Nonlinear Bogolyubov-Valatin transformations: 2 modes

TL;DR

This paper thoroughly studies nonlinear Bogolyubov-Valatin transformations for two fermionic modes, revealing their mathematical structure, algebraic properties, and applications to diagonalizing two-fermion and spin Hamiltonians.

Contribution

It extends previous work to two modes, characterizes the transformation group, and introduces new mathematical tools involving Clifford algebras and biparavectors.

Findings

The Bogolyubov-Valatin group for two modes is isomorphic to SO(6;R)/Z_2.

A novel mathematical framework using Clifford algebra and biparavectors is developed.

A method for diagonalizing arbitrary two-fermion Hamiltonians is presented.

Abstract

Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for n=2 fermionic modes which can be implemented by means of unitary SU(2^n = 4) transformations is isomorphic to SO(6;R)/Z_2. The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra C(0,4)] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington's E-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions H. From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonian. Relying on Jordan-Wigner transformations for two-spin-1/2 and single-spin-3/2 systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-1/2 and single-spin-3/2 Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of SU(4) and SO(6;R) matrices (of respective sizes 4x4 and 6x6) and their mutual relation.