# Hohenberg-Kohn theorems for interactions, spin and temperature

## TL;DR

This paper extends the Hohenberg-Kohn theorems to various quantum models, demonstrating how ground state properties encode information about interactions, external fields, and temperature, with some limitations.

## Contribution

It proves Hohenberg-Kohn theorems for models with interactions, spin, and temperature, including constraints and counterexamples, broadening their applicability.

## Key findings

- Pair correlation functions contain interaction and potential info.
- External fields are constrained by ground state properties with Zeeman interaction.
- Hohenberg-Kohn theorems generally hold at positive temperature.

## Abstract

We prove Hohenberg-Kohn theorems for several models of quantum mechanics. First, we show that the pair correlation function of any ground state contains 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. Moreover, we provide a counterexample in a setting involving non-local potentials. 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.