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.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.

Physical review. B, Condensed matterDec 1, 198037 citations

The Free Energy and the Scaling Function of the Ferromagnetic Heisenberg Chain in a Magnetic Field

Hiroaki Nakamura, Minoru Takahashi

A nonlinear susceptibilities (the third derivative of a magnetization $m_S$ by a magnetic field $h$ ) of the $S$ =1/2 ferromagnetic Heisenberg chain and the classical Heisenberg chain are calculated at low temperatures $T.$ In both chains the nonlinear susceptibilities diverge as $T^{-6}$ and a linear susceptibilities diverge as $T^{-2}.$ The arbitrary spin $S$ Heisenberg ferromagnet $[$ ${\cal H} = \sum_{i=1}^{N} \{ - J{\bf S}_{i} {\bf S}_{i+1} - (h/S) S_{i}^{z} \}$ $(J>0),$ $]$ has a scaling relation between $m_S,$ $h$ and $T:$ $m_S(T,h) = F( S^2 Jh/T^2).$ The scaling function $F(x)$ =(2 $x$ /3)-(44 $x^{3}$ /135) + O( $x^{5}$ ) is common to all values of spin $S.$ A nonlinear susceptibilities (the third derivative of a magnetization $m_S$ by a magnetic field $h$ ) of the $S$=1/2 ferromagnetic Heisenberg chain and the classical Heisenberg chain are calculated at low temperatures $T.$ In both chains the nonlinear susceptibilities diverge as $T^{-6}$ and a linear susceptibilities diverge as $T^{-2}.$ The arbitrary spin $S$ Heisenberg ferromagnet $[$ ${\cal H} = \sum_{i=1}^{N} \{ - J{\bf S}_{i} {\bf S}_{i+1} - (h/S) S_{i}^{z} \}$ $(J>0),$ $]$ has a scaling relation between $m_S,$ $h$ and $T:$ $m_S(T,h) = F( S^2 Jh/T^2).$ The scaling function $F(x)$=(2$x$/3)-(44$x^{3}$/135) + O($x^{5}$) is common to all values of spin $S.$

arXiv:cond-mat/9401074Journal of the Physical Society of JapanJul 15, 199420 citations

Spin dynamics of a Heisenberg antiferromagnetic chain

Roger Arthur Cowley, D. M. Tennant, T. G. Perring +2 more

Physica A Statistical Mechanics and its ApplicationsMar 1, 19933 citations

Dynamical Properties of the Diluted Heisenberg and XY Magnets at Infinite Temperature. I: Spin Diffusion Constant

T. Horiguchi, T. Morita

The exact short-time expansion coefficients of the two-time spin-pair correlation function σ(Rif,t)=≪siz(t)sfz(0)≫c at infinite temperature are used in order to estimate the spin diffusion constants of the diluted Heisenberg and XY magnets of spin 1/2. The used coefficients are those up to the term of t10 for the Heisenberg magnet on the linear chain, and those up to the term of t8 for the Heisenberg and XY magnets on the square, sc and bcc lattices. Our calculations are based on the “linear extrapolation approximation”, in which unknown coefficients in Mori's continued fraction representation for ~Is(k), the Laplace-time Fourier-space transform of σ(Rif,t), are linearly extrapolated from the last two of the known coefficients. The obtained values of the diffusion constant and their concentration dependences show a reasonable trend of convergence for successive approximations in a range of concentration not close to the percolation limit. Discussions are also given on the diffusion constants obtained by the “two-parameter Gaussian approximation in time” applied to the memory function.The exact short-time expansion coefficients of the two-time spin-pair correlation function σ(Rif,t)=≪siz(t)sfz(0)≫c at infinite temperature are used in order to estimate the spin diffusion constants of the diluted Heisenberg and XY magnets of spin 1/2. The used coefficients are those up to the term of t10 for the Heisenberg magnet on the linear chain, and those up to the term of t8 for the Heisenberg and XY magnets on the square, sc and bcc lattices. Our calculations are based on the “linear extrapolation approximation”, in which unknown coefficients in Mori's continued fraction representation for ~Is(k), the Laplace-time Fourier-space transform of σ(Rif,t), are linearly extrapolated from the last two of the known coefficients. The obtained values of the diffusion constant and their concentration dependences show a reasonable trend of convergence for successive approximations in a range of concentration not close to the percolation limit. Discussions are also given on the diffusion constants obtained by the “two-parameter Gaussian approximation in time” applied to the memory function.

Progress of Theoretical PhysicsOct 1, 19802 citations

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.

A nonlinear susceptibilities (the third derivative of a magnetization

m_S

by a magnetic field

h

) of the

S

=1/2 ferromagnetic Heisenberg chain and the classical Heisenberg chain are calculated at low temperatures

T.

In both chains the nonlinear susceptibilities diverge as

T^{-6}

and a linear susceptibilities diverge as

T^{-2}.

The arbitrary spin

S

Heisenberg ferromagnet

[

{\cal H} = \sum_{i=1}^{N} \{ - J{\bf S}_{i} {\bf S}_{i+1} - (h/S) S_{i}^{z} \}

(J>0),

]

has a scaling relation between

m_S,

h

and

T:

m_S(T,h) = F( S^2 Jh/T^2).

The scaling function

F(x)

=(2

x

/3)-(44

x^{3}

/135) + O(

x^{5}

) is common to all values of spin

S.