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1. Semiclassics: The hidden theory behind the success of DFT

   Okun, Pavel ; Burke, Kieron ( May 2021 , ArXivorg)

   We argue that the success of DFT can be understood in terms of a semiclassical expansion around a very specific limit. This limit was identified long ago by Lieb and Simon for the total electronic energy of a system. This is a universal limit of all electronic structure: atoms, molecules, and solids. For the total energy, Thomas-Fermi theory becomes relatively exact in the limit. The limit can also be studied for much simpler model systems, including non-interacting fermions in a one-dimensional well, where the WKB approximation applies for individual eigenvalues and eigenfunctions. Summation techniques lead to energies and densities that are functionals of the potential. We consider several examples in one dimension (fermions in a box, in a harmonic well, in a linear half-well, and in the Pöschl-Teller well. The effects of higher dimension are also illustrated with the three-dimensional harmonic well and the Bohr atom, non-interacting fermions in a Coulomb well. Modern density functional calculations use the Kohn-Sham scheme almost exclusively. The same semiclassical limit can be studied for the Kohn-Sham kinetic energy, for the exchange energy, and for the correlation energy. For all three, the local density approximation appears to become relatively exact in this limit. Recent work, both analytic and numerical, explores how this limit is approached, in an effort to deduce the leading corrections to the local approximation. A simple scheme, using the Euler-Maclaurin summation formula, is the result of many different attempts at this problem. In very simple cases, the correction formulas are much more accurate than standard density functionals. Several functionals are already in widespread use in both chemistry and materials that incorporate these limits, and prospects for the future are discussed. 

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2. Large N algebras and generalized entropy

   https://doi.org/10.1007/JHEP04(2023)009  

   Chandrasekaran, Venkatesa ; Penington, Geoff ; Witten, Edward ( April 2023 , Journal of High Energy Physics)

   We construct a Type II_∞von Neumann algebra that describes the largeNphysics of single-trace operators in AdS/CFT in the microcanonical ensemble, where there is no need to include perturbative 1/Ncorrections. Using only the extrapolate dictionary, we show that the entropy of semiclassical states on this algebra is holographically dual to the generalized entropy of the black hole bifurcation surface. From a boundary perspective, this constitutes a derivation of a special case of the QES prescription without any use of Euclidean gravity or replicas; from a purely bulk perspective, it is a derivation of the quantum-corrected Bekenstein-Hawking formula as the entropy of an explicit algebra in theG →0 limit of Lorentzian effective field theory quantum gravity. In a limit where a black hole is first allowed to equilibrate and then is later potentially re-excited, we show that the generalized second law is a direct consequence of the monotonicity of the entropy of algebras under trace-preserving inclusions. Finally, by considering excitations that are separated by more than a scrambling time we construct a “free product” von Neumann algebra that describes the semiclassical physics of long wormholes supported by shocks. We compute Rényi entropies for this algebra and show that they are equal to a sum over saddles associated to quantum extremal surfaces in the wormhole. Surprisingly, however, the saddles associated to “bulge” quantum extremal surfaces contribute with a negative sign.

    

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3. On the rigorous derivation of the incompressible Euler equation from Newton’s second law

   https://doi.org/10.1007/s11005-023-01630-w  

   Rosenzweig, Matthew ( January 2023 , Letters in Mathematical Physics)

   A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system ofNparticles interacting in$${\mathbb {T}}^d$$$T^d$,$$d\ge 2$$$d\ge 2$, via Newton’s second law through asupercritical mean-field limit. Namely, the coupling constant$$\lambda $$$\lambda$in front of the pair potential, which is Coulombic, scales like$$N^{-\theta }$$$N^{-\theta }$for some$$\theta \in (0,1)$$$\theta \in (0,1)$, in contrast to the usual mean-field scaling$$\lambda \sim N^{-1}$$$\lambda \sim N^{-1}$. Assuming$$\theta \in (1-\frac{2}{d(d+1)},1)$$$\theta \in (1-\frac{2}{d(d+1)},1)$, they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$N\rightarrow \infty $$$N\to \infty$. Han-Kwan and Iacobelli asked if their range for$$\theta $$$\theta$was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$N\rightarrow \infty $$$N\to \infty$for$$\theta \in (1-\frac{2}{d},1)$$$\theta \in (1-\frac{2}{d},1)$. Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta $$$\theta$. Additionally, we show that for$$\theta \le 1-\frac{2}{d}$$$\theta \le 1-\frac{2}{d}$, one cannot, in general, expect convergence in the modulated energy notion of distance.

    

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4. Semiclassical Measures for Higher-Dimensional Quantum Cat Maps

   https://doi.org/10.1007/s00023-023-01309-x  

   Dyatlov, Semyon ; Jézéquel, Malo ( April 2023 , Annales Henri Poincaré)

   Consider a quantum cat mapMassociated with a matrix $$A\in {{\,\textrm{Sp}\,}}(2n,{\mathbb {Z}})$$$A\in \text{Sp}(2n,Z)$, which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of Mon any nonempty open set in the position–frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue ofAof largest absolute value and (2) the characteristic polynomial ofAis irreducible over the rationals. This is similar to previous work (Dyatlov and Jin in Acta Math 220(2):297–339, 2018; Dyatlov et al. in J Am Math Soc 35(2):361–465, 2022) on negatively curved surfaces and (Schwartz in The full delocalization of eigenstates for the quantized cat map, 2021) on quantum cat maps with$$n=1$$$n=1$, but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.

    

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5. Deriving approximate functionals with asymptotics

   https://doi.org/10.1039/D0FD00057D  

   Burke, Kieron ( December 2020 , Faraday Discussions)

   Modern density functional approximations achieve moderate accuracy at low computational cost for many electronic structure calculations. Some background is given relating the gradient expansion of density functional theory to the WKB expansion in one dimension, and modern approaches to asymptotic expansions. A mathematical framework for analyzing asymptotic behavior for the sums of energies unites both corrections to the gradient expansion of DFT and hyperasymptotics of sums. Simple examples are given for the model problem of orbital-free DFT in one dimension. In some cases, errors can be made as small as 10 −32 Hartree suggesting that, if these new ingredients can be applied, they might produce approximate functionals that are much more accurate than those in current use. A variation of the Euler–Maclaurin formula generalizes previous results. 

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