We use quantum Monte Carlo (QMC) simulations to study the combined effects of harmonic confinement and temperature for bosons in a two dimensional optical lattice. The scale invariant, finite temperature, state diagram is presented for the Bose-Hubbard model in terms of experimental parameters -the particle number, confining potential and interaction strength. To distinguish the nature of the spatially separated superfluid, Mott Insulator and normal Bose liquid phases, we examine the local density, compressibility, superfluid density and Green's function. In the annular superfluid rings, as the width of the ring decreases, the long range superfluid correlations start to deviate from an equivalent homogeneous 2D system. At zero temperature, the correlation decay is intermediate between 1D and 2D, while at finite temperature, the decay is similar to that in 1D at a much lower temperature. The calculations reveal shortcomings of the local density approximation (LDA) in describing superfluid properties of trapped bosons. We also present the finite temperature phase diagram for the homogeneous two dimensional Bose-Hubbard model. We compare our state diagram with the results of a recent experiment at NIST on a harmonically trapped 2D lattice [Phys. Rev. Lett. 105, 110401 (2010)], and identify a finite temperature effect in the experiment.

We propose that the temperature dependence of a single quantity R = κi/δn 2 i , the ratio of the local compressibility to the local number fluctuations, can be used to map out the finite temperature phase diagram, diagnose the critical region around a quantum phase transition, and identify critical points belonging to different universality classes. We test our proposal using state-of-the-art largescale quantum Monte Carlo simulations of the two-dimensional Bose Hubbard model. Our results have implications for recently developed single site imaging experiments.

At a quantum phase transition, one ground state evolves into a different one by passing through a quantum critical region with enhanced spatial and temporal fluctuations. A method to map the quantum critical region using the single, local quantity R, the ratio of compressibility to local number fluctuations is proposed. R can be calculated from in situ experiments and also enables thermometry and phase diagnosis (for example whether superfluid or Mott insulating). The definition of R can be generalized to inhomogeneous systems and provides a powerful tool for experimentally mapping the finite temperature phase diagram demonstrated here for the two-dimensional Bose Hubbard model.

The equilibrium thermodynamics of bosons and fermions in optical lattices are considered in the single-band Hubbard regime, with an emphasis on interesting magnetic, superfluid, and spin liquid ground states. The parameters of the Hubbard model—the tunneling and interaction parameters—can be obtained quantitatively in terms of the strength and periodicity of the optical lattice potential, tuned by the laser intensity and wavelength. This direct link between the parameters of a theoretical model and the actual experimental optical lattice gives rise to the phase diagram of the Bose–Hubbard model. The definition of the order parameter as an expectation value of the annihilation operator in a coherent state is discussed. This phase-coherent superfluid state is contrasted with the phase-incoherent Mott state naturally defined in terms of number states. Following the repulsive Bose–Hubbard model, the phase diagram of the Fermi–Hubbard model is considered with both attractive and repulsive interactions.