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Physical review. B, Condensed matter

Dynamics of the s=12 linear Heisenberg antiferromagnet in a magnetic field: Experiment and theory

J. P. Groen, Tjeerd O. Klaassen, N.J. Poulis, Gerhard H. Muller, H. Thomas, H. Beck

Published Dec 1, 1980• 37 citations
Publisher

Abstract

Proton spin-lattice relaxation times have been measured in the quasi-one-dimensional $s=\frac{1}{2}$ Heisenberg antiferromagnetic compounds CuSe${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}5${\mathrm{H}}_{2}$O and CuS${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}5${\mathrm{H}}_{2}$O, in order to study the magnetic field dependence of the 1D quantum spin dynamics at various temperatures. Information is obtained on the low-frequency behavior of the spin autocorrelation functions for the linear chains. The transverse component shows at high $T$ a marked decrease with magnetic field above the saturation field ${B}_{c}$. Towards lower temperatures a pronounced maximum just below ${B}_{c}$ develops and the decrease above ${B}_{c}$ becomes more abrupt. The longitudinal component behaves more smoothly near and above ${B}_{c}$. The field dependence of the relaxation rates is well explained at all nonzero temperatures by finite-chain calculations. For $T=0$ we present analytical calculations of autocorrelation functions based on the predominance of a two-parameter spin-wave continuum for the quantum spin dynamics yielding a nondivergent behavior at $\ensuremath{\omega}=0$ and $B=0$. Exact results are obtained for fields above ${B}_{c}$. Further, an effective $\mathrm{XY}$ approach is used for low but finite $T$. Here, good agreement is found with the proton relaxation data for very low temperatures.

DOI
10.1103/physrevb.22.5369

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Proton spin-lattice relaxation times have been measured in the quasi-one-dimensional s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnetic compounds CuSeO4{\mathrm{O}}_{4}O4​\ifmmode\cdot\else\textperiodcentered\fi{}5H2{\mathrm{H}}_{2}H2​O and CuSO4{\mathrm{O}}_{4}O4​\ifmmode\cdot\else\textperiodcentered\fi{}5H2{\mathrm{H}}_{2}H2​O, in order to study the magnetic field dependence of the 1D quantum spin dynamics at various temperatures. Information is obtained on the low-frequency behavior of the spin autocorrelation functions for the linear chains. The transverse component shows at high TTT a marked decrease with magnetic field above the saturation field Bc{B}_{c}Bc​. Towards lower temperatures a pronounced maximum just below Bc{B}_{c}Bc​ develops and the decrease above Bc{B}_{c}Bc​ becomes more abrupt. The longitudinal component behaves more smoothly near and above Bc{B}_{c}Bc​. The field dependence of the relaxation rates is well explained at all nonzero temperatures by finite-chain calculations. For T=0T=0T=0 we present analytical calculations of autocorrelation functions based on the predominance of a two-parameter spin-wave continuum for the quantum spin dynamics yielding a nondivergent behavior at \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0 and B=0B=0B=0. Exact results are obtained for fields above Bc{B}_{c}Bc​. Further, an effective XY\mathrm{XY}XY approach is used for low but finite TTT. Here, good agreement is found with the proton relaxation data for very low temperatures.
A comprehensive comparison between the magnetic field- and temperature-dependent low-frequency spin dynamics in the antiferromagnetic spin-1/2 Heisenberg chain system copper pyrazine dinitrate, probed via the 13C^{13}\mathrm{C}13C nuclear magnetic resonance relaxation rate ${T}_{1}^{\ensuremath{-}1}$, and the field theoretical approach in the Luttinger liquid regime was performed. We found very good agreement between experiment and theory in the investigated temperature and field range. Our results demonstrate how spin-spin interactions strongly affect the spin dynamics of Heisenberg spin chain compounds in the vicinity of the quantum critical point.
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.
We present a 13C^{13}\text{C}13C NMR study of the magnetic field driven transition to complete polarization of the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg chain system copper pyrazine dinitrate Cu(C4H4N2)(NO3)2\text{Cu}({\text{C}}_{4}{\text{H}}_{4}{\text{N}}_{2}){({\text{NO}}_{3})}_{2}Cu(C4​H4​N2​)(NO3​)2​ (CuPzN). The static local magnetization as well as the low-frequency spin dynamics, probed via the nuclear spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}$, were explored from the low to the high field limit and at temperatures from the quantum regime (kBT⪡J)({k}_{B}T⪡J)(kB​T⪡J) up to the classical regime (kBT⪢J)({k}_{B}T⪢J)(kB​T⪢J). The experimental data show very good agreement with quantum Monte Carlo calculations over the complete range of parameters investigated. Close to the critical field, as derived from static experiments, a pronounced maximum in ${T}_{1}^{\ensuremath{-}1}$ is found which we interpret as the finite-temperature manifestation of a diverging density of zero-energy magnetic excitations at the field-driven quantum critical point.
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