Hohenberg–Kohn Theorems for Interactions, Spin and Temperature

Abstract

We prove Hohenberg-Kohn theorems for several models of quantum mechanics. First, we show that for possibly degenerate systems of several types of particles, the pair correlation functions of any ground state contain the information of the interactions and of the external potentials. Then, in the presence of the Zeeman interaction, a strong constraint on external fields is derived for systems having the same ground state densities and magnetizations. Next, we prove that the density and the entropy of a ground state contain the information of both the imposed external potential and temperature. Eventually, we conclude that at positive temperature, Hohenberg-Kohn theorems generically hold.

Unique continuation for the magnetic Schrödinger equation

“…Define Θ = V − + j e j + 6 A j j 2 + j rÁA j, from Equations (15), (16), and (17) we obtain an upper bound for the right hand side of Equation (14) given by…”

“…However, as first noted in Reference [11], the argument is at best incomplete.For more detailed accounts on the existence of generalized Hohenberg-Kohn theorems within CDFT see References [9,11,12], and for related and positive results within the Maxwell-Schrödinger theory and quantum-electrodynamical DFT see References [13][14][15]. An interesting and recent development is also given in Reference [16] where the existence of generalized Hohenberg-Kohn theorems is further explored. A different route, where a Hohenberg-Kohn result comes for free by virtue of the convex-analytic properties of a regularized energy functional was taken in References [17,18].…”

Mixed state representability of entropy-density pairs

“…We recall that the map (v, T ) → (ρ Γ , S Γ ), where Γ is the Gibbs state of (v, T ), is injective, this is the Hohenberg-Kohn theorem for systems at positive temperature, see [18] and [4,Theorem 4.1], and this exposes the (v, T ) − (ρ, S) duality. The information of the internal quantities (ρ, S) enables to recover the information of the external fields (v, T ).…”

Convergence of the regularized Kohn-Sham iteration in Banach spaces

“…Different physical effects, like coupling to internal magnetic fields [44] or finite temperatures [45][46][47], can have an effect analogous to regularization, most importantly the establishment of a Hohenberg-Kohn result.…”