Density-functional theory on graphs

Abstract

The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional. CONTENTSI. Introduction 1 II. Fermionic Hilbert space structures on graphs 2 A. General definitions 2 B. Fermionic Hilbert space 3 C. Densities and N -representability 4 D. Graph Laplacians 5 E. Simple graph examples 5 III. Violations of the Hohenberg-Kohn theorem 6 A. Unique v-representability 6 B. Condition for unique v-representability from Odlyzko's theorem 7 C. Examples for non-unique v-representability 8 D. Still almost all densities are uniquely v-representable 10 IV. Positive results from the Rellich theorem 11 A. Openness of the set of non-degenerate, uniquely v-representable ground-state densities 11 B. Degeneracy generates non-unique v-representability 12 V. Special results from the Perron-Frobenius theorem 13 A. Positivity of the non-interacting ground-state density 13 B. Unique v-representability for a linear chain with interacting fermions 14 VI. Constrained-search functionals and v-representability 14 A. General notions 14 B. Results on v-representability 16 C. Triangle graph: Full v-representability and explicit constrained-search functional 17 D. Complete graph: Counterexample to Lieb's non-convexity proof 19 E. Cuboctahedron graph: Pure-state v-representability counterexample 20 VII. Conclusions and open questions 21 Data Availability Statement 23 Acknowledgments 23 A. Expression for the constrained-search functional in the triangle example 23 References 25 a) Electronic mail:

Can the Hartree-Fock kinetic energy exceed the exact kinetic energy?

“…Our conjecture only pertains to real space Hamiltonians, unlike the case of the dimer [15,16]. Lattice systems require separate considerations, as many basic theorems do not hold as they would in real space [17].…”

Does Hohenberg-Kohn Theorem apply to a general quantum system?

“…If viewing this Hamiltonian as a single quantum particle hopping in a graph, this example falls in the scope of Ref. [15], for which the one-to-one correspondence has been shown to generically exists, but rare anomalies do exist. We will exploit the correlation matrix to reveal these anomalies.…”

“…Unlike in the continuous space [21,22], GS density can have extensive zero points in a graph, as stressed in Ref. [15]. Here, we choose the nearest-neightbor hopping model on the kagome lattice as an example [23], where |ψ L = 1 √ 6 i∈Hex (−1) i |i is a ground state of Ĥint .…”

“…In Ref. [15], Penz and van Leeuwen insightfully pointed out several misconceptions in formulating the lattice DFT in the discretized electron model, and presented explicit examples violating the HK theorem. Going beyond electron models, it is even easier to design the counterexamples.…”

Extensibility of Hohenberg–Kohn Theorem to General Quantum Systems

“…For example, in ref. [7], Penz and van Leeuwen insightfully pointed out several misconceptions in formulating the lattice DFT in the discretized electron model. A primary goal of this article is to provide a simple-to-use criterion to identify systems violating the HK theorem.…”