Nonmagnetic impurities in a $S=\frac{1}{2}$ frustrated triangular antiferromagnet: Broadening of ${}^{13}\text{C}$ NMR lines in $\kappa \text{−}{\left(\text{ET}\right)}_2{\text{Cu}}_2{\left(\text{CN}\right)}_3$

We study effects of nonmagnetic impurities in a spin-1/2 frustrated triangular antiferromagnet with the aim of understanding the observed broadening of ${}^{13}\text{C}$ NMR lines in the organic spin liquid material $\kappa \text{−}{\left(\text{ET}\right)}_2{\text{Cu}}_2{\left(\text{CN}\right)}_3$. For high temperatures down to $J/3$, we calculate local susceptibility near a nonmagnetic impurity and near a grain boundary for the nearest-neighbor Heisenberg model in high-temperature series expansion. We find that the local susceptibility decays to the uniform one in few lattice spacings, and for a low density of impurities we would not be able to explain the line broadening present in the experiments already at elevated temperatures. At low temperatures, we assume a gapless spin liquid with a Fermi surface of spinons. We calculate the local susceptibility in the mean field and also go beyond the mean field by Gutzwiller projection. The zero-temperature local susceptibility decays as a power law and oscillates at $2k_F$. As in the high-temperature analysis we find that a low density of impurities is not able to explain the observed broadening of the lines. We are thus led to conclude that there is more disorder in the system. We find that a large density of pointlike disorder gives broadening that is consistent with the experiment down to about 5 K, but that below this temperature additional mechanism is likely needed.

Figure 1

(Color online) The uniform susceptibility and the local susceptibilities at the first five inequivalent neighbors of nonmagnetic impurity (missing site, which is denoted by an open circle). The four curves for each ${\chi }_i$ are the Padé approximants [5,6],[5,7],[6,6],[6,7]. $\chi$ is in units of ${\left(g{\mu }_B\right)}^2/J$.

Figure 2

The histogram of local susceptibilities for a $30\times 30$ triangular lattice sample with 5% randomly placed vacancies, obtained from the high-temperature series expansion to 11th order in $J$. $\chi$ is in units of ${\left(g{\mu }_B\right)}^2/J$.

Figure 3

(Color online) The uniform susceptibility and the local susceptibilities at the first five inequivalent neighbors of the boundary. The four curves for each $\chi$ are the Padé approximants [5,6],[5,7],[6,6],[6,7]. $\chi$ is in units of ${\left(g{\mu }_B\right)}^2/J$.

Figure 4

(Color online) The uniform susceptibility and the local susceptibilities at the first five inequivalent neighbors of the impurity obtained for the system of free fermions hopping on the triangular lattice at half-filling. The spinon hopping $t$ is kept fixed and the susceptibilities are in units of ${\left(g{\mu }_B\right)}^2/t$.

Figure 5

Local susceptibility near impurity at $T=0.074t$ obtained from the free fermion calculation for size $80\times 80$.

Figure 6

(Color online) The uniform susceptibility and the local susceptibilities at the first five inequivalent neighbors of the boundary obtained for the system of free fermions. $\chi$ is in units of ${\left(g{\mu }_B\right)}^2/t$.

Figure 7

The local susceptibility as a function of the distance from the boundary $x$ (line index in Fig. 6) at $T=0.0364t$. Note that the vertical scale does not start at zero and that the $x\ge 2$ sites lie within $\pm 15\%$ window around the average.

Figure 8

Histogram of the susceptibilities for the system with grain boundaries separated by 100 lines of sites.

Figure 9

Histogram of the susceptibilities for the system with 5% of vacancies. Note especially that the peaks broaden as we lower the temperature. The curves are obtained for $60\times 60$ samples and are in the thermodynamic limit for temperatures shown.

Figure 10

Histogram of the susceptibilities for the system with random values of bonds with randomness $\pm 20\%$. Note especially that the peaks do not broaden significantly below $T\approx 0.44t$. The curves are obtained for $60\times 60$ samples and are in the thermodynamic limit for temperatures shown.

Figure 11

(Color online) Local magnetization profile in a 1D chain of length $L=129$ with open boundaries before and after Gutzwiller projection of a partially polarized spinon Fermi sea state, here with $N_{\uparrow }=65$, $N_{\downarrow }=64$. The excess up spin occupies orbital $\text{sin}\left(k_{F}x\right)=\text{sin}\left(\pi x/2\right)$, producing a staggered component in ${\chi }_{\text{loc}}$ with a constant amplitude. The projection dramatically enhances the staggered component, which now grows as $\sqrt{x}$ away from the boundary, in agreement with Ref. 38 for the Heisenberg chain. Such effects can produce broad ${\chi }_{\text{loc}}$ histogram even for small impurity density in the 1D chain. We want to contrast this with the 2D case, where we find only a fixed numerical enhancement.

Figure 12

(Color online) Local magnetization profile in a triangular lattice with grain boundaries before and after Gutzwiller projection of a partially polarized spinon Fermi sea state. The lattice is constructed of 24 chains of length 24, with periodic boundary conditions along the chains and open BC in the perpendicular direction; see the drawing in Fig. 6. The partial spin polarization is obtained starting from an unpolarized state and depopulating five spin-down orbitals and occupying five spin-up orbitals near the Fermi patch whose normal is perpendicular to the boundary.

Figure 13

Histogram of the susceptibilities in the metallic phase, where we model disorder by random on-site potentials uniformly distributed in an interval $\left[-W,W\right]$ with $W=0.3t_e$. The curves are obtained for $60\times 60$ samples and are in the thermodynamic limit for temperatures shown.