3. Dimerization, SSH bands, and edge states
Position in the course: Lesson 3 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
The SSH model uses two orbitals per unit cell and alternating nearest-neighbor hoppings. Here t1 couples sites within the chosen cell and t2 couples successive cells. Setting both onsite energies to zero imposes a chiral or sublattice symmetry. It is an idealized model that isolates dimerization and boundary effects rather than a complete account of every polymer or one-dimensional material.
The Bloch matrix has off-diagonal element q=t1+t2 exp(−ika). Taking its determinant gives E²−|q|²=0. Multiplying q by its conjugate yields t1²+t2²+2t1t2 cos ka. For positive hoppings, the gap at the zone boundary is 2|t1−t2|. At equal hoppings it closes; changing their order then changes the winding of q around the origin relative to a fixed cell and boundary convention.
An edge solution can be derived directly in real space. At zero energy set amplitudes on one sublattice to zero. The remaining site equation relates successive nonzero amplitudes by −t1/t2. The amplitude decays only when |t1/t2|<1 for the illustrated termination. Normalizability distinguishes an actual boundary-localized mode from a formal recurrence. On a finite chain opposite edges can hybridize and split the energies away from zero.
Bulk topology and boundary termination must be discussed together. Redefining the unit cell changes how intra- and intercell bonds are named; it does not create a new physical boundary. Staggered onsite energies break the chiral symmetry and move protected zero energies, while a finite bulk gap can remain. Edge states seen in an arbitrary finite model are not automatically topological: defects, dangling orbitals, and onsite potentials can also localize states. Inspect symmetry, bulk invariant, termination, and robustness before using that label.
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.
2.1. An edge recurrence is a boundary calculation
The zero-energy ansatz must satisfy every site equation, including the first and last site. A normalizable solution for one termination may disappear for another termination that pairs the end orbitals differently. For a finite chain, the recurrence from both ends overlaps and produces exponentially small splitting when the chain is long compared with the localization length. Near the gap closure the localization length grows, so that separation becomes poor. A numerical finite chain should therefore be compared across several lengths. Observe both the eigenvalue and spatial weight; an energy near zero by itself does not prove an edge-localized state.
3. Worked example
For t1=0.5 and t2=1 eV, the bulk gap is 1 eV. A zero-energy edge amplitude ratio is −0.5, so successive probabilities fall by a factor of 0.25. An infinite semi-chain has normalization sum 1+0.25+0.25²+…=4/3, giving initial amplitude magnitude √3/2.
4. Exercises with explained solutions
Exercise. Find the edge localization length from |A_j|∝exp(−ja/ξ).
Explained solution. Equate exp(−a/ξ)=|t1/t2|. Taking logs gives ξ=a/ln|t2/t1|. It diverges as the hoppings approach equality, consistent with the bulk gap closing.
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
A quantized winding requires the stated symmetry and a gap. A plotted edge mode alone is not a proof of quantization or experimental protection.
The illustration is an original teaching schematic. It is not output from a numerical materials simulation.
6. Connections and sources
Related: density-functional foundations · Interacting Monte Carlo methods
- Slater–Koster, original orbital-geometry construction (1954)
- Su–Schrieffer–Heeger, solitons in polyacetylene (1979)
- Marzari et al., maximally localized Wannier functions review (2012)
The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.
Original analytic teaching diagram under the lesson assumptions; no simulation results.
7. Related theory and practice
Quantum mechanics · Density functional theory · ABACUS · DFTB+