Loop Current Order on the Kagome Lattice

Abstract

Recent discoveries in kagome materials have unveiled their capacity to harbor exotic quantum states, including intriguing charge density wave (CDW) and superconductivity. Notably, accumulating experimental evidence suggests time-reversal symmetry breaking within the CDW, hinting at the long-pursued loop current order (LCO). Despite extensive research efforts, achieving its model realization and understanding the mechanism through unbiased many-body simulations have remained both elusive and challenging. In this Letter, we develop a microscopic model for LCO on the spinless kagome lattice with nonlocal interactions, utilizing unbiased functional renormalization group calculations to explore ordering tendencies across all two-particle scattering channels. At the Van Hove filling, we identify sublattice interference to suppress onsite CDW order, leaving LCO, charge bond order, and nematic CDW state as the main competitors. Remarkably, a $2\times2$ LCO emerges as the many-body ground state over a significant parameter space with strong second nearest-neighbor repulsion, stemming from the unique interplay between sublattice characters and lattice geometry. The resulting electronic model with LCO bears similarities to the Haldane model and culminates in a quantum anomalous Hall state. We also discuss potential experimental implications for kagome metals.

Figures (15)

• Figure 1: Representative FRG flow with 1nn repulsion and real-space CDW patterns. (a) FRG flow of the expectation values of CBO, LCO, nCDW, and $\mathrm{CDW}_\text{M}$ and for $V_1=t$ and $V_2=0$. (b) Real-space configurations of CBO and LCO on 1nn and 2nn bond as well as the onsite modulated CDW with a wave vector of $\bm{Q}_C$. The real-space pattern at two other nesting vectors can be obtained by the application of sixfold rotational operation. Red (blue) denotes positive (negative) onsite or bond order parameters, while arrows denote directions of current flows. Only the bond orders within the antisymmetric channel are depicted.
• Figure 2: Phase diagram of the spinless interacting kagome model at the p-type Van Hove filling. The critical scales $\Lambda_{\mathrm{c}}$ proportional to the expected transition temperature $T_{\mathrm{c}}$ are indicated by color. Gray dots indicate the calculated points in interaction parameter space.
• Figure 3: (a) Representative FRG flow of the expectation values of nCDW, 2nn CBO, 2nn LCO and fSC for $V_1=0$ and $V_2=1.1t$. (b) Variation of 1nn and 2nn contributions to the LCO and CBO phases as a function of changing $V_1$ and $V_2$ at fixed $V_1+V_2=1.2t$. (c) Real-space pattern of the representative $3\bm{Q}$ LCO on both 1nn and 2nn bonds. The enlarged $2 \times 2$ supercell is indicated by gray shading. (d) Emergent Chern bands of the LCO phase in the folded BZ with order parameters $\Delta_{\mathrm{1nn}}^{\mathrm{LCO}}=0.1t$ and $\Delta_{\mathrm{2nn}}^{\mathrm{LCO}}=0.15t$, exhibiting a full gap around the Fermi level. The filled bands feature a total Chern number of $C=1$. Gray curves indicate the backfolded dispersion.
• Figure 4: (a) FRG flow of leading instabilities for $V_1=0$ and $V_2=1.6\,t$. (b) Variation of 3nn and 6nn components in the $f$- and $p$-wave superconductivity as a function of changing $V_1$ and $V_2$. (c) Schematics for the real-space pairing configuration, mediated by a combination of high-order virtual $V_1$ and $V_2$ processes. (d) $f$-wave gap function on the FS transforming under the $B_{2u}$ representation in the $V_2$ dominant regime.
• Figure S1: Kagome lattice, electronic structure and sublattice-resolved Fermi surface. (a) Real space structure of the kagome lattice with three sublattices A, B and C indicated by different colors. (b) Fermi surface at the upper van-Hove filling and corresponding sublattice makeup of the FS states. (c) Energy dispersion of the tight binding model along the high symmetry path and density of state (DOS) peaking featured by van-Hove singularities.