The bifunctional formalism: an alternative treatment of density functionals

The bifunctional formalism presents an alternative how to obtain the functional value from its functional derivative by exploiting homogeneous density scaling. In the bifunctional formalism the density dependence of the functional derivative is suppressed. Consequently, those derivatives have to be treated as formal functional derivatives. For a pointwise correspondence between the true and the formal functional derivative, the bifunctional expression yields the same value as the density functional. Within the bifunctional formalism the functional value can directly be obtained from its derivative (while the functional itself remains unknown). Since functional derivatives are up to a constant uniquely defined, this approach allows for a pointwise comparison between approximate potentials and reference potentials. This aspect is especially important in the field of orbital-free density functional theory, where the burden is to approximate the kinetic energy. Since in the bifunctional approach the potential is approximated directly, full control is given over the latter, and consequently over the final electron densities obtained from variational procedure. Besides the bifunctional formalism itself another concept is introduced, dividing the total non-interacting kinetic energy into a known functional part and a remainder, called Pauli kinetic energy. Only the remainder requires further approximations. For practical purposes sufficiently accurate Pauli potentials for application on atoms, molecular and solid-state systems are presented.

Deformed $$\sigma $$-models, Ricci flow and Toda field theories

It is shown that the Pohlmeyer map of a $$\sigma $$ σ -model with a toric two-dimensional target space naturally leads to the ‘sausage’ metric. We then elaborate the trigonometric deformation of the $$\mathbb {CP}^{n-1}$$ CP n - 1 -model, proving that its T-dual metric is Kähler and solves the Ricci flow equation. Finally, we discuss a relation between flag manifold $$\sigma $$ σ -models and Toda field theories.

Lie polynomials and a twistorial correspondence for amplitudes

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of $$\mathcal {M}_{0,n}$$ M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle $$T^*_D\mathcal {M}_{0,n}$$ T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and $$\mathcal {K}_n$$ K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain $$n-3$$ n - 3 -forms on $$\mathcal {K}_n$$ K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $$n-3$$ n - 3 -planes in $$\mathcal {K}_n$$ K n introduced by ABHY.