Tóm tắt tiếng anh: Chuyển 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:
1. Assoc. Prof. Dr. Hoang Anh Tuan
2. 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


1
INTRODUCTION
1. 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.

2
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 do 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 do 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.


3
2. 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.
3. 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.
4. The scope of research
Some strongly correlated electronic systems in optical lattices in
paramagnetic and at the half-filling.


4
5. 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.
6. 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.
7. 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


5
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
2

|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.
8. 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.


6

Chapter 1
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.


7
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; Spindependent 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.


8
• 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 do not
affect the system's physical properties.
Applying CPA to approximate the simple Hubbard model can
get the critical potential value U=U C=1. It shows that CPA is an easyto-use 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 socalled 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


9
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 2

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)
+ ¿c i,σ

H=

∑

i ∈ A , j∈ B , σ

+¿ c j ,σ+ t ij c j,σ +

t ij c i , σ

UA
2

[∑n
i ∈ A ,σ

i,σ

ni,−σ −

∑

i∈ A ,σ

UB

] 2 [∑ n

ni, σ +

j∈ B, σ

j,σ

nj ,−σ −

∑

j ∈ B,σ

The CPA demands that the scattering matrix vanishes on
average ⟨ T n ⟩=0, this leads to a pair of equations for G A ( ω) and

G B (ω):

]

n j, σ −∑ ( μ+hσσ ) n
i,σ


10

2

U
G ( ω)
1 2 ( )
1
G α σ ω G ασ ( ω )− ω G α´ σ ( ω ) G ασ ( ω )+ ω 2− α G ασ ( ω ) + α´ σ −ω−
16
2
4
4

(

)

Equations (2.2) must now be solved with n A +nB =2 where
0

−2
nα=
∫ I G {ασ } (ω) dωω, (the chemical potential equals zero due to
π −∞
the electron-hole symmetry in the half-filled system). To study the
MIT in the system with alternating interactions, we calculate the

−1
I G α ( ω ) , DOS at the Fermi
π
level ρα (0) and double occupancy D α =〈 nα ↑ n α ↓ 〉. We have also

DOS for each sublattice ρα ( ω ) =

constructed the phase diagram for the homogeneous phases at T = 0
K.
2.1.2. Metal-insulator transition in Hubbard model with sitedependent interactions
The DOS is a very important sign because it tells us the
possibility of a carrier in each state. In Figure 2.1 we show the DOS
for each sublattice A (B) for UB = 0 and for two values of UA (=1; 2).
It can be seen that in both cases the sharp quasiparticle peaks for Bsublattice appear in the vicinity of the Fermi level (ω = 0), which
implies that the system is metallic.

Fig. 2.1. The DOS for the
sublattice A(B) for UB = 0 and
two values of UA.

Fig. 2.2. The DOS for the
sublattice A(B) for UB/UA = 2.


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Figure 2.2 shows the DOS for
each sublattice for fixed value
U B=2 U A . When U A =−0.6 the
DOS for both sublattices at Fermi
level is nonzero, which indicates
that system is in a metallic state. In
contrast, when U A =−1.5 the DOS
for both sublattices shows a gap
around ω = 0, indicating an
insulating phase. Comparing the

Fig. 2.3: The phase diagram of
cases where the metal-insulator
HMSDI at zero temperature at
phase transition occurs, we see that
half-filling
the phase transition occurs when |
UA| increase 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

U A U B=± W 2

(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

Fig. 2.5. The double occupancy
Dα as a function of UA for
different fixed values of


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
↑

2
↓

√

4
↑

4
↓

2 2
↑ ↓

U C = 2 ( t +t + {t + t + 14 t t

})

1

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 con 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 3
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 0 ασ ( ω )=ω+ μσ +

Uα
− Δασ ( ω ) .(3.7)
2

Here G 0 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 )=

1
2

G−1
0 ασ ( ω ) +

1
U α Δα σ ( ω )
G

−1

0 ασ

( ω ) −U α −2 Δ α σ ( ω )

+

1
2

1
G−1
0 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
0 ασ

G

( ω)