Quantum Phase Transition And Critical Properties For Tuned System Of Bonsons On Optical Lattices

Systems of ultracold bosonic gases in optical lattices represent nowadays a popularresearch topic, as they establish a versatile bridge between the field of ultracold quantummatter and solid-state systems. In particular, they can be experimentally controlled with ayet unprecedented level of precision. With this it allows for a clear theoretical analysis andwhich are even predestined as universal quantum simulators. Recent research efforts havetargeted more complex systems, because in this system the corresponding phase diagramsbecome richer and more complex. In this thesis, we study the quantum phase transitionand critical properties for Bosons on optical lattice, including the superlattice system andS-wave periodically driven system.For optical superlattice system, due to artificially tuned, the translational symmetryhas been broken, which induced more complex quantum phase diagram and critical be-havior. To study the phase diagram we generalize a recently established effective potentialLandau theory for a single component to the case of multi components. We find not onlythe characteristic incompressible solid phases with fractional filling, but also obtain theunderlying quantum phase diagram in the whole parameter region at zero temperature.The analytical of the critical points of the phase lobes shows the competition of Density-Wave phase and Mott phase. Also the comparison of the phase diagram that we got withthe high accuracy of the Quantum Monte Carlo simulation proves the e?ciency of thegeneralization.Then we use the Generalized Effective Potential Landau Theory method to studythe frustrated optical superlattice system. The tune of these more complicated systemsnot only make the Density Wave phase appear, but also shows the new phenomenon ofthe superfluid. The study of the critical behavior of Kagome optical superlattice revealsthat the bias of the anisotropy of the superfluid density is alternating between differentdirection while detuning the parameter of the system.Apart from the tune on the real space as superlattice system, another tuned opticallattice system that we study in this thesis is the Bosons on optical lattice with a periodicmodulation of the S-wave scattering length. At first we map the underlying periodicallydriven Bose- Hubbard model approximately to an effective time-independent Hamiltoni-an with a conditional hopping. Combining different analytical approaches with quantumMonte Carlo simulations then reveals that the superfluid-Mott insulator quantum phasetransition still exists despite the periodic driving and that the location of the quantumphase boundary turns out to depend quite sensitively on the driving amplitude. A moredetailed quantitative analysis shows even that the effect of driving can be described withinthe usual Bose-Hubbard model provided that the hopping is rescaled appropriately withthe driving amplitude when the driven amplitude is smaller than some certain value. Thisfinding indicates that the Bose-Hubbard model with a periodically driven s-wave scatter-ing length and the usual Bose- Hubbard model belong to the same universality class fromthe point of view of critical phenomena.For the system of Bosons on optical lattice witha periodic modulation of the s-wave scattering length, when the driven amplitude goeslarger, our analytical and numerical method will be not suitable. This made us turn ourinterest to the newly developed numerical method-process-chain method. We first checkour program by using it to calculate the strong coupling method and effective potentialtheory to high order for the Bose-Hubbard model. The result that we get comparing withthe numerical simulation result shows that we have very high accuracy. Then we use it tostudy the phase diagram of the system of Bosons on optical square lattice with a period-ic modulation of the s-wave scattering length. We found that when the driven amplitudegoes larger, the shape of phase diagram of the system will change, which indicates that wecan not use a Bose-Hubbard like effective Hamiltonian to describe the system anymore.