## Periodic Reporting for period 1 - GAPS (Spectral gaps in interacting quantum systems)

**Reporting period:**2015-09-01

**to**2017-02-28

## Summary of the context and overall objectives of the project

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.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The first main result of the project, published in Nature, is the proof that there exist quantum interactions 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.

The second 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.

Finally, the third 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.

The second 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.

Finally, the third 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.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

These three 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.

Moreover, the undecidability of the spectral gap problem has wider potential impact, since it has a couple of striking consequences. One of them is philosophical. The result shows the existence of microscopic interactions for which it is totally impossible to predict their macroscopic behavior. It then challenges the reductionists’ point of view. The other consequence is the appearance of a new type of quantum effect, that we name size-driven phase transition: 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).

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.

Moreover, the undecidability of the spectral gap problem has wider potential impact, since it has a couple of striking consequences. One of them is philosophical. The result shows the existence of microscopic interactions for which it is totally impossible to predict their macroscopic behavior. It then challenges the reductionists’ point of view. The other consequence is the appearance of a new type of quantum effect, that we name size-driven phase transition: 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).