The first main result of the project, published in Nature and Forum of Mathematics Pi, is the proof that there exist quantum interactions in 2D for which it is impossible to decide the existence of spectral gap (the spectral gap problem is undecidable). This result shows clearly the difficulty of the problem we are studying in this project. We have shown that the same result holds true even in 1D, a result published in Physical Review X. In a complementary direction, we have also shown that generic systems have typically a very large spectral gap, a result published in Annales Henri Poincare. These results have received extensive media coverage.
The second main result of the project is the prediction, based on the previous result, of a new quantum effect: the existence of materials whose properties depend dramatically on the size of the sample and for which the critical size where the properties change can be tuned to any desired value (no matter how large). We give also the first steps to observe such "size-driven quantum phase transitions". These results have been published in PNAS and have also reached the media.
The third main result of the project, published in Annals of Physics, is the characterization of suitable representatives, called renormalization fixed points, of all quantum phases of matter in 2D, as well as a procedure to distill from them the topological order present in the phase. This is done by exploiting a holographic principle that allows to study properties in the bulk analyzing the boundary of the system. The result has been presented at several research institutes, including IHP (Paris) or KITP (Santa Barbara).
The fourth main result of the project is the use of such holographic correspondence to conclude the existence of a spectral gap in the bulk just from the locality of the boundary Hamiltonian. The result has been published in Communications in Mathematical Physics. A consequence of this result is that the most paradigmatic topological models in 2D cannot be used as good quantum memories, even at very small temperatures. The result has also been presented at several research institutes, such as BIRS (Banff) or IPAM (Los Angeles).
The fifth main result of the project is to show that topological phase transitions and symmetry-enriched topological phases of matter are inextricably connected. Using this connection and the theory of group extensions we have given a complete characterization of those phases that come from groups, together with order parameters that detect all of them. Those results have been published in Physical Review B and in New Journal of Physics.
The sixth main result of the project, published in Quantum, is the definition of an adequate notion of phase for dissipative quantum systems. Moreover, we have analyzed in detail the 1D case. Invited talks on these results have been given at CRM (Montreal) and YITP (Kyoto).
The seventh main result of the project, published in Journal of Mathematical Physics, is a proof that quantum systems subjected to fast mixing dissipative noise have a bound on the amount of correlations present in the system. This result settles in the positive the so-called area law conjecture in the context of dissipative evolutions. We have also been able to show that thermal dissipative evolutions in 1D converge exponentially fast at any temperature, no matter how small.
The eight main result of the project is a proof that tensor network methods in machine learning have a much larger degree of privacy than standard neural network methods.
Finally, the last main result of the project is the use of totally new mathematical techniques (Geometric Banach space theory) in the area of holographic quantum gravity, based on a recent connection established with the area of position based quantum cryptography. The result has been presented at several research institutes like Oberwolfach (Germany) or IPAM (Los Angeles).
