We explore with exact diagonalization the propagation of a single hole in four magnetic phases of the $t\text{\ensuremath{-}}J$-like Kitaev-Heisenberg model on a honeycomb lattice: the N\'eel antiferromagnetic, stripe, zigzag, and Kitaev spin-liquid phases. We find coherent propagation of spin-polaron quasiparticles in the antiferromagnetic phase by a mechanism similar to that in the $t\text{\ensuremath{-}}J$ model for high-
Tc cuprates. In the stripe and zigzag phases clear quasiparticles features appear in spectral functions of those propagators where holes are created and annihilated on one sublattice, while they remain largely hidden in those spectral functions that correspond to photoemission experiments. As the most surprising result, we find a totally incoherent spectral weight distribution for the spectral function of a hole moving in the Kitaev spin-liquid phase in the strong-coupling regime $t\ensuremath{\gg}J$ relevant for iridates. At intermediate coupling the finite systems calculation reveals a well-defined quasiparticle at the $\ensuremath{\Gamma}$ point; however, we find that the gapless spin excitations wipe out quasiparticles at finite momenta. Also for this more subtle case we conclude that in the thermodynamic limit the lightly doped Kitaev liquid phase does not support quasiparticle states in the neighborhood of $\ensuremath{\Gamma}$, and therefore yields a non-Fermi liquid, contrary to earlier suggestions based on slave-boson studies. These observations are supported by the presented study of the dynamic spin-structure factor in the Kitaev spin liquid regime.
We present a controlled microscopic study of mobile holes in the spatially anisotropic (Abelian) gapped phase of the Kitaev honeycomb model. We address the properties of (i) a single hole (its internal degrees of freedom as well as its hopping properties); (ii) a pair of holes [their (relative) particle statistics and interactions]; and (iii) the collective state for a finite density of holes. We find that each hole in the doped model has an eight-dimensional internal space, characterized by three internal quantum numbers: the first two ``fractional'' quantum numbers describe the binding to the hole of the fractional excitations (fluxes and fermions) of the undoped model, while the third ``spin'' quantum number determines the local magnetization around the hole. The fractional quantum numbers also encode fundamentally distinct particle properties, topologically robust against small local perturbations: some holes are free to hop in two dimensions, while others are confined to hop in one dimension only; distinct hole types have different particle statistics, and in particular, some of them exhibit nontrivial (anyonic) relative statistics. These particle properties in turn determine the physical properties of the multihole ground state at finite doping, and we identify two distinct ground states with different hole types that are stable for different model parameters. The respective hopping dimensionalities manifest themselves in an electrical conductivity approximately isotropic in one ground state and extremely anisotropic in the other one. We also compare our microscopic study with related mean-field treatments, and discuss the main discrepancies between the two approaches, which in particular involve the possibility of binding fractional excitations as well as the particle statistics of the holes. On a technical level, we describe the hopping of mobile holes via a quasistationary approach, where effective hopping matrix elements are calculated between ground states with stationary holes at different positions. This approach relies on the fact that the model remains exactly solvable in the presence of stationary holes, and that the motion of sufficiently slow holes does not generate bulk excitations in a gapped phase. When the bare hopping amplitude is much smaller than the energy gap, many of our results, in particular those on the hopping properties and the particle statistics, are exact.
Numerical studies, from variational calculation to exact diagonalization, all indicate that the quasiparticle generated by introducing one hole into a two-dimensional quantum antiferromagnet has the same nature as a string state in the t-
Jz model. Based on this observation, we attempt to visualize the quasiparticle formation and subsequent coherent propagation at low energy by studying the generalized t-
Jz-${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ model in which we first diagonalize the t-
Jz model and then perform a degenerate perturbation in ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$. We construct the quasiparticle state and derive an effective Hamiltonian describing the coherent propagation of the quasiparticle and its interaction with the spin wave excitations in the presence of the N\'eel order. We expect that qualitative properties of the quasiparticle remain intact when analytically continuing ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ from the anisotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$
Jz to the isotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=
Jz limit, despite the fact that the spin wave excitations change from gapful to gapless. Extrapolating to ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=
Jz, our quasiparticle dispersion and spectral weight compare well with the exact numerical results for small clusters. \textcopyright{} 1996 The American Physical Society.
Abstract The dynamical response of a quantum spin liquid upon injecting a hole is a pertinent open question. In experiments, the hole spectral function, measured momentum-resolved in angle-resolved photoemission spectroscopy (ARPES) or locally in scanning tunneling microscopy (STM), can be used to identify spin liquid materials. In this study, we employ tensor network methods to simulate the time evolution of a single hole doped into the Kitaev spin-liquid ground state. Focusing on the gapped spin liquid phase, we reveal two fundamentally different scenarios. For ferromagnetic spin couplings, the spin liquid is highly susceptible to hole doping: a Nagaoka ferromagnet forms dynamically around the doped hole, even at weak coupling. By contrast, in the case of antiferromagnetic spin couplings, the hole spectrum demonstrates an intricate interplay between charge, spin, and flux degrees of freedom, best described by a parton mean-field ansatz of fractionalized holons and spinons. Moreover, we find a good agreement of our numerical results to the analytically solvable case of slow holes. Our results demonstrate that dynamical hole spectral functions provide rich information on the structure of fractionalized quantum spin liquids.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped
Z2 fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (
Kx=Ky=Kz). We explore the phase diagram along two axes, an external magnetic field
h, applied out of plane of the honeycomb, and anisotropic interactions
Kz larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and
h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped
Z2 toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (
e) and magnetic (
m) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower
Kz in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
We introduce a spin-
21 model in three dimensions which is a generalization of the well-known Kitaev model on a honeycomb lattice. Following Kitaev, we solve the model exactly by mapping it to a theory of noninteracting fermions in the background of a static
Z2 gauge field. The phase diagram consists of a gapped phase and a gapless one, similar to the two-dimensional case. Interestingly, unlike in the two-dimensional model, in the gapless phase the gap vanishes on a contour in the
k space. Furthermore, we show that the flux excitations of the gauge field, due to some local constraints, form looplike structures; such loops exist on a lattice formed by the plaquettes in the original lattice and is topologically equivalent to the pyrochlore lattice. Finally, we derive a low-energy effective Hamiltonian that can be used to study the properties of the excitations in the gapped phase.