The Quadratic Hamiltonians

P. Croxson

Ming-wen Xiao

A theory of transformation is presented for the diagonalization of a Hamiltonian that is quadratic in creation and annihilation operators or in coordinates and momenta. It is the systemization and theorization of Dirac and Bogoliubov-Valatin transformations, and thus provides us an operational procedure to answer, in a direct manner, the questions as to whether a quadratic Hamiltonian is diagonalizable, whether the diagonalization is unique, and how the transformation can be constructed if the diagonalization exists. The underlying idea is to consider the dynamic matrix. Each quadratic Hamiltonian has a dynamic matrix of its own. The eigenvalue problem of the dynamic matrix determines the diagonalizability of the quadratic Hamiltonian completely. In brief, the theory ascribes the diagonalization of a quadratic Hamiltonian to the eigenvalue problem of its dynamic matrix, which is familiar to all of us. That makes it much easy to use. Applications to various physical systems are discussed, with especial emphasis on the quantum fields, such as Klein-Gordon field, phonon field, etc..

A. Horzela

Patrycja Łydżba, Marcos Rigol, Lev Vidmar

In a recent Letter [Phys. Rev. Lett. 125, 180604 (2020)], we introduced a closed-form analytic expression for the average bipartite von Neumann entanglement entropy of many-body eigenstates of random quadratic Hamiltonians, namely, of Hamiltonians whose single-particle eigenstates have random coefficients in the position basis. A paradigmatic Hamiltonian for which the expression is valid is the quadratic Sachdev-Ye-Kitaev (SYK2) model in its Dirac fermion formulation. Here we show that the applicability of our result is much broader. Most prominently, it is also relevant for local Hamiltonians such as the three-dimensional (3D) Anderson model at weak disorder. Moreover, it describes the average entanglement entropy in Hamiltonians without particle-number conservation, such as the SYK2 model in the Majorana fermion formulation and the 3D Anderson model with additional terms that break particle-number conservation. We extend our analysis to the average bipartite second R\'enyi entanglement entropy of eigenstates of the same quadratic Hamiltonians, which is derived analytically and tested numerically. We conjecture that our results for the entanglement entropies of many-body eigenstates apply to quadratic Hamiltonians whose single-particle eigenstates exhibit quantum chaos, which we refer to as quantum-chaotic quadratic Hamiltonians.

Yujun Zheng

It is shown that the propagator and wave function recently found by are not complete and the results are not correct for constant coefficients of a time-dependent general quantum quadratic Hamiltonian for the case of . We presented the correct evolution operator, propagator, and wave function of the general time-dependent quantum quadratic Hamiltonian system.