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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 |U A U B|=( t ↑+ t↓ + √ t 2↑ +t 2↓+ 14 t↑ 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 H= ∑ ¿ i , j>, σ t ij ¿ ¿ + ¿(ai ,σ)¿ in there: a i ,σ is the operator that generates (annihilates) +¿ ai,σ ¿ electrons with spin σ on site i; ni , 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 12 (a), 2.0 (b), −0.6 (c), and −2.0 (d) U B /U A =1.0(a), 2.0(b) ,−0.6 (c) ,−2.0(dω The DOS at Fermi level for each sublattice ρ α(0) as a function of UA for different values of U A/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= ⟨ n α ↑ 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= ∑ ¿ i , j>, σ t ij ¿ ¿ 13 The asymmetric parameter is defined as r = t↓ with two limits: t↑ r =0 corresponding to the FKM and r =1 to the HM We should 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 [ ↑ ↓ √ ↑ ↓ 2 ↑ ↓ U C = ( t +t + {t + t + 14 t t }) ] (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 U C ≈ 1.22 D , 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 14 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 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] 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] We have used the equation of motion approach as an impurity solver for the DMFT to investigate the MIT in the AHM at halffilling The technique has been implemented directly on the realfrequency 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 16 H αimp =∑ ε αkσ c+kσ¿c + ∑ ¿ ¿ ¿ kσ kσ kσ The lattice Green function is then obtained via self-consistent conditions imposed on the impurity problem −1 G ασ ( ω )=ω+ μσ + Uα − Δασ ( ω ) (3.7) Here G ασ (ω) are the bare Green functions of the associated quantum impurity problem for the sublattices α 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: Gασ ( ω )= G−1 ασ ( ω ) + U α Δα σ ( ω ) G −1 ασ ( ω ) −U α −2 Δ α σ ( ω ) + G−1 ασ ( ω )−U α + In order to study the MIT in the system with alternating interactions, we calculate the spin-dependent DOSs for each −1 I G ασ ¿ ), DOSs at the Fermi level ρασ(0) and π double occupancy D α =〈 nα ↑ n α ↓ 〉 We then construct the phase diagrams for the homogeneous phases at T =0 K In order to show sublattice ρασ (ω)= 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 =1.5 D 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 UαΔ −1 ασ G ( ω)