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arXiv:0804.2170 · Physical Review B

Quantum critical dynamics of an S=12 antiferromagnetic Heisenberg chain studied by C13 NMR spectroscopy

H. Kühne, H.‐H. Klauß, Simon Grossjohann, W. Brenig, Fred Jochen Litterst, Arneil P. Reyes, P. L. Kuhns, Mark M. Turnbull, C.P. Landee

Published Jul 13, 2009• 32 citations
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Abstract

We present a $^{13}\text{C}$ NMR study of the magnetic field driven transition to complete polarization of the $S=\frac{1}{2}$ antiferromagnetic Heisenberg chain system copper pyrazine dinitrate $\text{Cu}({\text{C}}_{4}{\text{H}}_{4}{\text{N}}_{2}){({\text{NO}}_{3})}_{2}$ (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 $({k}_{B}T⪡J)$ up to the classical regime $({k}_{B}T⪢J)$. 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.

DOI
10.1103/physrevb.80.045110

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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 T1\ensuremath−1{T}_{1}^{\ensuremath{-}1}T1\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 T1\ensuremath−1{T}_{1}^{\ensuremath{-}1}T1\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.
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 ${T}_{1}^{\ensuremath{-}1}$, 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 are reported of the magnetic field dependence of excitations in the quantum critical state of the spin S=1/2 linear chain Heisenberg antiferromagnet copper pyrazine dinitrate (CuPzN). The complete spectrum was measured at k(B)T/J< or =0.025 for H=0 and H=8.7 T, where the system is approximately 30% magnetized. At H=0, the results are in agreement with exact calculations of the dynamic spin correlation function for a two-spinon continuum. At H=8.7 T, there are multiple overlapping continua with incommensurate soft modes. The boundaries of these continua confirm long-standing predictions, and the intensities are consistent with exact diagonalization and Bethe ansatz calculations.
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.
High-precision DC magnetization (M) measurements have been performed on a single crystal of copper pyrazine dinitrate (CuPzN) in magnetic fields H up to 14.7 T at temperatures down to 0.08 K. This material is one of the best realizations of a one-dimensional (1D) S = 1/2 Heisenberg antiferromagnet. At the base temperature, full saturation of M is observed above Hs = 13.97 T with the M(H) curve closely following a prediction based on the Bethe ansatz. The peak in M(T ), which appears at about 7 K in zero field, moves to lower temperatures with increasing H and vanishes into a temperature region well below 0.08 K at H ∼ Hs; as a result, M(T ) exhibits a strong upturn as T → 0 at this field, in agreement with a theoretical prediction. These results constitute an observation of field-induced quantum critical behavior in bulk magnetization.
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.
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.