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arXiv:1711.04654

Velocity operator approach to a Fermion system

Seiya Nishiyama, João da Providência

Published Nov 10, 2017• 0 citations
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Abstract

In this paper, we formulate a velocity operator approach to a three-dimensional (3D) Fermion system. Following Sunakawa, introducing density and velocity operators, we treat 3D quantum fluid dynamics in the system. We get a collective Hamiltonian in terms of collective variables. The lowest order collective Hamiltonian is diagonalized. This diagonalization leads us to a Bogoliubov transformation for Boson-like operators.

DOI
10.48550/arxiv.1711.04654

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In this paper, we formulate a velocity operator approach to a three-dimensional (3D) Fermion system. Following Sunakawa, introducing density and velocity operators, we treat 3D quantum fluid dynamics in the system. We get a collective Hamiltonian in terms of collective variables. The lowest order collective Hamiltonian is diagonalized. This diagonalization leads us to a Bogoliubov transformation for Boson-like operators.
In the preceeding paper, introducing [Formula: see text]-dependent density operators and defining exact momenta (collective variables), we could get an exact canonically momenta approach to a one-dimensional (1D) neutron–proton (NP) system. In this paper, we attempt at a velocity operator approach to a 3D NP system. Following Sunakawa, after introducing momentum density operators, we define velocity operators, denoting classical fluid velocities. We derive a collective Hamiltonian in terms of the collective variables.
A new well-defined velocity operator is proposed for the collective description of a many-boson system. It is shown that the new velocity operator is irrotational. The Hamiltonian of the system is successfully described in terms of the density-fluctuation operator and the velocity operator. The interaction term in the order of N-1 has a different form from that in previous papers. The range of validity of the present formalism is discussed in the final section in answer to the questions raised so far on the collective formalism of the many-boson system.
Green's functions of phonons for many-boson systems at high density are obtained from the phonon Hamiltonian given in a previous paper on the basis of the coherent-state representation. A dispersion equation derived from these Green's functions gives rise to the excitation spectrum leading to the Feynman energy in the low-momentum limit. It is shown that the dynamic structure function obtained here satisfies the sum rules in the low-momentum limit. As an illustrative example, the one-dimensional Lieb-Liniger model is considered. The relationships to the previous theories, in particular, to the velocity field approach of Yamasaki, Kebukawa and Sunakawa and to our density and phase operator approach are elucidated by the use of a unitary transformation. Discussion will be made on the D-wave portion of the roton-roton interaction in relation to the resonance state of roton pair studied in one of our previous papers. Finally, it is pointed out that the zeroth Fourier transform of the phase operator is related to the superfluid velocity in the two-fluid theory of liquid helium II.
We propose a new method that transcribes the fermion system to the boson-fermion system by introducing a unitary operator defined in the whole boson-fermion space. We obtain the boson-fermion expansion of the fermion pair operators such that only the collective pair modes are transformed into boson operators while the non-collective modes are unchanged in the lowest order approximation. The odd- A system is treated in the same formalism as the even-even nuclear system. We briefly discuss the relation between our method and the conventional boson expansion theory.
The three-dimensional (3D) Ising model is mapped into a 3D spinless fermionic model by the Jordan-Wigner transformation. The exact solution of the 3D model for spinless fermions is derived analytically by performing a diagonalization process consisting of the Clifford algebraic approach, the Fourier transformation and the Bogoliubov transformation. The Clifford algebraic approach is the same as that developed for the 3D Ising model, using a time average within the Jordan-von Neumann-Wigner framework, a linearization procedure and a local gauge transformation. The formulas for eigenvalues, partition function, subsequent thermodynamic properties and critical behaviors are presented. The dimensionality and the topological phases are investigated. The present results for many spinless fermions in a 3D lattice are applicable for studying the mechanisms of magnetism, superfluid, superconductors and topological materials.