RPA as a Hessian Closure: Effective Functionals and Source-Variable Duality Across DFT, LR-TDDFT, 1RDMFT, and MBPT

Density level At the density level, one works with a dual pair (Xρ ,X ∗ ρ ) of density and potential spaces. For Coulombic systems in three 10 dimensions, a standard choice is Xρ =L 1(R3)∩L 3(R3),X ∗ ρ =L 3/2(R3) +L ∞(R3),(A1) with pairing ⟨v,ρ⟩= Z drv(r)ρ(r).(A2) The variable-side functional is then the Lieb functional, un- derstood as a convex lower sem...

A convenient working convention is Xρ,β ∼L p per([0,β];X ρ ),X ∗ ρ,β ∼L p′ per([0,β];X ∗ ρ ),(A3) with p−1 +p ′−1 =1

Dynamical density level At the dynamical density level, the static density source– variable pairing is lifted to imaginary time. A convenient working convention is Xρ,β ∼L p per([0,β];X ρ ),X ∗ ρ,β ∼L p′ per([0,β];X ∗ ρ ),(A3) with p−1 +p ′−1 =1 . The subscript “per” indicates the bosonic imaginary-time periodicity appropriate for density sources and dens...

Equal-time bilocal level At the equal-time bilocal level, a convenient admissible set is the ensemble-representable set of self-adjoint one-body density matrices satisfying γ=γ †,0≤γ≤1,(A5) together with a finite-kinetic-energy condition such as Tr (1+ ˆT)γ <∞.(A6) If one wishes to restrict to a fixed particle-number sector, one further imposes Trγ=N.(A7)...

The physical time-ordered fermionic Green’s function is not most naturally viewed as an arbitrary regular bilocal kernel, because it has a fixed equal-time discontinuity

Spacetime-bilocal level At the Green’s function level, the corresponding functional- analytic structure is more delicate. The physical time-ordered fermionic Green’s function is not most naturally viewed as an arbitrary regular bilocal kernel, because it has a fixed equal-time discontinuity. With the convention G(1,2) = −⟨Tτ ˆψ(1) ˆψ†(2)⟩, one has G(x,τ +...

Pair fluctuation form of the adiabatic-connection integrand For a fixed coupling constantλ, let Γλ ρ be a minimizing state at fixed densityρ. Define δ ˆρ(r) = ˆρ(r)−ρ(r)(B1) and the equal-time density-fluctuation correlation function Cλ (r,r ′) = δ ˆρ(r)δ ˆρ(r ′) λ .(B2) Using ˆW= 1 2 ZZ drdr ′ v(r,r ′) ˆρ(r) ˆρ(r ′)−δ(r−r ′) ˆρ(r) ,(B3) one obtains Tr Γλ...

Since Z ∞ 0 dω π 2Ω Ω2 +ω 2 =1,(B7) the equal-time correlation function satisfies Cλ (r,r ′) =− Z ∞ 0 dω π χ λ (r,r ′;iω).(B8) Substituting Eq

Fluctuation-dissipation relation For a nondegenerate ground state, the imaginary-frequency density response has the Lehmann representation27,28 χ λ (r,r ′;iω) =−2 ∑ m>0 Ωλ m⟨0λ |δ ˆρ(r)|m λ ⟩⟨mλ |δ ˆρ(r ′)|0λ ⟩ (Ωλm)2 +ω 2 , (B6) whereΩ λ m =E λ m −E λ 0 . Since Z ∞ 0 dω π 2Ω Ω2 +ω 2 =1,(B7) the equal-time correlation function satisfies Cλ (r,r ′) =− Z ∞ ...

The source-dependent partition func- tional is Zβ [J] =Tr F e−β ˆK Tτ exp hZ d1d2J(2,1) ˆψ† H(2) ˆψH(1) i!

Green’s function source functional and finite-dimensional regularization Let ˆK:= ˆH−µ ˆN,(D1) and define the imaginary-time Heisenberg fields by ˆψH(1) =e τ1 ˆK ˆψ(r1,σ 1)e −τ1 ˆK, ˆψ† H(1) =e τ1 ˆK ˆψ†(r1,σ 1)e −τ1 ˆK, (D2) with 1≡( r1,τ 1,σ 1). The source-dependent partition func- tional is Zβ [J] =Tr F e−β ˆK Tτ exp hZ d1d2J(2,1) ˆψ† H(2) ˆψH(1) i! . ...

Quadratic reference theory, source derivatives, and concavity For the quadratic reference theory, ˆH0 = ∑ ab hab c† acb, ˆK0 := ˆH0 −µ ˆN,(D6) the regularized quadratic action takes the form S(N,M) 0 ( ¯ψ,ψ) = n ∑ α,β=1 ¯ψα A(N,M) 0,αβ ψβ ,(D7) and the free propagator is G(N,M) 0 := A(N,M) 0 −1.(D8) In continuum notation one writesA 0 =G −1 0 . With a bil...

Legendre transform, reference functional, and the Luttinger–Ward remainder At the finite-dimensional level, define Γ(N,M) β [G] =sup J n E(N,M) β [J]−Tr(JG) o .(D14) For the quadratic reference theory one can carry out this Leg- endre transform explicitly. Since G(N,M) = A(N,M) 0 −J (N,M) −1,(D15) 13 one has J(N,M) =A (N,M) 0 −(G (N,M) )−1.(D16) Substitut...

This is the kernel-level split used in the main text

The retained interaction part and the RPA kernel split For the direct RPA-type closure, one chooses the retained interaction kernel in the direct particle-hole channel to be the bare Coulomb kernel and writes Kirr =v+K rem,(D24) where Krem denotes the remaining irreducible contribution. This is the kernel-level split used in the main text. 4,19,20 It need...

Define the equal-time reduc- tion mapP et on bilocal kernels by (PetG)(x,x ′;τ):=−lim η↓0 G (x,τ),(x ′,τ+η) ,(D25) whenever the one-sided limit exists

Equal-time reduction and density-channel contraction Let x= ( r,σ) and 1= (x,τ) . Define the equal-time reduc- tion mapP et on bilocal kernels by (PetG)(x,x ′;τ):=−lim η↓0 G (x,τ),(x ′,τ+η) ,(D25) whenever the one-sided limit exists. In equilibrium, the right- hand side is τ-independent, and one identifies the one-body reduced density matrix as γ(x,x ′) =...

Density level For a fixed coupling constant λ∈[0,1] , define the grand- canonical operator ˆKλ [v,µ] = ˆT+λ ˆW+ ˆV[v]−µ ˆN, ˆV[v] = Z dr v(r) ˆρ(r). The grand potential is Ωλ β,µ [v] =− 1 β lnTr F e−β ˆKλ [v,µ].(E1) By the Gibbs variational principle, Ωλ β,µ [v] =inf Γ TrF Γ ˆKλ [v,µ] + 1 β TrF (ΓlnΓ) ,(E2) where the infimum is over grand-canonical densit...

Let ˆU[u] = ZZ drdr ′ u(r′,r) ˆψ†(r′) ˆψ(r), and define ˆKλ [u,µ] = ˆT+λ ˆW+ ˆU[u]−µ ˆN

Equal-time bilocal level For a fixed coupling constant λ∈[0,1] , the 1RDM formu- lation is parallel, with the local scalar source replaced by a Hermitian equal-time bilocal one-body source. Let ˆU[u] = ZZ drdr ′ u(r′,r) ˆψ†(r′) ˆψ(r), and define ˆKλ [u,µ] = ˆT+λ ˆW+ ˆU[u]−µ ˆN. The grand potential is Ωλ β,µ [u] =− 1 β lnTr F e−β ˆKλ [u,µ] .(E8) Again, the...

Relation to the main text The main text uses the zero-temperature ensemble func- tionals F[ρ] and F[γ] because they expose the static density and equal-time bilocal Hessian structures most directly. The present appendix shows that these formulations, together with their coupling-constant variants, may also be obtained from grand-canonical finite-temperatu...

2012