arXiv:cond-mat/0011453 · Physical review. B, Condensed matter

NMR relaxation rates in a spin- 12 antiferromagnetic chain

Victor Barzykin

Published Mar 21, 2001• 34 citations

Abstract

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

DOI: 10.1103/physrevb.63.140412

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arXiv:cond-mat/0011453 · Physical review. B, Condensed matter

NMR relaxation rates in a spin- 12 antiferromagnetic chain

Victor Barzykin

Published Mar 21, 2001• 34 citations

arXiv HTML PDF

Abstract

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}.

DOI: 10.1103/physrevb.63.140412

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.

arXiv:cond-mat/9509118Physical review. B, Condensed matterOct 1, 199546 citations

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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

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Subir Sachdev A

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Temperature dependence of the NMR spin-lattice relaxation rate for spin- 1 2 chains

Emanuele Coira, Peter Barmettler, Thierry Giamarchi +1 more

Recent developments in density matrix renormalization group/matrix product state techniques allow today for a quantitative computation of dynamical correlations at finite temperature in low-dimensional quantum spin systems. By combining numerical results and field theoretical asymptotic methods, e.g., bosonization, it is now possible to give a full description of the dynamical quantity one is interested in for the range of temperatures experimentally accessible. This enhances of course the possibility of one-to-one comparisons with experiments without having to resort to uncontrolled approximations. Here, the authors focus on the temperature dependence of the NMR spin-lattice relaxation rate $1/{T}_{1}$ in spin- $\frac{1}{2}$ chain systems. This quantity is directly related to local, dynamical spin-spin correlation functions. The results agree in the low-temperature regime with bosonization predictions, and show interesting deviations from this behavior at intermediate temperatures.Recent developments in density matrix renormalization group/matrix product state techniques allow today for a quantitative computation of dynamical correlations at finite temperature in low-dimensional quantum spin systems. By combining numerical results and field theoretical asymptotic methods, e.g., bosonization, it is now possible to give a full description of the dynamical quantity one is interested in for the range of temperatures experimentally accessible. This enhances of course the possibility of one-to-one comparisons with experiments without having to resort to uncontrolled approximations. Here, the authors focus on the temperature dependence of the NMR spin-lattice relaxation rate $1/{T}_{1}$ in spin-$\frac{1}{2}$ chain systems. This quantity is directly related to local, dynamical spin-spin correlation functions. The results agree in the low-temperature regime with bosonization predictions, and show interesting deviations from this behavior at intermediate temperatures.

arXiv:1606.03985Physical review. B./Physical review. BOct 11, 201624 citations

Quantum Critical Scaling and Temperature-Dependent Logarithmic Corrections in the Spin-Half Heisenberg Chain

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

Low temperature dynamics of the $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ Heisenberg chain is studied via a simple ansatz generalizing the conformal mapping and analytic continuation procedures to correlation functions with multiplicative logarithmic factors. Closed form expressions for the dynamic susceptibility and the NMR relaxation rates $1/{T}_{1}$ and $1/{T}_{2G}$ are obtained, and are argued to improve the agreement with recent experiments. Scaling in $q/T$ and $\ensuremath{\omega}/T$ are violated due to these logarithmic terms. Numerical results show that the logarithmic corrections are very robust. While not yet in the asymptotic low temperature regime, they provide striking qualitative confirmation of the theoretical results.Low temperature dynamics of the $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ Heisenberg chain is studied via a simple ansatz generalizing the conformal mapping and analytic continuation procedures to correlation functions with multiplicative logarithmic factors. Closed form expressions for the dynamic susceptibility and the NMR relaxation rates $1/{T}_{1}$ and $1/{T}_{2G}$ are obtained, and are argued to improve the agreement with recent experiments. Scaling in $q/T$ and $\ensuremath{\omega}/T$ are violated due to these logarithmic terms. Numerical results show that the logarithmic corrections are very robust. While not yet in the asymptotic low temperature regime, they provide striking qualitative confirmation of the theoretical results.

arXiv:cond-mat/9610015Physical Review LettersJan 20, 199739 citations

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},

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

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 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.

{\mathit{T}}_{1}

and 1/

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

is computed.

1/{T}_{1}

in spin-

\frac{1}{2}

chain systems. This quantity is directly related to local, dynamical spin-spin correlation functions. The results agree in the low-temperature regime with bosonization predictions, and show interesting deviations from this behavior at intermediate temperatures.

Low temperature dynamics of the

S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}

Heisenberg chain is studied via a simple ansatz generalizing the conformal mapping and analytic continuation procedures to correlation functions with multiplicative logarithmic factors. Closed form expressions for the dynamic susceptibility and the NMR relaxation rates

1/{T}_{1}

and

1/{T}_{2G}

are obtained, and are argued to improve the agreement with recent experiments. Scaling in

q/T

and $\ensuremath{\omega}/T$ are violated due to these logarithmic terms. Numerical results show that the logarithmic corrections are very robust. While not yet in the asymptotic low temperature regime, they provide striking qualitative confirmation of the theoretical results.