arXiv:cond-mat/9509118 · Physical review. B, Condensed matter

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

Anders W. Sandvik

Published Oct 1, 1995• 46 citations

Abstract

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.

DOI: 10.1103/physrevb.52.r9831

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Dynamics of the spin- Heisenberg chain at intermediate temperatures

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

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NMR relaxation in the spin-1 Heisenberg chain

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NMR relaxation rates in a spin- 12 antiferromagnetic chain

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Spin diffusion and the anisotropic spin- 12 Heisenberg chain

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

arXiv:cond-mat/9509118 · Physical review. B, Condensed matter

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

Anders W. Sandvik

Published Oct 1, 1995• 46 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.52.r9831

Related papers

Dynamics of the spin- Heisenberg chain at intermediate temperatures

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

Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/ ${\mathrm{T}}_{1}$ from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for ${\mathrm{Sr}}_{2}$ ${\mathrm{CuO}}_{3}$ and ${\mathrm{CuGeO}}_{3}$ . We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T) ${]}^{3\mathrm{/}2}$ and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T) ${]}^{1\mathrm{/}2}$ , expected from previous theoretical studies.Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/${\mathrm{T}}_{1}$ from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for ${\mathrm{Sr}}_{2}$${\mathrm{CuO}}_{3}$ and ${\mathrm{CuGeO}}_{3}$. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)${]}^{3\mathrm{/}2}$ and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)${]}^{1\mathrm{/}2}$, expected from previous theoretical studies.

arXiv:cond-mat/9701052Physical review. B, Condensed matterJun 1, 199766 citations

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

Masashi Takigawa, Oleg A. Starykh, Anders W. Sandvik +1 more

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.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.

arXiv:cond-mat/9706177Physical review. B, Condensed matterDec 1, 199764 citations

NMR relaxation in the spin-1 Heisenberg chain

Sylvain Capponi, Maxime Dupont, Anders W. Sandvik +1 more

We consider the isotropic $S=1$ Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate $1/{T}_{1}$ measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.We consider the isotropic $S=1$ Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate $1/{T}_{1}$ measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.

arXiv:1905.12697Physical review. B./Physical review. BSep 9, 201918 citations

NMR study of lineshifts and relaxation rates of the one-dimensional antiferromagnet LiCuVO4

C. Kegler, Norbert Büttgen, H.‐A. Krug von Nidda +5 more

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)].

Physical Review BMar 15, 200621 citations

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

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 spin-lattice relaxation rate 1/

{\mathit{T}}_{1}

and the spin-echo decay rate 1/

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

{\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}}

{\mathrm{T}}_{1}

from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for

{\mathrm{Sr}}_{2}

{\mathrm{CuO}}_{3}

and

{\mathrm{CuGeO}}_{3}

. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)

{]}^{3\mathrm{/}2}

and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)

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

, expected from previous theoretical studies.

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.

We consider the isotropic

S=1

Heisenberg chain with a finite Haldane gap $\mathrm{\ensuremath{\Delta}}$ and use state-of-the-art numerical techniques to investigate its dynamical properties at finite temperature, focusing on the nuclear spin-lattice relaxation rate

1/{T}_{1}

measured in nuclear magnetic resonance (NMR) experiments, for instance. In particular, we analyze the contributions from modes with momenta close to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as a function of temperature. At high-temperature we observe spin diffusion, while at low-temperature we argue that a simple activated behavior $1/{T}_{1}\ensuremath{\propto}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ can be observed only at temperatures much smaller than the gap $\mathrm{\ensuremath{\Delta}}$.

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}.

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.