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arXiv:cond-mat/9411079 · Physical review. B, Condensed matter

NMR relaxation in half-integer antiferromagnetic spin chains

Subir Sachdev A

Published Nov 1, 1994• 87 citations
arXivHTMLPDF

Abstract

Nuclear relaxation in half-integer spin chains at low temperatures (T\ensuremath{\ll}J, the antiferromagnetic exchange constant) is dominated by dissipation from a gas of thermally excited, overdamped, spinons. The universal low-temperature dependence of the relaxation rates 1/${\mathit{T}}_{1}$ and 1/${\mathit{T}}_{2\mathit{G}}$ is computed.

DOI
10.1103/physrevb.50.13006

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Nuclear relaxation in half-integer spin chains at low temperatures (T\ensuremath{\ll}J, the antiferromagnetic exchange constant) is dominated by dissipation from a gas of thermally excited, overdamped, spinons. The universal low-temperature dependence of the relaxation rates 1/T1{\mathit{T}}_{1}T1​ and 1/T2G{\mathit{T}}_{2\mathit{G}}T2G​ is computed.
We consider the low-temperature behavior of NMR relaxation rates T1{T}_{1}T1​ and T2G{T}_{2G}T2G​ in the spin-1/21/21/2 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 Sr2CuO3.{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}.Sr2​CuO3​.
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.
Quasi-one-dimensional (1D) antiferromagnets are good candidates to realize a Tomonaga-Luttinger liquid (TLL), describing 1D interacting quantum systems, provided 3D coupling remains small as compared to temperature. In systems such as NiCl2{}_{2}2​-4SC(NH2{}_{2}2​)2{}_{2}2​ (DTN), made of weakly coupled spin S=1S=1S=1 chains, interchain coupling plays an important role at low temperature as it leads to antiferromagnetic long-range order below 111 K, corresponding to a Bose-Einstein condensate of magnons. Using time-dependent numerical simulations at finite temperature, the authors compute here the nuclear magnetic resonance (NMR) relaxation rate 1/T11/{T}_{1}1/T1​, a key experimental quantity, and address the question of the low-temperature crossover between TLL predictions and the higher-temperature regime in purely 111D quantum spins chains. They also comment on the experimental observation of the TLL regime, possibly masked by 333D long-range order in materials such as DTN.
Low-dimensional quantum magnets, particularly those with strong spin frustration, are characterized by their notable spin fluctuations. Nuclear magnetic resonance (NMR) serves as a sensitive probe of low-energy fluctuations that offers valuable insight into rich magnetic phases and emergent phenomena in quantum magnets. Although experimentally accessible, the numerical simulation of NMR relaxation rates, specifically the spin-lattice relaxation rate 1/T11/T_11/T1​, remains a significant challenge. Analytical continuation based on Monte Carlo calculations are hampered by the notorious negative sign for frustrated systems, and the real-time simulations incur significant costs to capture low-energy fluctuations. Here we propose computing the relaxation rate using thermal tensor networks (TTNs), which provides a streamlined approach by calculating its imaginary-time proxy. We showcase the accuracy and versatility of our methodology by applying it to one-dimensional spin chains and two-dimensional lattices, where we find that the critical exponents ηηη and zνzνzν can be extracted from the low-temperature scalings of the simulated 1/T11/T_11/T1​ near quantum critical points. Our results also provide insights into the low-dimensional and frustrated magnetic materials, elucidating universal scaling behaviors in the Ising chain compound CoNb2_22​O6_66​ and revealing the renormalized classical behaviors in the triangular-lattice antiferromagnet Ba8_88​CoNb6_66​O24_{24}24​. We apply the approach to effective model of the family of frustrated magnets AYbCh2_22​ (A = Na, K, Cs, and Ch = O, S, Se), and find dramatic changes from spin ordered to the proposed quantum spin liquid phase. Overall, with high reliability and accuracy, the TTN methodology offers a systematic strategy for studying the intricate dynamics observed across a broad spectrum of quantum magnets and related fields.
The low-temperature properties of spin-1/2 one-dimensional (1D) Heisenberg antiferromagnetic (HAF) chains which have relatively small exchange couplings between the spins can be tuned using laboratory-scale magnetic fields. Magnetization measurements, made as a function of temperature, provide phase diagrams for these systems and establish the quantum critical point (QCP). The evolution of the spin dynamics behavior with temperature and applied field in the quantum critical (QC) region, near the QCP, is of particular interest and has been experimentally investigated in a number of 1D HAFs using neutron scattering and nuclear magnetic resonance as the preferred techniques. In the QC phase both quantum and thermal spin fluctuations are present. As a result of extended spin correlations in the chains, magnon excitations are important at finite temperatures. An expression for the NMR spin-lattice relaxation rate 1/T11/{T}_{1}1/T1​ of probe nuclei in the QC phase of 1D HAFs is obtained by considering Raman scattering processes which induce nuclear spin flips. The relaxation rate expression, which involves the temperature and the chemical potential, predicts scaling behavior of 1/T11/{T}_{1}1/T1​ consistent with recent experimental findings for quasi-1D HAF systems. A simple relationship between 1/T11/{T}_{1}1/T1​ and the deviation of the magnetization from saturation (${M}_{S}\ensuremath{-}M$) is predicted for the QC region.
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.