LESSON NOTES · 03

Dimerization, a gap, and an edge

Prerequisites: Cosine band; two-by-two matrices; geometric series

1. State the cell and hopping convention

The orthogonal spinless SSH toy model has cell length \(a\) with \(A_j,B_j\). Intracell hopping is \(-t_1\) and \(B_j\to A_{j+1}\) intercell hopping is \(-t_2\), with both positive \(t_1,t_2\). On-site energies equal \(\varepsilon_0\). Both orbitals in cell \(j\) receive phase \(e^{ikja}\). This fixed convention matters for winding and edges.

\[ H(k)=\begin{pmatrix}\varepsilon_0&-t_1-t_2e^{-ika}\\-t_1-t_2e^{ika}&\varepsilon_0\end{pmatrix},\qquad q(k)=t_1+t_2e^{-ika}. \]

2. Derive the bulk bands

The determinant is quadratic in \(E-\varepsilon_0\). Evaluating \(|q|^2\) yields two bands and a gap at the zone boundary. At equal hoppings the gap closes: this is the uniform chain with a doubled cell, not a gap created by folding. One spinless electron per cell fills the lower band; spin-degenerate electrons need two per cell.

\[ \begin{aligned}\det(H-EI)&=(\varepsilon_0-E)^2-|q|^2=0,\\E_\pm(k)&=\varepsilon_0\pm\sqrt{t_1^2+t_2^2+2t_1t_2\cos ka},\\E_{gap}&=2|t_1-t_2|\quad\text{at }k=\pi/a.\end{aligned} \]

3. Derive the specified left-edge state

Cut the chain with leftmost site \(A_1\) and first bond \(A_1\)–\(B_1\) of strength \(-t_1\). Subtract \(\varepsilon_0\) and seek zero energy with all \(B\) amplitudes zero. The equation at each \(B_j\) gives a recurrence. On a semi-infinite chain it is normalizable only when \(r=t_1/t_2<1\). Its amplitude alternates sign; its probability decays geometrically.

\[ \begin{aligned}-t_1A_j-t_2A_{j+1}&=0\Rightarrow A_{j+1}=-rA_j,\\A_j&=\sqrt{1-r^2}(-r)^{j-1},\quad r<1,\\\xi&=\frac{a}{\ln(t_2/t_1)}.\end{aligned} \]

4. Winding depends on a declared convention

The complex \(q(k)\) traces a circle centered at \((t_1,0)\) with radius \(t_2\). As \(k\) increases across the zone, it winds clockwise if \(t_2>t_1\). With the derivative-of-argument convention below this is \(\nu=-1\), while \(t_1>t_2\) gives \(0\). Other conventions reverse the sign. The distinction needs chiral/sublattice symmetry of \(H-\varepsilon_0I\) and a fixed unit cell and cut. Moving the cell changes bookkeeping; state the actual termination.

\[ \nu=\frac1{2\pi}\int_{-\pi/a}^{\pi/a}\partial_k\arg q(k)\,dk. \]

Dimerization, a gap, and an edge — schematic under the stated model assumptions

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

Worked comparison: same bulk, different edge

Toy \((t_1,t_2)=(0.5,1.0)\) and \((1.0,0.5)\) eV with \(\varepsilon_0=0\) both give \(\pm1.5\) eV at \(k=0\), \(\pm0.5\) eV at \(\pi/a\), and a \(1\) eV gap. Only the first has the normalizable semi-infinite edge state for the declared cut: \(P(A_j)=0.75(0.25)^{j-1}\) sums to one. A finite balanced chain has two edge modes which generally hybridize and split exponentially; do not call them both exactly zero. At \(t_1=0\), dangling endpoints are exactly decoupled.

Symmetry check and model limits

What if on-site energies become \(\varepsilon_0\pm\Delta\)? The bands become \(\varepsilon_0\pm\sqrt{|q|^2+\Delta^2}\); chiral symmetry breaks, so zero-energy pinning and this winding argument no longer directly apply. This minimal model omits lattice dynamics, interactions and real-material multi-orbital details; it teaches a mechanism rather than quantitatively predicting all polymers.

References and further reading

Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.

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