Analytical diagonalization of the open-boundary bosonic Kitaev chain: An asymmetric plane-wave ansatz approach

Source abstract

Abstract The bosonic Kitaev chain under open boundary conditions has attracted recent attention due to its realization in driven-dissipative systems and its intriguing non-Hermitian boundary physics. The model is known to be solvable via local squeezing transformations in the position-momentum representation. In this paper, we present an alternative, purely algebraic solution that relies entirely on the standard bosonic Bogoliubov transformation. For an $N$-site chain, we propose an asymmetric plane-wave ansatz with unequal left- and right-moving momenta to analytically solve the associated $2N\times 2N$ non-Hermitian ``associated matrix". The left eigenvalue problem yields $N$ distinct eigenvalues, each of which is twofold degenerate. By carefully resolving these degeneracies using the bosonic commutation relations, we construct the $N$ physical Bogoliubov quasiparticle operators. The construction reveals non-uniquenesses that in special cases exactly mirror the freedom in the local squeezing transformations of the original approach. The diagonal form of the Hamiltonian is obtained explicitly and is shown to be equivalent to the original Hamiltonian. The proposed asymmetric plane-wave ansatz and degeneracy-resolution technique are not limited to the present model and can be generalized to other bosonic pairing systems, including those with inhomogeneous pairing or hopping terms.