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arXiv:1509.05721 · Physical review. B./Physical review. B

Kondo route to spin inhomogeneities in the honeycomb Kitaev model

Shreya Das, Kusum Dhochak, Vikram Tripathi

Published Jul 8, 2016• 19 citations
arXivHTMLPDF

Abstract

Paramagnetic impurities in a quantum spin liquid give rise to Kondo effects with highly unusual properties. We have studied the effect of locally coupling a paramagnetic impurity with the spin-$\frac{1}{2}$ honeycomb Kitaev model in its gapless spin-liquid phase. The (impurity) scaling equations are found to be insensitive to the sign of the coupling. The weak and strong coupling fixed points are stable, with the latter corresponding to a noninteracting vacancy and an interacting, spin-1 defect for the antiferromagnetic and ferromagnetic cases, respectively. The ground state in the strong coupling limit in both cases has a nontrivial topology associated with a finite ${Z}_{2}$ flux at the impurity site. For the antiferromagnetic case, this result has been obtained straightforwardly owing to the integrability of the Kitaev model with a vacancy. The strong-coupling limit of the ferromagnetic case is, however, nonintegrable, and we address this problem through exact-diagonalization calculations with finite Kitaev fragments. Our exact diagonalization calculations indicate that the weak-to-strong coupling transition and the topological phase transition occur rather close to each other and are possibly coincident. We also find an intriguing similarity between the magnetic response of the defect and the impurity susceptibility in the two-channel Kondo problem.

DOI
10.1103/physrevb.94.024411

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Paramagnetic impurities in a quantum spin liquid give rise to Kondo effects with highly unusual properties. We have studied the effect of locally coupling a paramagnetic impurity with the spin-12\frac{1}{2}21​ honeycomb Kitaev model in its gapless spin-liquid phase. The (impurity) scaling equations are found to be insensitive to the sign of the coupling. The weak and strong coupling fixed points are stable, with the latter corresponding to a noninteracting vacancy and an interacting, spin-1 defect for the antiferromagnetic and ferromagnetic cases, respectively. The ground state in the strong coupling limit in both cases has a nontrivial topology associated with a finite Z2{Z}_{2}Z2​ flux at the impurity site. For the antiferromagnetic case, this result has been obtained straightforwardly owing to the integrability of the Kitaev model with a vacancy. The strong-coupling limit of the ferromagnetic case is, however, nonintegrable, and we address this problem through exact-diagonalization calculations with finite Kitaev fragments. Our exact diagonalization calculations indicate that the weak-to-strong coupling transition and the topological phase transition occur rather close to each other and are possibly coincident. We also find an intriguing similarity between the magnetic response of the defect and the impurity susceptibility in the two-channel Kondo problem.
We study the effect of coupling magnetic impurities to the honeycomb lattice spin-1/21/21/2 Kitaev model in its spin-liquid phase. We show that a spin-SSS impurity coupled to the Kitaev model is associated with an unusual Kondo effect with an intermediate coupling unstable fixed point ${K}_{c}\ensuremath{\sim}J/S$ separating topologically distinct sectors of the Kitaev model. We also show that the massless spinons in the spin-liquid mediate an interaction of the form ${S}_{i\ensuremath{\alpha}}^{2}{S}_{j\ensuremath{\beta}}^{2}/{R}_{ij}^{3}$ between distant impurities unlike the usual dipolar RKKY interaction ${S}_{i\ensuremath{\alpha}}{S}_{j\ensuremath{\alpha}}/{R}_{ij}^{3}$ noted in various 2D impurity problems with a pseudogapped density of states of the spin bath. Furthermore, this long-range interaction is possible only if the impurities (a) couple to more than one neighboring spin on the host lattice and (b) the impurity spin $S\ensuremath{\ne}1/2$.
Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here, we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it binds an emergent gauge flux, while otherwise it becomes free at low temperatures. We show how Majorana degrees of freedom determine the fixed-point properties, make contact with Kondo screening in pseudogap Fermi systems, and discuss effects away from the dilute limit.
We study the effect of adding disorder to the Kitaev model on the hyperhoneycomb lattice, which hosts both gapped and gapless spin liquid phases with an emergent Z2{\mathbb{Z}}_{2}Z2​ gauge field. The latter has an unusual gapless spectrum of Majorana fermion excitations, with a co-dimension-two Fermi ring. We thus address the question of the interplay of topological physics and disorder by considering the properties of isolated single and pairs of vacancies. We show that near the vacancies, the local magnetic response to a field hz{h}_{z}hz​ is parametrically enhanced in comparison to the pristine bulk. Unlike the previously studied case of the 2D honeycomb Kitaev model, the vacancies do not bind a flux of the Z2{\mathbb{Z}}_{2}Z2​ gauge field. In the gapped phase, an isolated vacancy gives rise to effectively free spin-half moments with a nonuniversal coupling to an external field. In the gapless phase, the low-field magnetization is suppressed parametrically to ${(\ensuremath{-}ln{h}_{z})}^{\ensuremath{-}1/2}$ because of interactions with the surrounding spin liquid. We also show that a pair of vacancies is subject to a sublattice-dependent interaction on account of coupling through the bulk spin liquid, which is spatially anisotropic even when all Kitaev couplings have equal strength. This coupling is thus exponentially suppressed with distance in the gapped phase. In the gapless phase, two vacancies on the same (opposite) sublattice exhibit an enhanced (suppressed) low-field response, amounting to an effectively (anti-)ferromagnetic interaction.
We study the response of a Kitaev spin liquid (KSL) to a local magnetic field perpendicular to the Kitaev honeycomb lattice. The local magnetic field induces a dynamical excitation of a flux pair in the spin liquid and the system can be described by a generally particle-hole asymmetric interacting resonant level model. The dynamical excitation of the flux pair closes the flux gap in the spectrum of the spin correlation function locally for the gapless KSL even from the perturbative response to a weak magnetic field. Beyond the perturbative regime, the p-h asymmetry competes with the magnetic field and results in a rich phase diagram. Moreover, the magnetic field breaks the gauge equivalence of the ferromagnetic and antiferromagnetic Kitaev couplings of the ground state and leads to very different behaviors for the two cases. The antiferromagnetic case experiences a first-order phase transition to the polarized state during magnetization, whereas the ferromagnetic case does not. This study can be generalized to the Kitaev model in a uniform magnetic field and may help understand issues in recent experiments on KSL candidates.
We study the dynamics of fractionalized excitations in a Kitaev spin liquid (KSL) induced by a local magnetic field perpendicular to the Kitaev honeycomb lattice. The local magnetic field induces a dynamic excitation of a pair of fluxes described by a generally particle-hole asymmetric interacting resonant level model and the dynamics resembles that of a Kondo problem. The p-h asymmetry competes with the magnetic field and results in a rich phase diagram. Moreover, the magnetic field breaks the gauge equivalence of the ferromagnetic (FM) and anti-ferromagnetic (AFM) Kitaev couplings of the ground state and leads to dramatically different behaviors for the two cases. The latter experiences a first order phase transition during magnetization whereas the former does not. By studying the response of the KSL to the magnetic field in different regimes, we obtain the RG flow and phase diagram of the system in the whole space. This study can be generalized to the KSL in a uniform magnetic field and help resolve issues in recent experiments on KSL candidates.
We address the effects of disorder in the Kitaev honeycomb model, focussing on the interplay of disorder and strong interactions in a quantum spin liquid. It is shown that nonmagnetic impurities bind a quantum of the emergent gauge flux. We find the formation of a local moment, which leads to a local magnetic response near the impurity which is parametrically larger than in the bulk. The magnetic response of a pair of impurities can be parametrically larger than that of an isolated one, and can in particular dominate the magnetic response of the system. The role of such impurities as diagnostic of the spin liquid state are emphasized – most remarkably, they provide a direct signature of the emergent gauge field.