Background and problem

The theoretical understanding and numerical simulation of strongly-correlated quantum materials has been one of the long-standing challenges in physics: Whereas the microscopic quantum-mechanical equations for describing e.g. electrons in a material have been known for over a century, solving these equations for a large number of them is prohibitively expensive. As a result, the use of approximate methods such as mean-field theory, Hartree-Fock theory, perturbation theory, etc have flourished and can actually yield very accurate descriptions of the relevant physics of many materials. For example, band theory often describes the electronic structure of many metallic or insulating systems, and Fermi-liquid theory describes the occurrence of quasiparticles in these systems.

One prominent example of such an approximate method is density functional theory (DFT), which describes the correlated nature of many-electron systems by designing an independent-particle model with effective exchange-correlation functionals. Of course, finding the exact functional is equally difficult as solving the quantum many-electron problem, but over the course of the last fifty years many proposals have been made for approximate functionals that allow to simulate electronic materials and large molecules with high precision.

Nonetheless, the search for numerical methods that capture correlated quantum many-body behavior continues, and recently the advent of tensor-network methods has made it possible to capture quantum correlations or entanglement in a new original way. These methods are a lot more expensive than an independent particle model such as Hartree-Fock theory or DFT, but they have the promise of truly describing the physics of strong quantum materials.

In the light of the emerging field of quantum science and technology, where strong quantum correlations are used as an essential resource for generating new exotic physical phenomena, the need for accurate and scalable numerical methods for simulating strong quantum correlations is more pressing than ever. Therefore, a fruitful combination of the the efficiency of DFT and the quantumness of tensor networks is highly needed.

Goal

In this master thesis proposal, we aim to realize this combination of DFT and tensor networks for simulating metal-organic frameworks (MOFs). These are materials that are built from inorganic building blocks (metal centers) that are connected by organic linkers, and have become extremely attractive in applications such as gas storage, separation, catalysis, drug delivery, etc. It appears that the quantum correlations between the electrons and spin configurations are very strong in these materials, and the accurate description or simulation goes beyond a simple DFT calculation.

We want to combine DFT and tensor networks in a specific way, where we start from the assumption that most of the electrons in these systems are weakly correlated, and that only a limited number of electron bands require a full quantum-mechanical treatment. We use the scalability of DFT to treat this first group of weakly-correlated electrons and design an effective model for the correlated bands, which we then tackle using tensor networks.

We will look at the class of MIL-53 type materials, where the metal atoms are aligned along one-dimensional metal-oxide chains. For these materials, the effective model will be a one-dimensional (1-D) electron system with hopping amplitudes and interaction terms. For 1-D models, we can use very powerful tensor network algorithms for obtaining the ground-state properties and compute the low-lying excitations that appear in spectroscopic measurements.

The main focus in this proposal is in the low-lying dynamics of the effective 1-D model, which can be studied using the time-dependent variational principle. Here we would like to investigate in how entanglement builds up through time after we suddenly quench the system, and whether we can understand this entanglement growth in terms of effective quasiparticles that propagate through the system. It would be very interesting to investigate the emergent scaling behaviour of the entanglement properties if we quench across a quantum phase transition.

In this proposal, the student has the opportunity to collaborate with the Center for Molecular Modeling, which has a great expertise in simulating MOF-type materials and DFT simulations, and the Quantum Group, which has a long-standing track record in developing tensor-network techniques.