Scale separation: Breaking down unsolvable problems into solvable ones – Phys.org

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In physics, one often has to deal with different scales that can be described separately from one another: For the Earth's orbit around the sun, it makes absolutely no difference whether an elephant in the zoo walks to the left or to the right. And the movement of the elephant can be described without having to know anything about the properties of the electrons in its ear. The world can be divided into different scales.

In materials research, too, it is important to describe the behavior of particles on the appropriate scales. However, you first have to find out which scales are the decisive onesa difficult task for which there was previously no clear solution strategy. One could only hope to guess the solution with a lot of experience.

However, a mathematical method has now been found by an international research cooperation with the participation of the TU Wien and Saitama University in Japan to calculate the appropriate scalesan important step in the search for better materials for different areas of application, from microchips to photovoltaics. The method has now been published in the journal Physical Review X.

"In materials physics, electrons often cannot be viewed separately from one another," says Anna Kauch, who heads an FWF research project on this topic. "Particularly exciting phenomena such as magnetism or superconductivity can only be understood if many particles and their complex interactions are described together."

However, this is usually not possible with complete accuracy: If many particles are involved, then the formulas of quantum theory quickly become so big and so complex that even the best supercomputers in the world cannot solve them exactlynot even the state of the particles can then be written down exactly, because that would require more storage space than we will ever have available.

One must therefore look for certain approximations. These approximations often consist in being able to disregard certain size scales in certain cases. "Sometimes you can find quite simple physical arguments for it," says Markus Wallerberger, one of the authors of the paper. "A typical example of this are electrons and atomic nuclei in a crystal: the electrons are very light and move quickly. The atoms are much heavier, so on the time scale used to describe the motion of the electrons, the atoms can be considered rigid and immobile.

"In this case, we have split a complicated problem into two much simpler problems: we can now think about the fast movement of the electrons on the one hand and the much slower movement of the atoms on the otherand think about how the two are related." (a) QTT representation in momentum space. The rightmost bits (indices) represent fine structures in momentum space. Low entanglement structures are assumed between different length scales. (b) Schematic illustration of the bond dimensions along the chain representing the momentum dependence. The dashed line indicates the maximum bond dimensions in maximally entangled cases. (c) Fourier transform from momentum space to real space by applying a MPO. The orange diamonds represent the MPO tensors. Credit: Physical Review X (2023). DOI: 10.1103/PhysRevX.13.021015

But what do you do if you cannot see such an intuitive solution? So far, one could only guess in this case. But now it has been possible to develop a mathematical recipe for this situation. "In our paper, we show how to break down the complete description of such a system into different scales," explains Hiroshi Shinaoka, professor at Saitama University in Japan and leader of the study.

"It then automatically shows which scales are important and which ones can be left out. At the same time, the calculation method also tells us what the coupling between the different scales looks like and how we can then use it for further calculations."

More information: Hiroshi Shinaoka et al, Multiscale Space-Time Ansatz for Correlation Functions of Quantum Systems Based on Quantics Tensor Trains, Physical Review X (2023). DOI: 10.1103/PhysRevX.13.021015

Journal information: Physical Review X

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Scale separation: Breaking down unsolvable problems into solvable ones - Phys.org