Emergence of nematic loop-current bond order in vanadium Kagome metals

TL;DR

The paper addresses the origin of nematic TRSB CDW order in the AV$_3$Sb$_5$ Kagome metals by combining a $9$-band tight-binding model with nearest-neighbor Coulomb interactions and an effective patch model around the $M$-point van Hove singularities. It shows that nematic loop-current bond order (NLCBO) emerges only in a narrow parameter window, as a concurrent real bond-order modulation on one Kagome bond and imaginary loop currents on the others, driven by phase frustration among the three $Q=0$ bond-order components and enhanced by interpatch coupling $oldsymbol{ extlambda}$. The analysis reveals that the nematic state features an elongated Fermi surface and a total phase $oldsymbol{ extPhi}=oldsymbol{ extpi}$, with specific phase choices $ig( heta_1, heta_2, heta_3 ig)=(0,oldsymbol{ extpi}/2,oldsymbol{ extpi}/2)$ that minimize the free energy. These findings illuminate how electronic structure near the $M$ points and symmetry-allowed couplings can stabilize nematic TRSB order, offering tunable routes via pressure or doping and guiding future experimental probes such as ARPES.

Abstract

The family of layered Kagome metals $\mathrm{A}\mathrm{V}_3\mathrm{Sb}_5$ $(\mathrm{A}=\mathrm{K,Rb,Cs})$ has recently attracted significant interest due to reports of charge-bond order, orbital magnetism, and superconductivity. Some of these phases may exhibit time-reversal symmetry breaking, as suggested by their response to magnetic fields. More recently, experiments have reported the emergence of nematic order that lowers the rotational symmetry of the system from sixfold to twofold. Here we investigate the mechanism behind a nematic phase that breaks both rotational and time-reversal symmetries. Starting from a nine-band tight-binding model and nearest-neighbour Coulomb interactions, we find nematic order to emerge in a narrow region of phase space within mean-field theory. The nematic state is a superposition of charge-bond order along one Kagome bond and loop-current order on the other two, preserving one of the three mirror planes. To understand this behaviour, we examine an effective patch model that captures one $p$-type and one $m$-type van Hove singularity at each $M$ point on the Brillouin zone boundary. Within the effective model, nematic order is stabilized by the coupling between the complex phases of the three bond order parameters. As a consequence, the nematic phase develops an elongated Fermi surface distinct from those of competing phases.

Figures (10)

• Figure 1: Band structure of the nine-band tight-binding model. Colour denotes the $d$-orbital weight.
• Figure 2: Fermi surfaces of the nine-band model. Colour as in Fig. \ref{['fig:bandstructure']}. The chosen chemical potentials are (a) $\mu<\rm{vH1},$ (b) $\mu=\rm{vH1},$ (c) $\mu=\rm{vH2},$ and (d) $\mu>\rm{vH2}.$
• Figure 3: Summary of bond-ordered phases. (a) For uniform phases preserving rotational symmetry, the order parameter can be represented on the complex plane. Real-valued $\Delta$ corresponds to modulations of the hopping amplitude (red positive, blue negative): (b) $\Delta>0$, (c) $\Delta<0$. Complex-valued $\Delta$ have non-zero expectation value of the current density $j_{\alpha\beta}\sim i\langle c_{\mathbf{R}\alpha}^\dagger c_{\mathbf{R'}\beta}-c_{\mathbf{R'}\beta}^\dagger c_{\mathbf{R}\alpha}\rangle.$ The direction of the current is represented by the arrows, and the TR partner is obtained by reversing the directions of the arrows. (d) Pure imaginary-valued $\Delta$ corresponds to a pattern of orbital currents without modulation of the hopping magnitude. (e) Coexisting loop-current and charge-bond order, possessing both amplitude and phase modulations. (f) Nematic loop-current bond order (ordering wavevector $\mathbf{Q}_{AB}=\mathbf{M}_C$ is chosen for example). The phase structure $(\phi_1,\phi_2,\phi_3)$ of each order is indicated (see text).
• Figure 4: Phase diagram of the tight-binding model at $T=90$ K. NLCBO emerges in a small pocket above vH2. The total phase transitions from $\Phi=0$ to $\Phi = \pi$ across $\mu\approx \rm{vH1}$. Fermi surfaces associated with points marked by crosses are shown in Fig. \ref{['FS3x']}.
• Figure 5: Fermi surfaces of the ordered phases. The chosen chemical potentials correspond to the black crosses in Fig. \ref{['fig:9x9a']}. The hexagon denotes the boundary of the folded Brillouin zone, where $\Gamma,M_A,M_B$ and $M_c$ are all mapped to the same point ($\tilde{\Gamma}$). Colour as in Fig. \ref{['fig:bandstructure']}.