Factorizing the wave function of large quantum systems

Project Information

FACT

Grant agreement ID: 788890

DOI

10.3030/788890

Project closed

EC signature date 10 August 2018

Start date 1 September 2019

End date 31 August 2025

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 2 443 932,00

EU contribution

€ 2 443 932,00

Coordinated by

THE HEBREW UNIVERSITY OF JERUSALEM
Israel

Periodic Reporting for period 4 - FACT (Factorizing the wave function of large quantum systems)

Reporting period: 2024-03-01 to 2025-08-31

Summary of the context and overall objectives of the project

One of the greatest challenges of condensed-matter physics is to predict the properties of large and complex quantum systems. In modern high-tech societies, essentially all sectors depend on materials with highly specific properties: For example, manmade materials harder than diamond are used in special-purpose drills, highly specific magnetic materials are necessary to make the read/write process in magnetic storage devices as fast as possible. For many purposes, especially in the IT sector, a thorough understanding on the level of the smallest constituents, the electrons and the nuclei, is indispensable. If one aims at a genuine first-principles description based on the fundamental equation of quantum mechanics, the Schrödinger equation, this goal is yet largely unattainable with present-day computing facilities. A variety of highly sophisticated methods such as quantum Monte Carlo, configuration interaction, coupled cluster, tensor networks, Feynman diagrams, dynamical mean-field theory have been developed to deal with the enormous complexity of the quantum many-body problem. For time-dependent scenarios, explicitly time-dependent variants of these methods are available as well. However, due to unfavorable scaling with the number of particles, all of these methods are applicable to small or medium-size systems only. The idea of this action is to attack the many-particle wave function with a novel theoretical approach, known as the exact factorization. In this approach one writes the wavefunction of the full quantum system as a single product of subsystem wavefunctions in such a way that the product is an exact representation of the complete wave function. No approximation is made in this first step. In the case of two subsystems, taking electrons and nuclei as prototypical example, the nuclear factor is a regular wavefunction, satisfying a standard time-dependent Schrödinger equation, while the electronic factor is a conditional probability amplitude that satisfies a more complicated Schrödinger-like equation with a non-linear and non-Hermitian Hamiltonian. The above-mentioned standard techniques can then be applied to the subsystem-Schrödinger equations, typically different standard techniques to different subsystems. The objective of this action is to exploit the exact factorization in various, particularly challenging situations such as solids with strong long-range correlations as well as molecular processes with strong non-adiabatic couplings.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The power of the exact factorization lies in its versatility. Throughout this ERC action, the technique has been applied to a variety of different scenarios. One very important case is the description of non-adiabatic effects in molecules and solids. Here we coined the concept of exact forces exerted by the electronic subsystem on the nuclear subsystem. These forces are N-body operators consisting of an electric-field-like component, as well as Lorentz-type forces involving a Berry-curvature-type magnetic field. While always there, in principle, it was not clear in the beginning how large (or small) these magnetic forces actually are. One important case could be identified where the magnetic forces can be very large: For a neutral atom moving in a constant external magnetic field, the Berry curvature exactly cancels the external field making the nucleus move as if there was no external magnetic field at all. Another manifestation of these forces has been studied in molecular junctions where an imposed AC current can make a molecule rotate as consequence of a subtle balance between current-induced forces on the nuclei and electronic friction. To describe non-adiabatic effects in solids, a “density-functionalization” of the exact factorization was formulated which allowed a systematic derivation of electron-phonon interactions in terms of (fractional) powers of a small parameter, the electronic-over-nuclear mass ratio. The same ordering principle in terms of powers in the small electronic-over-nuclear mass ratio has also been used in the construction of a novel functional used in the density functional theory of (phonon-driven) superconductors. The resulting theory is highly successful. Critical temperatures of phonon-driven superconductors can be predicted (without any adjustable parameters) within a few percent of experiment. A very different scenario can be studied by applying the factorization to electrons only, leading to a novel embedding scheme for strongly correlated electrons in an environment of weakly correlated ones. In contrast to previous embedding schemes, which are local in real space, the new scheme allows the description of truly long-range correlations in strongly correlated solids. A very successful application of this novel scheme was the accurate calculation of the topological phase diagram of a strongly correlated solid. The time-dependent version of the exact factorization was successfully employed to calculate decoherence times.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Significant progress beyond the state-of-the-art has been achieved in this action. In the laser-induced dynamics of molecules and solids we have identified a Lorentz-type N-body force that appears as consequence of the non-adiabatic interactions between electrons and nuclei. This Lorentz force not only cancels a constant external magnetic field in the case of a single neutral atom, it is also important in the dynamics of an "atomic water wheel" where an electronic AC current flowing through a molecular junction triggers a rotational motion of the molecule. A steady rotation can be achieved, but only when the frequency and amplitude of the AC current lie within certain “islands of stability”. Outside these islands, the motion of the molecule is chaotic. For phonon-driven superconductors, the developed density functional theory of superconductivity has been established as the gold standard for calculating material-specific superconducting properties: Critical temperatures of phonon-driven superconductors are reliably predicted within a few percent of experiment, as shown in Fig 1, and highly complex scenarios like the competition of superconductivity with a charge-density wave have been accurately described as well, see Fig 2. In normal metals, the exact-factorization-based density-functional theory of electrons and phonons yields the phonon-induced renormalization (the "wiggle") of the band structure near the Fermi energy, see Fig 3. Another important feature of systems consisting of two or more subsystems is the phenomenon known as decoherence which means that through the interaction of the quantum system of interest (say the electrons) with its environment (the moving nuclei), the original quantum system loses its quantumness: Typical quantum features like entanglement or interferences get lost on the scale of the so-called decoherence time. Decoherence is responsible for the fact that a scalable quantum computer still does not exist, in spite of enormous efforts worldwide. Hence it is desirable to have a predictive ab-initio theory of decoherence that allows one to reliably calculate the decoherence time and, ultimately, to find ways to make it very large, thereby controlling the decoherence. A first step in this direction was achieved by applying the exact factorization to the full wave function of the system plus environment, leading to a novel algorithm that successfully captures electronic decoherence along a single nuclear trajectory.

Fig 1: Predicted critical temperatures of superconductors vs experiment

Fig 2: Superconducting gap and electron-phonon coupling in 2H-NbSe2

Fig 3: Renormalization of band structure at Fermi surface in normal metal