetic field is computed for the first time. The result is compared with experimental neutron scattering data; good agreement is demonstrated between theory and experiments on rmCsCoBr3{rm CsCoBr}_3rmCsCoBr3​.Finally, the Green function approach is generalised to account for the interactions between spin chains in magnetic compounds described by a staggered field. Various dynamic structure factors are computed for this scenario with results compared to experiment; good agreement between the theory and experiments on rmRbCoCl3{rm RbCoCl}_3rmRbCoCl3​ is shown.