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Physical Review C

Fermion and boson condensates in a QCD-inspired model Hamiltonian

Osvaldo Civitarese, Peter Oto Hess, Jorge G. Hirsch, M. Reboiro

Published May 3, 2000• 4 citations
Publisher

Abstract

The interaction between bi-fermionic and bosonic degrees of freedom is described by using an effective Hamiltonian inspired in QCD. The Dyson boson mapping technique is utilized in conjunction with a coherent state basis to construct energy landscapes in a multiparametric space. The appearance of fermionic and bosonic condensates is studied by minimizing the potential energy surface. The relationship between the domains of fermion and boson condensates and the order parameters of the model is discussed.

DOI
10.1103/physrevc.61.064303

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The interaction between bi-fermionic and bosonic degrees of freedom is described by using an effective Hamiltonian inspired in QCD. The Dyson boson mapping technique is utilized in conjunction with a coherent state basis to construct energy landscapes in a multiparametric space. The appearance of fermionic and bosonic condensates is studied by minimizing the potential energy surface. The relationship between the domains of fermion and boson condensates and the order parameters of the model is discussed.
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This Brief Report deals with the solution of the shell-model Hamiltonian in the boson representation. In the present approach, the fermion Hamiltonian is exactly mapped to the boson space using the Dyson boson mapping. The problem of mapping the basis states is resolved by employing the intrinsic boson basis. The method has been applied to a model Hamiltonian consisting of a monopole pairing and a deformation force. It is observed that only s and d bosons are dominant in the intrinsic wave function for the model case studied. This is synonymous to the interacting boson model approach.
An algorithm for obtaining a recursive algebraic expression for the various expansion coefficients appearing in the physical boson basis states is presented, using the Dyson boson mapping. It is observed that only the first few terms of the physical boson basis states, expressed in the present form, contribute to the matrix element of the Hamiltonian. Explicit numerical calculations have been performed for Ni and Sn isotopes in the obtained physical boson basis states. The pairing Hamiltonian has been used as the model interaction. The results are more than 90% in agreement with exact shell model calculations.
Abstract Elementary quantum mechanics proposes that a closed physical system consistently evolves in a reversible manner. However, control and readout necessitate the coupling of the quantum system to the external environment, subjecting it to relaxation and decoherence. Consequently, system-environment interactions are indispensable for simulating physically significant theories. A broad spectrum of physical systems in condensed-matter and high-energy physics, vibrational spectroscopy, and circuit and cavity QED necessitates the incorporation of bosonic degrees of freedom, such as phonons, photons, and gluons, into optimized fermion algorithms for near-future quantum simulations. In particular, when a quantum system is surrounded by an external environment, its basic physics can usually be simplified to a spin or fermionic system interacting with bosonic modes. Nevertheless, troublesome factors such as the magnitude of the bosonic degrees of freedom typically complicate the direct quantum simulation of these interacting models, necessitating the consideration of a comprehensive plan. This strategy should specifically include a suitable fermion/boson-to-qubit mapping scheme to encode sufficiently large yet manageable bosonic modes, and a method for truncating and/or downfolding the Hamiltonian to the defined subspace for performing an approximate but highly accurate simulation, guided by rigorous error analysis. In this pedagogical tutorial review, we aim to provide such an exhaustive strategy, focusing on encoding and simulating certain bosonic-related model Hamiltonians, inclusive of their static properties and time evolutions. Specifically, we emphasize two aspects: (1) the discussion of recently developed quantum algorithms for these interacting models and the construction of effective Hamiltonians, and (2) a detailed analysis regarding a tightened error bound for truncating the bosonic modes for a class of fermion-boson interacting Hamiltonians.
The use of physical boson basis states is stressed for the calculations in the boson space. The explicit form of physical boson basis states in terms of bosons is derived for the nonunitary boson mapping of Dyson. The ambiguity in the normalization introduced due to the use of bi-orthonormal basis states is satisfactorily resolved, resulting in a Hermitian matrix. This Hermitian matrix is found to coincide with the Hamiltonian matrix in the fermion space. The model cases where the use of a boson basis is justified are shown to be consistent with our view of using physical boson basis states.
The interplay between fermionic and bosonic degrees of freedom, in the context of many-body systems and with reference to applications in nuclear and elementary particle physics, is revisited. Starting from the formulation due to Geyer, Hahne, and co-workers, bifermion operators which obey bosonic transformation rules are introduced. Fermion-boson coupling terms are included in the original Hamiltonian. Exact solutions to this Hamiltonian are compared to approximate ones. The occurrence of fermion and boson condensates is discussed.