One of the major differences between fermions and bosons is that fermionic states have a maximum occupation number of one, whereas the occupation number for bosonic states is in principle unlimited. For bosons that are made up of fermions, one could ask the question to what extent the Pauli principle for the constituent fermions would limit the boson occupation number. Intuitively one can expect the maximum occupation number to be proportional to the available volume for the bosons divided by the volume occupied by the fermions inside one boson, though a rigorous derivation of this result has not been given before. In this letter we show how the maximum occupation number can be calculated from the ground-state energy of a fermionic generalized pairing problem. A very accurate analytical estimate of this eigenvalue is derived. From that a general expression is obtained for the maximum occupation number of a composite boson state, based solely on the intrinsic fermionic structure of the bosons. The consequences for Bose–Einstein condensates of excitons in semiconductors and ultra cold trapped atoms are discussed.
One of the major differences between fermions and bosons is that fermionic states have a maximum occupation number of one, whereas the occupation number for bosonic states is in principle unlimited. For bosons that are made up of fermions, one could ask the question to what extent the Pauli principle for the constituent fermions would limit the boson occupation number. Intuitively one can expect the maximum occupation number to be proportional to the available volume for the bosons divided by the volume occupied by the fermions inside one boson, though a rigorous derivation of this result has not been given before. In this letter we show how the maximum occupation number can be calculated from the ground-state energy of a fermionic generalized pairing problem. A very accurate analytical estimate of this eigenvalue is derived. From that a general expression is obtained for the maximum occupation number of a composite boson state, based solely on the intrinsic fermionic structure of the bosons. The consequences for Bose–Einstein condensates of excitons in semiconductors and ultra cold trapped atoms are discussed.
One of the major differences between fermions and bosons is that fermionic states have a maximum occupation number of one, whereas the occupation number for bosonic states is in principle unlimited. For bosons that are made up of fermions, one could ask the question to what extent the Pauli principle for the constituent fermions would limit the boson occupation number. Intuitively one can expect the maximum occupation number to be proportional to the available volume for the bosons divided by the volume occupied by the fermions inside one boson, though a rigorous derivation of this result has not been given before. In this letter we show how the maximum occupation number can be calculated from the ground-state energy of a fermionic generalized pairing problem. A very accurate analytical estimate of this eigenvalue is derived. From that a general expression is obtained for the maximum occupation number of a composite boson state, based solely on the intrinsic fermionic structure of the bosons. The consequences for Bose–Einstein condensates of excitons in semiconductors and ultra cold trapped atoms are discussed.
Chapter 16 introduces the last part that addresses the bosonic condensation of composite bosons. To bridge the gap between elementary bosons and composite bosons, elementary fermion systems like semiconductor electrons and holes are also studied. Whereas the previous chapters provided a fairly precise knowledge of bosonic particles relevant for condensed matter physics, namely, Wannier excitons, Frenkel excitons, Cooper pairs, biexcitons, and polaritons, this chapter addresses an important question of current interest: what is the ground state of N composite bosons from the perspective of Bose-Einstein condensation? Or rather, how does the genuine Bose-Einstein condensation for elementary bosons persist in composite boson systems? This chapter lists important results with respect to condensate momentum fragmentation or spin fragmentation, correlation energy, and the dark nature of exciton condensate, to emphasize the connection between elementary bosons, elementary fermions, elementary fermion pairs, and composite bosons, from the composite boson perspective of their condensation.
Attractive p-wave one-dimensional fermions are studied in the fermionic Tonks-Girardeau regime in which the diagonal properties are shared with those of an ideal Bose gas. We study the off-diagonal properties and present analytical expressions for the eigenvalues of the one-body density matrix. One striking aspect is the universality of the occupation numbers which are independent of the specific shape of the external potential. We show that the occupation of natural orbitals occurs in pairs, indicating the formation of composite bosons, each consisting of two attractive fermions. The formation of composite bosons sheds light on the pairing mechanism of the system orbitals, yielding a total density equal to that of a Bose-Einstein condensate.
For a system of 2n fermions it is shown that the occupation number nα of any fermion-pair state φα(x1x2) is nα=n∫φα*(x1x2)ρ2(x1x2,x1′x2′)φα(x1′x2′) dx1 dx2 dx1′ dx2′, where ρ2, assumed normalized to unity, is the two-particle density matrix. The known upper bound on the largest eigenvalue of ρ2 implies that nα ≤ ½ in the thermodynamic limit n → ∞, equality being approached, with suitable φα, only for certain limiting BCS states ψBCS. Bose condensation of fermion pairs, in the sense of macroscopic occupation of any nα, is impossible. Fermi condensation into φα is defined to be present if nα > 0 in the thermodynamic limit. It occurs, for example, for suitable BCS states, but for a normal Fermi system all nα are of order n−1 or smaller. It is argued that the physical criterion for ψBCS exhibiting Fermi condensation is that the pair state from which it is constructed must have a bound-state component of range ≲kF−1, where kF is the Fermi momentum (3π2ρ)⅓. The maximally-occupied pair state is the eigenfunction of ρ2 belonging to the largest eigenvalue, associated with off-diagonal long-range order of the type defined by Yang. A formula for n0 for the original BCS state is derived. Some remarks are made concerning the interpretation of the Fermi condensation as a superconducting transition. The analysis is generalized to occupation of l-fermion states. It is conjectured that, when Fermi condensation first sets in at a given even l, this is associated with the formation of bound states of l fermions, and a formula for the maximal occupation of such states is exhibited. The implications, for the theory of liquid 4He, of the fact that a helium atom contains electrons are examined. It is shown that Bose condensation into a single-4He-atom state is impossible, but a Fermi condensation similar in some respects to that in a superconductor can and probably does occur. It is argued that the mechanism preventing Bose condensation in superconductors and liquid 4He lies in the effect of collisions, acting via the exclusion principle, in causing virtual internal excitations.
Abstract Composite particles made of two fermions can be treated as ideal elementary bosons as long as the constituent fermions are sufficiently entangled. In that case, the Pauli principle acting on the parts does not jeopardise the bosonic behaviour of the whole. An indic-ator for bosonic quality is the composite boson normal-isation ratio χN+1/χN of a state of N composites. This quantity is prohibitively complicated to compute exactly for realistic two-fermion wavefunctions and large com-posite numbers N. Here, we provide an efficient char-acterisation in terms of the purity P and the largest eigenvalue λ1 of the reduced single-fermion state. We find the states that extremise χN for given P and λ1, and we provide easily evaluable, saturable upper and lower bounds for the normalisation ratio. Our results strengthen the relationship between the bosonic qual-ity of a composite particle and the entanglement of its constituents. 1
The bosonic atoms used in present day experiments on Bose-Einstein condensation are made up of fermionic electrons and nucleons. In this Letter we demonstrate how the Pauli exclusion principle for these constituents puts an upper limit on the Bose-Einstein-condensed fraction. Detailed numerical results are presented for hydrogen atoms in a cubic volume and for excitons in semiconductors and semiconductor bilayer systems. The resulting condensate depletion scales differently from what one expects for bosons with a repulsive hard-core interaction. At high densities, Pauli exclusion results in significantly more condensate depletion. These results also shed a new light on the low condensed fraction in liquid helium II.