We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with
N particles. The pair distribution function
g(2)(r) of fermions clearly exhibits antibunching with
g(2)(0)=0 due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with
g(2)(0)≈2, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature
T0, an off-diagonal long-range order develops in the one-particle density matrix to reach
g(1)(r)=1 at
T=0, and the pair correlation starts to decrease towards
g(2)(r)≈1 at
T=0. The results for
N→∞ are seen to converge to those of the grand canonical ensemble obtained by assuming the average
⟨ψ^(r)⟩ of the field operator
ψ^(r) below
T0. This fact justifies the introduction of the "anomalous" average
⟨ψ^(r)⟩=0 below
T0 in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.
Abstract In Ref. Gaudin (Nuclear Physics 15 , 89 1960), Gaudin established a generalized version of Wick’s theorem at finite temperature for systems described by a diagonal quadratic Hamiltonian. In this work, we extend Gaudin’s proof to statistical density operators generated by a general quadratic Hamiltonian, prior to diagonalization, as commonly encountered in interacting Bose systems. We show explicitly that the resulting thermal contractions depend on the Bogoliubov transformation parameters and on the quasiparticle spectrum, providing a transparent framework for evaluating thermal averages of arbitrary operator products. As an illustration of the usefulness of this generalized formulation, we evaluate particle-number fluctuations in a homogeneous, weakly interacting Bose gas below the Bose–Einstein condensation temperature. We show that particle-number fluctuations are generically non-Poissonian as a consequence of interaction-induced correlations and quantum depletion, with the Poissonian limit recovered only for an ideal Bose gas at zero temperature. Our results clarify the role of quadratic Hamiltonians and Bogoliubov quasiparticles in shaping fluctuation statistics and highlight the generalized Wick’s theorem as a controlled tool for computing correlation functions in interacting quantum gases.
We consider Wick's Theorem for finite temperature and finite volume systems. Working at an operator level with a path ordered approach, we show that contrary to claims in the literature, expectation values of normal ordered products can be chosen to be zero and that results obtained are independent of volume. Thus the path integral and operator approaches to finite temperature and finite volume quantum field theories are indeed seen to be identical. The conditions under which normal ordered products have simple symmetry properties are also considered.
While the thermodynamics of a system of non-interacting fermions can be straightforwardly determined in the grand canonical ensemble, results for a specific number of N particles are more difficult to obtain. In this talk we will present a general recursion scheme for the canonical partition function of free fermions with a quadratic energy level spacing as might be present for ultracold fermionic atoms confined inside a box trap. Exact results for the entropy, specific heat and one and two particle occupation probabilities are numerically obtained to arbitrary precision and compared with their corresponding grand canonical values. The numerical stability of the recursion relation allows us to quantify deviations from Wicks theorem in the canonical ensemble for a variety of temperatures and densities.
We define a mean field approximation of the modified grand partition function of the pairing model, which has been defined on the modified grand canonical ensemble with even or odd particle number. Using numerical evaluations, it is shown that this approximation reproduces the even-odd effect exhibited in the case of the modified grand partition function at low temperature very well in a qualitative sense. The expectation values of one-body operators can be expressedsimply in terms of those obtainedwith the conventional mean field approximation when a Fermi-type distribution is replaced by a renormalized distribution. Although Wick’s theorm does not hold in this case, the dominant parts of the expectation values of two-body operators are also given using those obtained by the conventional mean fieldapproximation with the replacement mentionedabove.
By using the extension of the statistical Wick's theorem (Gaudin's theorem) to deal with generalized statistical density operators (those which can be expressed as the product of and operator carrying out a canonical transformations times a density operator) and using the appropriate limits we are able to rederive in a very simple way the standard generalized Wick's theorem for overlaps of mean-field wave functions. Due to the simplicity of the derivation it is now straightforward to consider more involved cases and some of them are discussed. The present derivation also allows us to obtain general and compact formulas for other particular cases of the generalized Wick theorem involving overlaps of multiquasiparticle excitations of product wave functions. The new expressions allow us to reduce the combinatorial complexity of the standard calculation of the above overlaps.