## Periodic Reporting for period 4 - GAPS (Spectral gaps in interacting quantum systems)

Reporting period: 2020-03-01 to 2021-08-31

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, like superconductivity, superfluidity, fractional statistics, topological order, etc. have attracted the attention of physicists for many years.

The main aim of this project is to catalogue, in a mathematically rigorous way, the possible global properties that may be present in a material at very low temperature and provide models that possess each one of them. In this way we would have a kind of periodic table of the quantum phases of matter.

In order to do that, we will deal with the mathematical problem of characterizing the existence or absence of a spectral gap in the operator (called the Hamiltonian) that models the interactions of a system and governs its evolution. 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. This guarantees that systems within the same phase have similar properties and phase transitions are unstable points where properties change abruptly.

The potential importance of the project for society is twofold. On the one hand, it may help to understand the mechanisms and properties of systems with topological order, which are expected to play a chief role in the emerging quantum technologies. On the other hand, it may lead to the discovery of new phases of matter.

The main aim of this project is to catalogue, in a mathematically rigorous way, the possible global properties that may be present in a material at very low temperature and provide models that possess each one of them. In this way we would have a kind of periodic table of the quantum phases of matter.

In order to do that, we will deal with the mathematical problem of characterizing the existence or absence of a spectral gap in the operator (called the Hamiltonian) that models the interactions of a system and governs its evolution. 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. This guarantees that systems within the same phase have similar properties and phase transitions are unstable points where properties change abruptly.

The potential importance of the project for society is twofold. On the one hand, it may help to understand the mechanisms and properties of systems with topological order, which are expected to play a chief role in the emerging quantum technologies. On the other hand, it may lead to the discovery of new phases of matter.

The first main result of the project, published in Nature, is the proof that there exist quantum interactions in 2D for which it is impossible to decide the existence or absence 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 shown that generic systems have typically a very large spectral gap and a very short correlation length, a result published in Annales Henri Poincare.

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.

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 two spatial dimensions, 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 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 have 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 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 a notion of phase for dissipative quantum systems, together with many results that indicate that it is indeed the appropriate definition for that context.

The seventh main result of the project, published in Journal of Mathematical Physics, is a proof that quantum systems subjected to sufficiently strong 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.

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, where such techniques allowed us to give exponentially better bounds on the quantum resources needed by an adversary to break it.

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.

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 two spatial dimensions, 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 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 have 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 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 a notion of phase for dissipative quantum systems, together with many results that indicate that it is indeed the appropriate definition for that context.

The seventh main result of the project, published in Journal of Mathematical Physics, is a proof that quantum systems subjected to sufficiently strong 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.

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, where such techniques allowed us to give exponentially better bounds on the quantum resources needed by an adversary to break it.

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 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. This result has been published in Nature Communications.

Other examples are the characterization of quantum cellular automata via tensor networks, published in JSTAT, the use of totally new mathematical techniques in holographic quantum gravity and quantum cryptography, or the impact in philosophy of the undecidability of the spectral gap problem (the result challenges the reductionists’ point of view, since it shows the existence of microscopic interactions for which it is totally impossible to predict their macroscopic behavior).

Expected results until the end of the project include an experimental proposal to observe size-driven quantum phase transitions, a classification of quantum phases in the presence of noise, or new applications to security and privacy in machine learning, where we have already obtained some preliminary promising results.

They are expected to have an 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. This result has been published in Nature Communications.

Other examples are the characterization of quantum cellular automata via tensor networks, published in JSTAT, the use of totally new mathematical techniques in holographic quantum gravity and quantum cryptography, or the impact in philosophy of the undecidability of the spectral gap problem (the result challenges the reductionists’ point of view, since it shows the existence of microscopic interactions for which it is totally impossible to predict their macroscopic behavior).

Expected results until the end of the project include an experimental proposal to observe size-driven quantum phase transitions, a classification of quantum phases in the presence of noise, or new applications to security and privacy in machine learning, where we have already obtained some preliminary promising results.