Real-space Loop Current Pattern in Time-reversal-symmetry Breaking Phase in Kagome Metals

The charge loop current (cLC) state has attracted increasing attention in kagome metals. Here, we calculate the spontaneous currents along the nearest sites $i$ and $j$, $J_{i,j}$, induced by the cLC order that is the imaginary and odd-parity hopping integral modulation $\delta t_{i,j}$. We reveal that the magnitude of $J_{i,j}$ strongly depends on the nearest sites $i$ and $j$ in the $2\times 2$ cLC state, where $\eta \equiv \mathrm{\mid }\delta t_{i,j}\mathrm{\mid }$ is equivalent for all nearest sites. The obtained $J_{i,j}$ becomes large near the van-Hove singularity (vHS) filling ($n\sim n_{vHS}$) even when $\eta$ is fixed. Interestingly, the obtained $J_{i,j}$ exhibits the logarithmic divergence behavior at low temperatures for $n\sim n_{vHS}$ with a fixed $\eta$, by reflecting the vHS points that are the characteristic of kagome metals. The present study provides useful information for local electronic state measurements, such as the site-selective NMR and STM experiments.