Macroscopic order appears as the collective behaviour of many interacting particles. Prime examples are superfluidity in helium 1 , atomic Bose-Einstein condensation 2 , s-wave 3 and d-wave superconductivity 4 and metal-insulator transitions 5. Such physical properties are tightly linked to spin and charge degrees of freedom and are greatly enriched by orbital structures 6. Moreover, high-orbital states of bosons exhibit exotic orders distinct from the orders with real-valued bosonic ground states 7. Recently, a wide range of related phenomena have been studied using atom condensates in optical lattices 8-10 , but the experimental observation of highorbital orders has been limited to momentum space 11,12. Here we establish microcavity exciton-polariton condensates as a promising alternative for exploring high-orbital orders. We observe the formation of d-orbital condensates on a square lattice and characterize their coherence properties in terms of population distributions both in real and momentum space. Exciton-polaritons emerge from the strong light-matter coupling in semiconductor quantum wells embedded in a planar microcavity structure. They behave as degenerate Bose gases in the low-density and low-temperature limit 13. Exciton-polaritons have undergone a dynamic phase transition, in which a macroscopic number of particles are accumulated in the lowest-energy single-particle state with a long-range order 14-17. Owing to their very light effective mass, the phase transition temperatures of exciton-polaritons are eight to nine orders of magnitude higher than those of atomic Bose-Einstein condensates. Coherence properties of exciton-polariton condensates have been characterized by the direct optical access in spatial and momentum spaces 14-16,18. Modern solid-state physics has studied quantum many-body phenomena whose properties reflect exotic orbital nature, another intrinsic degree of freedom, which interplays with charge and spin degrees of freedom. Its energy degeneracy and spatial anisotropy generate rich dynamics in weakly interacting many-body systems. For example, a key role of d-orbital nature has been actively investigated in salient phenomena including metal-insulator transitions 5 , colossal magnetoresistance 6,19 , and recently discovered iron-pnictide superconductors 20,21. These orbital ordering phenomena originate from the strong correlation effects of electrons in the anisotropic degenerate d-orbitals. Theoretical modelling of such

Microcavity exciton-polariton condensates, as coherent matter waves, have provided a great opportunity to investigate hydrodynamic vortex properties, superfluidity and low-energy quantum state dynamics. Recently, exciton condensates were trapped in various artificial periodic potential geometries: one-dimensional (1D), 2D square, triangular and hexagonal lattices. The 2D kagome lattice, which has been of interest for many decades, exhibits spin frustration, giving rise to magnetic phase order in real materials. In particular, flat bands in the 2D kagome lattice are physically interesting in that localized states in the real space are formed. Here, we realize exciton-polariton condensates in a 2D kagome lattice potential and examine their photoluminescence properties. Above quantum degeneracy threshold values, we observe 6

We propose a device for studying the Fermi-Hubbard model with long-range Coulomb interactions using an array of quantum dots defined in a semiconductor two-dimensional electron gas system. Bands with energies above the lowest energy band are used to form the Hubbard model, which allows for an experimentally simpler realization of the device. We find that depending on average electron density, the system is well described by a one-or two-band Hubbard model. Our device design enables the control of the ratio of the Coulomb interaction to the kinetic energy of the electrons independently to the filling of the quantum dots, such that a large portion of the Hubbard phase diagram may be probed. Estimates of the Hubbard parameters suggest that a metal-Mott insulator quantum phase transition and a d-wave superconducting phase should be observable using current fabrication technologies.