We investigate few body physics in a cold atomic system with synthetic dimensions (Celi et al., PRL 112, 043001 (2014)) which realizes a Hofstadter model with long-ranged interactions along the synthetic dimension. We show that the problem can be mapped to a system of particles (with SU (M ) symmetric interactions) which experience an SU (M ) Zeeman field at each lattice site and a non-Abelian SU (M ) gauge potential that affects their hopping from one site to another. This mapping brings out the possibility of generating non-local interactions (interaction between particles at different physical sites). It also shows that the non-Abelian gauge field, which induces a flavororbital coupling, mitigates the "baryon breaking" effects of the Zeeman field. For M particles, the SU (M ) singlet baryon which is site localized, is "deformed" to be a nonlocal object ("squished" baryon) by the combination of the Zeeman and the non-Abelian gauge potential, an effect that we conclusively demonstrate by analytical arguments and exact (numerical) diagonalization studies. These results not only promise a rich phase diagram in the many body setting, but also suggests possibility of using cold atom systems to address problems that are inconceivable in traditional condensed matter systems. As an example, we show that the system can be adapted to realize Hamiltonians akin to the SU (M ) random flux model. Celi et al. [38] proposed the concept of "synthetic dimensions" which achieves the goal of realizing finite sized "strip" of a Hofstadter model. Their idea, illustrated in Fig. 1, involves atoms with M internal states (labeled 1 . . . γ) in a 1D (this can also be in higher dimensions) optical lattice. The hopping of the atoms from a site j (with coordinate x j = jd, d is the spacing of the optical lattice) to its neighbour does not change its internal state, and the amplitude t is independent of γ. The internal states at a site j are now coherently coupled such that an atom in state γ at site j can "hop" to the state γ + 1 at j with an amplitude Ω j γ . This produces, as shown in Fig. 1, a square lattice strip of finite width with M sites along the "synthetic dimension". Since the coherent coupling is produced by a light of wavenumber k , we have Ω Another interesting aspect of the problem is that the SU (M ) symmetric interactions between the atoms at a site j manifest as "infinite-ranged" (distanceindependent) interactions along the synthetic dimension. For example, two atoms at site j (see Fig. 1) with γ = 1 and γ = 2 will interact with the same strength as γ = 1 and γ = 4(M ). It is the physics of such a system that is the subject of this paper, i. e., to understand interplay between the flux p/q and the SU (M ) interactions. It is essential to focus, as we do, on the physics of few particles since it provides crucial insights into constructing a many body phase diagram of the system. Previous studies [23,[25][26][27] of fermionic atoms with attractive SU (M )-interactions in a simple 1d lattice (no flux, i. e., p q = 0, Ω γ =...

An innovative way to produce quantum Hall ribbons in a cold atomic system is to use M hyperfine states of atoms in a one-dimensional optical lattice to mimic an additional "synthetic dimension." A notable aspect here is that the SU(M) symmetric interaction between atoms manifests as "infinite ranged" along the synthetic dimension. We study the many-body physics of fermions with SU(M) symmetric attractive interactions in this system using a combination of analytical field theoretic and numerical density-matrix renormalization-group methods. We uncover the rich ground-state phase diagram of the system, including unconventional phases such as squished baryon fluids, shedding light on many-body physics in low dimensions. Remarkably, changing the parameters entails interesting crossovers and transition; e.g., we show that increasing the magnetic field (that produces the Hall effect) converts a "ferrometallic" state at low fields to a "squished baryon superfluid" (with algebraic pairing correlations) at high fields. We also show that this system provides a unique opportunity to study quantum phase separation in a multiflavor ultracold fermionic system.

The combined effect of frustration and correlation in electrons is a matter of considerable interest lately. In this context a Falicov-Kimball model on a triangular lattice with two localized states, relevant for certain correlated systems, is considered. Making use of the local symmetries of the model, our numerical study reveals a number of orbital ordered ground states, tuned by the small changes in parameters while quantum fluctuations between the localized and extended states produce homogeneous mixed valence. The inversion symmetry of the Hamiltonian is broken by most of these ordered states leading to orbitally driven ferroelectricity. We demonstrate that there is no spontaneous symmetry breaking when the ground state is inhomogeneous. The study could be relevant for frustrated systems like GdI2, NaTiO2 (in its low-temperature C2/m phase) where two Mott localized states couple to a conduction band.

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Correlated systems with hexagonal layered structures have come to fore with renewed interest in Cobaltates, transition-metal dichalcogenides and GdI2. While superconductivity, unusual metal and possible exotic states (prevented from long range order by strong local fluctuations) appear to come from frustration and correlation working in tandem in such systems, they freeze at lower temperature to crystalline states. The underlying effective Hamiltonian in some of these systems is believed to be the Falicov-Kimball model and therefore, a thorough study of the ground state of this model and its extended version on a non-bipartite lattice is important. Using a Monte Carlo search algorithm, we identify a large number of different possible ground states with charge order as well as valence and metal-insulator transitions. Such competing states, close in energy, give rise to the complex charge order and other broken symmetry structures as well as phase segregations observed in the ground state of these systems.