Similarity hypothesis in the theory of phase transitions

А.З. Паташинский

Peter Kopietz, Lorenz Bartosch, Florian Schütz

L. Wojtczak, B. Mrygoń

Shijie Wei, Yue Zhai, Quanfeng Lu +7 more

Abstract The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Linking these zeros to physical phenomena offers new perspectives on their origin and verification. Here we establish a direct correspondence between these zeros and dynamical quantum phase transitions in two quantum many-body systems, characterized by the average accumulated phase factor and the Loschmidt amplitude respectively. This precise correspondence recasts the RH as the emergence of phase transitions at a unique temperature and identifies it as a previously unknown transition mechanism. We demonstrate this correspondence in a proof-of-principle experiment on a quantum processor. Moreover, we propose a quantum computational framework that implements both systems with polynomial resources, suggesting quantum advantage in probing the hypothesis. Our work bridges nonequilibrium quantum dynamics and number theory, positioning quantum computing as a powerful platform for exploring mathematical conjectures, phase transitions and beyond.

G. Toulouse

Всеволод Феликсович Гантмахер, Valerii T. Dolgopolov

Metal--insulator transitions and transitions between different quantum Hall liquids are used to describe the physical ideas forming the basis of quantum phase transitions and the methods of application of theoretical results in processing experimental data. The following two theoretical schemes are discussed and compared: the general theory of quantum phase transitions, which has been developed according to the theory of thermodynamic phase transitions and relies on the concept of a partition function, and a theory which is based on a scaling hypothesis and the renormalization-group concept borrowed from quantum electrodynamics, with the results formulated in terms of flow diagrams.