## Periodic Reporting for period 5 - GAPS (Spectral gaps in interacting quantum systems)

Reporting period: 2021-09-01 to 2022-02-28

When talking about different phases in Physics, the first thing that comes to mind is the division in solid, liquid and gas, where temperature is the varying parameter which connects them through phase transition points. At zero (or close to zero) temperature, where quantum mechanics is the physical law that governs the system, there are also different phases interconnected via phase transitions. The exotic and unexpected properties of some of these quantum phases have attracted the attention of physicists for many years.

This project has been motivated by the challenge to catalogue, in a mathematically rigorous way, the possible global properties that may be present in a material at very low temperature, very much like a “periodic table” for the quantum phases of matter.

In order to address that challenge, one needs to characterize the existence or absence of a spectral gap in the operator (Hamiltonian) that models the interactions of a system. The spectral gap represents the energy the system needs to change its properties. One can then formalize the definition of phases as regions in parameter space where the spectral gap is positive, and phase transitions as points where the spectral gap vanishes.

In this project we have proven that the existence or absence of spectral gap is an undecidable property, formalizing the inherent complexity of deriving the desired periodic table. An implication of this fact is the discovery of a new phase of matter: systems whose properties depend on their size.

Despite that, we have been able to obtain a sharp sufficient condition for the existence of spectral gap in a class of systems, called PEPS, which are complex enough to cover all possible phases of matter. The condition exploits a bulk-boundary correspondence.

Using such correspondence, we have also shown that the mathematical structure - fusion categories - expected to classify all phases of matter in 2D, emerges naturally from a renormalization fixed point condition, covering in this way all quantum phases with a renormalization fixed point representative.

Finally, the techniques developed in the project have proven rich enough to influence other areas of science. Together with the discovery of new phases of matter, this is probably the main impact of the project for society.

This project has been motivated by the challenge to catalogue, in a mathematically rigorous way, the possible global properties that may be present in a material at very low temperature, very much like a “periodic table” for the quantum phases of matter.

In order to address that challenge, one needs to characterize the existence or absence of a spectral gap in the operator (Hamiltonian) that models the interactions of a system. The spectral gap represents the energy the system needs to change its properties. One can then formalize the definition of phases as regions in parameter space where the spectral gap is positive, and phase transitions as points where the spectral gap vanishes.

In this project we have proven that the existence or absence of spectral gap is an undecidable property, formalizing the inherent complexity of deriving the desired periodic table. An implication of this fact is the discovery of a new phase of matter: systems whose properties depend on their size.

Despite that, we have been able to obtain a sharp sufficient condition for the existence of spectral gap in a class of systems, called PEPS, which are complex enough to cover all possible phases of matter. The condition exploits a bulk-boundary correspondence.

Using such correspondence, we have also shown that the mathematical structure - fusion categories - expected to classify all phases of matter in 2D, emerges naturally from a renormalization fixed point condition, covering in this way all quantum phases with a renormalization fixed point representative.

Finally, the techniques developed in the project have proven rich enough to influence other areas of science. Together with the discovery of new phases of matter, this is probably the main impact of the project for society.

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).

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).

These main results represent significant advances beyond the state of the art in the three main research lines of the project: (1) the mathematical problem of characterizing the existence of spectral gaps in Hamiltonians, (2) the construction of a periodic table for all quantum phases of matter in two spatial dimensions and (3) the extension of those results to the presence of noise.

They are expected to have an important impact in the understanding of noise and topological order in two-dimensional quantum systems, and hence in the context of quantum technologies and in the study of exotic materials. Indeed, size-driven quantum phase transitions constitute already a very exotic quantum effect.

Also the techniques and ideas are expected to cross-fertilize other disciplines. As an example, it is shown in the project that memory effects can make the transmission capacity of a communication channel uncomputable. Other examples are the characterization of quantum cellular automata via tensor networks, the use of totally new mathematical techniques in holographic quantum gravity and quantum cryptography, or a proof that tensor network methods protect against privacy leaks in machine learning.

They are expected to have an important impact in the understanding of noise and topological order in two-dimensional quantum systems, and hence in the context of quantum technologies and in the study of exotic materials. Indeed, size-driven quantum phase transitions constitute already a very exotic quantum effect.

Also the techniques and ideas are expected to cross-fertilize other disciplines. As an example, it is shown in the project that memory effects can make the transmission capacity of a communication channel uncomputable. Other examples are the characterization of quantum cellular automata via tensor networks, the use of totally new mathematical techniques in holographic quantum gravity and quantum cryptography, or a proof that tensor network methods protect against privacy leaks in machine learning.