pythtb.models.fu_kane_mele#

fu_kane_mele(t, soc, dt=[0, 0, 0, 0])[source]#

Fu-Kane-Mele tight-binding model.

This function creates a Fu-Kane-Mele tight-binding model on a diamond lattice. The lattice vectors are given by,

\[\mathbf{a}_1 = (0, 1, 1), \quad \mathbf{a}_2 = (1, 0, 1), \quad \mathbf{a}_3 = (1, 1, 0)\]

and the orbital positions are given by,

\[\mathbf{\tau}_1 = (0, 0, 0), \quad \mathbf{\tau}_2 = \frac{1}{4} \mathbf{a}_1 + \frac{1}{4} \mathbf{a}_2 + \frac{1}{4} \mathbf{a}_3\]

The second-quantized Hamiltonian can be written as:

\[H = t \sum_{\langle ij \rangle} c_i^{\dagger} c_j + i \lambda_{SO} \sum_{\langle\langle ij \rangle\rangle} c_i^{\dagger} \vec{\sigma} \cdot (\mathbf{d}_{ij}^{1} \times \mathbf{d}_{ij}^{2}) c_j\]

where the first term is a nearest-neighbor hopping term connecting the two fcc sublattices of the diamond lattice, and the second term is a spin-orbit coupling term connecting second-neighbor sites within the same sublattice. Here, \(\mathbf{d}_{ij}^{1,2}\) are the two nearest-neighbor bond vectors connecting sites \(i\) and \(j\).

Due to inversion symmetry, each band is doubly degenerate. The degeneracy is lifted by symmetry lowering perturbations of the four nearest-neighbor hoppings \(t \rightarrow t + \delta t_p\) with \(p = 1, 2, 3, 4\) indexing the four bonds connected to each site.

Added in version 2.0.0.

Parameters:
tfloat

Spin-independent nearest-neighbor hopping amplitude.

socfloat

Spin-orbit coupling strength. Modulates next-nearest neighbor hopping amplitudes.

dtlist[float, float, float, float], optional

Offsets added to the four nearest-neighbor hoppings along the bonds connected to each site. The entries are applied in the following order:

  • dt[0] : bond along R = [0, 0, 0]

  • dt[1] : bond along R = [-1, 0, 0]

  • dt[2] : bond along R = [0, -1, 0]

  • dt[3] : bond along R = [0, 0, -1]

The default is [0, 0, 0, 0], which corresponds to uniform hopping amplitudes. This parameter allows for symmetry-lowering perturbations to the nearest-neighbor hoppings.

Returns:
TBModel

An instance of the model.

Notes

  • The Fu-Kane-Mele model describes a three-dimensional topological insulator with a non-trivial band structure. It is characterized by a strong \(\mathbb{Z}_2\) invariant and exhibits surface Dirac cones that are protected by time-reversal and inversion symmetry [1].

References

[1]

L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett., 98, 106803 (2007).

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