Applications of Hafnians I am learning about Hafnians but I am struggling to find real-world applications of them. I understand the applications of determinants, permanents, and even pfaffians but I am at a loss for Hafnians. I know they can be used in nonbipartite graph matching but where would this be useful in the real world?
Thanks in advance.
 A: In the context of the fractional quantum Hall effect, the ground state wave function $\Psi$ of electrons in a two-dimensional layer (coordinates $z_n=x_n+iy_n$, $n=1,2,\ldots$) in a strong perpendicular magnetic field can, for certain electron densities, be described by a Hafnian, see From Irrational to Non-Unitary: on the Haffnian and Haldane-Rezayi wave functions (2011):
$$\Psi(z_1,z_2,\ldots)={\rm Hf}\,(1/(z_i-z_j)^2)\prod_{i<j}(z_i-z_j)^2$$
A different application is to spin liquids, see pages 37 and following of Physics of Resonating Valence Bond Spin Liquids (2013).

Incidentally, most physics papers I have seen write Haffnian, mistakenly, with two f's, probably confused by Pfaffian.
A: In fact Hafnians were introduced by Eduardo Caianiello for a "real-world application", namely to simplify calculations in renormalized quantum field theory. As he writes, 

Fermi and Bose statistics require, respectively, the definition and study of pfaffians and determinants, and of hafnians and permanents: any expectation value of any product of free Fermi fields is in fact a pfaffian (which often, as we shall see, reduces to a determinant, e.g., in the Fredholm case and in electrodynamics); of Bose fields, a hafnian (which often reduces to a permanent). Tremendous simplifications are trivially seen to occur if we start with some preliminary (and quite elementary) mathematical considerations concerning these objects and their main properties.

See his book "Combinatorics and Renormalization in Quantum Field Theory" https://www.amazon.com/Combinatorics-Renormalization-Quantum-Frontiers-Physics/dp/0805316450
Interestingly, "The "Hafnian" denomination was coined by Caianiello to mark the fruitful period of stay in Copenhagen (Hafnia in Latin)" - see p.98 in the book " Imagination and Rigor: Essays on Eduardo R. Caianiello's Scientific Heritage" https://books.google.ru/books?id=K3xBKR2HiT8C
