Band-Structure Numerics

Band-structure numerics diagonalizes a Bloch Hamiltonian over momentum space. The energy calculation is often easy; reliable eigenvector tracking, degeneracies, and Brillouin-zone integration require more care. This page concerns prescribed lattice Hamiltonians, while material-specific electronic-structure validation belongs to Quantum Matter.

Helpful background. Tight-binding assembly defines the Fourier and orbital conventions.

Hermitian eigenproblems over momentum

For each $\boldsymbol k$, solve

$$H(\boldsymbol k)U(\boldsymbol k) =U(\boldsymbol k)\operatorname{diag}(E_n(\boldsymbol k)).$$

Check Hermiticity before using a Hermitian solver, then verify eigenpair residuals and $U^\dagger U=I$. If the basis is nonorthogonal, solve the generalized problem $Hc=ESc$ with a positive-definite overlap matrix $S$; ordinary normalization is then inappropriate.

A path through high-symmetry momenta is a visualization. It is not a quadrature mesh and can miss an indirect gap or an extremum away from the path. For integrated quantities use a full Brillouin-zone mesh with normalized weights.

For a candidate insulator with $r$ occupied bands, distinguish the minimum direct gap $\min_{\boldsymbol k}[E_{r+1}(\boldsymbol k)-E_r(\boldsymbol k)]$ from the indirect gap $\min_{\boldsymbol k}E_{r+1}(\boldsymbol k)-\max_{\boldsymbol k}E_r(\boldsymbol k)$. The former isolates a fixed-rank band subspace; a positive latter value permits a common chemical potential between the two groups. For $H(k)=2\cos k\,I+\sigma_z$, the direct gap is two everywhere, but the indirect gap is $-2$. Smooth and separated curves at each momentum therefore need not describe an insulating filling.

Energy sorting versus state tracking

Sorting eigenvalues independently at every momentum gives energy order, not continuous band identity. At a crossing, a physical branch can change its position in the sorted list. To track isolated bands, compare neighboring overlaps

$$M_{mn}=\langle u_m(\boldsymbol k)|u_n(\boldsymbol k+\Delta\boldsymbol k)\rangle.$$

Find a one-to-one assignment maximizing overlap, rather than greedily choosing each row’s largest element. Near degeneracies track the whole subspace: individual eigenvectors are arbitrary unitary mixtures and cannot be assigned uniquely.

Subspace continuity can be measured by singular values of $U_{\rm old}^\dagger U_{\rm new}$ or projector differences. A tiny singular value indicates a coarse mesh, wrong subspace, or physical gap closure.

These coefficient overlaps assume a common orthonormal orbital basis. In a nonorthogonal or momentum-dependent basis, construct the actual cross-overlap matrix between basis functions at the two momenta. The required matrix element is $c_m(\boldsymbol k)^\dagger S(\boldsymbol k,\boldsymbol k')c_n(\boldsymbol k')$; the equal-momentum normalization matrix $S(\boldsymbol k)$ alone does not specify it.

Dimerized-chain benchmark

For

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

with real $t_1,t_2$, energies are $\pm\sqrt{t_1^2+t_2^2+2t_1t_2\cos k}$. For positive hoppings the direct gap is $2|t_1-t_2|$ at $k=\pi$.

A mesh omitting $k=\pi$ overestimates a small gap. Refinement or targeted minimization around the apparent minimum is necessary. At $t_1=t_2$, the degeneracy makes the separate eigenvectors ill-defined at the crossing even though the Hamiltonian and two-dimensional subspace remain regular.

Compare the periodic finite-chain spectrum against the sampled Bloch eigenvalues to test Fourier assembly independently.

Derivatives and density of states

For an isolated normalized band in a fixed orthonormal basis, Hellmann–Feynman gives

$$\partial_{k_i}E_n =\langle u_n|\partial_{k_i}H|u_n\rangle.$$

At degeneracy, diagonalize the derivative operator within the degenerate subspace to obtain directional slopes. Taking finite differences of energy-sorted labels can mix branches.

For $Hc_n=E_nSc_n$ with $c_n^\dagger Sc_n=1$, differentiate the eigenproblem and multiply by $c_n^\dagger$. The terms containing $\partial c_n$ cancel, leaving

$$\partial_{k_i}E_n =c_n^\dagger(\partial_{k_i}H-E_n\partial_{k_i}S)c_n.$$

Omitting the overlap derivative gives a wrong band slope even if every generalized eigenvalue is accurate. Compare this expression with centered finite differences for a smooth isolated band. It computes the energy derivative; identifying it with a physical velocity still requires the representation assumptions below.

A histogram or broadened density of states depends on both momentum resolution and energy-bin width. Refine the momentum mesh at fixed broadening first, then reduce broadening with adequate sampling. Check that the integrated density of states equals the number of bands under the chosen normalization.

In a momentum-dependent orbital basis, velocity and geometric quantities can require basis-connection terms. A numerical derivative of a Hamiltonian matrix is not automatically the full physical velocity; preserve the Quantum Matter conventions.

Exercise

Why can a smooth band plot incorrectly suggest an insulator?

Solution

The chosen path may miss conduction minima or valence maxima elsewhere in the Brillouin zone. Even if every plotted direct gap is positive, the global conduction minimum may lie below the global valence maximum. Determine the indirect gap on a converged full-zone search.

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.
• Vanderbilt, D., Berry Phases in Electronic Structure Theory, Cambridge University Press (2018), doi:10.1017/9781316662205 — Bloch conventions, subspaces, and momentum-space calculations.