Motivated by the appearance of Majorana fermions in a broad range of correlated and topological electronic systems, we develop a general method to compute the dynamical response of interacting Majorana fermions in the random-phase approximation (RPA). This can be applied self-consistently on top of Majorana mean-field theory backgrounds, thereby in particular providing a powerful tool to analyze behavior in the vicinity of (various heavily studied) exactly soluble models. Prime examples are quantum spin liquids (QSL) with emergent Majorana excitations, with the celebrated exact solution of Kitaev. We employ the RPA to study in considerable detail phase structure and dynamics of the extended Kitaev honeycomb K J Γ model, with and without an applied field. First, we benchmark our method with Kitaev's exactly soluble model, finding a remarkable agreement. The interactions between Majorana fermions even turn out to mimic the effect of local Z 2 flux excitations, which we explain analytically. Second, we show how small non-Kitaev couplings J and Γ induce Majorana bound states, resulting in sharp features in the dynamical structure factor in the presence of fractionalization: such “spinon excitons” naturally appear, and can coexist and interact with the broad Majorana continuum. Third, for increasing couplings or field, our theory predicts instabilities of the Kitaev QSL (KQSL) triggered by the condensation of the sharp modes. From the high-symmetry momenta of the condensation we can deduce which magnetically ordered phases surround the KQSL, in good agreement with previous finite-size numerics. We discuss implications for experiments and the broad range of applicability of our method to other QSL and Majorana systems.
We present an augmented parton mean-field theory which (i) reproduces the exact ground state, spectrum, and dynamics of the quantum spin-liquid phase of Kitaev's honeycomb model, and (ii) is amenable to the inclusion of integrability breaking terms, allowing a perturbation theory from a controlled starting point. Thus, we exemplarily study dynamical spin correlations of the honeycomb Kitaev quantum spin liquid within the $K\ensuremath{-}J\ensuremath{-}\mathrm{\ensuremath{\Gamma}}$ model, which includes Heisenberg and symmetric-anisotropic (pseudodipolar) interactions. This allows us to trace changes of the correlations in the regime of slowly moving fluxes, where the theory captures the dominant deviations when integrability is lost. These include an asymmetric shift together with a broadening of the dominant peak in the response as a function of frequency, the generation of further-neighbor correlations and their structure in real and spin space, and a resulting loss of an approximate rotational symmetry of the structure factor in reciprocal space. We discuss the limitations of this approach and also view the neutron-scattering experiments on the putative proximate quantum spin-liquid material $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ in the light of the results from this extended parton theory.
The Kitaev honeycomb model realizes a quantum spin liquid where spin degrees of freedom are fractionalized into itinerant Majorana fermions and localized vortices. The entanglement among these particles holds promise for executing fault-tolerant topological quantum computation. While several materials have been proposed to host a quantum spin liquid phase, concrete evidence confirming its existence remains elusive. To aid in identifying this phase, we propose an electrical test using the charge profile, offering a signature for the Z2 vortices in Kitaev materials. This study explores the electric charge response in Kitaev materials, considering a generalized spin model including Heisenberg, off-diagonal exchange interactions, and magnetic Zeeman field. Majorana mean-field theory is employed to analyze spin correlations around vortices, revealing spatial anisotropy in the charge distribution, quantified by the electric quadrupole moment. This mean-field approach remains exact in the pure Kitaev limit and captures the fundamental physics across the Kitaev spin liquid phase. The quadrupole-quadrupole interaction between distant vortices is shown to be either repulsive or attractive, depending on the parameters. We predict that electrically biased scanning probe tips can help create vortices at predetermined locations. The findings of our study pave the way for electrically manipulating Ising anyons in Kitaev spin liquids, offering a promising path for technological applications of Kitaev materials.
Three-dimensional Kitaev models -- generalizations of the well known exactly solvable Kitaev honeycomb model -- exhibit a rich variety of quantum spin liquid phases. Their fractionalized excitations are Majorana fermions that may form topological band structures. These can be enriched by lattice symmetries, as shown in the present work. In particular, the authors propose two new instances of Kitaev spin liquids that exhibit nodal chains and 3D Dirac nodes in the bulk. Such crystalline Kitaev spin liquids are direct generalizations of crystalline topological insulators, albeit in an interacting spin system. A potential material realization in metal-organic frameworks is also discussed.
Experimental identification of quantum spin liquids remains a challenge, as the pristine nature is to be seen in asymptotically low temperatures. We here theoretically show that the precursor of quantum spin liquids appears in the spin dynamics in the paramagnetic state over a wide temperature range. Using the cluster dynamical mean-field theory and the continuous-time quantum Monte Carlo method, which are newly developed in the Majorana fermion representation, we calculate the dynamical spin structure factor, relaxation rate in nuclear magnetic resonance, and magnetic susceptibility for the honeycomb Kitaev model whose ground state is a canonical example of the quantum spin liquid. We find that dynamical spin correlations show peculiar temperature and frequency dependence even below the temperature where static correlations saturate. The results provide the experimentally accessible symptoms of the fluctuating fractionalized spins evincing the quantum spin liquids.
A prominent feature of quantum spin liquids is fractionalization of the spin degree of freedom. Fractionalized excitations have their own dynamics in different energy scales, and hence, affect finite-temperature (
T) properties in a peculiar manner even in the paramagnetic state harboring the quantum spin liquid state. We here present a comprehensive theoretical study of the spin dynamics in a wide
T range for the Kitaev model on a honeycomb lattice, whose ground state is a quantum spin liquid. In this model, the fractionalization occurs to break up quantum spins into itinerant matter fermions and localized
Z2 fluxes, which results in two crossovers at very different
T scales. Extending the previous study for the isotropic coupling case [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016)], we calculate the dynamical spin structure factor $S(\mathbf{q},\ensuremath{\omega})$, the NMR relaxation rate
1/T1, and the magnetic susceptibility $\ensuremath{\chi}$ while changing the anisotropy in the exchange coupling constants, by using the dynamical mean-field theory based on a Majorana fermion representation. We describe the details of the methodology including the continuous-time quantum Monte Carlo method for computing dynamical spin correlations and the maximum entropy method for analytic continuation. We confirm that the combined method provides accurate results in a wide
T range including the region where the spins are fractionalized. We find that also in the anisotropic cases the system exhibits peculiar behaviors below the high-
T crossover whose temperature is comparable to the average of the exchange constants: $S(\mathbf{q},\ensuremath{\omega})$ shows an inelastic response at the energy scale of the averaged exchange constant,
1/T1 continues to grow even though the equal-time spin correlations are saturated and almost
T independent, and $\ensuremath{\chi}$ deviates from the Curie-Weiss behavior. In particular, when the exchange interaction in one direction is stronger than the other two, the dynamical quantities exhibit qualitatively different
T dependences from the isotropic case at low
T, reflecting the opposite parity between the flux-free ground state and the flux-excited state, and a larger energy cost for flipping a spin in the strong interaction direction. On the other hand, when the exchange anisotropy is in the opposite way, the results are qualitatively similar to those in the isotropic case. All these behaviors manifest the spin fractionalization in the paramagnetic region. Among them, the dichotomy between the static and dynamical spin correlations is unusual behavior hardly seen in conventional magnets. We discuss the relation between the dichotomy and the spatial configuration of the
Z2 gauge fields. Our results could stimulate further experimental and theoretical analyses of candidate materials for the Kitaev quantum spin liquids.