**Permanent researchers:**

P. Azaria, S. Camalet, N. Dupuis, J.-N. Fuchs, L. Messio, L. Pricoupenko, R. Rossi, P. Sindzingre, J. Vidal

**Theme 1 : Lattice spin systems **(L. Messio, P. Sindzingre)

Mott insulators are crystalline materials with a half-filled conduction band. According to the band theory, they should be conductors, but are nevertheless insulators because of the strong Coulomb repulsions between electrons. These are thus blocked on their atom and cannot carry any current. Only their spin degree of freedom survives. Such magnetic systems are well described by a Heisenberg Hamiltonian (sum of the scalar products of neighboring spin pairs). The system is said to be frustrated when its energy is greater than the sum of the minimum energies of each link, such as in antiferromagnetic models on a crystal lattice comprising triangles (such as triangular or kagome lattices). Then, surprising phenomena arise, for example the appearance of phases with no classical equivalent, not only phases with symmetry breaking (dimer crystal, bond nematics, etc.), but also phases without symmetry breaking up to the absolute zero in temperature, called "spin liquids" but presenting a "topological" order. These "liquid spin" phases have been widely studied in recent years, with the realization and experimental study of many frustrated compounds. Our work focuses on the study of these unconventional phases (of which spin liquids are a part) and is guided both by experiments on many compounds (frustrated systems, nematics, ladders and spin tubes, etc.) and through intimate links with quantum information (entanglement, topological order, etc.). We study in particular the structural changes of the ground states responsible for the transitions of quantum phases (at zero temperature), as well as the excitations which in spin liquids with topological order are very often fractional. We also develop interactions with experimenters to elucidate the properties of new materials. The tools used to carry out these various studies are not only numerical (exact diagonalizations, Monte Carlo simulations, variational approaches) but also analytical and semi-analytical (continuous perturbative unitary transformations, low energy effective theories, high temperature series). Schwinger's theory of bosons (or fermions) in the mean field makes it possible in particular to describe numerous liquid spin phases. It is at first sight difficult to distinguish different phases not breaking any symmetry. But it is possible within the framework of mean field theories thanks to the groups of projective symmetries, characterizing the behavior of fractional excitations (spinons and visons) under the effect of the symmetries of the model.

**Theme 2: ****Topological quantum order** (J.-N. Fuchs, J. Vidal)

The concept of topological quantum order was introduced by X. G. Wen at the end of the 1980s. In a very schematic, two-dimensional way, a system exhibits topological order if the degeneracy of its ground state depends on the topology of the surface (sphere, torus, etc.). This topological degeneracy finds its origin in the nature of the excitations called anyons. For twenty years, topologically ordered systems have been at the heart of many studies, especially since the work of A. Kitaev who revealed their potential for quantum computing as well as information storage. The essential idea of this founding work is that the non-locality of this order makes these systems insensitive to external disturbances. Most of our research work concerns the study of phase transitions in topologically ordered models such as "string networks".

**Theme 3: ****Electron crystals and electrons in crystals** (J.-N. Fuchs, R. Rossi, J. Vidal)

In the dense phase, an electron gas forms a Fermi liquid; this is the case, for example, of the valence electrons of solid sodium. Low-energy excitations are characterized by the effective mass of the quasiparticles, their spectral weight Z, etc. In less dense phase, the electron gas is more correlated and eventually forms a Wigner crystal. We are interested in the quantitative description of the ground state and gas excitations as a function of its density, in two and three dimensions. Are there different exotic phases of Fermi's liquid and Wigner's crystal? Within the framework of the Hartree-Fock approximation (for a homogeneous electron gas), we find a richer scenario than a first-order transition between liquid and crystal: the electron density in the ground state is always periodic and evolves continuously between the crystal and the Fermi liquid, giving rise to anisotropic densities of states. We verified that these states persist when taking into account the correlations (Hartree-Fock and Jastrow). The next step is to verify that these states still persist with the more precise method of Diffusion Monte Carlo. We are also interested in another type of electron gas. These are crystals in which the low-energy quasi-particles are Dirac fermions. This is particularly the case for graphene, but also for other band structures, such as those of topological insulators. These systems have the particularity of involving a hidden geometry of the band structure (Berry curvature and quantum metric) whose topology can be non-trivial (Chern number). In particular, the response (orbital susceptibility) to a magnetic field is studied.

**Theme 4****: Ultracold atomic gases** (P. Azaria, N. Dupuis, L. Pricoupenko, R. Rossi)** **

Ultracold atomic gases offer an experimental realization of strongly correlated quantum fluids and are therefore of interest to condensed matter theorists. These systems are characterized by a remarkable control of the experimental parameters and the possibility of a quantitative comparison between theory and experiment. They allow not only the simulation of Hamiltonian models of solids (fermionic gases in reduced dimensions, quantum particles moving on a lattice, etc.) but also the realization of systems without equivalent in "traditional" condensed matter (fermion-boson mixtures, bosons or fermions with high spin quantum number, etc.). Part of our activity concerns the study of systems with a small number of bodies within the framework of effective low-energy approaches (study of bound states of Efimov with three or four bodies, reduced dimension of waveguides) . Another part concerns the study of the collective low-energy properties of ultracold gases from numerical or semi-analytical methods (low-energy effective theories, non-perturbative renormalization group): one-dimensional fermionic gases with several spin components (diagram phases, exotic superfluid phases, confinement and analogy with QCD), intrinsic superconductivity and topological phases in one-dimensional systems, spin one boson gas, superfluid-Mott insulator transition of a boson gas in an optical lattice (thermodynamics , transport, entanglement), spin waves in gases of pseudo-spin 1/2 bosons (e.g. in atomic clocks), gases of fermions or bosons in optical lattices leading to non-trivial band structures (e.g. with Dirac dots), etc.

**Theme 5****:** **Quantum information** (S. Camalet)

Quantum information processing seeks to take advantage of all the possibilities offered by quantum physics to process information more efficiently. In this context, the study of entanglement and its consequences is of great interest. Our activity on this topic develops along two axes: (1) After having worked on a detailed description of the Hilbert space, with regard to an appropriate measure of this entanglement, we now project, within the framework of topological quantum computation , to analyze the entanglement of anyons in interaction. (2) We study the relationship between bipartite entanglement and other quantum characteristics, either of the two entangled systems, such as nonlocality, or of only one, such as contextuality or a quantum resource. We have obtained an inequality showing that such a local quantum property and the entanglement limit each other. Concerning nonlocality, we have shown that a local state in Bell's sense can be more entangled, for any measure, than nonlocal states. Finally, we are going to develop another research theme, that of lattice quantum walks, quantum analogues of classical random walks, which can be the support of quantum algorithms.