Perfect State Transfer from a Localised Two-Excitation State to a Dicke State via Static Spin-Network Hamiltonians

Abstract: I construct a family of time-independent, excitation-preserving spin Hamiltonians that realises perfect state transfer from a localised two-excitation state to the symmetric two-excitation Dicke state, for every system size $N \ge 4$. The Hamiltonian has the physical form $H = \sum_{i&lt;j} J_{ij} (σ<em>i⁺ σ_j^- + σ_j⁺ σ_i^-) + \sum_i ε_i n_i$ with real couplings, and satisfies $$e^{-iHt} |110\cdots0\rangle = e^{iφ} |D_N<sup>{(2)}\rangle$$ at a finite time. The construction exploits an $S</em>{N-2}$ permutation symmetry acting on the initially unoccupied spins, which reduces the dynamics to a four-dimensional invariant subspace. Requiring $(|ψ_0\rangle + |D_N<sup>{(2)}\rangle)/2$ to be a zero eigenvector determines the on-site energies in closed form and leaves three coupling parameters free. The remaining inverse spectral problem reduces to two polynomial equations in two dimensionless coupling ratios; eliminating one ratio yields a degree-six reciprocal polynomial, which the substitution $z = x + x<sup>{-1}$ converts to a cubic. For the spectral family $(-n,-1,1)$, with $n$ an odd integer, an explicit factorisation of the leading coefficient together with a boundary evaluation at $z=-2$ shows that for every $N \ge 4$ an odd $n$ can be chosen large enough that a real root with $z &lt; -2$ exists; a subresultant argument supplies a real lift of that root to the original system. The existence proof is entirely symbolic and does not rely on numerical optimisation; direct propagation is used only as independent validation. The result is a constrained analogue of perfect state transfer: unlike the general real-state problem, where an unconstrained real symmetric matrix suffices, here the Hamiltonian is required to arise from an excitation-preserving spin-network form.