## Periodic Reporting for period 1 - FLATOPS (Flat bands and topology in superconductive materials)

Reporting period: 2016-04-01 to 2018-03-31

A superconductor has the property that below a certain temperature, called the critical temperature, the electric resistivity vanishes and the material becomes a perfect conductor, that is the current flow is not accompanied by the dissipation of energy. Thanks to this unique property, superconductivity has found important applications, for example it is used to create the large and uniform magnetic fields required for Magnetic Resonance Imaging, nowadays an essential medical tool. However many more applications on a much larger scale would likely open up if materials were found whose critical temperature is higher than the current records. Unfortunately our understanding of the origin of superconductivity in currently known high-temperature superconductors is still rather incomplete and there is no clear established receipt for engineering materials with even higher critical temperatures.

One approach is to increase the density of states, a quantity which measures the amount of quantum states available for electrons in a given energy range since this has the effect of increasing the critical temperature. The density of states is proportional to the effective mass of the electrons in a crystal. Therefore to increase the critical temperature one should find a material where electrons have an effective mass as large as possible. In the limit of infinite effective mass the electronic states form a so-called flat band.

A crucial question is what are the transport properties in the presence a flat band. At the classical level an electron with an infinite effective mass is stuck in place and cannot be moved by any external force, no matter how large. This means that the material cannot conduct any current and is an insulator rather than a superconductor. Thus it may appear that the idea of using the high density of states of a flat band to increase the critical temperature is bound to fail.

The central result of FLATOPS is that the situation is completely different if quantum effects are taken into account. Whereas in a flat band single particles are localized due to the infinite effective mass, in the presence of interactions the effective mass of composite particles, such as Cooper pairs, can be finite and thus allow for the transport of current. This quantum effect where two single particles, which are immobile due to the diverging effective mass of the flat band, are combined into a mobile Cooper pair with finite effective mass has no classical analogue and is the key to achieve high-temperature superconductivity.

In this project it has also been established that the effective mass of Cooper pairs is controlled by a geometric property of the flat band quantum wave functions, the so-called quantum metric, which essentially measures the overlap between the flat band wave functions. The higher the overlap the higher the chance for a Cooper pair to jump from one wave function to the other as a consequence of interactions and thus the mobility of the Cooper pair is higher. This corresponds to a higher critical temperature. Therefore, using flat bands with large quantum metric is a new promising route to engineer high-temperature superconducting materials. This new route to high-temperature superconductivity has been established in FLATOPS and has been proved to be physically sound using a number of different approaches. These results will be important in the effort of engineering novel materials with increasingly higher critical temperatures.

One approach is to increase the density of states, a quantity which measures the amount of quantum states available for electrons in a given energy range since this has the effect of increasing the critical temperature. The density of states is proportional to the effective mass of the electrons in a crystal. Therefore to increase the critical temperature one should find a material where electrons have an effective mass as large as possible. In the limit of infinite effective mass the electronic states form a so-called flat band.

A crucial question is what are the transport properties in the presence a flat band. At the classical level an electron with an infinite effective mass is stuck in place and cannot be moved by any external force, no matter how large. This means that the material cannot conduct any current and is an insulator rather than a superconductor. Thus it may appear that the idea of using the high density of states of a flat band to increase the critical temperature is bound to fail.

The central result of FLATOPS is that the situation is completely different if quantum effects are taken into account. Whereas in a flat band single particles are localized due to the infinite effective mass, in the presence of interactions the effective mass of composite particles, such as Cooper pairs, can be finite and thus allow for the transport of current. This quantum effect where two single particles, which are immobile due to the diverging effective mass of the flat band, are combined into a mobile Cooper pair with finite effective mass has no classical analogue and is the key to achieve high-temperature superconductivity.

In this project it has also been established that the effective mass of Cooper pairs is controlled by a geometric property of the flat band quantum wave functions, the so-called quantum metric, which essentially measures the overlap between the flat band wave functions. The higher the overlap the higher the chance for a Cooper pair to jump from one wave function to the other as a consequence of interactions and thus the mobility of the Cooper pair is higher. This corresponds to a higher critical temperature. Therefore, using flat bands with large quantum metric is a new promising route to engineer high-temperature superconducting materials. This new route to high-temperature superconductivity has been established in FLATOPS and has been proved to be physically sound using a number of different approaches. These results will be important in the effort of engineering novel materials with increasingly higher critical temperatures.

The idea that the finite effective mass of Cooper pairs in a flat band is the combined result of interactions and the geometric properties of the localized wave functions of the flat band, encoded in the quantum metric, has been put forward in S. Peotta and P. Törmä, Nature Communications 6, 8944 (2015). In FLATOPS the results of this first work have been applied to a number of lattice models with flat bands relevant both for experiments in ultracold gases and for solid state materials. An important example is the Lieb lattice. The superfluid weight for the Lieb lattice is shown in Fig 1. For fillings in between one and two the flat band is partially filled. Precisely for a partially filled flat band the superfluid weight reaches its maximum. This is a striking example how flat bands can enhance the superfluid weight, which in turn controls the critical temperature. The Lieb lattice and other models studied in this project have been recently realized with ultracold gases in optical lattices, where the results of FLATOPS are likely to find a first immediate application. The results shown in Fig. 1 are obtained using mean-field theory, which is the simplest approximation that can be used to describe the superconducting state. However, in this project mean-field results have been systematically checked using beyond mean-field methods. An example is shown in Fig. 2, where the superfluid weight of the Creutz ladder, another lattice model with flat bands, obtained from mean-field is compared to the essentially exact result calculated using Density Matrix Renormalization Group, which is an important beyond mean-field method used for one-dimensional systems. An important goal of FLATOPS was to further explore the relation between superfluidity and topological properties of the band structure. Indeed the superfluid weight of a flat band is bounded from below by the Chern number, a topological invariant of the band structure as shown in S. Peotta and P. Törmä, Nature Communications 6, 8944 (2015). It is a natural question to ask whether similar bounds can be found also in the case of other topological invariants that are currently known. Within FLATOPS it has been shown that also the winding number, an important topological invariant for one-dimensional systems, provides a bound from below to the superfluid weight, in the same way as the Chern number.

A major goal of FLATOPS was to verify that the effect of the quantum metric on the superfluid weight is important in multiband/multiorbital systems and cannot be neglected. This goal has been successfully accomplished by analyzing representative lattice models with flat bands or quasi-flat bands. Ultracold gas experiments with interesting multiband/multiorbital lattices featuring flat bands in their band structure are under way in several groups worldwide, therefore the theoretical predictions put forward during this project may soon find experimental confirmation.

The second major goal of FLATOPS has been to validate mean-field theory, which was the initial approach used for deriving the contribution to the superfluid weight associated to the quantum metric. For this purpose a number of numerical beyond mean-field methods have been employed during this project. Moreover it has also been possible to provide exact analytical results. Therefore at present the geometric effect on the superfluid properties in multiband/multiorbital systems can be regarded as an established theoretical paradigm that has passed numerous consistency checks and only awaits for an experimental confirmation.

It is fair to say that FLATOPS has brought a whole new level of understanding of the physics of flat band systems and their transport properties, which will be crucial for a proper interpretation of future experiments in ultracold gases and may provide a viable route to engineer superconducting materials with increasingly higher critical temperature.

The second major goal of FLATOPS has been to validate mean-field theory, which was the initial approach used for deriving the contribution to the superfluid weight associated to the quantum metric. For this purpose a number of numerical beyond mean-field methods have been employed during this project. Moreover it has also been possible to provide exact analytical results. Therefore at present the geometric effect on the superfluid properties in multiband/multiorbital systems can be regarded as an established theoretical paradigm that has passed numerous consistency checks and only awaits for an experimental confirmation.

It is fair to say that FLATOPS has brought a whole new level of understanding of the physics of flat band systems and their transport properties, which will be crucial for a proper interpretation of future experiments in ultracold gases and may provide a viable route to engineer superconducting materials with increasingly higher critical temperature.