Abstract:
Introducing the Dyson-Maleev transformation. the coherent state ansatz and the time-dependent variation principle, we obtain two partial different equations of motion from Hamiltonian. Employing the method of multiple scales. we reduce these equations into the envelope function equations and force the amplitude function to satisfy a nonlinear Schrodinger equation. Using the inverse- scat-tering transformation, we obtain the single soliton solution and discuss the solitary magnon localization in antiferromagnet RbFeBr3

Abstract:
Employing the coherent state ansatz and the time-dependent variational principle, we obtain a partial differential equation of motion from Hamiltonian in inharmonic molecuIar crystals. By using the method of multiple scales. we reduce this equation into the envelope function and find that the amplitude function satisfied a nonlinear Schrodinger equation. Introducing the inverse scattering transformation, we gain the single-, twry and N- soliton solutions. The energy and the spatial configurations of the system are given. We also acquire the periodic wave solution and analyze its stability.

Abstract:
The exact eigenstates of the Hamiltonian of a continuum model for heavy-electron metal are constructed by using the Bethe ansatz. The Bethe ansatz equations are obtained from the periodic boundary conditions. The results show that this system is also completely integrable.

Abstract:
Since 2008, many new families of iron-based high temperature (high-$T_c$) superconductors have been discovered \cite{Hosono,ChenXH,FeTe,ChenXL}. Unlike all parent compounds of cuprates that share a common antiferromagnetically (AF) ordered ground state, those of iron-based superconductors exhibit many different AF ordered ground states, including collinear-AF (CAF) state in ferropnictides \cite{caf}, bicollinear-AF (BCAF) state in 11-ferrochalcogenide $FeTe$ \cite{bcaf,bcaf2}, and block-AF (BAF) state in 122-ferrochalcogenide $K_{0.8}Fe_{1.6}Se_2$ \cite{baf}. While the universal presence of antiferromagnetism suggests that superconductivity is strongly interrelated with magnetism, the diversity of the AF ordered states obscures their interplay. Here we show that all magnetic phases can be unified within an effective magnetic model. This model captures three incommensurate magnetic phases, two of which have been observed experimentally. The model characterizes the nature of phase transitions between the different magnetic phases and explains a variety of magnetic properties, such as spin-wave spectra and electronic nematism. Most importantly, by unifying the understanding of magnetism, we cast new insight on the key ingredients of magnetic interactions which are critical to the occurrence of superconductivity.

Abstract:
Goldstone and Higgs modes have been detected in various condensed matter, cold atom and particle physics experiments. Here, we demonstrate that the two modes can also be observed in optical systems with only a few (artificial) atoms inside a cavity. We establish this connection by studying the $ U(1)/Z_2 $ Dicke model where $ N $ qubits (atoms) coupled to a single photon mode. We determine the Goldstone and Higgs modes inside the super-radiant phase and their corresponding spectral weights by performing both $ 1/J=2/N $ expansion and exact diagonization (ED) study at a finite $ N $. We find nearly perfect agreements between the results achieved by the two approaches when $ N $ gets down even to $ N = 2 $. The quantum finite size effects at a few qubits make the two modes quite robust against an effectively small counter-rotating wave term. We present a few schemes to reduce the critical coupling strength, so the two modes can be observed in several experimental systems of (artificial) atoms inside a cavity by just conventional optical measurements.

Abstract:
We investigate the fidelity susceptibility (FS) of a two-dimensional spin-orbit coupled (SOC) Fermi superfluid and the topological phase transition driven by a Zeeman field in the perspective of its ground-state wave-function. Without Zeeman coupling, FS shows new features characterizing the BCS-BEC crossover induced by SOC. In the presence of a Zeeman field, the topological phase transition is explored using both FS and the topological invariant. In particular, we obtain the analytical result of the topological invariant which explicitly demonstrates that the topological phase transition corresponds to a sudden change of the ground state wave-function. Consequently, FS diverges at the phase transition point with its critical behavior being: $\chi \propto ln|h-h_{c}|$ . Based on this observation, we conclude that the topological phase transition can be detected by measuring the momentum distribution in cold atoms experiment.

Abstract:
The Photoluminescent (PL) measurements on photons and the transport measurements on excitons are the two types of independent and complementary detection tools to search for possible exciton superfluids in electron-hole semi-conductor bilayer systems. In fact, it was believed that the transport measurements can provide more direct evidences on superfluids than the spectroscopic measurements. It is important to establish the relations between the two kinds of measurements. In this paper, using quantum Heisenberg-Langevin equations, we establish such a connection by calculating various exciton correlation functions in the putative exciton superfluids. These correlation functions include both normal and anomalous Greater, Lesser, Advanced, Retarded, and Time-ordered exciton Green functions and also various two exciton correlation functions. We also evaluate the corresponding normal and anomalous spectral weights and the Keldysh distribution functions. We stress the violations of the fluctuation and dissipation theorem among these various exciton correlation functions in the non-equilibrium exciton superfluids. We also explore the input-output relations between various exciton correlation functions and those of emitted photons such as the angle resolved photon power spectrum, phase sensitive two mode squeezing spectrum and two photon correlations. Applications to possible superfluids in the exciton-polariton systems are also mentioned. For a comparison, using conventional imaginary time formalism, we also calculate all the exciton correlation functions in an equilibrium dissipative exciton superfluid in the electron-electron coupled semi-conductor bilayers at the quantum Hall regime at the total filling factor $ \nu_T= 1 $. We stress the analogies and also important differences between the correlations functions in the two exciton superfluid systems.

Abstract:
The double or extended main-sequence turnoffs (MSTOs) in the color-magnitude diagram (CMD) of intermediate-age massive star clusters in the Large Magellanic Cloud are generally interpreted as age spreads of a few hundred Myr. However, such age spreads do not exist in younger clusters (i.e., 40-300 Myr), which challenges this interpretation. The effects of rotation on the MSTOs of star clusters have been studied in previous works, but the results obtained are conflicting. Compared with previous works, we consider the effects of rotation on the MS lifetime of stars. Our calculations show that rotating models have a fainter and redder MSTO with respect to non-rotating counterparts with ages between about 0.8 and 2.2 Gyr, but have a brighter and bluer MSTO when age is larger than 2.4 Gyr. The spread of the MSTO caused by a typical rotation rate is equivalent to the effect of an age spread of about 200 Myr. Rotation could lead to the double or extended MSTOs in the CMD of the star clusters with ages between about 0.8 and 2.2 Gyr. However, the extension is not significant; and it does not even exist in younger clusters. If the efficiency of the mixing were high enough, the effects of the mixing would counteract the effect of the centrifugal support in the late stage of evolution; and the rotationally induced extension would disappear in the old intermediate-age star clusters; but younger clusters would have an extended MSTO. Moreover, the effects of rotation might aid in understanding the formation of some "multiple populations" in globular clusters.

Abstract:
The purpose of this work is to find out the best-fit models of KIC 11081729, and then test whether the special oscillation characteristics of KIC 11081729 can be directly reproduced by the models. After constructing a grid of evolutionary tracks, the chi squared method of spectroscopic constraints and individual frequencies is used to obtain a set of solutions. Then the observed frequency ratios $r_{01}$ and $r_{10}$ and those calculated from theoretical models are compared. The frequency ratios $r_{01}$ and $r_{10}$ of KIC 11081729 decrease firstly and then increase with frequency. With different spectroscopic constraints, all models with overshooting parameter $\delta_{\mathrm{ov}}$ less than 1.7 can not reproduce the distributions of the observed ratios. However, the distributions of the observed ratios can be directly reproduced by the models with $\delta_{\mathrm{ov}}$ in the range of about $1.7-1.8$. The estimations of mass and age of the star can be affected by spectroscopic results, but the determination of the $\delta_{\mathrm{ov}}$ is not dependent on the spectroscopic results. The characteristics of $r_{01}$ and $r_{10}$ of KIC 11081729 may result from the effects of a large overshooting of convective core. With the constraints of the ratios and spectroscopic results, the estimated age for KIC 11081729 is obviously larger than that determined in previous work. The observed ratios and those calculated from theoretical models can be reproduced well by a function of sine. The function aids in understanding the ratios and the size of overshooting region of convective core from observed frequencies.

Abstract:
We review our recent theoretical advances in quantum information and many body physics with cold atoms in various external potential, such as harmonic potential, kagome optical lattice, triangular optical lattice, and honeycomb lattice. The many body physics of cold atom in harmonic potential is investigated in the frame of mean-field Gross-Pitaevskii equation. Then the quantum phase transition and strongly correlated effect of cold atoms in triangular optical lattice, and the interacting Dirac fermions on honeycomb lattice, are investigated by using cluster dynamical mean-field theory and continuous time quantum Monte Carlo method. We also study the quantum spin Hall effect in the kagome optical lattice.