LESSON NOTES · 10

10. Disorder, boundaries, and model validation

Position in the course: Lesson 10 of 10. Complete the preceding derivation and use the explained exercises to check understanding.

1. Purpose and assumptions

Finite tight-binding models make boundary and disorder effects explicit. Open chains remove end bonds; periodic chains reconnect them. Random onsite energies model one class of disorder, while random hopping represents another. The disorder distribution, correlations, and strength are physical assumptions. A single random realization is not a statistically representative ensemble.

The inverse participation ratio measures concentration of a normalized orthogonal-basis eigenvector. Uniform weight across N sites gives 1/N; a state on one site gives one. Scaling with system size helps distinguish extended and localized behavior, but finite-size effects and boundary modes can mimic localization. In a nonorthogonal basis the coefficient norm is not a physical probability, so the simple formula needs an appropriate metric or spatial-density definition.

Validate a model in layers. Algebraic checks include Hermiticity, state counts, overlap positivity, symmetry, and known limiting cases. Numerical checks include matrix-size convergence, hopping-range convergence, independent eigensolver residuals, and ensemble uncertainty. Physical checks compare reference band character, energies, charge distributions, forces, or response quantities within the intended domain. These checks are complementary; passing a matrix symmetry test does not establish chemical accuracy.

For an uncertainty study, average the chosen observable over independent disorder realizations and track the standard error; do not use spread among correlated adjacent eigenvalues as a substitute. Compare multiple sizes and boundary conditions. Distinguish a formal teaching model from a fitted predictive model, and label every schematic example. Connect interacting questions to the quantum Monte Carlo curriculum, orbital projection questions to molecular methods and DFT, and practical self-consistent-charge models to the DFTB+ software entry.

2. Derivation step by step

Read each equality with its assumptions. Atomic units are used for DFT equations unless another unit is stated; TB parameters retain explicit energy and length units. The conjugate transpose is denoted by a dagger, and a prime on a coordinate denotes a separate integration variable.

\[ \begin{aligned} H&=\sum_i\epsilon_i|i\rangle\langle i|+\sum_{ij}t_{ij}|i\rangle\langle j|,\\ \mathrm{IPR}&=\sum_i|c_i|^4\quad\text{for normalized orthogonal coefficients},\\ \mathrm{IPR}_{\mathrm{uniform}}&=N(1/N)^2=1/N,\\ \mathrm{IPR}_{\mathrm{one\ site}}&=1. \end{aligned} \]

2.1. A reproducible model includes its random experiment

Record the random-number seed or full disorder realization, distribution, number of independent samples, boundary conditions, and system sizes. A seed allows reconstruction but does not make one sample representative. Check whether the observable distribution has rare tails; the mean alone can hide large realization-to-realization variability. For a deterministic benchmark, first test clean-chain energies, trace, and uniform IPR before adding randomness. Store eigenvector conventions and normalization when computing participation measures. The present arithmetic examples contain no executed disorder simulation, and their role is to supply checks that a future implementation must pass.

3. Worked example

An eight-site uniform state has coefficients 1/√8 and IPR=1/8. A state equally distributed over two sites has IPR=1/2. The latter value indicates concentration, but does not reveal whether it arises from disorder, a chemical defect, or a symmetry-protected edge.

4. Exercises with explained solutions

Exercise. What is the standard error of the mean for M independent realizations with sample deviation s?

Explained solution. It is s/√M under finite variance and independence. Reducing it by two requires about four times as many independent samples. Correlated samples reduce effective sample size, so this formula must not be used blindly.

Further check. State the units and the allowed regime for every parameter in the worked example. Change one assumption and identify which derivation step must be revisited. A correct explanation names the affected constraint, operator, or boundary condition rather than merely saying that the answer changes.

5. Misconceptions and limitations

IPR is basis dependent and finite-size-sensitive. A localization claim should combine size scaling, boundary inspection, and suitable physical response evidence.

Nearest-neighbor chain and its cosine dispersion

The illustration is an original teaching schematic. It is not output from a numerical materials simulation.

6. Connections and sources

DFTB+

The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.

Quantum mechanics · Density functional theory · ABACUS · DFTB+

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