Constructing Tight-Binding Matrices Numerically

A tight-binding implementation starts from orbitals, directed hopping amplitudes, and boundary conditions. Its real-space and Bloch forms must represent the same operator. Here we construct and test both forms; tight-binding models explains their physical assumptions and material interpretation.

Hopping dictionary and Hermiticity

Let $c_{R\alpha}$ annihilate orbital $\alpha$ in cell $R$. Define

$$H=\sum_{R,\Delta,\alpha\beta} c^\dagger_{R\alpha}h_{\alpha\beta}(\Delta)c_{R+\Delta,\beta}.$$

Hermiticity requires $h(-\Delta)=h(\Delta)^\dagger$. Store either the complete directed set or an independent half-set with explicit conjugate completion; do not apply both conventions simultaneously.

In a finite open system, omit bonds whose destination is outside the sample. With periodic boundaries, wrap cells and apply any boundary twist exactly once. Keep lattice geometry separate from orbital indexing.

Using $c_{R\alpha}=N^{-1/2}\sum_k e^{ikR}c_{k\alpha}$ gives the cell-periodic convention

$$H_{\alpha\beta}(k)=\sum_\Delta h_{\alpha\beta}(\Delta)e^{ik\Delta}.$$

If orbital positions are included in the Fourier phase, extra factors involving their intracell coordinates appear. Spectra agree after the appropriate unitary transformation, but eigenvectors, derivatives, and boundary sewing must follow the selected convention.

Dimerized-chain example

For two orbitals per cell, intracell hopping $t_1$ and intercell hopping $t_2$, choose

$$H(k)= \begin{pmatrix} 0&t_1+t_2e^{-ik}\\ t_1^*+t_2^*e^{ik}&0 \end{pmatrix}.$$

For real hoppings the bands are

$$E_\pm(k)=\pm\sqrt{t_1^2+t_2^2+2t_1t_2\cos k}.$$

At $t_2=0$, each cell is an isolated dimer with energies $\pm|t_1|$. At $t_1=t_2$, the two bands touch at $k=\pi$. These checks expose bond-counting errors, but the cosine dispersion alone cannot distinguish $k$ from $-k$.

For $N$ periodic cells, diagonalize the $2N$ real-space matrix and compare its spectrum with the union of the two-band eigenvalues at $k=2\pi m/N$. This is a stringent real-space/Bloch consistency check. It is stronger than checking Hermiticity alone.

To test the Fourier convention itself, retain the momentum labels and compare matrix blocks, not only the combined spectrum. The displayed model corresponds to $h_{AB}(0)=t_1$ and $h_{AB}(-1)=t_2$, together with their Hermitian partners. Form the Fourier embedding

$$F_{R\alpha,k\beta}=\frac{e^{ikR}}{\sqrt N}\delta_{\alpha\beta}.$$

Then $F^\dagger H_{\rm real}F$ must have block $H(k)$ at the declared $k$, with all off-block entries zero. Choose complex, unequal $t_1,t_2$ for this test. An overall Fourier-sign mistake can permute the $k$ blocks and leave the entire unordered spectrum unchanged, while changing the momentum associated with an eigenvector or current.

Phases, flux, and gauge tests

A site-dependent basis change $c_i'=e^{i\theta_i}c_i$ gives $h'_{ij}=e^{i(\theta_i-\theta_j)}h_{ij}$ when the same Hamiltonian is written in the primed operators. It redistributes hopping phases while leaving oriented loop products and spectra invariant. Test this invariance with random phases. A phase on an isolated bond is removable; a nonzero sum around a loop can be physical.

For magnetic fields, a Peierls phase depends on the directed path and charge convention. Verify the accumulated phase around an elementary loop against the intended flux. Reversing a bond must complex-conjugate its amplitude.

Do not infer a topological index from a finite open-chain spectrum alone. Edge levels depend on termination and finite-size hybridization. Bulk geometry requires eigenvector-consistent algorithms such as Wilson loops.

Numerical error controls

For a finite-orbital lattice model there is no grid-spacing limit unless it approximates a continuum system. Increase sample size for finite-size effects and refine the momentum mesh for Brillouin-zone integrals. These are different convergence parameters.

Track the basis convention in saved hopping files. Comparing two eigenvector arrays without orbital order, phase convention, or positions is not meaningful, even if the eigenvalues match.

Exercise

For a one-orbital chain with real nearest-neighbor hopping $-t$ and onsite energy $\epsilon$, derive the periodic dispersion.

Solution

The hopping dictionary has $h(0)=\epsilon$ and $h(\pm1)=-t$, so $E(k)=\epsilon-2t\cos k$. A finite ring of $N$ cells samples $k=2\pi m/N$. Counting both directed bonds and then adding their adjoints again incorrectly doubles the cosine term.

References

• Groth, C. W., Wimmer, M., Akhmerov, A. R., and Waintal, X., “Kwant: a software package for quantum transport,” New Journal of Physics 16, 063065 (2014), doi:10.1088/1367-2630/16/6/063065.
• Kwant developers, “First steps: setting up a simple system and computing conductance”, Kwant documentation — sites, directed hoppings, and finite systems.