One machine, all the energies: perturbation theory with matrices
A compact matrix recursion gives every eigenvalue and eigenstate at once, to any order. We run it, and show exactly when and why it breaks.
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The previous article built the energy by hand, order by order. The appendix of our paper turns the whole procedure into a machine that only uses matrix operations, and that gives every state at once.
The problem as a table of numbers
We write the Hamiltonian as H = Λ + λΔ: a diagonal, easy part Λ with entries εn, and a small off-diagonal push Δ of size λ.
Store every correction in a matrix
Energies and states are expanded as series in λ. Each correction to state n is a mix of the simple states, |ψn(k)⟩ = Σj Aj,n(k) |j⟩. Stacking these numbers gives one matrix A(k) per order, with one column per state. Two rules fix everything: A(0) is the identity, and corrections never point back along the original state, so the diagonal of A(k) is empty. The energy corrections are then read off a diagonal:
The machine
With the gap table θj,n = 1/(εj − εn) (zero on the diagonal), the next correction follows from the previous ones:
Here ∘ is the cell-by-cell (Hadamard) product and D keeps only the diagonal. In words: take the previous correction, mix it with Δ using ordinary matrix products, multiply cell by cell by the gap table, and repeat.
Turning the crank
On our six-site chain with λ = 0.2, the error drops by orders of magnitude with each even order and reaches about 10−13 after 30 orders; odd orders change nothing, as expected from the bipartite structure.
When the machine breaks
Push harder, λ = 0.6, and the error first decreases, then explodes. The reason is a near-resonance: two levels only 0.033 apart give a huge entry in θ, which is amplified at every order. Perturbation theory is a theory for small pushes, and the closest gaps set its range.
