arXiv:cond-mat/9904250 · Journal of Physics Condensed Matter

Temperature-dependent logarithmic corrections in the spin-1/2 Heisenberg chain

Victor Barzykin

Published Feb 17, 2000• 15 citations

Abstract

We obtain the logarithmic corrections to the dynamic response function and nuclear magnetic resonance (NMR) T 1 - and T 2 G -rates in the spin-1/2 antiferromagnetic Heisenberg chain using the perturbative renormalization group for the leading irrelevant operator. The result is compared with NMR experiments on Sr 2 CuO 3 .

DOI: 10.1088/0953-8984/12/9/309

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arXiv:cond-mat/9904250 · Journal of Physics Condensed Matter

Temperature-dependent logarithmic corrections in the spin-1/2 Heisenberg chain

Victor Barzykin

Published Feb 17, 2000• 15 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0953-8984/12/9/309

Related papers

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arXiv:cond-mat/0604327Physical Review BJun 16, 200635 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 ${T}_{2G}\ensuremath{\propto}\sqrt{T},$ 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}.

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.

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.

^{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.

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.