Momentum Matrix Elements of Bloch Eigenfunctions

• Momentum matrix elements of Bloch eigenfunctions quantify quantum expectation values of the momentum operator between periodic states, underpinning key electrical and optical responses.
• Their decomposition into band-structure and Berry connection components distinguishes intraband and interband contributions while addressing nonlocal corrections.
• Computational strategies using Wannier interpolation and smooth subspaces enable accurate evaluation, impacting predictions of optical absorption and topological invariants.

Momentum matrix elements of Bloch eigenfunctions quantify the quantum mechanical expectation values of the canonical momentum operator between periodic eigenstates in a crystalline solid. These matrix elements are central to the study of electronic and optical properties, underpin the calculation of physical observables such as electrical conductivity, optical absorption, and photoemission intensities, and form the basis for probing the geometric and topological features encoded in Bloch wavefunctions. Their rigorous definition, calculation, and interpolation strategies involve subtleties arising from gauge freedom, basis choice, nonlocal terms in the Hamiltonian, and the interplay with the Berry connection.

1. Formal Definition and Operator Structure

The canonical momentum matrix element between two Bloch states $|\psi_{n\mathbf{k}}\rangle$ and $|\psi_{m\mathbf{k}}\rangle$ at crystal momentum $\mathbf{k}$ is given by

$$p_{mn}(\mathbf{k}) = \langle \psi_{m\mathbf{k}}|\hat{\mathbf{p}}|\psi_{n\mathbf{k}}\rangle$$

where $$|\psi_{n\mathbf{k}}\rangle = e^{i\mathbf{k}\cdot\mathbf{r}}|u_{n\mathbf{k}}\rangle$$ and $|u_{n\mathbf{k}}\rangle$ is the cell-periodic part.

For a general single-particle Hamiltonian in a periodic potential,

$$\hat H = \frac{\hat{\mathbf p}^2}{2m_e} + V(\hat{\mathbf r}) + \hat H'$$

where $V(\hat{\mathbf r})$ is the local lattice potential and $\hat H'$ comprises nonlocal components (such as pseudopotentials, spin-orbit, or Hartree–Fock exchange), one has the operator identities

$$\hat{\mathbf p} = -i\hbar\nabla_{\mathbf r} = i[\hat H_L,\hat{\mathbf r}]$$

$$\hat{\mathbf v} = \frac{i}{\hbar}[\hat H, \hat{\mathbf r}] = \frac{\hat{\mathbf p}}{m_e} + \frac{i}{\hbar}[\hat H',\hat{\mathbf r}]$$

Within this structure, $p_{mn}(\mathbf{k})$ is generally not simply proportional to $v_{mn}(\mathbf{k})$ when $\hat H'$ is nonzero, as nonlocal contributions and finite-basis corrections must be considered (Esteve-Paredes et al., 2022).

2. Decomposition: Band-structure, Berry Connection, and Nonlocal Corrections

The matrix elements admit a decomposition reflecting the underlying geometry and band structure: $$v_{mn}(\mathbf{k}) = \delta_{mn}\nabla_{\mathbf{k}}\epsilon_n(\mathbf{k}) + i\omega_{mn}(\mathbf{k})\mathbf{A}_{mn}(\mathbf{k})$$ where $\epsilon_n(\mathbf{k})$ are the band energies, $$\omega_{mn} = [\epsilon_m(\mathbf{k})-\epsilon_n(\mathbf{k})]/\hbar$$, and the non-Abelian Berry connection

$$\mathbf{A}_{mn}(\mathbf{k}) = i\langle u_{m\mathbf{k}}|\nabla_{\mathbf{k}}u_{n\mathbf{k}}\rangle_{\Omega}$$

Projecting the velocity-momentum correspondence gives, restoring physical units,

$$p_{mn}(\mathbf{k}) = m_e v_{mn}(\mathbf{k}) + \frac{i m_e}{\hbar}\langle u_{m\mathbf{k}}|[\hat H'_{\mathbf{k}}, \hat{\mathbf r}]|u_{n\mathbf{k}}\rangle + \Delta_{mn}$$

$\Delta_{mn}$ are small trace corrections vanishing in the complete basis limit, and the additional nonlocal correction term is essential in the presence of $\hat H'$. This formalism ensures that both the "Peierls" term ($m_e\nabla_{\mathbf{k}}\epsilon_n$) and the geometric Berry connection term ($i m_e \omega_{mn} \mathbf{A}_{mn}$) are explicitly included, as well as any nonlocal or finite-basis effects (Esteve-Paredes et al., 2022).

3. Gauge Dependence, Representations, and Physical Invariance

Bloch functions exhibit gauge freedom: under unitary, $\mathbf{k}$-dependent transformations among the relevant bands, matrix elements generally transform covariantly: $$|\tilde{\psi}_{n\mathbf{k}}\rangle = \sum_{m=1}^{N_w} |\psi_{m\mathbf{k}}\rangle U_{mn}(\mathbf{k})$$

$$p^{\text{new}}(\mathbf{k}) = U^\dagger(\mathbf{k})p(\mathbf{k})U(\mathbf{k})$$

The Berry connection transforms accordingly, but the sum in the momentum matrix element remains gauge-invariant. In a local-orbital basis, the distinction between “cell” and “atomic” gauge arises, but the full $p_{mn}(\mathbf{k})$ expression is strictly independent of this choice. This invariance is critical for the calculation of physical observables derived from momentum matrix elements (Esteve-Paredes et al., 2022, Volpato et al., 2024).

4. Computational Strategies: Wannier Interpolation and Smooth Subspaces

Accurate evaluation and interpolation of $p_{mn}(\mathbf{k})$ on dense $\mathbf{k}$-grids are enabled by use of maximally localized Wannier functions (MLWFs) and the construction of the optimally smooth subspace (OSS). The matrix elements are rotated into the OSS using unitary matrices $U(\mathbf{k})$ that minimize the total Wannier spread functional,

$$\Omega[U] = \sum_{n=1}^{N_w}[\langle r^2\rangle_n - |\langle r\rangle_n|^2]$$

Once represented in the OSS, $p_{mn}^\text{oss}(\mathbf{k})$ becomes a smooth function of $\mathbf{k}$, allowing efficient interpolation via either

• Fourier–Wannier interpolation: a real-space FFT-based approach, efficient for periodic matrix elements.
• Direct multidimensional spline interpolation: particularly suitable for both periodic and non-periodic operators.

Algorithmically, one first computes raw $p_{mn}(\mathbf{k})$ on a coarse grid (preferably via the more stable $m_e\langle u|\partial_k H|u\rangle$ form), rotates to OSS, interpolates in reciprocal space, and finally rotates back if required. For periodic $p(\mathbf{k})$, both interpolation methods are equally accurate; for non-periodic observables, only the direct approach is unbiased (Volpato et al., 2024).

5. Experimental Access: Photoemission and Bloch Wavefunction Reconstruction

Angle-resolved photoemission spectroscopy (ARPES) accesses the electronic structure and, through the photoemission or dipole (length gauge) matrix elements,

$$M(E, \mathbf{k}, \theta) = \langle \mathbf{k}, E | \hat{\boldsymbol{\epsilon}}(\theta)\cdot\mathbf{\hat{r}} | \psi_{\mathbf{k}, \alpha}\rangle$$

relates directly to the momentum matrix elements of Bloch eigenstates. Polarization modulation of the ionizing field enables extraction of both the amplitude and phase of the complex photoemission matrix elements. In minimal two-band models, these can be inverted to reconstruct the orbital pseudospin $\boldsymbol{\tau}(\mathbf{k})$, which parametrizes the Bloch eigenfunction in orbital space. This pseudospin governs the local Berry curvature via

$$\Omega(\mathbf{k}) = \frac{1}{2} \boldsymbol{\tau}\cdot (\partial_{k_x} \boldsymbol{\tau} \times \partial_{k_y} \boldsymbol{\tau})$$

enabling the extraction of Chern numbers and topological invariants from momentum-resolved spectroscopic data (Schüler et al., 2021).

6. Relations, Limitations, and Physical Interpretation

The canonical relation between velocity and momentum breaks down in the presence of nonlocal $\hat H'$, necessitating explicit evaluation of the commutator $[\hat H', \hat{\mathbf r}]$ and finite basis corrections. The Berry term $i m_e \omega_{mn} \mathbf{A}_{mn}$ describes interband transition amplitudes, while the $m_e \nabla_\mathbf{k} \epsilon_n$ term reflects intraband processes such as Drude response. The total physical matrix element is always gauge-invariant, but its decomposition into band-structure and Berry terms is gauge-dependent.

Nonlocal corrections $\Delta p^{\rm nonloc}$ are especially relevant in calculations involving modern pseudopotentials, spin–orbit coupling, or hybrid exchange. Incomplete treatment of these effects can lead to significant errors, particularly in properties sensitive to interband coherence or topological indices (Esteve-Paredes et al., 2022). A plausible implication is that rigorous benchmarking against real-space and reciprocal-space evaluation is required for high-fidelity property prediction.

7. Applications and Practical Considerations

Momentum matrix elements of Bloch eigenfunctions are requisite for ab initio optical response calculations, Wannier interpolation of response tensors, and in the analysis and interpretation of photoemission and pump-probe experiments. Their accurate computation underpins ab initio modeling of nonlinear optics, dynamical Berry curvature effects, and the ab initio construction of tight-binding models for topological band theory. The combination of MLWF-based OSS construction and direct multidimensional interpolation supports computation on ultradense momentum grids with substantial savings over direct DFT evaluation, and uniquely enables interpolation of both periodic (e.g., $p_{mn}(\mathbf{k})$) and non-periodic observables (e.g., oscillator strengths, derivative couplings) in a consistent framework (Volpato et al., 2024).

These theoretical and computational strategies form the foundation for systematic exploration of geometric and topological aspects of condensed matter systems, enabling detailed links between microscopic wavefunction structure, measurable physical properties, and emergent quantum phenomena.