Graphene has attracted enormous attention over the past years in condensed matter physics. The most interesting feature of graphene is that its low-energy excitations are relativistic Dirac fermions. Such feature is the origin of many topological properties in graphene-like physics. On the other hand, ultracold quantum gas trapped in an optical lattice has become a unique setting for quantum simulation of condensed matter physics. Here, we mainly review our recent work on quantum simulation of graphene-like physics with ultracold atoms trapped in a honeycomb or square optical lattice, including the simulation of Dirac fermions and quantum Hall effect with and without Landau levels. We also present the related experimental advances.
Based on the rapid experimental developments of circuit QED,we propose a feasible scheme to simulate the spin-boson model with superconducting circuits,which can be used to detect quantum Kosterlitz-Thouless(KT) phase transition.We design the spinboson model by using a superconducting phase qubit coupled to a semi-infinite transmission line,which is regarded as a bosonic reservoir with a continuum spectrum.By tuning the bias current or the coupling capacitance,the quantum KT transition can be directly detected through tomography measurement on the states of the phase qubit.We also estimate the experimental parameters using the numerical renormalization group method.
