Magnetoelectric generation of a Majorana-Fermi surface in Kitaev's honeycomb model

We study the effects of static magnetic and electric fields on Kitaev's honeycomb model. Using the electric polarization operator appropriate for Kitaev materials, we derive the effective Hamiltonian for the emergent Majorana fermions to second order in both the electric and magnetic fields. We find that while individually each perturbation does not qualitatively alter Kitaev spin liquid, the cross term induces a finite chemical potential at each Dirac node, giving rise to a Majorana-Fermi surface. We argue this gapless phase is stable and exhibits typical metallic phenomenology, such as linear in temperature heat capacity and finite, but nonquantized, thermal Hall response. Finally, we speculate on the potential for realization of this physics in Kitaev materials.

Figure 1

Illustration of the spectrum of the Kitaev model in the presence of both electric ($\mathit{E}$) and magnetic ($\mathit{B}$) fields. An effective chemical potential is generated by the combination $\widehat{\mathit{n}}·(\mathit{E}\times \mathit{B})$ where $\widehat{\mathit{n}}$ is the direction perpendicular to the honeycomb plane. Near the Dirac points this chemical potential induces Majorana-Fermi surfaces (see inset).

Figure 2

Definition of the nearest-neighbor bond types, sublattices, and the numbering convention for a hexagonal plaquette $p$.

Figure 3

Illustration of the two contributions to the $O\left(EB\right)$ part of the effective Hamiltonian for the $x$-type second-nearest-neighbor bond from $i$ to $k$ (via $j$) shown in red. The piece of the electric polarization operator is indicated by a thick bond with filled circles, while the piece from the magnetic field is indicated by an open circle. The hexagons that carry flux excitations in the corresponding virtual state are indicated.

Figure 4

Illustration of two contributions to the $O\left(E^2\right)$ part of the effective Hamiltonian for a four-spin coupling along a $z$-type third-nearest-neighbor bond from $i$ to $l$ shown in red. The pieces of the electric polarization operator are indicated by thick bonds with filled circles. The hexagons that carry flux excitations in the corresponding virtual state are indicated.

Figure 5

Illustration of a contribution to the $O\left(E^2\right)$ part of the effective Hamiltonian for a four-spin coupling along a $zy$-type fourth-nearest-neighbor bond from $i$ to $l$ shown in red. The pieces of the electric polarization operator are indicated by thick bonds with filled circles. The hexagons that carry flux excitations in the corresponding virtual state are indicated.

Figure 6

Illustration of notation for the second- (left), third-, and fourth-nearest-neighbor bonds (right) of the honeycomb lattice, as defined in Secs. 4b and 4c.

Figure 7

Illustration of the Majorana-Fermi surface for crossed electric and magnetic fields $\mathit{E}=E\widehat{\mathit{X}}$ and $\mathit{B}=B\widehat{\mathit{Y}}$ [Eq. (57)], with $E=B=0.1$. See Sec. 7 for the choice of electric polarization parameters and magnetic $g$ factors.

Figure 8

Illustration of the electric ($\mathit{E}$) and magnetic field configurations ($\mathit{B}$) considered in Sec. 7, relative to the honeycomb plane. These fields are parametrized by an angle $\psi$, as given in Eq. (57).

Figure 9

Illustration of the band gap [${min}_{\mathit{k}}|{\epsilon }_{+}\left(\mathit{k}\right)-{\epsilon }_{-}\left(\mathit{k}\right)|$], spectral gap [${min}_{\mathit{k},\pm }\left|{\epsilon }_{\pm }\left(\mathit{k}\right)\right|$], and thermal Hall conductivity [Eq. (60)] as a function of the angle $\psi$ [Eq. (57)] for $E=B=0.1$. We identify three regions: region I where the band gap is finite and the spectral gap is zero, region II where the thermal Hall conductivity has changed sign, and region III where the spectral gap has become finite. The boundary between regions I and II is denoted as ${\psi }_1\approx 0.282047\pi$, and that between II and III is denoted as ${\psi }_2\approx 0.3487\pi$. See Sec. 7 for the choice of electric polarization parameters and magnetic $g$ factors.

Figure 10

(a)–(f) (Top) Illustration of the spectrum of the Majorana fermions as a function of $\mathit{E}$ and $\mathit{B}$ (parametrized by $\psi$ [Eq. (57)]) near the location of the (minimal) band gap at ${\mathit{K}}^{*}$. Gapless phases with Majorana-Fermi surfaces (a)–(d) and gapped phases (e), (f) are shown. (a)–(f) (Bottom) Illustration of the Majorana-Fermi surfaces as a function of $\mathit{E}$ and $\mathit{B}$ (parametrized by $\psi$ [Eq. (57)]). The location of the band gap (${\mathit{K}}^{*}$) is indicated. For $\psi >{\psi }_2\approx 0.3487\pi$ (e), (f) the system is gapped and there is no Majorana-Fermi surface. Throughout we choose field strengths $E=B=0.1$. See Sec. 7 for the choice of electric polarization parameters and magnetic $g$ factors.

Figure 11

Illustration of $z$-type coverings. A product of $W_p$ over either the shaded plaquettes (or the unshaded plaquettes) yields the operator ${\mathrm{\Sigma }}^z$ used in Eq. (61). The ${\mathrm{\Sigma }}^x$ and ${\mathrm{\Sigma }}^y$ operators are defined similarly, with the plaquettes in the product sharing $x$-type or $y$-type bonds.

Figure 12

Illustration of a contribution to the $O\left(B^3\right)$ part of the effective Hamiltonian, for $i\in A$. The relevant pieces of the magnetization operator are indicated by thick bonds.