Magnetic field induced phenomena in Kitaev spin liquids

Abstract

Quantum spin liquids (QSLs) host a variety of fractionalized particles. In Kitaev's paradigmatic honeycomb model a spin-$\tfrac{1}{2}$ fractionalizes into $Z_2$ flux due to emergent $Z_2$ gauge field and matter Majorana fermions. Although these excitations have well-defined dynamics in the integrable limit, their direct experimental identification is notoriously challenging: realistic materials inevitably host additional symmetry-allowed interactions that break integrability and hybridize gauge and matter sectors, while magnetic fields, which are often required to suppress competing order and stabilize a putative QSL regime, further entangle the responses of different fractionalized quasiparticles and may even drive the system into field-induced spin-liquid phases that are not adiabatically connected to the integrable limit. A prominent example is the quantum Majorana metal, in which the distinct dynamics of fractionalized Majorana fermions can become directly visible in scattering. This report highlights recent progress on these related questions: in which field-stabilized QSL regimes and nearby emergent phases, and under what conditions, can the response of a specific fractionalized quasiparticle be isolated and positively understood, thereby clarifying the existence and the experimental scope of putative spin liquids? We review the progress on these questions across Abelian, non-Abelian, and an emergent quantum phases under magnetic field that are not perturbatively connected to the integrable limit. We connect these field-induced dynamical phenomena to concrete experimental observables, relevant for neutron scattering, resonant inelastic X-ray scattering, and pump-probe spectroscopy that are capable of resolving specific types of fractionalized particles, including Majoranas and $Z_2$ fluxes.

Figures (27)

• Figure 1: Bifurcation of quantum spin liquids in 2D into gapped (top) and gapless (bottom) classes. Top left: Nonchiral gapped (string-net states or short-range RVB, adapted from Wen19): intrinsic topological order with a finite anyon set, long-range entanglement, and no chiral edge; includes Abelian and non-Abelian examples Levin05. Top right: Gapped chiral RVB: time-reversal broken with a chiral edge mode (pink) and quantized thermal Hall conductance. Bottom row (gapless): spinon Fermi-surface liquid, Dirac/nodal spin liquid, and thermal metal (disorder-enabled spinon metal). Other exotic examples and formal classification such as symmetry-enriched topological order are beyond the scope of this article, and are not shown.
• Figure 2: (a) Illustration of spin-orbital frustration. (b) Spin-orbital frustrated honeycomb model with anisotropic compass interactions along the $x$, $y$, and $z$ axes, depicted by distinct colors. Each local spin degree of freedom (gray dots or circles) fractionalizes into four Majorana fermions, indicated by four colors inside larger gray circles. Three Majoranas combine into conserved localized bond fermions (colored ovals), determining the local $Z_2$ gauge field on each bond. (c) The remaining itinerant Majorana fermion (purple dot) propagates in a static background of $Z_2$ gauge fields, with positive directions indicated by arrows. Flipping an odd number of bonds around a hexagonal plaquette (red arrows) excites a $Z_2$ flux within the plaquette (orange). These fluxes sit at the endpoints of dashed purple string operators, and all link variables intersected by the string operator change sign. (d) In the ground-state zero-flux sector, itinerant Majorana fermions form Dirac cones in the absence of time-reversal symmetry-breaking perturbations.
• Figure 3: (a) Dirac cone in absence of perturbation. (b) with $\lambda \sim h_x h_y h_z$ perturbation, the Dirac point is gapped out by $Q(\mathbf{k})$, resulting in a chiral spin liquid (CSL). (c) A single flux (shown in orange) is created in an infinite CSL by flipping a line of bond operators starting at the flux and extending horizontally to infinity. The system can be rewritten as a pair of Chern insulators which are then coupled together using the remaining horizontal line of bonds. Each of the two systems contributes an edge mode traveling with velocity $\pm v$ at the boundary, shown in blue. The black thick lines marks out the boundaries of two patches of Chern insulators, whose chiral modes are coupled via $t(x)$ in between the two edges.
• Figure 4: The zero-temperature dynamical spin structure factor $S_1(\mathbf{k}, \omega)$ for the antiferromagnetic isotropic Kitaev honeycomb model along the $\rm M\Gamma K M$ path in the Brillouin zone. Figure adapted from Ref. hermanns2018physics.
• Figure 5: Zero-temperature dynamical spin structure factor $S_2(\mathbf{k}, \omega)$ for the isotropic antiferromagnetic Kitaev model. (a) The momentum-resolved spectrum $S_2^z(\mathbf{k}, \omega)$ in the time-reversal (TR) symmetric case ($\lambda = 0$). The dotted white line highlights the gapless spectrum at the $\Gamma$ point. (b) The normalized equal-momentum cut at the $\Gamma$ point, $S_2^z(\Gamma, \omega)$, indicated by the vertical dashed white line in panel (a), is plotted as a red solid line. The blue solid line shows the corresponding density of states (DOS) of the Dirac Majorana fermions. Both intensities are rescaled to emphasize the exact correspondence between the spin and Majorana spectra at low energies, that $S_2^z(\Gamma, \omega)$ directly reflects the Majorana DOS up to a scaling factor in energy. (c) $S_2^z(\mathbf{k}, \omega)$ with time-reversal symmetry explicitly broken ($\lambda > 0$). The dotted white line indicates the opening of a finite spectral gap at the $\Gamma$ point due to the TR-breaking perturbation. (d) The rescaled Majorana DOS (blue line) and spin structure factor $S_2^z(\Gamma, \omega)$ (red line) for the TR-broken case. The similar profiles of DOS and $S_2^z(\Gamma, \omega)$ shows that the Majorana DOS can be directly observed from the dimer spectral function, up to a scaling factor of $2$ in energy.