  Although free-fermion systems are considered exactly solvable, they
generically do not admit closed expressions for nonlocal quantities such as
topological string correlations or entanglement measures. We derive closed
expressions for such quantities for a dense subclass of certain classes of
topological fermionic wires (classes BDI and AIII). Our results also apply to
spin chains called generalised cluster models. While there is a bijection
between general models in these classes and Laurent polynomials, restricting to
polynomials with degenerate zeros leads to a plethora of exact results: (1) we
derive closed expressions for the string correlation functions - the order
parameters for the topological phases in these classes; (2) we obtain an exact
formula for the characteristic polynomial of the correlation matrix, giving
insight into ground state entanglement; (3) the latter implies that the ground
state can be described by a matrix product state (MPS) with a finite bond
dimension in the thermodynamic limit - an independent and explicit construction
for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021),
033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer
topological invariant, all non-zero eigenvalues of the transfer matrix are
identified as products of zeros and inverse zeros of the aforementioned
polynomial. General models in these classes can be obtained by taking limits of
the models we analyse, giving a further application of our results. To the best
of our knowledge, these results constitute the first application of Day's
formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum
physics.
