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Physical review. B, Condensed matter

Anyons in spin liquids

K. Hallberg, C. A. Balseiro

Published May 1, 1991• 1 citation
Publisher

Abstract

We use a variational procedure to show, starting with the flux-phase state of a Heisenberg model, that holes added to the system behave as anyons. The hole distorts the gauge flux in its surroundings. The amount of flux carried by the hole depends on the ratio between the magnetic- (J) and the kinetic- (t) energy parameters. For J/t\ensuremath{\rightarrow}0 the quasiparticles are fermions, while for J/t\ensuremath{\rightarrow}\ensuremath{\infty} they are semions.

DOI
10.1103/physrevb.43.10289

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We use a variational procedure to show, starting with the flux-phase state of a Heisenberg model, that holes added to the system behave as anyons. The hole distorts the gauge flux in its surroundings. The amount of flux carried by the hole depends on the ratio between the magnetic- (J) and the kinetic- (t) energy parameters. For J/t\ensuremath{\rightarrow}0 the quasiparticles are fermions, while for J/t\ensuremath{\rightarrow}\ensuremath{\infty} they are semions.
We study in detail the properties of $\ensuremath{\pi}$ fluxes embedded in a state with a finite density of anyons that form either a Fermi liquid or a Bose-Einstein condensate. By employing a recently developed exact lattice bosonization in 2D, we demonstrate that such $\ensuremath{\pi}$ flux remains a fully deconfined quasiparticle with a finite energy cost in a Fermi liquid of emergent fermions coupled to a Z2{\mathbb{Z}}_{2}Z2​ gauge field. This $\ensuremath{\pi}$ flux is accompanied by a screening cloud of fermions, which in the case of a Fermi gas with a parabolic dispersion binds exactly 1/81/81/8 of a fermionic hole. In addition, there is a long-ranged power-law oscillatory disturbance of the liquid surrounding the $\ensuremath{\pi}$ flux akin to Friedel oscillations. These results carry over directly to the $\ensuremath{\pi}$ flux excitations in orthogonal metals. In sharp contrast, when the $\ensuremath{\pi}$ flux is surrounded by a Bose-Einstein condensate of particles coupled to a Z2{\mathbb{Z}}_{2}Z2​ gauge field, it binds a superfluid half-vortex, becoming a marginally confined excitation with a logarithmic energy cost divergence.
The extended t - J model has been studied in the slave boson scheme. It is found that there exists some range of parameters where the chiral spin state is stabilized within a mean field approximation. The Chern-Simons Lagrangian for the gauge field, which represents the fluctuation of the phases of the order parameters and the local constraint, is obtained by integrating out spin fermion fields as in the undoped case [Wen, Wilczek and Zee: Phys. Rev. B 39 (1989) 11413]. The coefficient of this term turns out to depend on doping. Due to the coupling to this Chern-Simons gauge field, hole bosons are transmuted into anyons.
We present an exactly solvable model for synthetic anyons carrying non-Abelian flux. The model corresponds to a two-dimensional electron gas in a magnetic field with a specific spin interaction term, which allows only fully aligned spin states in the ground state; the ground state subspace is thus twofold degenerate. This system is perturbed with identical solenoids carrying a non-Abelian gauge potential. We explore dynamics of the ground state as these solenoids are adiabatically braided and show they behave as anyons with a non-Abelian flux. Such a system represents a middle ground between the ordinary Abelian anyons and the fully non-Abelian anyons.
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{T}_{c}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 describe a theory of the two dimensional Heisenberg-Hubbard model working in the spin and hole fermion representation based on the use of a singular gauge transformation on the spins. We are motivated by a new mean field theory for a system of strongly coupled fermions and bosons which is also described. In the latter case we obtain the bose condensed—fermi liquid phase; in the former we obtain equations governing generalized flux phases. We do not find superconducting solutions to these equations in the atomic limit, J=0. If solutions exist for finite J case then it is argued that an energy gap is present in that case.