[Paper Review] Expressing Products of Fermi Fields in terms of Fermi Sea Displacements

Girish S. Setlur|ArXiv.org|Jan 29, 1997

Cold Atom Physics and Bose-Einstein Condensates1 references3 citations

TL;DR

This paper generalizes Fermi surface bosonization by introducing Bose fields that describe displacements of the entire Fermi sea, not just the surface. It expresses number-conserving products of Fermi fields as combinations of these Bose fields, reproducing most commutation relations and dynamical correlation functions exactly in the free theory, enabling the study of high-energy single-particle excitations and short-wavelength Fermi surface fluctuations.

ABSTRACT

An attempt is made to generalise the ideas introduced by Haldane and others regarding Bosonizing the Fermi surface. The present attempt involves introduction of Bose fields that correspond to displacements of the Fermi sea rather than just the Fermi surface. This enables the study of short wavelength fluctuations of the Fermi surface and hence the dispersion of single particle excitations with high energy. The number conserving product of two Fermi fields is represented as a simple combination of these Bose fields. It is shown that most(!) commutation rules involving these number conserving products are reproduced exactly, as are the dynamical correlation functions of the free theory. Also the work of Sharp, Menikoff and Goldin has shown that the field operator may be viewed as a unitary representation of the current algebra. An explicit realisation of this unitary representation is given in terms of canonical conjugate of the density operator.

Motivation & Objective

• To extend Haldane's Fermi surface bosonization by incorporating Fermi sea displacements rather than just surface modes.
• To enable the description of short-wavelength fluctuations in the Fermi surface and high-energy single-particle excitations.
• To provide a field-theoretic realization of the current algebra via unitary representations of density conjugate operators.
• To reproduce key commutation relations and dynamical correlation functions of the free Fermi gas exactly.
• To generalize the framework of Sharp, Menikoff, and Goldin on field operators as unitary representations of current algebra.

Proposed method

• Introduces Bose fields associated with collective displacements of the Fermi sea, defined via the canonical conjugate of the density operator.
• Constructs number-conserving products of Fermi fields as exponential functionals of these Bose fields.
• Uses canonical commutation relations between the Bose fields and the density conjugate to derive algebraic consistency.
• Applies the unitary representation of the current algebra to express the Fermi field products in terms of displacement operators.
• Performs explicit algebraic manipulations to verify that most commutation rules and correlation functions match the free theory.
• Relies on a formalism where the field operator is realized as a unitary transformation generated by the conjugate of the density operator.

Experimental results

Research questions

• RQ1 Can Fermi field products be expressed in terms of Bose fields that describe collective displacements of the entire Fermi sea?
• RQ2 Do these Bose fields reproduce the essential algebraic structures—especially commutation relations—of the original Fermi field theory?
• RQ3 Can this formalism describe high-energy single-particle excitations and short-wavelength Fermi surface fluctuations?
• RQ4 To what extent can the dynamical correlation functions of the free Fermi gas be recovered using this displacement-based bosonization?
• RQ5 How does this approach generalize the current algebra realization of the Fermi field operator as a unitary transformation?

Key findings

• The number-conserving product of two Fermi fields is expressed as a simple exponential combination of Bose fields associated with Fermi sea displacements.
• Most commutation rules involving the number-conserving Fermi field products are reproduced exactly within the new formalism.
• The dynamical correlation functions of the free Fermi gas are exactly recovered using this displacement-based bosonization approach.
• The framework provides a realization of the field operator as a unitary representation of the current algebra, explicitly constructed from the canonical conjugate of the density operator.
• The method enables the study of short-wavelength Fermi surface fluctuations and high-energy single-particle excitations, extending beyond standard Fermi surface bosonization.
• The work generalizes earlier results by Haldane and others by including bulk Fermi sea degrees of freedom, not just surface modes.