Nonuniqueness and derivative discontinuities in density-functional theories for current-carrying and superconducting systems

Abstract

Current-carrying and superconducting systems can be treated within density-functional theory if suitable additional density variables ͑the current density and the superconducting order parameter, respectively͒ are included in the density-functional formalism. Here we show that the corresponding conjugate potentials ͑vector and pair potentials, respectively͒ are not uniquely determined by the densities. The Hohenberg-Kohn theorem of these generalized density-functional theories is thus weaker than the original one. We give explicit examples and explore some consequences. DOI: 10.1103/PhysRevB.65.113106 PACS number͑s͒: 71.15.Mb, 31.15.Ew, 75.20.Ϫg, 74.25.Jb Today, density-functional theory 1 ͑DFT͒ is an indispensable tool for the investigation of the electronic structure of matter in atomic, molecular, or extended systems. The theory rests on the celebrated Hohenberg-Kohn ͑HK͒ theorem, 2 which guarantees that the (v-representable͒ ground-state density n(r) uniquely determines the ground-state manybody wave function 0 (r 1 , . . . ,r N ). This theorem on its own is a very powerful result, but in the original formulation 2,3 of DFT one can prove even more: the external potential v(r) ͑e.g., the nuclear charge distribution in a molecule or a solid͒, too, is a functional of the density, and is unique up to an additive constant. Since this external potential in turn determines all eigenstates of the many-body Hamiltonian, this implies that all observables ͑and not only ground state ones͒ are functionals of the ground-state density.Following original ideas of von Barth and Hedin, 4 it has recently been shown by Eschrig and Pickett 5 and by the present authors 6 that in spin-DFT ͑SDFT͒ the situation is not that simple: while the wave function is still uniquely determined by the spin densities n ↑ (r) and n ↓ (r), the external potentials v ↑ (r) and v ↓ (r) ͓or v(r) and B(r)͔ are not. This implies that SDFT functionals are not always differentiable, and has far-reaching consequences for the construction of better exchange-correlation ͑XC͒ functionals, and for applications to systems such as half-metallic ferromagnets. 5,6 SDFT is not the only instance at which the original HK theorem has been generalized. In the present work we extend the analysis of Ref. 6 to two other generalizations of DFT, namely, current-DFT 7,8 ͑CDFT͒ and DFT for superconductors. [9][10][11][12] The discovery of nonuniqueness in these generalized DFTs deepens our understanding of the respective XC functionals and flags a warning signal to alltoo-immediate generalizations of the original HK theorem to more complex situations.The basic physics of nonuniqueness is simple. When a sufficiently small change in one of the external fields does not change the corresponding density distribution, the associated susceptibility vanishes. The search for, and the interpretation of, nonuniqueness in DFT is thus guided by investigations of the circumstances under which some response function becomes zero.We first consider current-carrying systems. The appropriat...

Nonexistence of a Hohenberg-Kohn variational principle in total current-density-functional theory

“…the counterexample in [7] shows that the densities ρ(r) and j(r) cannot satisfy a variational principle [8]. We shall here give a mathematical proof of this claim.…”

“…For instance, Vignale and Capelle [7] have constructed a counterexample that shows that different Hamiltonians can share a common ground-state for systems with magnetic fields.…”

Intrinsic exchange-correlation magnetic fields in exact current density functional theory for degenerate systems

“…While the KS wavefunctions are uniquely determined by (n, j p ), the potentials (v KS , A KS ) are not 23 and so one cannot generally speak of the exact KS vector potential per se. Fig.…”

Hohenberg-Kohn theorem including electron spin

“…For completeness, we note that in the literature [21][22][23] it is thought that the basic variables for the Hamiltonian of Eq. (10) are the ground state density ρ(r), the magnetization density m(r), and the gauge variant paramagnetic current density j p (r).…”