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arXiv:1007.3608 · Physical Review B

Dynamics of a Heisenberg spin chain in the quantum critical regime: NMR experiment versus effective field theory

H. Kühne, Andrei A. Zvyagin, M. Günther, Arneil P. Reyes, P. L. Kuhns, Mark M. Turnbull, C.P. Landee, H.‐H. Klauß

Published Mar 14, 2011• 26 citations
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Abstract

A comprehensive comparison between the magnetic field- and temperature-dependent low-frequency spin dynamics in the antiferromagnetic spin-1/2 Heisenberg chain system copper pyrazine dinitrate, probed via the $^{13}\mathrm{C}$ 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.

DOI
10.1103/physrevb.83.100407

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A comprehensive comparison between the magnetic field- and temperature-dependent low-frequency spin dynamics in the antiferromagnetic spin-1/2 Heisenberg chain system copper pyrazine dinitrate, probed via the 13C^{13}\mathrm{C}13C nuclear magnetic resonance relaxation rate T1\ensuremath−1{T}_{1}^{\ensuremath{-}1}T1\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.
We present a 13C^{13}\text{C}13C NMR study of the magnetic field driven transition to complete polarization of the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg chain system copper pyrazine dinitrate Cu(C4H4N2)(NO3)2\text{Cu}({\text{C}}_{4}{\text{H}}_{4}{\text{N}}_{2}){({\text{NO}}_{3})}_{2}Cu(C4​H4​N2​)(NO3​)2​ (CuPzN). The static local magnetization as well as the low-frequency spin dynamics, probed via the nuclear spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}$, were explored from the low to the high field limit and at temperatures from the quantum regime (kBT⪡J)({k}_{B}T⪡J)(kB​T⪡J) up to the classical regime (kBT⪢J)({k}_{B}T⪢J)(kB​T⪢J). The experimental data show very good agreement with quantum Monte Carlo calculations over the complete range of parameters investigated. Close to the critical field, as derived from static experiments, a pronounced maximum in ${T}_{1}^{\ensuremath{-}1}$ is found which we interpret as the finite-temperature manifestation of a diverging density of zero-energy magnetic excitations at the field-driven quantum critical point.
We present an extensive NMR study of the spin-1/2 antiferromagnetic Heisenberg ladder Cu2(C5H12N2)2Cl4\mathrm{Cu}{}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}\mathrm{Cl}{}_{4}Cu2​(C5​H12​N2​)2​Cl4​ in a magnetic field range 4.5--16.7 T. By measuring the proton NMR relaxation rate 1/T11/{T}_{1}1/T1​ and varying the magnetic field around the critical field ${H}_{c1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Delta}/g{\ensuremath{\mu}}_{B}\ensuremath{\approx}7.5\mathrm{T}$, we have studied the transition from a gapped spin liquid ground state to a gapless magnetic regime which can be described as a Luttinger liquid. We identify an intermediate regime $T\ensuremath{\ge}|{H\ensuremath{-}H}_{c1}|$, where the spin dynamics is (possibly) only controlled by the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 critical point Hc1{H}_{c1}Hc1​.
Low-dimensional quantum magnets promote strong correlations between magnetic moments that lead to fascinating quantum phenomena. A particularly interesting system is the antiferromagnetic spin-1/2 Heisenberg chain because it is exactly solvable by the Bethe-Ansatz method. It is approximately realized in the magnetic insulator copper pyrazine dinitrate, providing a unique opportunity for a quantitative comparison between theory and experiment. We investigate its thermodynamic properties with a particular focus on the field-induced quantum phase transition. Thermal expansion, magnetostriction, specific heat, magnetization, and magnetocaloric measurements are found to be in excellent agreement with exact Bethe-Ansatz predictions. Close to the critical field, thermodynamics obeys the expected quantum critical scaling behavior, and in particular, the magnetocaloric effect and the Grüneisen parameters diverge in a characteristic manner. Beyond its importance for quantum magnetism, our study establishes a paradigm of a quantum phase transition, which illustrates fundamental principles of quantum critical thermodynamics.
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.
We demonstrate quantum critical scaling for an S=1/2S=1/2S=1/2 Heisenberg antiferromagnetic chain compound Cu(C4H4N2)(NO3)2\mathrm{Cu}({\mathrm{C}}_{4}{\mathrm{H}}_{4}{\mathrm{N}}_{2}){({\mathrm{NO}}_{3})}_{2}Cu(C4​H4​N2​)(NO3​)2​ in a magnetic field around saturation, by analyzing previously reported magnetization [Y. Kono et al., Phys. Rev. Lett. 114, 037202 (2015)], thermal expansion [J. Rohrkamp et al., J. Phys.: Conf. Ser. 200, 012169 (2010)], and NMR relaxation data [H. K\"uhne et al., Phys. Rev. B 80, 045110 (2009)]. The scaling of magnetization is demonstrated through collapsing the data for a range of both temperature and field onto a single curve without making any assumption for a theoretical form. The data collapse is subsequently shown to closely follow the theoretically predicted scaling function without any adjustable parameters. Experimental boundaries for the quantum critical region could be drawn from the variable range beyond which the scaled data deviate from the theoretical function. Similarly to the magnetization, quantum critical scaling of the thermal expansion is also demonstrated. Further, the spin dynamics probed via NMR relaxation rate 1/T11/{T}_{1}1/T1​ close to the saturation is shown to follow the theoretically predicted quantum critical behavior as $1/{T}_{1}\ensuremath{\propto}{T}^{\ensuremath{-}0.5}$ persisting up to temperatures as high as ${k}_{\mathrm{B}}T\ensuremath{\simeq}J$, where JJJ is the exchange coupling constant.
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.