Study of single impurity Anderson model and dynamical mean field theory based on equation-of-motion method [Elektronische Ressource] / von Qingguo Feng

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Study of single impurity Anderson model anddynamical mean eld theory based onequation-of-motion methodDissertationzur Erlangung des Doktorgradesder Naturwissenschaftenvorgelegt beim Fachbereich Physikder Goethe{Universit at Frankfurtin Frankfurt am MainvonQingguo Fengaus Shandong (China)Frankfurt am Main 2009(D30)vom Fachbereich Physik der Goethe{Universit at Frankfurt als Dissertation angenommen.Prufungsk ommission:Prof. Dr. Claudius Gros (Vorsitz)Dr. Harald O. JeschkeProf. Dr. Jens Muller (Protokoll)Prof. Dr. Jochim MaruhnPrufungszeit: 16. November, 2009Dekan: Prof. Dr. Dirk-Hermann RischkeGutachter: Dr. Harald O. Jeschke,Prof. Dr. Claudius Gros,Prof. Dr. Fakher Assaad (Universit at Wurzburg)to my parentsAbstractIn this thesis, we studied the single impurity Anderson model and developed a new and fastimpurity solver for the dynamical mean eld theory (DMFT). Using this new impurity solver,we studied the Hubbard model and periodic Anderson model for various parameters. This workis motivated by the fact that the dynamical mean eld theory is widely used for the studies ofstrongly correlated systems, and the most frequently used methods, e.g. the quantum Monte-Carlomethod (QMC), and the exact digonalization method are much CPU time consuming and usuallylimited by the available computers. Therefore, a fast and reliable impurity solver is needed.

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Study of single impurity Anderson model and
dynamical mean eld theory based on
equation-of-motion method
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Physik
der Goethe{Universit at Frankfurt
in Frankfurt am Main
von
Qingguo Feng
aus Shandong (China)
Frankfurt am Main 2009
(D30)vom Fachbereich Physik der Goethe{Universit at Frankfurt als Dissertation angenommen.
Prufungsk ommission:
Prof. Dr. Claudius Gros (Vorsitz)
Dr. Harald O. Jeschke
Prof. Dr. Jens Muller (Protokoll)
Prof. Dr. Jochim Maruhn
Prufungszeit: 16. November, 2009
Dekan: Prof. Dr. Dirk-Hermann Rischke
Gutachter: Dr. Harald O. Jeschke,
Prof. Dr. Claudius Gros,
Prof. Dr. Fakher Assaad (Universit at Wurzburg)to my parentsAbstract
In this thesis, we studied the single impurity Anderson model and developed a new and fast
impurity solver for the dynamical mean eld theory (DMFT). Using this new impurity solver,
we studied the Hubbard model and periodic Anderson model for various parameters. This work
is motivated by the fact that the dynamical mean eld theory is widely used for the studies of
strongly correlated systems, and the most frequently used methods, e.g. the quantum Monte-Carlo
method (QMC), and the exact digonalization method are much CPU time consuming and usually
limited by the available computers. Therefore, a fast and reliable impurity solver is needed.
This new impurity solver was explored based on the equation-of-motion method (also called
Green’s function and decoupling method in some literature). Using the retarded Green’s function,
we rst derived the equations of motion of Green’s functions. Then, we employed a decoupling
scheme to close the equations. By solving self-consistently the obtained closed set of integral
equations, we obtained the single particle Green’s function for the single impurity Anderson model.
After that, the single impurity Anderson model was solved along with self-consistency conditions
within the framework of DMFT. In this work, we studied and compared two decoupling schemes.
Moreover, we also derived possible higher order approximations which will be tested in future work.
Besides the theoretical work, we tested the method in numerical calculations. The integral
equations are rst solved by iterative methods with linear mixing and Broyden mixing, respectively.
However, these two methods are not su cient for nding the self-consistent solutions of the DMFT
equations because converged results are di cult to obtain. Moreover, the computing speed of the
two methods is also not satisfactory. Especially the iterative method with linear mixing costs
always a lot of CPU time due to the required small mixing. Hence, we developed a new method,
which is a combination of genetic algorithm and iterative method. This new method converges
very fast and removes artifacts appearing in the results from the iterative method with linear
and Broyden mixing. It can directly operate on the real axis, where no numerical error from the
high frequency tail corrections and the analytical continuation is introduced. In addition, our new
technique strongly improves the precision of the numerical results by removing the broadening.
With this newly developed impurity solver and numerical technique, we studied the single im-
purity Anderson model, the single band Hubbard model and the periodic Anderson model with
arbitrary spin and orbital degeneracy N on the real axis. For the single impurity Anderson model,
the spectral functions are calculated for the in nite and nite Coulomb interaction strength. Wealso studied the spectral functions in dependence of the parameters of impurity position and hy-
bridization. For the Hubbard model, we studied the bandwidth control and lling control Mott
metal-insulator transition for spin and orbital degeneracy N = 2. It gives qualitatively the critical
value of Coulomb interaction strength for the Mott metal-insulator transition, and the spectral
functions which are comparable to those obtained in QMC and numerical renormalization group
methods. We also studied the quasiparticle weight and the self-energy in metallic states. The latter
shows almost Fermi liquid behavior. At last we calculated the densities of states for the Hubbard
model with arbitrary spin and orbital degeneracy N. The periodic Anderson model (PAM) is also
studied as another important lattice model. It was solved for various combinations of parameters:
the Coulomb interaction strength, the impurity position, the center position of the conduction
band, the hybridization, the spin and orbital degeneracy. The PAM results represents the physics
of impurities in a metal. In short, our method works for the Hubbard model and the periodic
Anderson model in a large range of parameters, and gives good results. Therefore, our impurity
solver could be very useful in calculations within LDA+DMFT.
Finally, we also made a preliminary investigation of the multi-band system based on the success
in single band case. We rst studied the two-band system in a simpli ed treatment by neglecting
the interaction between the two bands through the bath. This has given promising numerical results
for the two-band Hubbard model. Moreover, we have studied theoretically the two-band system
with mean eld approximation and Hubbard-I approximation in dealing with the higher order cross
Green’s functions which are related to both the two bands. In the mean eld approximation, we
even generalized the two-band system to arbitrary M =N=2 band system. Potential improvement
can be carried out on the basis of this work.Contents
1 Introduction 1
1.1 Mott transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Dynamical Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Single impurity Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Periodic Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Theory and method 19
2.1 Why have we chosen the EOM method? . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Calculation of equations of motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Analytical calculation of equations of motion . . . . . . . . . . . . . . . . . . 20
2.2.2 Machine and symbol manipulation . . . . . . . . . . . . . . . . . . 27
2.3 Decoupling Scheme and approximations . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Mean eld approximation and Hubbard-I approximation . . . . . . . . . . . . 28
2.3.2 Lacroix’s decoupling scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 Wang’s scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Calculation of integrals on the real axis: analytical method . . . . . . . . . . . . . . 34
~2.4.1 and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
y y
0 02.4.2 hd c i andhc c i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 0 0k k k
2.4.3 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Calculation on the Matsubara axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Higher approximations beyond Lacroix’s level 45
3.1 Equations of motion for three-particle Green’s function . . . . . . . . . . . . . . . . . 45
3.2 EOMs for multi-c operator Green’s functions . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Extended Lacroix’s decoupling scheme . . . . . . . . . . . . . . . . . . . . . . 47
iii CONTENTS
3.2.2 Wang’s decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Multi-band system 55
4.1 The Hamiltonian and EOMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Approximation without inter-band hybridization . . . . . . . . . . . . . . . . . . . . 58
4.3 Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Hubbard-I approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Code realization and optimization 69
5.1 Technique in calculating the integral terms . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.1 Analytical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2 Lorentzian broadening method . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Iterative method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Iterative method with linear mixing . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 Broyden mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.1 Introduction of genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 Physical results and Discussion 85
6.1 Single impurity Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Periodic Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Multi-band system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4.1 Two-band Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7 Conclusions 117
Appendices 120
A An example of FORM code 121CONTENTS iii
B Separating the Green’s function into two parts 129
C Calculation of two-particle correlations 133
Bibliography 136