Origin of switchable quasiparticle-interference chirality in loop-current phase of kagome metals measured by scanning-tunneling-microscopy

TL;DR

The work addresses the origin of QPI chirality in the chiral loop-current phase of kagome metals by constructing a giant-unit-cell tight-binding model that embeds coexisting loop-current and bond-order (LC+BO) orders, yielding a $Z_3$ nematic state. Through GL theory and symmetry analysis, it shows that single impurities can generate a chiral QPI signal ($\chi_Z=\pm1,0$) whose sign is dictated by the $Z_3$ director and can be switched by a small out-of-plane field $B_z$, with chirality amplified in dilute impurity regimes. The study also explores Sb-site impurities and stripe CDW order, demonstrating robust and configurable QPI chirality across energies and linking the phenomena to observations of $B_z$-switchable QPI chirality and eMChA in experiments. Altogether, the results provide a concrete mechanism tying LC+BO nematic order to QPI chirality and offer a framework for interpreting STM-based QPI measurements in kagome AV$_3$Sb$_5$ systems.

Abstract

The chiral loop-current (LC) phase in kagome metals AV3Sb5 (A = Cs, Rb, K) has attracted considerable attention as a novel quantum state driven by electron correlations. Scanning tunneling microscopy (STM) experiments have provided strong evidence for the chiral LC phase through the detection of chirality in the quasiparticle interference (QPI) signal. However, the fundamental relationship between ``QPI chirality'' and ``LC chirality'' remains unexplored. For instance, the QPI signal is unchanged even when all LC orders are inverted. Furthermore, only the chiral LC order cannot induce QPI chirality. At present, the true essence of kagome metals that we should learn from the remarkable QPI experiments remains elusive. To address this, we investigate the origin of the QPI signal in the LC phase using a large unit-cell tight-binding model for kagome metals. The LC phase gives rise to a $Z_3$ nematic phase, characterized by three distinct directors, under the Star-of-David bond order. Our findings demonstrate that the QPI chirality induced by a single impurity at site Z, denoted as $χ_Z$, can take values of $\pm1$ (chiral) or 0 (achiral), depending on the direction of the $Z_3$ nematic order. Prominent QPI chirality originates from extremely dilute impurities ($\lesssim$0.1%) in the present mechanism. Notably, $χ_Z$ ($=\pm1$, 0) changes smoothly with minimal free-energy barriers by applying a small magnetic field $B_z$, accompanied by a switching of the $Z_3$ nematic director. This study provides a comprehensive explanation for the observed ``$B_z$-switchable QPI chirality'' in regions with dilute impurities, offering fundamental insight into the chiral LC in kagome metals.

Figures (9)

• Figure 1: (a) Kagome lattice model. The unit cell is composed of the sublattices a, b, and c. $2\boldsymbol{a}_{\mathrm{ab}}$, $2\boldsymbol{a}_{\mathrm{bc}}$ give the two primitive vectors. Fermi surfaces without density waves are also shown. Red, blue, and green colors represent the weight of the sublattices a, b, and c, respectively. (b) Nematic LC + BO existing state with ${\bm\phi}_0$ and ${\bm\eta}_{\rm I}$. The director is along $y$ (or equivalently $x$) axis. The $2\times2$ unit cell is composed of 12 sites ${\rm Z}$, ${\rm Z}'$, ${\rm Z}"$, and ${\rm Z}"'$ with Z = A, B, C. The $C_6$ center of ${\bm\eta}_{\alpha}$ is denoted as $O_{\alpha}$. (c) Band structure and (d) DOS in the folded BZ for $-\phi=\eta=0.01$. (e) Equal-energy surface for $E=0$ (Fermi level) and $E=-0.04$.
• Figure 2: (a) Modulation of the LDOS $\delta\rho_i$ ($E=-0.04$) induced by Imp A in $N=1452$-site unit cell under the LC + BO state with (${\bm\eta}_{\rm I}$, ${\bm\phi}_0$). Single impurity gives drastic long-range modulation, reflecting the chirality of the LC phase. (Inset) LDOS of the four sites around Imp A. (b) QPI signals $I_{ {\bm q} }$ for $V=0$. $2{ {\bm q} }_n$ is the Bragg peak of the 3-site unit-cell, and ${ {\bm q} }_n$ is the Bragg peak of the 12-site due to LC + BO order. (c) QPI signals for $V=\infty$ given by the Fourier transformation of the LDOS shown in (a). (d) $E$-dependence of the QPI signal ($I_1,I_2,I_3$) due to a single Imp A, whose chirality is defined as $\chi_{\rm A}\equiv\varepsilon_{n_1 n_2 n_3}$ for $I_{n_1}>I_{n_2}>I_{n_3}$.
• Figure 3: (a) LDOS under the LC order $(\eta,\phi)=(0.01,0)$ and (b) LDOS under the BO order $(\eta,\phi)=(0,-0.01)$ for $V=0$ at $E=-0.05$. Both states have $C_6$ symmetry. (c) LDOS under the LC + BO order [$\eta=-\phi=0.01$] with $C_2$ symmetry. All mirror symmetries are violated by a single impurity at A-site or B-site. Therefore, the chirality is induced by a single impurity. In contrast, a mirror symmetries survive by the C-site impurity. (d) Obtained QPI signal at $E=-0.05$, which is chiral for A-site impurity ($\chi_{\rm A}=-1$) and B-site impurity ($\chi_{\rm B}=+1$), while it is achiral for C-site impurity ($\chi_{\rm C}=0$).
• Figure 4: (a) Adiabatic change from ${\bm\eta}_{\rm I}$ to $-{\bm\eta}_{\rm III}$ under the constant $|{\bm\eta}|=\eta_0=\sqrt{6}$ and $\eta_1+\eta_2+\eta_3=0$. The corresponding $M_{\rm orb}\propto \eta_1 \eta_2 \eta_3$ exhibits a monotonic change. Therefore, the transition from ${\bm\eta}_{\rm I}$ to $-{\bm\eta}_{\rm III}$ (or $-{\bm\eta}_{\rm II}$) can be induced by the outer magnetic field $B_z$. (b) Adiabatic change from ${\bm\eta}_{\rm I}$ to $-{\bm\eta}_{\rm I}$ under the constant $|{\bm\eta}|$. Because $M_{\rm orb}\propto \eta_1 \eta_2 \eta_3$ is non-monotonic, the transition from ${\bm\eta}_{\rm I}$ to $-{\bm\eta}_{\rm I}$ is not realized.
• Figure S1: (a) BO parameter ${\bm\phi}_0$ in real space for TrH pattern ($\phi>0$) and SoD pattern ($\phi<0$). (b) LC parameter ${\bm\eta}_{{\alpha}}$ (${\alpha}$=I, II, III) in real space ($\eta>0$). Sites A, B, and C in the main text (in Fig. 1 (b) and Fig. 3 (c)) respectively correspond to sites 10, 8, and 12.