Impurity quadrupole moments as local probes of flux sectors in the Kitaev spin liquid

Abstract

Emergent fluxes play a central role in the low-energy properties of quantum spin liquids (QSLs), where they encode the underlying gauge structure and fractionalization of spins. Here, we show that the quadrupole moment of magnetic impurities provides a direct probe of these flux configurations in QSLs. Employing the SO(6) Majorana representation for spin-3/2 impurity operators in the isotropic Kitaev spin liquid together with a self-consistent mean-field approximation for impurity-related terms, we show that the ground-state flux sector can be identified by discontinuous jumps of the impurity quadrupole moment at the flux sector transition points. We also demonstrate that the quadrupole correlations between impurities under a magnetic field exhibit exponential decay, with decay rates that depend sensitively on the flux sector. Furthermore, we discuss the stability of pi fluxes bound to impurities with respect to model parameters and internal flux configurations, and relate our findings to Lieb's conjecture on flux configurations. These results establish the quadrupole moments of magnetic impurities as a sensitive tool to study fractionalized excitations and flux physics in Kitaev magnets.

Figures (10)

• Figure 1: Kitaev spin liquid with randomly distributed spin-3/2 impurities. Yellow plaquettes denote the presence of $\pi$-fluxes bound at impurity sites. Site labels are used for defining the flux operator on plaquette $p$.
• Figure 2: Internal flux operators.a Schematic illustration of the internal and triple-plaquette operators. Site labels correspond to those used in Eq. \ref{['eq: def_internal_W']}. b Four patterns of internal flux configurations within a given flux sector. Yellow plaquette denotes internal $\pi$-flux. Thick red bonds indicate example choices of flipped gauge fields for each configuration; these choices are not unique in general. c Majorana representation for both spin-1/2 and spin-3/2 operators, showing the static gauge field on an $x$-bond as an example.
• Figure 3: Majorana hopping lattice around an impurity. Each arrow indicates the positive hopping direction. Solid (dashed) lines correspond to NN (NNN) hoppings, respectively. Orange and blue arrows require a mean-field decomposition. Red arrows do not require such a decomposition; instead, the associated perturbative processes involve internal flux configurations, leading to an enhanced amplitude compared with that in the bulk.
• Figure 4: Internal flux excitations in a $L=32$ cluster.a A finite-size cluster with two impurities that preserve reflection symmetry. Two impurities are located on different sublattices. Yellow plaquettes indicate the presence of $\pi$-fluxes, which corresponds to the internal flux configuration with Pattern (IV, IV) in the bound-flux sector. Red bonds represent gauge-flipped bonds that realize the corresponding flux configurations. The green dashed line penetrating the cluster denotes the reflection symmetry arising from the impurity configuration. Periodic boundary conditions are imposed. b, c Flux gaps of the internal flux configurations obtained in the zero-flux and bound-flux sectors, respectively. The red dashed line in each panel indicates the bulk flux gap. We set $D_z=0.1$ in both cases. The inset in c corresponds to the small coupling regime.
• Figure 5: a Difference in the ground-state energies between the zero-flux and bound-flux sectors $\Delta E$. b Local quadrupole moments of $|Q_j^x|$ with $j=$imp1 for small coupling regime. c Local quadrupole moments of $Q^z\equiv Q_{\textsf{imp1}}^z=Q_{\textsf{imp2}}^z$. The inset corresponds to the small coupling regime.