There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Journal of Statistical Mechanics: theory and experiment

We use the density-matrix renormalization group, applied to a one-dimensional model of continuum Hamiltonians, to accurately solve chains of hydrogen atoms of various separations and numbers of atoms. We train and test a machine-learned approximation to $F[n]$, the universal part of the electronic density functional, to within quantum chemical accuracy. We also develop a data-driven, atom-centered basis set for densities which greatly reduces the computational cost and accurately represents the physical information in the machine-learning calculation. Our calculation (a) bypasses the standard Kohn-Sham approach, avoiding the need to find orbitals, (b) includes the strong correlation of highly stretched bonds without any specific difficulty (unlike all standard DFT approximations), and (c) is so accurate that it can be used to find the energy in the thermodynamic limit to quantum chemical accuracy.

Pure density functional for strong correlation and the thermodynamic limit from machine learning

odinger equation many times. The density matrix renormalization group method makes this possible for one-dimensional, real-space systems of more than two interacting electrons. We illustrate and explore the convergence properties of the exact KS scheme for both weakly and strongly correlated systems. We also explore the spin-dependent generalization and densities for which the functional is ill defined.

/pdf/kohn-sham-calculations-with-the-exact-functional-1t8t3kq0pj.pdf

Kohn-Sham calculations with the exact functional

Historical methods of functional development in density functional theory have often been guided by analytic conditions that constrain the exact functional one is trying to approximate. Recently, machine-learned functionals have been created by interpolating the results from a small number of exactly solved systems to unsolved systems that are similar in nature. For a simple one-dimensional system, using an exact condition, we find improvements in the learning curves of a machine learning approximation to the non-interacting kinetic energy functional. We also find that the significance of the improvement depends on the nature of the interpolation manifold of the machine-learned functional.

Can exact conditions improve machine-learned density functionals?

An exponential interaction is constructed so that one-dimensional atoms and chains of atoms mimic the general behavior of their three-dimensional counterparts. Relative to the more commonly used soft-Coulomb interaction, the exponential greatly diminishes the computational time needed for calculating highly accurate quantities with the density matrix renormalization group. This is due to the use of a small matrix product operator and to exponentially vanishing tails. Furthermore, its more rapid decay closely mimics the screened Coulomb interaction in three dimensions. Choosing parameters to best match earlier calculations, we report results for the one dimensional hydrogen atom, uniform gas, and small atoms and molecules both exactly and in the local density approximation.

/pdf/one-dimensional-mimicking-of-electronic-structure-the-case-3hy0ojk3zh.pdf

One Dimensional Mimicking of Electronic Structure: The Case for Exponentials

The order parameter of superconducting pairs penetrating an inhomogeneous magnetic material can acquire a long range triplet component (LRTC) with non-zero spin projection. This state has been predicted and generated in proximity systems and Josephson junctions. We show, using a realistic domain wall of an exchange spring bilayer, how the LRTC emerges and can be tuned with the twisting of the magnetization. We also introduce a new kind of Josephson current reversal, the singlet-LRTC 0–π transition, that can be observed in one and the same system either by tuning the domain wall or by varying temperature.

Thomas E. Baker

Papers

Pure density functional for strong correlation and the thermodynamic limit from machine learning

Kohn-Sham calculations with the exact functional

Can exact conditions improve machine-learned density functionals?

One Dimensional Mimicking of Electronic Structure: The Case for Exponentials

Long range triplet Josephson current and 0−π transitions in tunable domain walls