Interesting properties of materials are all characterized by strongly interacting electrons, whose low-energy physics cannot accurately be obtained through conventional techniques.

Interesting properties of materials are all characterized by strongly interacting electrons, whose low-energy physics cannot accurately be obtained through conventional techniques.A novel approach is to try to use the optical lattice system as an emulator for such models. An emulator is an artificial material whose behavior is governed by the same underlying mathematical description as the material of interest. Over the past 10 years, the development of the field of cold gases was spectacular and one has realized such models as the Hubbard model in the atomic physics lab, but at relatively high temperatures compared to the Fermi temperature.

The physics of those models is reasonably well under control theoretically and numerically, but this will no longer be the case at lower temperatures. Presently, major efforts are being carried out in order to lower the temperature and improve experimental detection tools. These experimental developments have sparked a renewed interest in studying those models numerically with extremely high precision, and has also led to new ideas for algorithms that would work at lower temperature.

This workshop focuses on three specific topics where the experimental and numerical developments will be addressed, and which matter greatly for understanding complex materials:

1) The Fermi-Hubbard model, which is the cornerstone for high-Tc superconductivity and frustrated magnetism. It remains one of the most studied and most elusive models in condensed matter physics.

2) Impurity models and disorder : A single impurity interacting with its environment quite often determines the low-temperature physics of a system. The classic example is an electron moving in a crystal lattice, where the electron gets dressed by the phonons (lattice distortions) and acquires a mass and a dispersion of a quasi-particle that is different from that of the bare electron. Since impurities and disorder are present in any material, studying those effects in a controlled fashion is important.

3) Criticality: phase transitions are at the heart of our interest and understanding of materials. The systems built with cold gases are inherently mesoscopic and feature an external potential. Is it possible to study criticality in those systems? Can new tools such as in-situ density measurements with sinlge-site resolution help?

**1. The Fermi-Hubbard model with cold gases**

The Fermi-Hubbard model is challenging both from theoretical and experimental point of view. There is no numerically exact method that can deal with many fermions in the general case and the methods that were applied successfully for the bosons will not work at low temperatures [1]. Experimentally, one has been able to cool fermions to degeneracy in the optical lattice and observe the Fermi surface [2]. More recently, the Mott phase in the Fermi-Hubbard model has been detected [3, 4]. It has been identified by looking at the probability that two fermions (up and down spins) occupy the same state [3]. When this probe tends to zero, it is indicative of a Mott state. The ’core compressibility’, which looks at the compressibility of the atoms in the center and filters out the edges which, by experimental design, should always be in a Fermi liquid regime, is a useful indicator [5]. Other experiments providing novel physics include the ’shaking’ of the lattice [3] and the build-up rate of the number of doublons [3, 6–9], the in-trap expansion of the cloud [10], and the lifetime of a doublon [11, 12]. The use of high temperature series expansions in combination with a local density approximation to describe the experiment was also suggested [5]. Thermodynamic equilibrium over at least most of the lattice has been established, while temperature, entropy, a confirmation of the Mott insulator phase, and an error budget for the Fermi Hubbard system have also been pointed out [13]. We also showed that DMFT [14] and the series expansions give exactly the same result at the present experimental temperatures. There are a few exact methods available when there is a special symmetry. Quantum Monte Carlo simulations are possible for attractive interactions when the population of up and down spins is equal, and for repulsive interactions at half filling only when there is a particle-hole symmetry. The BCS-BEC crossover problem in the lattice description belongs to this class and has been mapped out exactly at unitarity [15] , and also away from unitarity [16]. A novel method, diagrammatic Monte Carlo, was recently proposed [17, 18] and validated DMFT and DCA in the U/t = 4 regime [19]. All of these are the first steps towards the

study of quantum magnetism, which will be reached at a temperature that is not too far from presently achieved temperatures [20]. The virtual exchange mechanism has already been observed for a two-well system [21].

**2. Impurities, mixtures, and disorder**

Present cold-gas experiments are in a temperature regime where the physics of impurities and disorder is accessible. The so-called Fermi polaron problem in which a single fermion is (resonantly) coupled to the Fermi sea consisting of fermions with a different spin, has been studied experimentally [22, 23] (it is a limiting case of the experiments on spin-imbalanced Fermi gases, which were studied in great detail in the past [24]). The transition to molecular binding was observed [22], as well as a measurement of the polaron energy and the polaron residue at unitarity [22, 23]. The equation of state of an (imbalanced) Fermi gas has been measured at unitarity, and the mixed normal state interpreted in terms of an ideal gas of polarons [25, 26].

Fixed-node diffusion Monte Carlo studies found binding energies and an effective mass in good agree- ment with experiment [27, 28]. N. V. Prokof’ev and B. V. Svistunov used their recently developed diagrammatic Monte Carlo approach to sample Feynman diagrams, and could obtain in principle unbi- ased answers in excellent agreement with experiment [29, 30]. The answers had reached convergence in low expansion orders, where the sign problem was still tolerable. On the theory side, F. Chevy has pro- duced a variational ansatz for the polaron that is in remarkable quantitative agreement with experiment and exact numerical calculations [31]. This ansatz wavefunction sums all particle-hole diagrams on top of the Fermi sea, and it can be shown that contributions with more than one particle-hole diagram interfere destructively almost exactly for the groundstate energy, especially at unitarity [32]. An analogous Ansatz wavefunction was also put forward for the molecular side [33, 34].

Disordered systems have been studied experimentally for bosons in three dimensions [35, 36], revealing a superfluid and an insulating phase by looking at time-of-flight images and measuring the center-of- mass speed after giving a kick to the atomic cloud. In one dimension, the first indications of Anderson localization were observed [37, 38]. The Florence group used a quasi-periodic potential that constitutes the exactly solvable Aubry-Andre model, which predicts a transition from extended to localized states [37]. The Paris group instead used a speckle pattern to generate uncorrelated disorder [38]. The localization length was determined by measuring the size of the cloud as a function of expansion time after switching off the longitudinal confinement.

The experiments on the three-dimensional disordered bose systems quite well [39, 40]. We computed the full phase diagram and explained that the Bose Glass phase always has to intervene between the Mott insulator and the superfluid, also at commensurate densities, [39], and we also explained why experiment is seeing a transition between an insulating and a superfluid state at outrageously strong disorder by invoking a percolation argument [40]. We could also give strong reasons based on temperature and finite size effects why present experiments are rather limited in exploring the full phase diagram. There have also been simulations on disordered Bose systems in continuous space also showing indications of a Bose glass phase [41].

The one-dimensional experiments on Anderson localization were in good agreement with theory.

**3. single-site resolution tools and Criticality**

The study of criticality in cold gases is best known for measuring the 3d XY exponent of the normal-to- superfluid Bose gas [42], and of the Berezinskii-Kosterlitz-Thouless transition in a trapped 2d atomic Bose gas [43]. The mesoscopic size and confinement potential pose fundamental challenges though. Recently, advances such as the cold gas microscope [44] have led to the ability to observe individual atoms with single site resolution using optical means [45–48] or with electrons [49] for bosonic systems. The density profile of the Mott insulator has been observed, as well as the superfluid edges, from which temperature could be determined in fair agreement with estimates based on time-of-flight images [50].

Single site resolution tools are a new probe to study criticality. In the large-trap limit one can perform trap-size scaling [51], analogous to conventional finite size scaling but with an additional parameter

describing the curvature of the trap. For present traps, the curvature and finite size put a limit on the accuracy on which the critical point can be determined parametrically and numerically [52], from which it follows that experiments will be challenging. Time-of-flight images are still the best tool when condensation is involved [52]. Single site detection tools are also useful in obtaining the full equation of state.