Site-selective observation of spin dynamics of a Tomonaga-Luttinger liquid in frustrated Heisenberg chains

The ³⁵Cl NMR spectrum of CROC consists of extremely sharp resonance lines coming from two Cl sites, Cl(1) and Cl(2), under magnetic field along the $a$ axis, as shown in Fig. 2. For $I=3/2$, the spectrum from each Cl site splits into three due to the electric-quadrupole interaction between the nuclear quadrupole moment $Q$ and the EFG at the Cl site. The number of resonance lines doubles as the inversion symmetry is broken away from the $ab$ and $bc$ planes, which allows us the accurate field alignment along the crystal axes within $1^{\circ}$. The quadrupole splitting $\delta\nu$ and the central frequency are plotted in Fig. 8 of Appendix A. The angular dependence is compared with the LDA calculation based on the crystal structure of CROC, leading to the unambiguous site assignment of the resonance lines into Cl(1) and Cl(2), as shown in Fig. 2.

The NMR spectrum displays a site-dependent shift upon cooling, as shown in Fig. 3(a). The low-frequency spectrum from Cl(1) exhibits a downward shift, while the higher one from Cl(2) stays at nearly the same position. Since $\chi$ exhibits Curie-Weiss paramagnetic behavior at high temperatures, the downward shift indicates the negative hyperfine coupling constant for Cl(1). Surprisingly, the linewidth remains sharp down to low temperatures, indicating a high-quality crystal free from the magnetic inhomogeneity arond free spins.

The Knight shift $K$ obtained from the central resonance line scales to the local spin susceptibility $\chi_{ii}$ along the crystal $i$ axis; $K_{ii}=\frac{A_{ii}}{N\mu_{\rm B}}\chi_{ii}$ ($i=a,b,c$), where $A_{ii}$ is the diagonal hyperfine coupling, $N$ the Avogadro number, and $\mu_{\rm B}$ the Bohr magneton. Figure 4 shows the $T$ dependence of $K$ for Cl(1) and Cl(2), denoted as $K(1)$ and $K(2)$, respectively. $-K(1)$ increases upon cooling, showing a broad peak around 28 K. It becomes nearly constant at low temperatures. The temperature dependence of $-K(1)$ for each crystal axis fits to the Bonner-Fisher model with $J/k_{\rm B}$ = 38 K ($7<T<70$ K) [47] in agreement with the result of $\chi$ ($J/k_{\rm B}=41.3$ K) [39]. In contrast with the bulk $\chi$, the local spin susceptibility obtained from $K(1)$ excludes an impurity contribution and extracts the intrinsic residual spin susceptibility down to low temperatures ($3.4\times 10^{-3}$ emu/mol), which corroborates the gapless TLL state. The spin susceptibility of TLL is expressed as $\chi(0)=(g\mu_{\rm B})^{2}K_{\sigma}/(\pi v)$ using the spin velocity $v=J\pi/2$ at $T=0$ [48, 49]. Using the $T$-linear term of the specific heat, $\gamma=115$ mJ/(K mol) [39], the Wilson ratio $R_{W}=(\pi k_{\rm B}/g\mu_{\rm B})^{2}(4\chi/3\gamma)$ is obtained as 2.46. It is compared with $R_{W}=4K_{\sigma}$ expected for TLL spin chains [49]. Using $K_{\sigma}=0.61-0.87$ for Cl(1) from $T_{1}^{-1}$ as described below (Table. 1), we can independently obtain $R_{W}=2.44$–3.48, consistent with the Knight shift.

Since $\chi$ is nearly isotropic [39], the anisotropy of $K(1)$ comes from the anisotropic hyperfine coupling constant governed by transfer or dipole interactions. $K(1)$ linearly scales to $\chi$ above 28 K, as seen in the $K(1)-\chi$ plot [Fig. 4(b)]. The components of the hyperfine coupling tensor are evaluated from the linearity as ($A_{aa}$, $A_{bb}$, $A_{cc}$, $A_{ac}$) = ($-0.12$, $-0.13$, $-0.058$, 0.13)T/$\mu_{\rm B}$ for Cl(1). Here, $A_{ac}$ is evaluated from the angular dependence of Knight shifts. The diagonalization yields the principal components of the hyperfine coupling ($A_{XX}$, $A_{YY}$, $A_{ZZ}$) = (0.04, $-0.13$, $-0.22$)T/$\mu_{\rm B}$ for Cl(1). The anisotropy is not explained by the dipolar interaction ($\sum_{\alpha}A_{\alpha\alpha}=0$), and hence comes from the transferred hyperfine interaction. In contrast, $K(2)$ along the $a$ and $b$ axes very weakly depends on $T$, indicating the tiny hyperfine coupling constants. We obtained ($A_{aa}$, $A_{bb}$, $A_{cc}$, $A_{ac}$) = ($-0.002$, 0.01, $-0.03$, 0.14)T/$\mu_{\rm B}$ and ($A_{XX}$, $A_{YY}$, $A_{ZZ}$) = (0.12, 0.01, $-0.15$)T/$\mu_{\rm B}$ for Cl(2). The remarkable site dependence reflects the transferred hyperfine paths originating the anisotropic Re $5d_{xy}$ orbital, as discussed in Appendixes.

Below $T_{\rm N}$ = 1.1 K, the NMR spectrum broadens and changes to a double horn shape, as shown in Fig. 3(b). It shows an emergence of the spontaneous local field below $T_{\rm N}$. One can exclude a possible collinear magnetic ordering that involves a discrete splitting of the NMR spectrum by the staggered hyperfine field. Despite a hyperfine coupling constant of Cl(1) greater than Cl(2), the splitting amplitude is smaller for Cl(1) along the $a$ axis. This means that the local fields at Cl sites are generated through the off-diagonal hyperfine coupling from the Re magnetic moments oriented perpendicular to the $a$ axis. The double horn shape is typically observed in incommensurate magnetic orders such as corn and spiral orders [35, 50]. In the present case with strong 1D anisotropy, weakly incommensurate modulation of the wave vector can be induced by DM interactions due to the lack of inversion symmetry along the chain [44, 41]. For the $D$-vector ${\bf D}$ parallel to the $c$ axis, the DM interaction forces the magnetic moments to align perpendicular to D. Therefore, we conclude that the moment direction should be close to the $b$ axis.

The nuclear spin-lattice relaxation rate $T_{1}^{-1}$ measures low-energy excitations in the NMR frequency window through magnetic and electric hyperfine interactions. As shown in Fig. 5(a), $T_{1}^{-1}$ exhibits sharp and broad peaks around 1.1 and 50 K, respectively, for $H_{0}$ = 9.13 T along the $a$ axis. The sharp peak manifests the critical slowing-down of spin fluctuations toward long-range magnetic ordering at $T_{\rm N}$ = 1.1 K. $T_{1}^{-1}$ drops steeply ($\sim T^{2}$) below $T_{\rm N}$ where the excitation is dominated by gapless magnons.

The broad $T_{1}^{-1}$ peak around 50 K can be attributed to structural fluctuations such as atomic motions instead of phase transition, since there is no signature of symmetry breaking in the NMR spectrum (Fig. 3) and specific heat [39]. For confirmation, we measured $T_{1}^{-1}$ of two isotopes ³⁵Cl and ³⁷Cl having distinct nuclear quadrupole moments $Q=-8.2$ and $-6.5\times 10^{-26}{\rm cm}^{2}$, respectively. As shown in Fig. 5(b), $T_{1}^{-1}$ approximately scales to $Q^{2}$ around 50 K, consistent with the predominant EFG fluctuations [51].

As the resonance frequency decreases from 38 MHz at 9.13 T to 20 MHz at 4.95 T, the $T_{1}^{-1}$ peak shifts to 43 K, as shown in Fig. 6(a). Further low frequency ($\sim$ kHz) fluctuations can be observed through $T_{2}^{-1}$, where the spin-echo intensity decays exponentially with the pulse interval time $\tau=100-1000$ $\mu$s. As seen from the sharp resonance line, $T_{2}$ is extremely long, despite a paramagnetic Mott insulator, and reaches $\sim 20$ ms for Cl(1). As shown in Fig. 6(b), $T_{2}^{-1}$ is independent of $T$ above 30 K and exhibits a sharp peak at $T^{*}$ = 18 K, indicating the freezing of atomic motions. Upon cooling, $T_{2}^{-1}$ remains constant for Cl(2), while it gradually increases for Cl(1) toward the magnetic order.

From a structural point of view, the x-ray diffraction measurement on CROC shows a large thermal structure factor of Cl sites at room temperature [38]. Although the structural data of CROC are absent at low temperatures, the metastable atomic positions may induce a persistent structural instability due to the large open space. Interestingly, the structural freezing coincides with the onset of antiferromagnetic correlation, as manifested in the $\chi$ peak at 20 K.

The system crossovers from a quantum TLL regime ($T\ll J$) into a classical diffusive regime ($T\gg J$) with increasing temperature. In an intermediate temperature range of $T\sim J$, spin dynamics of 1D chains is expected to obey the superdiffusive Kardar-Parisi-Zhang (KPZ) universality with characteristic spin correlation [52, 53]. To extract the spin dynamics of the crossover regime, we subtract the structural contribution from $T_{1}^{-1}$ by assuming the Lorentzian correlation function in the Bloembergen-Purcell-Pound (BPP) model [54]:

where $\tau_{c}$ is the correlation time, $\tau_{c}=\tau_{0}\exp{(\frac{E_{a}}{k_{\rm B}T})}$, with the activation energy $E_{a}$ and the constant $\tau_{0}$. The experimental result is well fitted by Eq. (2) at the measured frequency (38 and 20 MHz at 9.13 and 4.95 T, respectively), as shown in Figs. 5 and 6, which yields $E_{a}/k_{\rm B}\approx$ 300 K. The fitting extracts a power law $\sim T^{n}$ term with the exponent $n=0.5$ in the temperature range above 20 K. It differs from those of the local spin fluctuations ($n=0$) [55] and the KPZ universality ($n=2$) [52, 53]. The exponent is rather close to that expected in the spin-drag relaxation of the spin-diffusion regime [56].

At low temperatures ($T\ll J$), $T_{1}^{-1}$ would be dominated by spin fluctuations along the chain, where structural fluctuations are exponentially suppressed. According to the fluctuation-dissipation theorem, the imaginary part of dynamical spin susceptibility $\chi({\bf q})$ relates to the dynamical spin structure factor $S^{\pm}({\bf q})$. $T_{1}^{-1}$ is expressed as [55, 57, 50, 58]

where $F_{\alpha\beta}({\bf q})$ ($\alpha,\beta=x,y,z$) is the form factor at the wave vector q and defined by $F_{\alpha\beta}({\bf q})=A_{\alpha\beta}({\bf q})A_{\alpha\beta}(-{\bf q})$ using the Fourier transformed hyperfine coupling $A_{\alpha\beta}({\bf q})=A_{\alpha\beta}({\bf r}){\rm e}^{i{\bf q\cdot r}}$. $\chi^{\prime\prime}_{\perp}$ and $\chi^{\prime\prime}_{\parallel}$ are the imaginary part of the dynamical spin susceptibility perpendicular and parallel to the magnetic field. Here, the $z$ axis is taken along the applied magnetic field direction. In 1D Mott insulators the low-energy $\chi({\bf q})$ is dominated by specific wave vectors close to ${\bf Q_{0}}$ = (0, $\pi$, 0) [2, 4, 10]. A shift from ${\bf Q_{0}}$ by the magnetic field would be negligible when the Zeeman energy is much lower than the exchange interaction $J$. Instead, the DM interaction induces a small splitting of the spinon dispersion by $\Delta q=0.07\pi$ [44], which is also omitted in the following analysis for simplicity.

In a TLL regime, $\chi({\bf q})$ obeys a power law in an anisotropic manner, $\chi_{\perp}\propto T^{1/2K_{\sigma}-2}$ and $\chi_{\parallel}\propto T^{2K_{\sigma}-2}$ [4, 7]. For an isotropic case, both transverse and longitudinal components of $T_{1}^{-1}$ are independent of temperature ($K_{\sigma}=0.5$), as expected for the antiferromagnetic Heisenberg chain [10, 4]. They become anisotropic as the Ising anisotropy $\Delta$ of the $XXZ$ model deviates from unity. For $K_{\sigma}>0.5$, the transverse component is enhanced at low temperatures, whereas the parallel component vanishes and thus becomes negligible. Furthermore, as seen from Eq. (3), $\chi_{\parallel}$ contributes to $T_{1}^{-1}$ only through the off-diagonal hyperfine coupling. Therefore, $T_{1}^{-1}$ would be dominated by $\chi_{\perp}(\mathbf{q})$, as shown in Eq.(1).

Below 10 K, $T_{1}^{-1}$ increases with a power law $\sim T^{n}$, as shown in Fig. 5. The fitting yields $n=-0.36$ for Cl(1) and $-0.59$ for Cl(2) in a magnetic field of 9.13 T along the $a$ axis. Applying the relation $n=1/(2K_{\sigma})-1$ in Eq. (1), we obtained $K_{\sigma}$ = 0.79 and 1.21 for Cl(1) and Cl(2), respectively. Since the dynamical spin susceptibility is carried by Re spins, $K_{\sigma}$ should be independent of the probe nuclear spins. Therefore, the observed Cl site dependence comes from the form factor dependence that filters antiferromagnetic fluctuations at the specific wave vector, as typically observed in tetragonal cuprate and pnictide superconductors [11, 58].

Referring to the crystal structure of CROC in Figs. 1 and 9, Cl(1) is located between neighboring Re atoms, while Cl(2) is on the Re atom along the $a$ axis. Here we consider antiferromagnetic spin fluctuations at the wave vector $\mathbf{Q_{0}}$ = (0, $\pi$, 0) along the chain. The staggered hyperfine fields from the second-neighbor Re sites are canceled out at Cl(1), as shown by the site-symmetry analysis in Appendix B. Then, the antiferromagnetic correlation at ${\bf q}$ = ${\bf Q_{0}}$ is filtered through the form factor at Cl(1), while it remains at Cl(2). Therefore, $T_{1}^{-1}$ measured at Cl(1) is strongly suppressed in contrast to that of Cl(2), despite the uniform hyperfine coupling of Cl(1) greater than that of Cl(2). $T_{1}^{-1}$ is indeed reversed between Cl(1) and Cl(2) at high temperatures above 50 K, where the uniform ${\bf q}=0$ mode becomes dominant.

In the TLL regime, the external magnetic field works as chemical potential and enhances magnetization or $K_{\sigma}$, as observed in the spin-1/2 ladder system with the spin gap [15]. We have investigated the field dependence of $K_{\sigma}$ for the Cl sites at $H_{0}$ = 4.95 T, as shown in Fig. 6. There is no significant field effect on $K_{\sigma}$, consistent with the negligible effect of the Zeeman energy in the gapless TLL, as listed in Table 1. Instead, the energy scale of the DM interaction is comparable to the magnetic field and thus impacts on the anisotropy.

Finally, we investigate the anisotropy of $T_{1}^{-1}$ by applying the magnetic field $H_{0}$ = 9.13 T along the $b$ and $c$ axes, as shown in Fig. 7. The result along the $b$ axis is similar to that for the $a$ axis in Figs. 5(a) and 6, where $T_{1}^{-1}$ is dominated by atomic fluctuations at high temperatures and then by electric spin fluctuations below 20 K. At low temperatures, we obtain $K_{\sigma}=0.87$ and 1.13 for Cl(1) and Cl(2), respectively, as listed in Table 1. In contrast, the anisotropy of $T_{1}^{-1}$ for the $c$ axis is not reversed at low temperatures, suggesting that the ${\bf q}={\bf Q_{0}}$ mode is less dominant in $T_{1}^{-1}$ for Cl(1) and Cl(2). Interestingly, $T_{1}^{-1}$ remains anisotropic in the intermediate temperature range where $T_{1}^{-1}$ exhibits the broad peak. It may indicate significant spin-lattice coupling through the DM interaction. In the TLL regime below 20 K, we obtained $K_{\sigma}$, 0.61 for Cl(1) and 0.70 for Cl(2), which is much lower than those in the other directions. The reason can be attributed to the anisotropy of dynamical spin susceptibility, as discussed below.