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arXiv:2610.27294

Counterexamples to the total-current Hohenberg-Kohn conjecture

Andre Laestadius, Mihály A. Csirik

math-phphysics.chem-phquant-phSubmitted Oct 19, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

The total-current Hohenberg-Kohn conjecture asserts that the ground-state particle density and total current determine the scalar potential up to an additive constant and the vector potential up to a gauge transformation. We give three counterexamples for two spinless fermions in the physically relevant setting of three spatial dimensions. In each, a Hamiltonian $H^{v,\mathbf{A}}$ has a nondegenerate ground state with identically zero total current but nonzero magnetic field. Complex conjugation then yields a second Hamiltonian $H^{v,-\mathbf{A}}$, with the same scalar potential and interaction, whose ground state has the same density and the same zero current, while the magnetic field is reversed. The first two examples are noninteracting. One is explicit, with a smooth bounded vector potential and a Gaussian density. The other has smooth bounded scalar and vector potentials in the standard potential spaces. The third example has Coulomb repulsion with unit coupling, an isotropic harmonic scalar potential, and Dirac monopole fields of magnetic charge $\pm 1/2$. The monopole example uses vector potentials defined on separate spatial patches, since these fields are not curls of globally defined smooth vector potentials.

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The total-current Hohenberg-Kohn conjecture asserts that the ground-state particle density and total current determine the scalar potential up to an additive constant and the vector potential up to a gauge transformation. We give three counterexamples for two spinless fermions in the physically relevant setting of three spatial dimensions. In each, a Hamiltonian Hv,AH^{v,\mathbf{A}}Hv,A has a nondegenerate ground state with identically zero total current but nonzero magnetic field. Complex conjugation then yields a second Hamiltonian Hv,−AH^{v,-\mathbf{A}}Hv,−A, with the same scalar potential and interaction, whose ground state has the same density and the same zero current, while the magnetic field is reversed. The first two examples are noninteracting. One is explicit, with a smooth bounded vector potential and a Gaussian density. The other has smooth bounded scalar and vector potentials in the standard potential spaces. The third example has Coulomb repulsion with unit coupling, an isotropic harmonic scalar potential, and Dirac monopole fields of magnetic charge ±1/2\pm 1/2±1/2. The monopole example uses vector potentials defined on separate spatial patches, since these fields are not curls of globally defined smooth vector potentials.
In this article, we examine Hohenberg–Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg–Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble‐representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble‐representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians. © 2014 Wiley Periodicals, Inc.
For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density j(r)\mathbf{j}(\mathbf{r})j(r) are sufficent to determine the one-body potential V(r)V(\mathbf{r})V(r) and vector potential A(r)A(\mathbf{r})A(r) is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials V0{V}_{0}V0​ and A0{\mathbf{A}}_{0}A0​. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and j\mathbf{j}j, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.
The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby, a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle in terms of these densities is also developed. These theorems are further generalized to electrons with spin by imposing the added constraint of fixed canonical orbital and spin angular momenta. The proofs differ from the original HK proof and explicitly account for the many-to-one relationship between the potentials and the nondegenerate ground state wave function. A Percus-Levy-Lieb constrained-search proof expanding the domain of validity to N-representable functions, and to degenerate states, again for fixed electron number and angular momentum, is also provided.
Abstract According to the Hohenberg–Kohn theorem, there is an invertible one‐to‐one relationship between the Hamiltonian of a system and the corresponding ground‐state density ρ(r). The extension of the theorem to the time‐dependent case by Runge and Gross states that there is an invertible one‐to‐one relationship between the density ρ(rt) and the Hamiltonian (t). In the proof of the theorem, Hamiltonians / (t) that differ by an additive constant C/function C(t) are considered equivalent. Because the constant C/function C(t) is extrinsically additive, the physical system defined by these differing Hamiltonians / (t) is the same. Thus, according to the theorem, the density ρ(r)/ρ(rt) uniquely determines the physical system as defined by its Hamiltonian / (t). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians { }/{ (t)} that differ by an intrinsic constant C/function C(t) but which represent different physical systems and yet possess the same density ρ(r)/ρ(rt). The intrinsic constant C/function C(t) contains information about the different physical systems and helps differentiate between them. In such a case, the density ρ(r)/ρ(rt) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians { }/{ (t)}. Thus, although the proof of the Hohenberg–Kohn theorem is independent of whether the constant C/function C(t) is additive or intrinsic, the applicability of the theorem is restricted to excluding the case of the latter. The corollary is as follows: degenerate Hamiltonians { }/{ (t)} that represent different physical systems, but differ by a constant C/function C(t), and yet possess the same density ρ(r)/ρ(rt), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for NNN-representable density kernels, that any finite non-negative differentiable function is an NNN-representable particle density.
A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of vvv representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form ${e}^{\ensuremath{-}ar}$ for two-particle systems are not ground-state vvv representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.