chuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang họcchuyển pha kim loại – điện môi trong một số hệ tương quan mạnh trên mạng quang học
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ––––––––––––––– TRAN THI THU TRANG METAL-INSULATOR TRANSITION IN SOME STRONGLY CORRELATED ELECTRONIC SYSTEMS IN OPTICAL LATTICES Major: Theoretical and Mathematical Physics Code: 9.44.01.03 SUMMARY OF DOCTORAL THESIS THEORETICAL PHYSICS Ha Noi - 2023 The thesis was conducted at the faculty of Physics, Hanoi National University of Education, and Institute of Physics, Vietnam academy of science and technology Science supervisor: Assoc Prof Dr Hoang Anh Tuan Assoc Prof Dr Le Đuc Anh Referee 1: Assoc Prof Dr Phan Van Nham Duy Tan University Referee 2: Assoc Prof Dr Đo Van Nam Phenikaa University Referee 3: Assoc Prof Dr Pham Van Hai Hanoi National University of Education The dissertation will be defended at Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Ha Noi, Viet Nam Time: ., , 2023 This thesis can be studied at: - National Library of Vietnam - Library of Hanoi National University of Education INTRODUCTION Motivation Metal-Insulator Transition (MIT) is one of the fundamental research problems of condensed matter physics Both theoretical and experimental research has been carried out and obtained many results to understand why partially filled materials can be insulators or insulators might become metals when control parameters are changed Theoretically, Mott gave the first explanation of the insulator state formed by electron-electron correlation, and since it’s called the Mott insulator Two commonly used models to describe correlated electron systems are the standard Hubbard model and the Falicov-Kimball model The Hubbard model is a simple but limited model that only is solved exactly in the case of a one-dimensional system or infinite dimensions system The Falicov – Kimball model (FKM) is a reduced Hubbard model when particles with up spin not move As per the Hubbard model, MIT in a half-filled system also occurs at FKM when U changes The fundamental difference between the two models is that the metallic phase in the Hubbard model is a Fermi liquid while it is a non-Fermi liquid in the FKM A natural combination of these two models is the Asymmetric Hubbard Model (AHM), where each particle with a different spin has different jumping parameters Most of the theoretical works on AHM focus on establishing phase diagrams in one-dimensional systems for both attractive and repulsive interactions The ground states are diverse, including Mott insulators, charge order waves, and superconductors However, it should be noted that the main problem when analyzing Hubbard-type models is the lack of reliable analytical methods for the large U Numerical methods such as Monte-Carlo simulation or exact crossover will work well for systems of small size but consume so much time and computational resources Analytical approximations not require much computation time Still, each method has its limited range of application, and in many cases, we don't know whether the results obtained from the approximation reveal the physical nature of the model Therefore, it is necessary to use many different methods to study, approach, and clarify the phase diagram of AHM In addition to AHM, we also study MIT in a more comprehensive model when simultaneously having asymmetric integral-hopping and site-dependent interactions The research models in the thesis have been established on optical networks The goals of the research Clarifying conditions for MIT occurrence and insulator phase characteristics in some strongly correlated systems, specifically: • Hubbard model with site-dependent interactions, • Asymmetric Hubbard model, • Mass-imbalanced Hubbard model with site-dependent interactions The objects of research Hubbard model with site-dependent interactions, Asymmetric Hubbard model, Mass-imbalanced Hubbard model with sitedependent interactions Metal-Insulator Transition in the strongly correlated electronic systems The scope of research Some strongly correlated electronic systems in optical lattices in paramagnetic and at the half-filling The methods of research Analytical methods: • Combining Green function formalism and CPA to find equations that allow exploring MIT in the Hubbard model with site-dependent interactions • Using DMFT and EMM to study MIT in the asymmetric Hubbard model • Using 2S-DMFT and DMFT in combination with EMM to study the mass-imbalanced Hubbard model with site-dependent interactions Numerical method: Use mathematic software (Fortran) to calculate the density of states, phase diagram, and double occupancy that are the basis for assessing phase transition conditions and characteristics of Mott phases Scientific meanings of the thesis The research results have shown the influence of the mass imbalance and the site-dependence of the alternative interactions on the metal-insulator phase transition conditions and the nature of the Mott insulator phases in the mixture of two-component fermion gas on an optical system with heterogeneous interactions In the limited cases when the system only has mass imbalances or heterogeneous interactions, the obtained results are in good agreement with the previous results The theoretical models studied in the thesis can be verified, tested, and compared by quantum simulations in an optical lattice for more understanding of the rule of Mott insulator phases New contributing of the thesis The thesis establishes the phase diagram, and the conditions of MIT and clarifies the nature of the Mott insulator phases through the number of double occupancy of the strongly correlated system at the half-filling Using 2S-DMFT, we study an asymmetric Hubbard model with site-dependent interactions, in a half-filled state, at a temperature of T=0K We obtain the phase diagram of the system analytically, where the metallic region reduces as the mass imbalance increases | 𝑈𝐴 𝑈𝐵 | = (𝑡↑ + 𝑡↓ + √𝑡↑2 + 𝑡↓2 + 14𝑡↑ 𝑡↓ ) We also show the ground-state properties of the system: when the mass imbalance is significant, the light fermions are more renormalized than the heavy fermions, and even when they are in a sublattice with the smaller on-site interaction, the strength of the critical interactions decreases as the mass imbalance increases The results are reliable when finding the individual results in the limited cases, showing that 2S-DMFT is a good method to study MIT in a strongly correlated system The structure of the thesis In terms of layout, in addition to the introduction, conclusion, and references, our thesis has four chapters with the following specific contents: Chapter 1: Overview of the metal-Mott insulator transition Chapter 2: Metal-insulator transition in the Hubbard model with one additional parameter Chapter 3: Metal-insulator transition in the mass-imbalanced Hubbard model with site-dependent interactions Chapter OVERVIEW OF THE METAL-MOTT INSULATOR TRANSITION 1.1 Insulator classification According to the energy band theory, the filling determines the conductive properties of a solid In the half-filled state or when the number of electrons on site is odd, solid is metal because of having an unfilled region However, sometimes this is not true The limitation of the band theory is based on the single-particle picture, ignoring the repulsive interactions between electrons in the crystal Thus, when the Coulomb interaction between electrons in a crystal is weak, the energy band theory can describe it quite well But when it increases and cannot be ignored, this picture is no longer available to respond Since Mott's in-depth studies at MIT were motivated by electron-electron interactions, many insulators formed by this type of interaction are called "Mott insulators" 1.2 Hubbard model and Metal-Mott insulator A landmark for theoretical research at MIT based on a simple model for fermion systems - the Hubbard Model - a very successful model used to solve this problem, proposed by J Hubbard in the 1960s The model is interested in electrons in a single band with a simple Hamiltonian 𝑈 + 𝐻 = ∑ 𝑡𝑖𝑗 (𝑎𝑖,𝜎 𝑎𝑗,𝜎 + ℎ 𝑐 ) + ∑ 𝑛𝑖,𝜎 𝑛𝑖,−𝜎 , (1.1) ,𝜎 𝑖,𝜎 + in there: 𝑎𝑖,𝜎 (𝑎𝑖,𝜎 ) is the operator that generates (annihilates) + electrons with spin σ on site i; 𝑛𝑖,𝜎 = 𝑎𝑖,𝜎 𝑎𝑖,𝜎 is the particle number operator with spin σ on site i; tij is the hopping integral, which characterizes the mobility of electrons; U is the Coulomb interaction potential on a site, which determines the localization of electrons The two core parameters in the Hubbard model are the electron correlation strength U/t and the fill number n MIT in extended Hubbard models has recently been studied with many different methods, but there is still no complete theory 1.3 Optical lattice Optical lattices are crystals of light formed by the interference of laser beams creating a periodic effective potential that can trap neutral atoms if it is cold enough In the optical lattice, atoms are trapped at positions of minimum potential energy like a real lattice Real crystals are complex because of many competing interactions, disorder, and lattice vibrations Therefore, it is very difficult to calculate the observed phenomena from the experiment Meanwhile, optical lattice provides an ideal lattice without error or vibration in which the interactions can be finely tuned It can provide a test model for theoretical studies of solid crystal physics There are many new methods to control the parameters of the ultracold atomic system in optical crystal lattice: Geometry and lattice size; Phonon; Tunneling (hop integral - t); interaction on-site – U; Proximity interactions and long-range interactions; Spin-dependent optical network; Multi-particle interaction (plaquette); Interaction potential; Temperature; Time-dependent With the achievement of laser cooling technology, optical lattices of ultracold neutral atoms can be established and they can simulate the Hubbard model Simulating the Hubbard model allows us to easily control and adjust the model's parameters 1.4 Some methods of studying metal-Mott insulators transition 1.4.1 The coherent potential approximation method The essence of CPA is: • Replace the random system described by the Green function G with an effective periodic system with the Green function Gp such that ⟨G⟩ = Gp • The effective periodic system is built to ensure the requirement of self-assembly, and the physical quantity measured in a random system must have zero fluctuations around the corresponding value in its effective system (⟨T⟩ = 0) The CPA is a simple approximation that works well in the case of narrow bands or low impurity densities where fluctuations not affect the system's physical properties Applying CPA to approximate the simple Hubbard model can get the critical potential value U=UC=1 It shows that CPA is an easy-touse method without complicated calculations, and the obtained results have little difference from some other popular methods (Using random dispersion approximation (RDA) Noack also found the result UC≈1) 1.4.2 DMFT method The dynamical mean-field equations are derived using the so-called cavity method This derivation starts by removing one lattice site together with its bonds from the rest of the lattice The remaining lattice, which now contains a cavity, is replaced by a particle bath which plays the role of the dynamical mean field So far the derivation and the underlying physical picture coincide with that of the CPA approach described in the previous section Now comes a new, physically motivated idea: the bath is coupled, via a hybridization, to the cavity In the DMFT, two approximations are applied: first, the solution is assumed to be translational invariant and homogeneous; second, the ground energy is assumed to be localized These approximations are correct in the infinite-dimensional case, but they give many good approximations in the finite-dimensional, even three-dimensional case for small spatial fluctuations By replacing the multi-particle lattice model with a single impurity model, the number of freedom degrees is significantly reduced, the problem is simpler with DMFT Besides, the single-particle model has been studied for a long time, and all the methods of solving Anderson's mixed models can be used to solve the DMFT equations 1.4.3 The two-site DMFT method The two-site dynamical mean field theory which proposed by Potthoff in 2001 is a simplification of the DMFT model by mapping the correlated network model to a one-site model with a noninteractive bath containing only one site It is the simplest bath case Although the mapping is approximate, the 2-site model can be solved correctly 2S-DMFT provides a simple, fast way to solve without compromising the mean-field approach to the correlated network model Some numerical results for the Mott phase transition in a simple Hubbard model will allow comparison of results obtained between methods Chapter METAL INSULATOR TRANSITION IN HUBBARD MODEL WITH ONE ADDITIONAL PARAMETER 2.1 Hubbard model with site-dependent interactions 2.1.1 Models and formalisms We consider the following Hubbard model with alternating interactions on a bipartite lattice (sublattices A and B) 𝐻= ∑ + + 𝑡𝑖𝑗 𝑐𝑖,𝜎 𝑐𝑗,𝜎 + 𝑡𝑖𝑗 𝑐𝑗,𝜎 𝑐𝑖,𝜎 𝑖 ∈𝐴,𝑗 ∈𝐵,𝜎 + 𝑈𝐴 [ ∑ 𝑛𝑖,𝜎 𝑛𝑖,−𝜎 − ∑ 𝑛𝑖,𝜎 ] 𝑖 ∈𝐴,𝜎 + 𝑖 ∈𝐴,𝜎 𝑈𝐵 [ ∑ 𝑛𝑗,𝜎 𝑛𝑗,−𝜎 − ∑ 𝑛𝑗,𝜎 ] 𝑗 ∈𝐵,𝜎 − ∑(𝜇 + ℎ𝜎)𝑛𝑖,𝜎 , 𝑖,𝜎 𝑗 ∈𝐵,𝜎 (2.1) 10 Figure 2.2 shows the DOS for each sublattice for fixed value 𝑈𝐵 = 2UA When 𝑈𝐴 = −0.6 the DOS for both sublattices at Fermi level is nonzero, which indicates that system is in a metallic state In contrast, when 𝑈𝐴 = −1.5 the DOS for both sublattices shows a gap around ω = 0, indicating an insulating phase Comparing the cases where the metal-insulator phase transition Fig 2.3: The phase diagram of occurs, we see that the phase HMSDI at zero temperature at transition occurs when |UA| increase half-filling and the |UB| decrease and vice versa The ground state phase diagram is shown in Figure 2.3 The phase boundary between the metallic and insulating phases in the system with alternating interactions at half-filling is given as 𝑈𝐴 𝑈𝐵 = ±𝑊 (2.11) Expression (2.11) is our main result In the case of the usual Hubbard model UA = UB, we have well-known result: UC = W for the Bethe lattice Fig.2.4 The DOS at Fermi energy ρα(0) as a function of UA for different fixed values of UB/UA = 1.0 (a), 2.0 (b), −0.6 (c), and −2.0 (d) Fig 2.5 The double occupancy Dα as a function of UA for different fixed values of 𝑈𝐵 / 𝑈𝐴 = 1.0(𝑎), 2.0(𝑏), −0.6(𝑐), −2.0(𝑑) 11 The DOS at Fermi level for each sublattice ρα(0) as a function of UA for different values of UA/UB are shown in Figure 2.4 One can see that exclusive of the vicinity of UA = UB = 0, ρα(0) is larger in the sublattice with a smaller local interaction As in DMFT with the NRG method Next, to clarify these Mott states we calculate the double occupancy 𝐷𝛼 = 〈𝑛𝛼↑ 𝑛𝛼↓ 〉 The numerical results are plotted in Figure 2.5 It can be seen that the double occupancy in each sublattice approaches zero when the local repulsive interaction is large, and this quantity approaches half as the local attractive interaction increases Phase transition region Sublattice A Sublattice B Mott i Mott transition Mott transition Mott ii Pairing transition Mott transition Mott iii Pairing transition Pairing transition Mott iv Mott transition Pairing transition The same results were obtained within the two-site DMFT, and here we confirm these by using the CPA Our results are in good agreement and published We believe that CPA is a simple method that allows us to find the analytic results, explain essential physical properties at low temperatures, and give correct qualitative results about the MIT in the system with alternating interactions 2.2 The asymmetric Hubbard model 2.2.1 Models and formalisms In the AHM, each spin species has a different hopping integral and a different value of the chemical potential The Hamiltonian of the model is + H = ∑ 𝑡𝑖𝑗 (𝑐𝑖,𝜎 𝑐𝑗,𝜎 + ℎ 𝑐 ) − ∑ 𝜇𝜎 𝑛𝑖,𝜎 + 𝑈 ∑ 𝑛𝑖,↑ 𝑛𝑖,↓ , (2.13) ,𝜎 𝑖,𝜎 𝑖 The asymmetric parameter is defined as 𝑟 = 𝑡↓ 𝑡↑ with two limits: 𝑟 = corresponding to the FKM and 𝑟 = to the HM We should 12 note that the AHM is also used for a description of two-component fermionic mixtures loaded in an optical lattice We used DMFT combined with the equations of motion method to solve and finally have a biquadratic equation for critical interaction with the solution 𝑈𝐶 = [ ( 𝑡↑2 + 𝑡↓2 + √{ 𝑡↑4 + 𝑡↓4 + 14 𝑡↑2 𝑡↓2 })] (2.44) The above expression for UC was obtained by using the projecting technique on the basis of fermionic Hubbard operators Here, we reproduce it in a simple manner 2.2.2 MIT in asymmetrical Hubbard model Figure 2.7 shows the density of states (DOS) for each spin species for three values of the on-site Coulomb interaction, the Mott transition in the system occurs at U = 1.22D Because the DOS at the Fermi level indicates the duction properties of the system, we calculate this value and show it in Figure 2.8 One can see that both ρσ(0) simultaneously vanish in the strong coupling region In the case of r = 0.4, by using a simple spline extrapolation from the data for U < 1.1D, we obtain 𝑈𝐶 ≈ 1.22𝐷, which is shown in Figure 2.9 Repeating this with many different values of r, we get the critical interaction as a function of r, which is presented in Figure 3.4 and is almost identical with the analytic result of Eq.(2.44) over the whole r range The numerical results of double occupancy are plotted in Figure 2.11 for various values of U and r In the noninteracting case (U = 0), the double occupation is 0.25, and it quickly decreases when U increases A metal is characterized by a linear decrease in the double occupation with increasing interaction U while in the insulating region, at a larger value of the interaction, the double occupation remains small and weakly depends on U 13 Fig 2.7: DOS for spin up and spin down for r = 0.4 and various values of U Fig 2.8: DOS at the Fermi level as a function of the on-site Coulomb repulsion for various values of r Fig 2.9: Total DOS ρ(0) = ρ↑(0)+ρ↓(0) at the Fermi level as a function of the on-site Coulomb repulsion for r = 0.4 Fig 2.10: Critical interaction as a function of r (numerical results (green dashed line) and analytical results (2.44) (red solid line)) according to EMM are compared with the results calculated by DMFT [85] and L-DMFT [15] Fig 2.11: Double occupation < 𝑛↑ 𝑛↓ > as a function of U according to EMM are compared with the results calculated by DMFT [85] 14 We have used the equation of motion approach as an impurity solver for the DMFT to investigate the MIT in the AHM at half-filling The technique has been implemented directly on the real-frequency axis, which turns out to be computationally efficient Our main results have been published, compares and shows good agreement with the results obtained by using exact crossover and quantum Monte Carlo techniques The EMM approach is a simple method but effective and trusted for research about MIT in AHM Chapter METAL INSULATOR TRANSITION IN MASS-IMBALANCED HUBBARD MODEL WITH SITE-DEPENDENT INTERACTIONS 3.1 Model We study two-component mass-imbalanced fermions in an optical lattice with spatially modulated interactions described by the following asymmetric Hubbard model on a bipartite lattice made of two interpenetrating (A; B) sublattices arranged such that the neighbors of A sites are all B sites and vice versa We use DMFT, the original lattice model is mapped onto an effective single-impurity Anderson model embedded in an uncorrelated bath of fermions + + ∗ 𝐻𝛼𝑖𝑚𝑝 = ∑ 𝜀𝛼𝑘𝜎 𝑐𝑘𝜎 𝑐𝑘𝜎 + ∑( 𝑉𝛼𝑘𝜎 𝑐𝑘𝜎 𝑑𝜎 + 𝑉𝛼𝑘𝜎 𝑑𝜎+ 𝑐𝑘𝜎 ) 𝑘𝜎 𝑘𝜎 − ∑ 𝜇𝜎 𝑛𝑑𝜎 𝜎 + 𝑈𝛼 [𝑛𝑑↑ 𝑛𝑑↓ + (𝑛𝑑↑ + 𝑛𝑑↓ )], (3.2) The lattice Green function is then obtained via self-consistent conditions imposed on the impurity problem 15 𝑈𝛼 − Δ𝛼𝜎 (𝜔) (3.7) Here 𝐺0𝛼𝜎 (𝜔) are the bare Green functions of the associated quantum impurity problem for the sublattices α 𝐺0−1𝛼𝜎 (𝜔) = 𝜔 + 𝜇𝜎 + 3.2 Approach by EMM In order to calculate the Green function of the single impurity Anderson model we make use of the equation of motion method Decoupling the equations of motion of the single impurity Anderson model (4.2) to the second order, one yields the following approximation for the impurity Green function: 1 𝐺𝛼𝜎 (𝜔) = 𝑈𝛼 Δ𝛼𝜎 (𝜔) −1 𝐺0𝛼𝜎 (𝜔) + −1 𝐺0𝛼𝜎 (𝜔) − 𝑈𝛼 − Δ𝛼𝜎 (𝜔) 1 + (3.26) 𝑈 Δ (𝜔) −1 𝐺0𝛼𝜎 (𝜔) − 𝑈𝛼 + −1 𝛼 𝛼𝜎 𝐺0𝛼𝜎 (𝜔) − Δ𝛼𝜎 (𝜔) In order to study the MIT in the system with alternating interactions, we calculate the spin-dependent DOSs for each sublattice 𝜌𝛼𝜎 (𝜔) = − π ℑGασ (ω), DOSs at the Fermi level ρασ(0) and double occupancy 𝐷𝛼 = 〈 𝑛𝛼↑ 𝑛𝛼↓ 〉 We then construct the phase diagrams for the homogeneous phases at 𝑇 = 𝐾 In order to show how the mass imbalance affects the stability of the normal metallic states, we plot the spin-dependent density of states for each sublattice with 𝑈 = 1.5𝐷 and 𝛾 = 0.8 for different values of 𝑟 (Figure 3.1) We plotted the density of states of a metallic state, a state right at the MIT, and an insulating state DOSs at the Fermi level indicates the conduction properties of the system, we calculate these values and show them in Figure 3.2 One can see that both ρασ(0) simultaneously vanish at the Mott transition When U is fixed, the smaller spatial modulation parameter γ, the easier the mass imbalanced system is driven from a metallic state to the Mott phase 16 Fig 3.1 Spin-dependent density of states for the sublattices for U= 1.5, spatial modulation parameter γ = 0.8 and various values of the mass imbalanced parameter r Fig 3.2 DOSs at the Fermi level as a function of the mass imbalanced parameter r for U= 1.5 and various values of the spatial modulation parameter γ Fig 3.3 Spin-dependent density of states for the sublattices for U= 1.5, r=0.8 and various values of the spatial modulation parameter γ Fig 3.4 Spin-dependent density of states at the Fermi level as a function of the mass imbalanced parameter r for U= 1.5 and various values of the spatial modulation parameter γ We discuss how the spatial modulation in the interactions affects the stability of the normal metallic states In Fig 3.3, we plot the spindependent density of states for each sublattice with 𝑈 = 1.5𝐷 and 𝑟 = 0.8 for different values of γ In order to find the critical value of the spatial modulation parameter 𝛾𝐶 , we show the DOSs at the Fermi level as a function of the spatial modulation parameter γ for 𝑈 = 1.5𝐷 17 and various values of r in Figure 3.4 One can see that both 𝜌𝛼𝜎 (0) simultaneously vanish at the Mott transition When U is fixed, the smaller mass-imbalanced parameter r, the more symmetric interaction must be to drive the system from a metallic state to the Mott phase We calculate the double occupation and the numerical results are plotted in Figure 4.5 for 𝛾 = 0.5 and various values of 𝑟 In the noninteracting case (U=0), the double occupation is 0.25, and it quickly decreases when U increases A metal is characterized by a linear decrease in the double occupation with increasing interaction U while in the insulating region, at a larger value of the interaction, the double occupation remains small and weakly depends on U At smaller values of r, the double occupation more rapidly decreases, and the value of the critical interaction is reduced Fig 3.5: Double occupation < n↑n↓ > as a function of U for γ = 0.5 and different fixed values of r Fig 3.6: The critical spatial modulation interaction 𝛾𝐶 for the half -filled model as a function of the mass imbalanced parameter r for different values of local interaction U We present the critical spatial modulation interaction 𝛾𝐶 (𝑟, 𝑈) for the half-filled model as a function of the mass imbalanced parameter r for different values of local interaction 𝑈 = 1.0𝐷, 1.5𝐷 2.0𝐷 (Fig 3.6) The more U increases, the more the metallic region is reduced as a results of strong correlation 18 Fig 3.7: The critical interaction UC for the half-filled model as a function of the mass imbalanced parameter r for different values of spatial modulation interaction γ Figure 3.8: Ground state phase diagram for the half-filled model as a function of the interactions UA and UB for different values of r = 1.0, 0.8 and 0.4 In Figure 3.7, we show the critical interaction UC for the half-filled model as a function of the mass imbalanced parameter r for different values of spatial modulation interaction γ = 1.0,0.8 0.6 For homogeneous interactions, we reproduce known results where 𝑈𝐶 = ( 𝑡↑2 + 𝑡↓2 + √𝑡↑4 + 𝑡↓4 + 14 𝑡↑2 𝑡↓2 ) The results are summarized into the phase diagram in Figure 3.7 For the mass balanced case (r=1), our results are in good agreement with those obtained from DMFT with NGR method The expression for the critical values is equivalent with |𝑈𝐴 𝑈𝐵 | = ( 𝑡↑2 + 𝑡↓2 + √𝑡↑4 + 𝑡↓4 + 14 𝑡↑2 𝑡↓2 ) (3.17) The expression (4.17) is our main result In the case of the usual asymmetric Hubbard model UA=UB, we reproduce known results 3.3 Approach by 2S-DMFT Using 2S-DMFT, apply to the AHM model to obtain the following two-site Anderson model: 𝐻𝛼2−𝑠𝑖𝑡𝑒 = ∑ 𝜀𝑐𝛼𝜎 𝑐𝜎+ 𝑐𝜎 + ∑ 𝑉𝛼𝜎 (𝑐𝜎+ 𝑑𝜎 + 𝑑𝜎+ 𝑐𝜎 ) + ∑ 𝜀𝑑𝛼𝜎 𝑑𝜎+ 𝑑𝜎 𝜎 𝜎 + 𝑈𝛼 𝑛𝑑↑ 𝑛𝑑↓ , 𝜎 (3.20) 19 We compute the quadratic term of Vσ The eigenvectors and corresponding eigenwavefunctions of an electron, leads to four linear equations for Vασ with (α = A,B;σ =↑,↓): 2𝑡𝜎 ( 2𝑉𝛼𝜎 + 𝑉𝛼 𝜎 ) 𝑉𝛼𝜎 = (3.30) |𝑈𝛼 | Within DMFT, the localization is associated with a vanishing quasi-particle weight or hybridization A nonzero value of 𝑉𝛼𝜎 is the signature of a metal, therefore the determinant of Eq.(3.30) must vanish when the Mott transition occurs, giving the following expression for the critical interactions: |𝑈𝐴 𝑈𝐵 | = (𝑡↑ + 𝑡↓ + √{𝑡↑2 + 𝑡↓2 + 14𝑡↑ 𝑡↓ }) (3.32) Equation (3.32) is one of the main results of this chapter In Fig 3.9 we present the phase diagram as a function of the interactions UA and UB for different values of the mass imbalance parameter r at zero temperature For the mass-balanced system (r = 1), the 2S-DMFT result Eq (332) is in very good agreement with those obtained from DMFT with the numerical renormalization group (NRG) method When the mass imbalance increases (r decreases), the metallic region is reduced, because at fixed D or t↑, the larger the difference in the bare mass, the easier it is to localize the system Fig 3.9: Ground-state phase diagram for the half-filled model as a function of the interactions UA and UB Fig 3.10: Density of states at T = for the sublattice (A, B) for r = 0.4,UB = −2UA and ratios UA 20 In order to confirm the result obtained from Eq (3.32), we also calculate the density of states (DOS) for each sublattice, 𝜌𝛼 (𝜔) = − Σσ ℑGασ (ω) π , and the DOS at the Fermi level 𝜌𝛼𝜎 (0) Figure 4.10 shows the DOS for each sublattice of the system with r = 0.4,UB = −2UA , and for various values of UA It is seen that at half-filling, the DOS for each sublattice is symmetric In a metallic state resonance peaks for both sublattices, which are a consequence of quasi-particle excitation, appear in the vicinity of the Fermi energy Their widths become smaller with increasing strength of the local interactions, and at the critical values, the quasi-particle peaks completely disappear Gap structure is formed around 𝜔=0 in both sublattices Its formation is connected with the vanishing of the quasi-particle weights and its value increases when the local interactions become larger The transition from the metal to an insulating state resembles that found within DMFT applied to the usual paramagnetic Hubbard model, except here the insulating state is induced by the Mott and pairing transitions Figure 3.11 presents the DOS at the Fermi level for each component and sublattice for the same model parameters we conclude that a single metal-insulator transition indeed occurs in the system with spatial modulated interactions Fig 3.11: Density of states at the Fermi level as a function of UA Fig 3.12: Quasi-particle at the Fermi level for the model as a function of UA 21 In Figure 3.12, we plot the quasi-particle weight for the model Across the cases, we find that light fermions are more renormalized than heavy ones, and renormalization occurs more strongly in the sublattice with stronger interactions Both mass imbalance and sitedependent interactions affect the stability of the conventional metallic state, but their effects are different In the Fermi system with repulsive and attractive interactions at half-filling, the explanation is analogous to the one above: when the system is at (or close to) half-filling and the correlation is strong, we essentially have one fermion on each A-sublattice site, Figure 3.13: Double occupancy whereas empty and doubly dA(dB) for the A(B) sublattice in the occupied states are equally system with UB = −2UA as a function realized in the B-sublattice of the interaction UA for r = (Fig 3.13) 1.0,0.4,0.05 In this chapter, we study a mass-imbalanced two-component fermion system in an optical crystal lattice with site-dependent interactions using DMFT in combination with EMM and 2S-DMFT With the first method, we also calculated the density of states at the Fermi level and the number of double occupancies that help experimentally verify the physical properties of the system With the second approach, the ground state properties of the fermion system are discussed from the quasi-particle weight at the Fermi level, and the number of double occupancies for each sublattice as functions of the site-dependent interaction for different values of mass imbalance Our results are published, showing good agreement with the present study in experiments on extra cool fermions The results are known in special cases such as the mass balance limit (r = 1) as well as the homogeneous interaction case (UA = UB) 22 CONCLUSION In this thesis, we have used three methods commonly used to study strongly correlated electronic systems to study the metaldielectric phase transition: CPA - an easy-to-use method with zero calculations Too complicated but still get relatively consistent results; DMFT and 2S-DMFT give satisfactory results for Mott phase transition and Fermi liquid phase in single-region Hubbard model with minimal computational effort We have studied MIT in the half-filled Hubbard model with spatially alternating interactions using the coherent potential approximation Within this approximation, in combination with the semi-elliptical model DOS, we derive the phase boundary between metallic and insulating phases at zero temperature We calculate the double occupancy and clarify the Mott states, as well as find continuous phase transition between the metallic and insulating phases Comparing our results with the ones obtained by DMFT, we believe that the CPA can catch the essential physics at low temperature and gives a correct qualitative picture of the MIT in the system with spatially alternating interactions We have used the equation of motion approach as an impurity solver for the DMFT to investigate the MIT in the AHM at half-filling The technique has been implemented directly on the real-frequency axis, which turns out to be computationally efficient In addition, it allows an explicit expression for the critical interaction in the system to be obtained as an increasing function of the hopping asymmetry We also numerically computed the DOS at the Fermi level and the double occupation that may permit the experimental identification of this remarkable physical behavior The main results have been compared with the results obtained by using the exact diagonalization and the quantum Monte Carlo techniques and were found to be in good agreement This work demonstrates that the equation of motion approach is a simple, but reliable, impurity solver for studying MIT in the AHM This method has the great advantage of performing simple calculations, giving an analytic expression explicitly for the phase transition condition But the result in the special cases are only approximated by Hubbard III and the double occupancies are not equal to zero at the critical point 23 Finally, we investigate the Mott metal-insulator transition in the asymmetric Hubbard model with site-dependent interactions at half-filling and zero temperature by using two approaches: EMM and 2S-DMFT At half-filling and zero temperature, when the repulsive and attractive interactions alternate in the system, the phase boundary between a metal and a Mott insulator is analytically derived By calculating the spin-dependent quasi-particle weights at the Fermi level for each sublattice as functions of the local interaction strength for various values of the hopping asymmetry, we clarify how the mass imbalance and spatial modulation of the interactions affect the stability of the normal metallic state In particular, we show that for a large mass imbalance the light fermions are more renormalized than the heavy ones It is also found that in the strong coupling region of a twocomponent fermionic system with repulsive and attractive interactions, the Mott and pairing transitions occur simultaneously in the corresponding sublattices and the phase transitions at zero temperature are continuous The critical interaction of the system as a function of the model parameters describing the analysis, according to the EMM approach: |𝑈𝐴 𝑈𝐵 | = ( 𝑡↑2 + 𝑡↓2 + √𝑡↑4 + 𝑡↓4 + 14 𝑡↑2 𝑡↓2 ), and the 2S-DMFT approach: |𝑈𝐴 𝑈𝐵 | = (𝑡↑ + 𝑡↓ + √𝑡↑2 + 𝑡↓2 + 14𝑡↑ 𝑡↓ ) The research results show the effectiveness of different approximation methods applied to study the metal-insulator phase transition problem in the strongly correlated system The calculation methods and techniques we have used are not sophisticated technics demanding or expensive computation resources but still give good results, the correlation comparison is consistent with previously published results using more sophisticated computational methods In this thesis, we have limited our research to the non-magnetic state at temperature T=0 K, extended research in the magnetic ordering state, the charge-ordering state, or the superfluid has not been considered and might be studied in the future LIST OF PULICATIONS USED FOR THE THESIS Publications in specialized journals on the list of ISI: [1] Le Duc Anh, Tran Thi Thu Trang, Hoang Anh Tuan (2013), “Coherent potential approximation study of the Mott transition in optical lattice system with site-dependent interactions”, The European Physical Journal B, 86(12), 503 [2] Hoang Anh Tuan, Tran Thi Thu Trang, Le Duc Anh (2016), “Mott transition in the asymmetric hubbard model at half-filling: Equation of motion approach”, Journal of the Korean Physical Society,68(2), pp.238-242 [3] Hoang Anh Tuan, Nguyen Thi Hai Yen, Tran Thi Thu Trang, Le Duc Anh (2016), "Two-component Fermions in Optical Lattice with Spatially Alternating Interactions”, Journal of the Physical Society of Japan,85(10), 104702 [4] Le Duc Anh, Tran Thi Thu Trang, Hoang Anh Tuan, Nguyen Toan Thang, Tran Minh Tien (2018), “Mass-imbalanced Hubbard model in optical lattice with site-dependent interactions”, Physica B: Condensed Matter, 532, pp.204-209