A Quantum Analogue of the Pfaffian–Determinant Identity

Alexey Orlovsky

Yuri N. Berdinsky, Alexander S. Ushakov

We unify the Henstock-Kurzweil treatment of open quantum systems with bosonic Gaussian reservoirs and fermionic Grassmann reservoirs. Finite cylindrical gauge integrals yield determinant and covariance identities for bosons, while Berezin integration yields Pfaffian identities for fermions. In both sectors the reservoir is eliminated into an influence functional and the reduced dynamics is organised by Chernoff splitting. The accompanying Lean 4 developments verify the finite-dimensional Gaussian, operator, exterior-algebra, and Pfaffian steps. This upload contains the paper and the accompanying Lean 4 formalisation. The Lean development is free of sorry, admit, and newly introduced axioms. Kernel axiom inspection of the headline results reports only propext, Classical.choice, and Quot.sound. The finite-mode formulas are exact and machine-checked; continuum and unbounded-generator qualifications are kept explicit in the paper.

Boris Maulén, Sergio M. Davis, Daniel Pons

T. Mizusaki, Makito Oi, Noritaka Shimizu

For the overlap matrix element between Hartree–Fock–Bogoliubov states, there are two analytically different formulae: one with the square root of the determinant (the Onishi formula) and the other with the Pfaffian (Robledo's Pfaffian formula). The former formula is two-valued as a complex function, hence it leaves the sign of the norm overlap undetermined (i.e., the so-called sign problem of the Onishi formula). On the other hand, the latter formula does not suffer from the sign problem. The derivations for these two formulae are so different that the reasons are obscured why the resultant formulae possess different analytical properties. In this paper, we discuss the reason why the difference occurs by means of the consistent framework, which is based on the linked cluster theorem and the product-sum identity for the Pfaffian. Through this discussion, we elucidate the source of the sign problem in the Onishi formula. We also point out that different summation methods of series expansions may result in analytically different formulae.

Matthew Lagro, Wei-Shih Yang, Sheng Xiong

Jiangke Yang

Recently we have shown that the anti-Pfaffian state and a state described by the second Landau level (SLL) projection of antiholomorphic Pfaffian wavefunctions have large overlap and almost identical low-energy orbital entanglement structures, suggesting they have the same topological order. In this paper, we further show that the similar entanglement structure observed in the SLL projected state is also identifiable in other Landau level projected states, indicating it is not the result of the SLL projection but rather a "fingerprint" of the PH-Pfaffian topological order of the original unprojected antiholomorphic Pfaffian wavefunctions. Consequently, we argue that the SLL projected state, and therefore the anti-Pfaffian state, is a PH-Pfaffian topological order state. We also discuss the implications on the edge physics.