In Slave-Boson formalism we define
ci+=bifi+
then this yields,
=~
with a constrain
Σ(fifj++bibj)=1
where
=δ.
- Does δ physically correspond to the number of density of “boson” or “vacancy” (namely “holes”)?
- In strong U limit does the above constrain physically mean “the total number of holes and electrons in a site allways equals to one”?
On appropriately modifying your notation, the answers to your two questions are in the positive.
Some remarks are in order. First, the slave-boson formalism to which you refer is due to F Barnes (J. Phys. F 6, 1375 (1976); 7, 2637 (1977)) and P Coleman (Phys. Rev. B 29, 3035 (1984)), discussed in detail in Sections 7.5, 8.3 and 10.5 of the book The Kondo Problem to Heavy Fermions, by AC Hewson (Cambridge University Press, 1993). Insofar as the Anderson model in the infinite-U limit is concerned, you can find the answers to your both questions in Eqs (10.30) and (10.34) of this book, taking into account that = = √z. Note that the results in the above-mentioned Sections 7.5 and 10.5 are mean-field ones.
By considering the functional integral representations of the Hubbard and the Anderson model for arbitrary on-site interaction energy U, G Kotliar and AE Ruckenstein (Phys. Rev. Lett. 57, 1362 (1986)) have investigated these models within a saddle-point approximation, taking advantage of a freedom in the choice of the relevant Hamiltonians whereby their results are exact in the limit of U = 0. This approach is reviewed in Section 12.7 of the book Electron Correlations in Molecules and Solids, 3rd edition (Springer, Berlin, 1995), by P Fulde.
As can be verified, the two slave-boson approaches differ: in the one as employed in particular by Coleman, the slave bosons are introduced in order to dispose of the Hubbard X operators, which do not satisfy canonical commutation relations [consult Sec. 5.1.3 of the book Lecture Notes on Electron Correlation and Magnetism (World Scientific, Singapore, 1999)] and thus are not amenable to conventional diagrammatic perturbation calculations; in that by Kotliar and Ruckenstein, the slave bosons serve a different purpose (they turn the relevant Hamiltonians in the extended Hilbert space quadratic in the fermion operators); in both treatments the physical Hilbert space is enlarged so that physical results are to be obtained through a projection process of the calculated results onto the physical Hilbert space.
I have expanded on the formalism by Kotliar and Ruckenstein since it deals with in principle arbitrary values of U and provides an explicit expression concerning the expectation value as encountered in Eq. (12.7.9) of the above-mentioned book by Fulde (here, the quantity m can be expressed as 1-δ (1+δ), where δ > 0 is the fraction of hole (electron) doping away from the half-filling). This expectation value is encountered in the effective Hamiltonian in Eq. (12.7.7) of the same book. The square root of this expectation value can be seen to be the equivalent of the expectation values = = √z, to which I have referred above. On the basis of the expression in the above-mentioned Eq. (12.7.9), one can calculate and see how it varies as a function of for instance the doping fraction δ. □
Dear Behnam Farid, Thank you very much for your detailed explanations. Best regards.
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