arXiv:cond-mat/9706177 · Physical review. B, Condensed matter

Nuclear relaxation in the spin- 1/2 antiferromagnetic chain compound Sr2CuO3: Comparison between theories and experiments

Masashi Takigawa, Oleg A. Starykh, Anders W. Sandvik, Rajiv R. P. Singh

Published Dec 1, 1997• 64 citations

Abstract

The NMR relaxation data on ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ [Phys. Rev. Lett. 76, 4612 (1996)] are reexamined and compared with the analytic theory of the dynamic susceptibility in the $S=1/2$ antiferromagnetic Heisenberg chain including multiplicative logarithmic corrections [Phys. Rev. Lett. 78, 539 (1997)]. Comparisons of the spin-lattice and the Gaussian spin-echo decay rates (${1/T}_{1}$ and ${1/T}_{2G}$) and their ratio all show good quantitative agreement. Our results demonstrate the importance of the logarithmic corrections in the analysis of experimental data for quasi-one-dimensional systems and indicate that the dynamics of ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ is well described by an $S=1/2$ one-dimensional Heisenberg model with a nearest-neighbor exchange.

DOI: 10.1103/physrevb.56.13681

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arXiv:cond-mat/9706177 · Physical review. B, Condensed matter

Nuclear relaxation in the spin- 1/2 antiferromagnetic chain compound Sr2CuO3: Comparison between theories and experiments

Masashi Takigawa, Oleg A. Starykh, Anders W. Sandvik, Rajiv R. P. Singh

Published Dec 1, 1997• 64 citations

arXiv HTML PDF

Abstract

The NMR relaxation data on

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}

[Phys. Rev. Lett. 76, 4612 (1996)] are reexamined and compared with the analytic theory of the dynamic susceptibility in the

S=1/2

antiferromagnetic Heisenberg chain including multiplicative logarithmic corrections [Phys. Rev. Lett. 78, 539 (1997)]. Comparisons of the spin-lattice and the Gaussian spin-echo decay rates (

{1/T}_{1}

and

{1/T}_{2G}

) and their ratio all show good quantitative agreement. Our results demonstrate the importance of the logarithmic corrections in the analysis of experimental data for quasi-one-dimensional systems and indicate that the dynamics of

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}

is well described by an

S=1/2

one-dimensional Heisenberg model with a nearest-neighbor exchange.

DOI: 10.1103/physrevb.56.13681

Related papers

NMR relaxation rates for the spin-1/2 Heisenberg chain

Anders W. Sandvik

The spin-lattice relaxation rate 1/ ${\mathit{T}}_{1}$ and the spin-echo decay rate 1/ ${\mathit{T}}_{2\mathit{G}}$ for the spin-1/2 antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum-entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor ${\mathit{A}}_{\mathit{q}}$ is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/ ${\mathit{T}}_{1}$ \ensuremath{\sim} ${\mathrm{ln}}^{1/2}$ (1/T), 1/ ${\mathit{T}}_{2\mathit{G}}$ \ensuremath{\sim} ${\mathrm{ln}}^{1/2}$ (1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If ${\mathit{A}}_{\mathit{q}}$ has significant weight for q\ensuremath{\approxeq}0 there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic T\ensuremath{\rightarrow}0 forms.The spin-lattice relaxation rate 1/${\mathit{T}}_{1}$ and the spin-echo decay rate 1/${\mathit{T}}_{2\mathit{G}}$ for the spin-1/2 antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum-entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor ${\mathit{A}}_{\mathit{q}}$ is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/${\mathit{T}}_{1}$\ensuremath{\sim}${\mathrm{ln}}^{1/2}$(1/T), 1/${\mathit{T}}_{2\mathit{G}}$\ensuremath{\sim}${\mathrm{ln}}^{1/2}$(1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If ${\mathit{A}}_{\mathit{q}}$ has significant weight for q\ensuremath{\approxeq}0 there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic T\ensuremath{\rightarrow}0 forms.

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NMR measurements were performed on single-crystalline samples of the one-dimensional Heisenberg antiferromagnet (1D HAF) ${\mathrm{LiCuVO}}_{4}$ . We investigated $^{7}\mathrm{Li}$ and $^{51}\mathrm{V}$ NMR spectra, deduced the respective lineshifts $K(T)$ , spin-lattice relaxation rates $1∕{T}_{1}$ , and spin-spin relaxation rates $1∕{T}_{2}$ and $1∕{T}_{2G}$ . The results are compared to theoretical predictions for the relaxation rates $1∕{T}_{1}(T)$ and $1∕{T}_{2G}(T)$ deduced from the dynamic susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ of a 1D HAF spin chain as proposed by Sachdev [Phys. Rev. B 50, 13006 (1994)]. We document a correspondence between the anisotropic relaxation rates derived from electron paramagnetic resonance linewidth $\mathrm{\ensuremath{\Delta}}H(T)$ and nuclear magnetic resonance $1∕{T}_{1}(T)$ , respectively, the former being well described in the framework of symmetric anisotropic exchange interactions for ${\mathrm{LiCuVO}}_{4}$ by Krug von Nidda et al. [Phys. Rev. B 65, 134445 (2002)].NMR measurements were performed on single-crystalline samples of the one-dimensional Heisenberg antiferromagnet (1D HAF) ${\mathrm{LiCuVO}}_{4}$. We investigated $^{7}\mathrm{Li}$ and $^{51}\mathrm{V}$ NMR spectra, deduced the respective lineshifts $K(T)$, spin-lattice relaxation rates $1∕{T}_{1}$, and spin-spin relaxation rates $1∕{T}_{2}$ and $1∕{T}_{2G}$. The results are compared to theoretical predictions for the relaxation rates $1∕{T}_{1}(T)$ and $1∕{T}_{2G}(T)$ deduced from the dynamic susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ of a 1D HAF spin chain as proposed by Sachdev [Phys. Rev. B 50, 13006 (1994)]. We document a correspondence between the anisotropic relaxation rates derived from electron paramagnetic resonance linewidth $\mathrm{\ensuremath{\Delta}}H(T)$ and nuclear magnetic resonance $1∕{T}_{1}(T)$, respectively, the former being well described in the framework of symmetric anisotropic exchange interactions for ${\mathrm{LiCuVO}}_{4}$ by Krug von Nidda et al. [Phys. Rev. B 65, 134445 (2002)].

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NMR relaxation rates in a spin- 12 antiferromagnetic chain

Victor Barzykin

We consider the low-temperature behavior of NMR relaxation rates ${T}_{1}$ and ${T}_{2G}$ in the spin- $1/2$ antiferromagnetic chain. We find that ${T}_{1}\ensuremath{\propto}\mathrm{const}$ and ${T}_{2G}\ensuremath{\propto}\sqrt{T},$ with logarithmic corrections. We determine both constants and logarithmic terms by matching perturbative renormalization-group with exact results, so that the final expressions for the relaxation rates do not contain any free parameters. Our theoretical results are in excellent agreement with NMR experiments in ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}.$ We consider the low-temperature behavior of NMR relaxation rates ${T}_{1}$ and ${T}_{2G}$ in the spin-$1/2$ antiferromagnetic chain. We find that ${T}_{1}\ensuremath{\propto}\mathrm{const}$ and ${T}_{2G}\ensuremath{\propto}\sqrt{T},$ with logarithmic corrections. We determine both constants and logarithmic terms by matching perturbative renormalization-group with exact results, so that the final expressions for the relaxation rates do not contain any free parameters. Our theoretical results are in excellent agreement with NMR experiments in ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}.$

arXiv:cond-mat/0011453Physical review. B, Condensed matterMar 21, 200134 citations

Accurate determination of the exchange constant in Sr2 CuO3 from recent theoretical results

Sebastian Eggert

Data from susceptibility measurements [Ami et al., Phys. Rev. B 51, 5994 (1995)] on ${\mathrm{Sr}}_{2}$ ${\mathrm{CuO}}_{3}$ are compared with recent theoretical predictions [Eggert et al., Phys. Rev. Lett. 73, 332 (1994)] for the magnetic susceptibility of the antiferromagnetic spin-1/2 Heisenberg chain. The experimental data fully confirms the theoretical predictions and in turn we establish that ${\mathrm{Sr}}_{2}$ ${\mathrm{CuO}}_{3}$ behaves almost perfectly like a one-dimensional antiferromagnet with an exchange coupling of J=${1700}_{\mathrm{\ensuremath{-}}100}^{+150}$ K. \textcopyright{} 1996 The American Physical Society.Data from susceptibility measurements [Ami et al., Phys. Rev. B 51, 5994 (1995)] on ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$ are compared with recent theoretical predictions [Eggert et al., Phys. Rev. Lett. 73, 332 (1994)] for the magnetic susceptibility of the antiferromagnetic spin-1/2 Heisenberg chain. The experimental data fully confirms the theoretical predictions and in turn we establish that ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$ behaves almost perfectly like a one-dimensional antiferromagnet with an exchange coupling of J=${1700}_{\mathrm{\ensuremath{-}}100}^{+150}$ K. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9508037Physical review. B, Condensed matterMar 1, 199641 citations

Spin diffusion and the anisotropic spin- 12 Heisenberg chain

Jesko Sirker

Measurements of the spin-lattice relaxation rate $1∕{T}_{1}$ by nuclear magnetic resonance for the one-dimensional Heisenberg antiferromagnet ${\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{3}$ have provided evidence for a diffusionlike contribution at finite temperature and small wave vector. By analyzing real-time data for the auto- and nearest-neighbor spin-spin correlation functions obtained by the density-matrix renormalization group I show that such a contribution indeed exists for temperatures $T>J$ , where $J$ is the coupling constant, but that it becomes exponentially suppressed for $T⪡J$ . I present evidence that the frequency dependence of $1∕{T}_{1}$ in the Heisenberg case is smoothly connected to that in the free fermion case where the exponential suppression of the diffusionlike contribution is easily understood.Measurements of the spin-lattice relaxation rate $1∕{T}_{1}$ by nuclear magnetic resonance for the one-dimensional Heisenberg antiferromagnet ${\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{3}$ have provided evidence for a diffusionlike contribution at finite temperature and small wave vector. By analyzing real-time data for the auto- and nearest-neighbor spin-spin correlation functions obtained by the density-matrix renormalization group I show that such a contribution indeed exists for temperatures $T>J$, where $J$ is the coupling constant, but that it becomes exponentially suppressed for $T⪡J$. I present evidence that the frequency dependence of $1∕{T}_{1}$ in the Heisenberg case is smoothly connected to that in the free fermion case where the exponential suppression of the diffusionlike contribution is easily understood.

arXiv:cond-mat/0604327Physical Review BJun 16, 200635 citations

The NMR relaxation data on

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}

[Phys. Rev. Lett. 76, 4612 (1996)] are reexamined and compared with the analytic theory of the dynamic susceptibility in the

S=1/2

antiferromagnetic Heisenberg chain including multiplicative logarithmic corrections [Phys. Rev. Lett. 78, 539 (1997)]. Comparisons of the spin-lattice and the Gaussian spin-echo decay rates (

{1/T}_{1}

and

{1/T}_{2G}

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}

is well described by an

S=1/2

one-dimensional Heisenberg model with a nearest-neighbor exchange.

The spin-lattice relaxation rate 1/

{\mathit{T}}_{1}

and the spin-echo decay rate 1/

{\mathit{T}}_{2\mathit{G}}

for the spin-1/2 antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum-entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor

{\mathit{A}}_{\mathit{q}}

is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/

{\mathit{T}}_{1}

\ensuremath{\sim}

{\mathrm{ln}}^{1/2}

(1/T), 1/

{\mathit{T}}_{2\mathit{G}}

\ensuremath{\sim}

{\mathrm{ln}}^{1/2}

(1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If

{\mathit{A}}_{\mathit{q}}

has significant weight for q\ensuremath{\approxeq}0 there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic T\ensuremath{\rightarrow}0 forms.

^{13}\mathrm{C}

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.

NMR measurements were performed on single-crystalline samples of the one-dimensional Heisenberg antiferromagnet (1D HAF)

{\mathrm{LiCuVO}}_{4}

. We investigated

^{7}\mathrm{Li}

and

^{51}\mathrm{V}

NMR spectra, deduced the respective lineshifts

K(T)

, spin-lattice relaxation rates

1∕{T}_{1}

, and spin-spin relaxation rates

1∕{T}_{2}

and

1∕{T}_{2G}

. The results are compared to theoretical predictions for the relaxation rates

1∕{T}_{1}(T)

and

1∕{T}_{2G}(T)

deduced from the dynamic susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ of a 1D HAF spin chain as proposed by Sachdev [Phys. Rev. B 50, 13006 (1994)]. We document a correspondence between the anisotropic relaxation rates derived from electron paramagnetic resonance linewidth $\mathrm{\ensuremath{\Delta}}H(T)$ and nuclear magnetic resonance

1∕{T}_{1}(T)

, respectively, the former being well described in the framework of symmetric anisotropic exchange interactions for

{\mathrm{LiCuVO}}_{4}

by Krug von Nidda et al. [Phys. Rev. B 65, 134445 (2002)].

We consider the low-temperature behavior of NMR relaxation rates

{T}_{1}

and

{T}_{2G}

in the spin-

1/2

antiferromagnetic chain. We find that ${T}_{1}\ensuremath{\propto}\mathrm{const}$ and ${T}_{2G}\ensuremath{\propto}\sqrt{T},$ with logarithmic corrections. We determine both constants and logarithmic terms by matching perturbative renormalization-group with exact results, so that the final expressions for the relaxation rates do not contain any free parameters. Our theoretical results are in excellent agreement with NMR experiments in

{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}.

Data from susceptibility measurements [Ami et al., Phys. Rev. B 51, 5994 (1995)] on

{\mathrm{Sr}}_{2}

{\mathrm{CuO}}_{3}

are compared with recent theoretical predictions [Eggert et al., Phys. Rev. Lett. 73, 332 (1994)] for the magnetic susceptibility of the antiferromagnetic spin-1/2 Heisenberg chain. The experimental data fully confirms the theoretical predictions and in turn we establish that

{\mathrm{Sr}}_{2}

{\mathrm{CuO}}_{3}

behaves almost perfectly like a one-dimensional antiferromagnet with an exchange coupling of J=${1700}_{\mathrm{\ensuremath{-}}100}^{+150}$ K. \textcopyright{} 1996 The American Physical Society.

Measurements of the spin-lattice relaxation rate

1∕{T}_{1}

by nuclear magnetic resonance for the one-dimensional Heisenberg antiferromagnet

{\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{3}

have provided evidence for a diffusionlike contribution at finite temperature and small wave vector. By analyzing real-time data for the auto- and nearest-neighbor spin-spin correlation functions obtained by the density-matrix renormalization group I show that such a contribution indeed exists for temperatures

T>J

, where

J

is the coupling constant, but that it becomes exponentially suppressed for

T⪡J

. I present evidence that the frequency dependence of

1∕{T}_{1}

in the Heisenberg case is smoothly connected to that in the free fermion case where the exponential suppression of the diffusionlike contribution is easily understood.