I recently proposed a method of bosonization valid for systems of an even number of fermions whose partition function is dominated at low energy by bosonic composites. This method respects all symmetries, in particular fermion number conservation. I extend it to treat odd systems and excitations involving unpaired fermions.
We derive a boson Hamiltonian from a Nuclear Hamiltonian whose potential is expanded in pairing multipoles and determine the fermion-boson mapping of operators. We use a new method of bosonization based on the evaluation of the partition function restricted to the bosonic composites of interest. By rewriting the partition function so obtained in functional form we get the euclidean action of the composite bosons from which we can derive the Hamiltonian. Such a procedure respects all the fermion symmetries.
We test a new approach to bosonization in relativistic field theories and many-body systems, based on the use of fermionic composites as integration variables in the Berezin integral defining the partition function of the system. The method appears promising since at zeroth order it correctly describes the propagators of the composites, which can be evaluated in a number of significant cases. Still to be established is a general procedure for deriving the free action of the composites starting from the one of the constituents. To shed light on this problem and to explore further features of the method we study a simplified version of the BCS model. In this simple case the action of the composites can indeed be obtained: whether this result can be generalized it remains hewever to be seen. Yet an interesting property of the wave operators appearing in the free actions of bilinear composites already emerges from the simple problem we have treated: here the wave operators do not describe any time evolution, even though they generate the right propagators. This outcome relates to the basic properties of the integrals over the even elements of a Grassmann algebra where the composites live, which entails that the propagators are no longer the inverse of the wave operators.
We investigate odd-in-time—or —pairing of fermions in equilibrium systems within the particle-number-conserving framework of Penrose, Onsager, and Yang, where superfluid order is defined by macroscopic eigenvalues of reduced density matrices. We show that odd-frequency pair correlations are synonymous with even fermion-exchange symmetry in a time-dependent correlation function that generalises the two-body reduced density matrix. Macroscopic even-under-fermion-exchange pairing is found to emerge from conventional Penrose-Onsager-Yang condensation in two-body or higher-order reduced density matrices through the symmetry-mixing properties of the Hamiltonian. We identify and characterize a matrix responsible for producing macroscopic even fermion-exchange correlations that coexist with a conventional Cooper-pair condensate, while a matrix is shown to be responsible for creating macroscopic even fermion-exchange correlations from hidden orders such as a multiparticle condensate. The transformer scenario is illustrated using the spin-balanced s -wave superfluid with Zeeman splitting as an example. The generator scenario is demonstrated by the composite-boson condensate arising for itinerant electrons coupled to magnetic excitations. Structural analysis of the transformer and generator matrices is shown to provide general conditions for odd-frequency pairing order to arise in a given system. Our formalism facilitates a fully general derivation of the Meissner effect for odd-frequency superconductors that holds also beyond the regime of validity for mean-field theory. Published by the American Physical Society 2024