Cumulant expansions for thermal density matrices and ground state logarithmic perturbation series

Abstract

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〉.