Quantum spin dynamics studied by the nuclear magnetic relaxation of protons in the Haldane-gap system $\left({\mathrm{CH}}_3{)}_4\mathrm{NNi}\right({\mathrm{NO}}_2{)}_3$

The nuclear spin-lattice relaxation time $T_1$ of ${}^1\mathrm{H}$ in the Haldane-gap system $\left({\mathrm{CH}}_3{)}_4\mathrm{NNi}\right({\mathrm{NO}}_2{)}_3$ has been measured at the temperatures down to $50\mathrm{mK}$ in the gapped and gapless phases with an external field $H$ up to $8.5\mathrm{T}$. In the gapless phase for $H>H_{\mathrm{C}1}=2.7\mathrm{T}$, the relaxation rate $T_1^{-1}$ exhibited, below about $3\mathrm{K}$, a divergent behavior with decreasing temperature. Such a feature is described by an equation, such as $T_1^{-1}\sim T^{-\alpha }$ with $\alpha =1-\eta$, where $\eta$ is the exponent of the power-law decay for the staggered mode of transverse correlation with respect to the distance in the one-dimensional chain. It has been found that the exponent $\alpha$ takes the values around 0.6, which are consistent with the numerical evaluation for $S=1$ Heisenberg antiferromagnetic linear chain (HALC). This yields evidence for the realization of Tomonaga-Luttinger liquid phase. At very low temperatures, $T_1^{-1}$ changes into a remarkable decrease after taking a round peak, accompanying an appreciable broadening in the NMR spectrum, and the temperature for such a peak has a trend to increase with increasing field. In the gapped phase for $H<H_{\mathrm{C}1}$, $T_1^{-1}$ showed an exponential decrease with decreasing temperature. The experimental results were well interpreted in terms of the intrabranch process associated with the scattering of the magnons within each of $S_z=-1$ and $+1$ branches of the first excited state. Anomalous hump of $T_1^{-1}$ appeared at low fields below $1\mathrm{T}$ and at low temperatures below $1\mathrm{K}$. This was reasonably explained considering the fractional-spin effect in the $S=1$ HALC.

Figure 1

Crystal structure of TMNIN. The ${\mathrm{Ni}}^{2+}$ linear chains extend along the $c$ axis. The ${\mathrm{Ni}}^{2+}$ ions, which are coordinated octahedrally with six nitrogens or six oxygens, appear alternately. The lattice constants are given in the text.

Figure 2

Temperature dependence of $T_1^{-1}$ for various fields. For the convenience, the data taken in the different temperature ranges with only slightly different operating frequencies were plotted using the same symbol.

Figure 3

Examples of NMR spectra for (a) the gapped phase $(H<H_{\mathrm{C}1})$ obtained at the operating frequency of ${\nu }_N=101.1\mathrm{MHz}$ and (b) the gapless phase $(H\gg H_{\mathrm{C}1})$ obtained at ${\nu }_N=286.7\mathrm{MHz}$. The spectra were obtained by plotting the spin-echo height by changing the applied field $H$. The solid line in (b) represents the NMR spectra calculated by assuming field-induced uniform moments (see Sec. 5).

Figure 4

The temperature dependence of $T_1^{-1}$ for $H=1.0$ and $1.2\mathrm{T}$, and $T>1.5\mathrm{K}$. The solid line represents the best fit of Eq. (6) to the experimental results obtained using the experimental values of ${\chi }_0$ and normalized at $T=20\mathrm{K}$.

Figure 5

Numerical calculation for the field dependence of $\eta$ for the staggered mode and the exponent $\alpha$ of the relaxation rate with the relation of $\alpha =1-\eta$.[16]

Figure 6

Temperature dependence of $T_1^{-1}$ for $H>H_{\mathrm{C}1}$. The solid lines represent the curve of $T_1^{-1}\sim T^{-\alpha }$ for the TL phase, where the values of $\alpha$ are taken from Fig. 5.

Figure 7

Excitation spectrum for the $S=1$ HALC in the presence of the external field, which is expressed in terms of wave vector $k$ with respect to twice of the lattice spacing along the $c$ axis. The thick, dotted, and solid arrows indicate intrabranch, interbranch, and staggered processes, respectively.

Figure 8

Comparison between the experiment and the calculation for the intra branch process [Eq. (13) in the text] below $H<H_{\mathrm{C}1}$. The value of the prefactor $A$ is chosen so as to obtain the best fitting.

Figure 9

The temperature-field diagram for various features in TMNIN revealed from the present relaxation measurements. The closed circles and the open squares indicate, respectively, the temperatures for the round minimum and the maximum of $T_1^{-1}$ for the applied external fields. The light solid line represents the phase boundary for the 3D long-range order evaluated numerically by Sakai.[23] The plus sign represents the temperature where the increase and the change of the NMR spectrum begins.

Figure 10

Examples of the recoveries of the normalized nuclear magnetization $m\left(t\right)$ as functions of time $t$ after the saturation the nuclear magnetization: (a) anomalous recovery for low fields below $1\mathrm{T}$ and the low temperatures below about $2\mathrm{K}$, and (b) normal recovery. Straight lines in (a) and (b) correspond to the single exponential recoveries, which yield the time constants $T_1$, and the dotted lines in (a) represent the curve $m\left(t\right)=\exp (-\sqrt{t∕{\tau }_1})$ with another time constant ${\tau }_1$.

Figure 11

Comparison between the experimental results at $H=1\mathrm{T}$ and the theoretical treatment for the fractional-spin effect. The closed and open circles below $1.7\mathrm{K}$ indicate the inverses of the time constants $T_1$ and ${\tau }_1$, which characterize the recovery of nuclear magnetization for the initial short-time and long-time durations, respectively. The dotted-dashed and dotted lines represent the theoretical curves for $T_1^{-1}$ and ${\tau }_1^{-1}$ characterizing the fractional-spin effect, which were calculated respectively from Eqs. (15, 16) using relevant numerical values given in the text. The closed circles above $1.7\mathrm{K}$ indicate usual relaxation rate. The solid and dashed lines represent the theoretical curves for the usual paramagnetic fluctuation and intrabranch process expressed by Eqs. (6, 13), which have already been given in Figs. 4, 8, respectively.