The behavior of the one-particle reduced density matrix ρ(1)(r) and the two-particle distribution function ρ(2)(r) of the ideal Fermi gas are investigated for various limiting cases. These cases are classified by the relative magnitude of the following parameters; the distance r, the thermal wave length λT and an “effective length” R0 = (6π2ρ/2S+1)-1/3, where ρ is the density and S is spin. At T = 0, R0-1 is reduced to the Fermi wave number k0 and the obtained long distance behaviors are well known; for instance, R0-2r-4[1+cos (2k0r)] for ρ(2)(r), whereas for T ≠0, this well-known behavior is shown to be valid only when r is smaller than λT and the asymptotic behavior is found to be λT-4r-2[1+cos (2k0r)]exp (-arλT), where a is a constant. This is a sum of terms which shows exponential decay and oscillatory exponential decay, respectively. An explicit form of the three-particle distribution function ρ(3)(r12, r23, r31) of the ideal Fermi gas is also given and its behaviors are discussed.
The long-distance behavior of the pair distribution function of a hard-sphere Fermi gas, ρ(2)(r) is investigated. It is shown that its distance dependence at zero temperature is the same as that of the ideal Fermi gas, though the amplitudes are different; namely ρ(2)(r)∼k0-2r-4[C1+C2 cos2k0r]. where k0 is the Fermi wave number and Ci's are constants. Corrections arise from the temperature dependence of the chemical potential and from the deformation of the Fermi distribution function fp. The temperature dependence of the terms due to these corrections is estimated to be of the order O(T2), while the distance behavior is found to be O(1/r3) and O(1/r2), respectively. Comments are given on higher order contributions. It is argued that the exchange and the direct contribution to the interaction parts of ρ(2)(r) have the same long-distance behavior for an electron gas as for a hard-sphere Fermi gas.
The singlet and pair distribution functions of a Fermi gas are evaluated for low temperatures. It is assumed that the particles interact with a hard-sphere potential and have an arbitrary spin, and the evaluation is made to first order in the hard-sphere diameter. The binary-kernel method developed by Siegert, Lee, and Yang is used and the cluster-expansion theory given recently by Isihara and Yee for the distribution functions is applied. The pair distribution function is determined by a dimensionless variable $\frac{r}{\ensuremath{\lambda}}$, with $\ensuremath{\lambda}$ being the thermal de Broglie wavelength. Thus, changing the distance corresponds to varying the temperature. Moreover, the pair distribution function depends on the density. Therefore, its asymptotic expression for $T\ensuremath{\rightarrow}0$ depends on the ways to approach the limit. If we keep the density finite and reduce the temperature, the gas will be degenerate when $\ensuremath{\lambda}$ is of order of ${n}^{\ensuremath{-}\frac{1}{3}}$, where
n is the number density. The pair distribution function for this case is an oscillating function of the Fermi momentum. On the other hand, if we reduce the density first and then decrease the temperature, the pair distribution function decays as ${r}^{\ensuremath{-}4}$, and the free-particle diagrams give a contribution of the same decay.
We compute exactly the one particle reduced density matrix ρ (r) of a system of impenetrable bosons in one dimension at zero temperature. We do this by relating ρ (r) to a certain double scaling limit of the transverse correlation function of the one-dimensional spin 1/2 X–Y model. We study the asymptotic behavior of ρ (r) for large r. This expansion contains oscillatory terms which arise due to the intrinsic quantum mechanical nature of the problem. We use these results to discuss the analytic structure of the momentum density function n (k).
The one-particle density matrix in the coordinate representation is calculated for the ideal Fermi gas in general D dimensions and a slow decay form of off-diagonal long-range order (ODLRO) is demonstrated therein. The pair distribution function and the structure factor are deduced from the one-particle density matrix. Their deviations from the classical values are viewed as a measure of the quantum phase, manifesting the existence of ODLRO. An upper bound on the structure factor due to Price [Phys. Rev. 94, 257 (1954)] is tested with our general solution and is also applied to the interacting electron gas.
We derive exact general relations between various observables for
N spin-1/2 fermions with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalize known relations between the large-momentum behavior of the momentum distribution, the short-distance behaviors of the pair distribution function and of the one-body density matrix, the norm of the regular part of the wave function, the derivative of the energy with respect to the scattering length or to time, and the interaction energy (in the case of finite-range interactions). The expression relating the energy to a functional of the momentum distribution is also generalized. Moreover, we find expressions (in terms of the regular part of the wave function) for the derivative of the energy with respect to the effective range
re in three dimensions (3D), and to the effective range squared in two dimensions (2D). They express the fact that the leading corrections to the eigenenergies due to a finite-interaction range are linear in the effective range in 3D (and in its square in 2D) with model-independent coefficients. There are subtleties in the validity condition of this conclusion, for the 2D continuous space (where it is saved by factors that are only logarithmically large in the zero-range limit) and for the 3D lattice models (where it applies only for some magic dispersion relations on the lattice that sufficiently weakly break Galilean invariance and that do not have cusps at the border of the first Brillouin zone; an example of such relations is constructed). Furthermore, the subleading short-distance behavior of the pair distribution function and the subleading
1/k6 tail of the momentum distribution are related to $\ensuremath{\partial}E/\ensuremath{\partial}{r}_{e}$ [or to $\ensuremath{\partial}E/\ensuremath{\partial}({r}_{e}^{2})$ in 2D]. The second-order derivative of energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length is found to be expressible for any eigenstate in terms of the eigen-wave-function's regular parts; this implies that, at thermal equilibrium, this second-order derivative, taken at fixed entropy, is negative. Applications of the general relations are presented: We compute corrections to exactly solvable two-body and three-body problems and find agreement with available numerics; for the unitary gas in an isotropic harmonic trap, we determine how the finite-
1/a and finite-range energy corrections vary within each energy ladder (associated with the SO(2,1) dynamical symmetry) and we deduce the frequency shift and the collapse time of the breathing mode; for the bulk unitary gas, we compare to fixed-node Monte Carlo data, and we estimate the deviation from the Bertsch parameter due to the finite interaction-range in typical experiments.
To obtain an expression for the Fermi energy εF, dened in Section 12.3, consider a system of N fermions of spin 12 and mass m in a container of volume V at T = 0 K. In zero-applied magnetic eld, the mS = ± 12 spin states are degenerate, and it is convenient to introduce a degeneracy factor in expressions that involve the density of single particle states. e fermions ll the lowest energy states up to the energy εF = µ. By equating the number of particles to the number of states in the range 0 to εF, an expression for εF is obtained. In general, for fermions, N n er r r= ∑ = ∑ +−1 1/( ).( )b e m Conversion of the sum to an integral gives N f= ∞∫2 0 r e e e( ) ( ) ,d (13.1) where ρ(ε) is the density of states and the factor 2 is the spin degeneracy factor. f ee b e m( ) = +−1 1/( )( ) is the Fermi function, which at T = 0 K has the form f(ε) = 1 for e eʺ F and f(ε) = 0 for e e> F. is may be seen in Figure 12.1. At T = 0 K, Equation 13.1 may therefore be written as N = ∫2 0 r e ee ( ) .dF (13.2) From Equation 4.14, the density of single particle states for particles in a box is r e p e( ) ( / )( / ) / /= V m4 22 2 3 2 1 2 , and substitution in Equation 13.2 gives N V m V m = ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ e pF = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 m N V .