Cumulant Expansions and Energy Shifts Obtained by Using a Quantum Physical Perturbation

R. Fauser, H.H. Wolter

H. A. Gersch

Two examples are given of the use of cumulant expansions for systems of particles with hard-core two-body interactions. One of these, in classical statistical mechanics, is shown to lead to virial coefficients. An analogous procedure in quantum mechanics produces energy level shifts due to hard-core interactions that are expressed in terms of the t matrix.

Yu. M. Shirokov

Michele Parrinello, Tadashi Arai

The cumulant expansion for spins developed previously is rearranged and extended up to an infinite order in perturbation. A pair of spin-deviation lines is considered as a propagator and, as an additional propagator appears and interacts with the original one, that part of the diagram is defined as a self-energy correction. All cumulant corrections are decomposed and added to the corresponding self-energy corrections, so that self-energy corrections and their cumulant corrections are automatically calculated together exactly. The contribution of a self-energy correction to a diagram depends on the geometry of the incoming and outgoing spin-deviation lines of the self-energy diagram, yielding a new type of perturbation expansion. Numerical results suggest that the nature of the spin-spin correlation in the spin-1/2 case is distinct from that in other cases and that spins behave like fermions rather than bosons in this limit.

Antoine Royer

We express the logarithm of the thermal density matrix ρH,β (X,X′)≡〈X‖e−βH‖X′〉, where H=H0+V, as a cumulant expansion in powers of the perturbation V, by associating with {H0,‖X〉,‖X′〉} a stochastic process. If the ground state of H0 is isolated, this stochastic process is of finite memory; it then follows from the properties of cumulants that the above expansion of ln ρ is nonsecular as β→∞ (all its terms ∼β), and is thus usable down to zero temperature, unlike the direct expansion of ρ in powers of V, whose nth term ∼βn. To determine explicitly the low-temperature behavior, we apply an analysis familiar in the theory of relaxation, and obtain the form ln ρH,β(X,X′)=h(X,X′,β) +a(X)+a(X′)°−bβ, where a,b,h are cumulant expansions in powers of V; a(X), b are independent of β; and h(X,X′,β)→0 as β→∞. Comparing with ρ→〈X‖ψ0〉e−βE0〈ψ0‖X′〉 as β→∞, where ψ0 and E0 are the ground state and energy of H, we deduce E0=b, ln〈X‖ψ0〉=a(X), i.e., the Rayleigh–Schrödinger perturbation series for E0 and ln〈X‖ψ0〉 (with 〈ψ0‖ψ0〉=1) emerge as cumulant expansions. In the case of a many-body system, the properties of cumulants immediately imply linked cluster theorems for ln ρ, as well as for E0 and ln〈X‖ψ0〉.