For his innovative research results, Chokri Manai has been presented the TopMath Award 2021.
The TopMath Award is a prize for PhD Students within the TopMath Program, in recognition of exceptional achievements in research. It is awarded once a year. Upon recommendation from their mentor or the subject study advisor, the candidates are requested to apply for the award. Annually in November, the TopMath Board meets to discuss the applications and to nominate the awardee. The TopMath Award includes prize money of 500 Euros, and is usually presented within the graduation ceremony of the Department of Mathematics.
Winner of the TopMath Award 2021 is Chokri Manai. He receives the award for his research on quantum spin glasses, a topic that is new in mathematics. Although he has only been working on his dissertation since the winter semester 2019/20, he has already written several articles.
We congratulate Chokri on his achievements and wish him every success in attaining his next goals!
Quantum Spin Glasses
Text: Chokri Manai
Spin glasses have been studied for several decades by numerous physicist and mathematicians. Their research lead to very important and deep mathematical results.
A particular highlight was the derivation of the free energy for the Sherrington-Kirkpatrick (SK) model by Giorgio Parisi and the rigorous proof of the so-called Parisi formula by Guerra and Talagrand around the year 2006.
Quantum spin glasses are spin glasses where additionally quantum effects are taken into account. They have been studied for about 30 years, but they appear to be barely accessible to an analytic or rigorous treatment. Our research focuses on hierarchical models as the Quantum Random Energy Model (QREM) and its generalization, the QGREM.
We succeeded by virtue of spectral methods to derive a formula for the limit of the free energy, which has been not known in the physics literature in the case of the QGREM. We also discussed the influence of longitudinal magnetic fields on the free energy. Moreover, we have established a precise characterization of the low energy spectrum of the QREM Hamiltonian.
A collaboration with Hajo Leschke and Rainer Ruderer lead to a proof of the existence of replica-symmetry breaking for low temperatures in the quantum SK model. The quantum SK model is considered as one of the most important quantum spin glass models, but a rigorous analysis seemed to be a very challenging problem. Our proof, however, which is based on the Falk-Bruch inequality , is surprisingly simple.
© Chokri Manai
Phase diagram of a 3-level Quantum GREM, where each of the three corresponding spin blocks undergoes a quantum and a freezing transition. The bold lines represent magnetic first order transitions, while the dashed lines indicate the already classically present second order glass transitions.
Publications
All of Chokri's articles can be found on his arXiv page:
https://arxiv.org/search/?searchtype=author&query=Manai%2C+C
Acknowledgement of the prize winners and their achievements
"Chokri Manai works on establishing techniques and results on so-called quantum spin glasses — a timely subject given the 2021 Nobel prize in physics to Parisi for his research into complex disordered systems. Both his master and the forthcoming PhD thesis concern mathematically uncharted territory at the interface of probability theory and mathematical physics, which is motivated by conjectures and approximate results by physicists on the fate of the classical spin glass phase under quantum fluctuations. Despite his young age, he already coauthored 5 partially trailblazing publications in this novel area during the last 3 years. This is extremely rare in mathematical physics, especially since these papers contain ideas and questions that originated on his initiative. International recognition by both physicists (in the form of an APS press release) and mathematicians (in the form of acceptance of papers at high-ranking journals) is just arriving." - Prof. Dr. Simone Warzel (TUM), Mentor of Chokri Manai