SSH chain and winding number

Take t1 as intracell hopping and t2 as hopping from one cell to the next, both real and positive. Measure momentum in dimensionless inverse-cell units. The two Pauli coefficients trace a circle centred at t1 on the horizontal axis with radius t2. Defining winding using dx plus i dy makes the loop counterclockwise as k increases. It encloses the origin once when t2 exceeds t1 and not at all when t1 exceeds t2.

The Hamiltonian anticommutes with the sublattice Pauli matrix, placing its spectrum symmetrically about zero. Its two bands have energies plus and minus the distance of that circle from the origin. At equal hoppings the loop touches the origin, the gap closes, and the winding is not defined. Adding a staggered sublattice potential moves the Pauli vector out of the plane, allowing interpolation without this chiral winding protection.

Open the chain by ending on an A site and removing the intercell bond that would have connected it to a cell on its left. At zero energy a candidate state supported only on A sites makes each B-site equation a cancellation between neighbouring A amplitudes. The resulting ratio is minus intracell hopping divided by intercell hopping. It produces an exponentially localized semi-infinite state only when its magnitude is below one.

The opposite end supports the complementary sublattice state under the matching termination. In a finite balanced chain these modes can overlap and split into a symmetric positive/negative energy pair, with splitting exponentially small at long length. Chiral symmetry pairs energies but does not magically forbid overlap across a narrow sample. A different end cut can leave no exposed weakly bonded site for the same physical bulk pattern, so the termination must be specified alongside the claimed edge state.

For this two-band chiral convention, the lower-band Zak phase is pi times the winding modulo two pi. The Zak phase is a Brillouin-cycle Berry phase, not automatically a cell-independent absolute polarization. Changing the unit cell or shifting orbital embeddings changes the representation and how boundary charge is assigned. Comparing two written winding numbers therefore requires matching conventions, just as comparing two edge charges requires matching the termination and ionic background.

Disorder that preserves the relevant chiral structure can retain an appropriate real-space topological distinction without a momentum loop. Generic onsite potentials break that protecting symmetry, and sufficiently strong changes can eliminate the simple zero-energy statement. Interacting chains also require a many-body formulation rather than a literal single-particle amplitude recurrence. The clean SSH model teaches a controlled mechanism: a gap, a symmetry constraint, and a localized solution tied to a concrete boundary.