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arXiv:cond-mat/0604327 · Physical Review B

Spin diffusion and the anisotropic spin- 12 Heisenberg chain

Jesko Sirker

Published Jun 16, 2006• 35 citations
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Abstract

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.

DOI
10.1103/physrevb.73.224424

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Measurements of the spin-lattice relaxation rate 1∕T11∕{T}_{1}1∕T1​ by nuclear magnetic resonance for the one-dimensional Heisenberg antiferromagnet Sr2CuO3{\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{3}Sr2​CuO3​ 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>JT>JT>J, where JJJ is the coupling constant, but that it becomes exponentially suppressed for T⪡JT⪡JT⪡J. I present evidence that the frequency dependence of 1∕T11∕{T}_{1}1∕T1​ 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.
We consider the isotropic S=1S=1S=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/T11/{T}_{1}1/T1​ 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 $\mathrm{\ensuremath{\Delta}}$.
Proton spin-lattice relaxation times have been measured in the quasi-one-dimensional s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnetic compounds CuSeO4{\mathrm{O}}_{4}O4​\ifmmode\cdot\else\textperiodcentered\fi{}5H2{\mathrm{H}}_{2}H2​O and CuSO4{\mathrm{O}}_{4}O4​\ifmmode\cdot\else\textperiodcentered\fi{}5H2{\mathrm{H}}_{2}H2​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 TTT a marked decrease with magnetic field above the saturation field Bc{B}_{c}Bc​. Towards lower temperatures a pronounced maximum just below Bc{B}_{c}Bc​ develops and the decrease above Bc{B}_{c}Bc​ becomes more abrupt. The longitudinal component behaves more smoothly near and above Bc{B}_{c}Bc​. The field dependence of the relaxation rates is well explained at all nonzero temperatures by finite-chain calculations. For T=0T=0T=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=0B=0B=0. Exact results are obtained for fields above Bc{B}_{c}Bc​. Further, an effective XY\mathrm{XY}XY approach is used for low but finite TTT. Here, good agreement is found with the proton relaxation data for very low temperatures.
The NMR relaxation data on Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ [Phys. Rev. Lett. 76, 4612 (1996)] are reexamined and compared with the analytic theory of the dynamic susceptibility in the S=1/2S=1/2S=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/T1{1/T}_{1}1/T1​ and 1/T2G{1/T}_{2G}1/T2G​) 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 Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ is well described by an S=1/2S=1/2S=1/2 one-dimensional Heisenberg model with a nearest-neighbor exchange.
We used 17O{}^{17}\mathrm{O}17O NMR to probe the uniform (wave vector q=0q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0q=0) electron spin excitations up to 800 K in Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ and separate the q=0q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0q=0 from the $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ifmmode\pm\else\textpm\fi{}\frac{\ensuremath{\pi}}{a}$ staggered components. Our results support the logarithmic decrease of the uniform spin susceptibility below $T\ensuremath{\sim}0.015J$, where J=2200KJ\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2200\mathrm{K}J=2200K. From measurement of the dynamical spin susceptibility for q=0q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0q=0 by the spin-lattice relaxation rate 1/T1{1/T}_{1}1/T1​, we demonstrate that the q=0q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0q=0 mode of spin transport is ballistic at the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 limit, but has a diffusion-like contribution at finite temperatures even for $T\ensuremath{\ll}J$.
The spin-lattice relaxation rate 1/T1{\mathit{T}}_{1}T1​ and the spin-echo decay rate 1/T2G{\mathit{T}}_{2\mathit{G}}T2G​ 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 Aq{\mathit{A}}_{\mathit{q}}Aq​ is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/T1{\mathit{T}}_{1}T1​\ensuremath{\sim}ln1/2{\mathrm{ln}}^{1/2}ln1/2(1/T), 1/T2G{\mathit{T}}_{2\mathit{G}}T2G​\ensuremath{\sim}ln1/2{\mathrm{ln}}^{1/2}ln1/2(1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If Aq{\mathit{A}}_{\mathit{q}}Aq​ 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.
We study dynamical properties at finite temperature (T)(T)(T) of Heisenberg spin chains with random antiferromagnetic exchange couplings, which realize the random singlet phase in the low-energy limit, using three complementary numerical methods: exact diagonalization, matrix-product-state algorithms, and stochastic analytic continuation of quantum Monte Carlo results in imaginary time. Specifically, we investigate the dynamic spin structure factor $S(q,\ensuremath{\omega})$ and its $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit, which are closely related to inelastic neutron scattering and nuclear magnetic resonance (NMR) experiments (through the spin-lattice relaxation rate 1/T1)1/{T}_{1})1/T1​). Our study reveals a continuous narrow band of low-energy excitations in $S(q,\ensuremath{\omega})$, extending throughout the qqq space, instead of being restricted to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as found in the uniform system. Close to $q=\ensuremath{\pi}$, the scaling properties of these excitations are well captured by the random-singlet theory, but disagreements also exist with some aspects of the predicted qqq dependence further away from $q=\ensuremath{\pi}$. Furthermore we also find spin diffusion effects close to q=0q=0q=0 that are not contained within the random-singlet theory but give non-negligible contributions to the mean 1/T11/{T}_{1}1/T1​. To compare with NMR experiments, we consider the distribution of the local relaxation rates 1/T11/{T}_{1}1/T1​. We show that the local 1/T11/{T}_{1}1/T1​ values are broadly distributed, approximately according to a stretched exponential. The mean 1/T11/{T}_{1}1/T1​ first decreases with TTT, but below a crossover temperature it starts to increase and likely diverges in the limit of a small nuclear resonance frequency ${\ensuremath{\omega}}_{0}$. Although a similar divergent behavior has been predicted and experimentally observed for the static uniform susceptibility, this divergent behavior of the mean 1/T11/{T}_{1}1/T1​ has never been experimentally observed. Indeed, we show that the divergence of the mean 1/T11/{T}_{1}1/T1​ is due to rare events in the disordered chains and is concealed in experiments, where the typical 1/T11/{T}_{1}1/T1​ value is accessed.