Quantum critical dynamics of an $S=\frac{1}{2}$ antiferromagnetic Heisenberg chain studied by ${}^{13}\text{C}$ NMR spectroscopy

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 $\text{Cu}\left({\text{C}}_4{\text{H}}_4{\text{N}}_2\right){\left({\text{NO}}_3\right)}_2$ (CuPzN). The static local magnetization as well as the low-frequency spin dynamics, probed via the nuclear spin-lattice relaxation rate $T_1^{-1}$, were explored from the low to the high field limit and at temperatures from the quantum regime $\left(k_{B}T⪡J\right)$ up to the classical regime $\left(k_{B}T⪢J\right)$. 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 $T_1^{-1}$ 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.

Figure 1

(a) Crystal structure of CuPzN. (b) ${}^{13}\text{C}$ NMR spectrum for $B_0=2\text{T}$, $T=7\text{K}$, and $\beta =50°$ in the $b\text{−}c$ plane. (c) NMR center shift $\delta$ of spectral lines versus $\beta$ compared with a simulation of the local magnetic field. (Ref. 26)

Figure 2

Comparison of temperature-dependent NMR shift $\delta$ with magnetization data calculated by QMC. For clarity, an offset has been added to the data at 13.8 T ($+1000\text{ppm}$ to NMR shift and $0.0099\left({\mu }_B/\text{T}\right)$ to calculated magnetization) and 28 T ($-1000\text{ppm}$ and $-0.0099\left({\mu }_B/\text{T}\right)$, respectively). The QMC errors are within symbol size. All solid lines are a guide to the eye.

Figure 3

Field dependence of the nuclear spin-lattice relaxation rate of ${}^{13}\text{C}$ in the critical regime. Left inset: The log-scale plot demonstrates the linear opening of the spin gap with field. Right inset: the full-scale plot highlights the maximum of $T_1^{-1}\left(B\right)$ near the $T=0\text{K}$ critical field. All solid lines are a guide to the eye.

Figure 4

Temperature dependence of the nuclear spin-lattice relaxation rate of ${}^{13}\text{C}$ for different external fields. The solid QMC data lines are each polynomial fits to 50 analytic continuations of a 128 site system and the error tube was chosen to contain all data points within a range of 2 $\sigma$. The log-scale inset shows the exponential decrease in $1/T_1$ with $1/T$ above $B_c$.