Discovery of BKT correlations in the quantum kagome compound Cs$_2$Cu$_3$SnF$_{12}$

Abstract

We investigate the microscopic properties of the kagome compound Cs$_2$Cu$_3$SnF$_{12}$ using $^{63,65}$Cu nuclear quadrupolar resonance (NQR). Analysis of the local hyperfine fields below the Néel temperature $T_N = 20$ K indicates a spin structure consistent with $P2_1/n$ symmetry of negative vector chirality. Measurements of the spin-lattice relaxation rate $T_1 ^{-1}$ reveal signatures of a gapless ground state and two-dimensional Berezinskii-Kosterlitz-Thouless (BKT)-type correlations above $T_N$, extending over a broad temperature range of approximately 130 K in zero magnetic field. Within the same temperature range, the observed increase in the NQR linewidth is consistent with short-range chiral order recently identified by neutron scattering. Our results establish Cs$_2$Cu$_3$SnF$_{12}$ as a unique quantum kagome system exhibiting BKT behavior.

Figures (3)

• Figure 1: (a) NQR line is split below the structural transition temperature $T_t=186$ K. Below $T_\text{N} = 20$ K magnetic order onsets and the spectrum is further split. Circles and triangles mark the signals of $^{63,65}$Cu isotopes. Purple marks the signal of a single crystallographic site above $T_t$, while red and blue mark the signals of Cu(1) and Cu(2) sites, respectively. Lines are guides for the eyes. (b) $T$-dependence of NQR linewidths of Cu(1,2) sites show an increase below $\sim 150$ K. Note that temperature scales in (a) and (b) are different.
• Figure 2: (a) Depiction of the hyperfine coupling (EFG) tensor tilting for angles $\alpha_{1,2}$ with respect to the kagome plane for Cu(1) and Cu(2) sites in red and blue, respectively. Tilting angles are emphasized for clarity. Principal axes $A_\parallel$ ($\textbf{e}_\text{EFG}$) are marked with dashed black lines, kagome plane normal $\textbf{n}$ with full lines, $A_\perp$ component in cyan and red dashed circles. 3D arrows show the spin orientation on the Cu(1,2) sites determined by Ref. Matan2019 (gray) and this work (green). See also Fig. S5. (b) Fitting of spectra below $T_\text{N}$ of Fig. \ref{['fig1']}(a) results with local magnetic fields on Cu(1) (red circles) and Cu(2) (blue circles). Blue squares are additional points for Cu(2) obtained by measuring only the highest-frequency line and fixing the fitting parameters obtained from the spectra.
• Figure 3: (a) $T_2 ^{-1}$ temperature dependence of both Cu sites. Below $T_\text{N} = 20$ K only the Cu(2) site was measured because of lower intensity of the Cu(1) site. Vertical dotted (dashed) line marks the magnetic (structural) transition temperature $T_\text{N}$ ($T_t$). (b) $T_1 ^{-1}$ temperature dependence of Cu(2) site. Only a small increase is seen at $T_t$. Below $150$ K $T_1 ^{-1}$ starts to monotonously increase down to $T_\text{N}$. Full red line is a fit $T_1^{-1} \propto\xi_\text{BKT}(T)^{z-\eta} + a_1 T+a_2 T^2$ (see main text). Below $T_\text{N}$, $T_1 ^{-1}$ drops first rapidly ($\propto T^6$) and below 9 K $\propto T^3$ indicating a gapless ground state. For comparison, we also show relaxation of Cu in RCSF which has a singlet ground state and shows no BKT divergence (yellow squares), but a gapped relaxation $T_1^{-1} \propto \exp(-\Delta/T)$ (red dashed line). (c) $\xi_\text{BKT}(T)$ calculated from (b). Full red line is the fit to $\xi_0\exp(0.5 \pi ⁄\sqrt{T⁄T_\text{BKT} -1})$.