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Theoretical background
Greens functions and self energy
When performing QFT calculations for materials it is often the goal to calculate the Green functions i.e. transition amplitudes and probabilities associated to certain pro-cesses. In a standard prescription this can be done order by order (orders being powers of the coupling constants) using Feynman diagrammatics. This is not what DMRG does but from a pedagogical point of view it is more intuitive to start with this prescription and then contrast it to DMRG.
Performing such calculations order by order a lot of diagrams need to be calculated but there are tricks to include higher order diagrams, especially when doing calculations iteratively using computational power. In general there are three types of diagrams (See Figure 3.1):
Type A: One particle reducible diagrams i.e. diagrams where processes are repeated n times.
Type B: Diagrams where internal lines are themselves corrected by processes of a certain order.
Type C: Diagrams of a certain order.
Type A Type B Type C
Figure 3.1: The di erent types of diagrams
Type A can be taken care of by introducing the self energy . In order to do so one has to decide which diagrams should be included in the self energy i.e. the self energy is a sum of diagrams itself. Then it is possible to diagrammatically create a \dressed » propagator (including all type A diagrams) from the \bare » one. This is shown in Figure 3.2. It shows a diagrammatic way of deriving the Dyson equation
Now diagrams of Type A are taken care of, but what about Type B? Those diagrams can be included into a calculation by simply recursively applying the Dyson equation. If and G0 is known a loop calculating G via the Dyson equation using this G as new G0 will take care of all Type B diagrams. Now in reality it is not this simple because neither G nor is known.
The Type C diagrams are actually part of the Type A diagrams. They are listed sepa-rately because in general one could only include diagrams to a certain order into the self energy. Then all higher order diagrams would not be included in the Dyson equation and would not be taken care of.
Now the DMRG procedure is di erent in the sense that is is an iterative and quasi exact procedure. Starting from an initial guess it can self-consistently approximate the ground state Green function very accurately using the (reduced-)system Hamiltonian directly. It is therefore a non-perturbative procedure and includes all orders. This is one of the reasons why it became so important. Nevertheless the Dyson equation appears in the DMFT cycle.
The self-energy of a strongly correlated system, e.g. the Hubbard model, for certain values of the local potential, shows an interesting property named kinks [3]. Those are abrupt changes near ! = 0 (where behaves linear) in the slope of the self-energy. Part of this project was to track down their rst appearance going from Falicov-Kimball to Hubbard model.
Hubbard and Falicov-Kimball model
In order to gain a deeper understanding of solid state physic problems it is important to come up with simpli ed models which still capture the essential physics. One such model is the Hubbard model with the Hamiltonian [11, 17, 1] H = t cy c j + U n i » n i# (3.4) where i; j denote the sites, the spin, h:::i next neighbours, t the hoping amplitude from one site to the next, cyi; and ci; creation and annihilation operator of an electron at site i with spin , ni; » and ni;# the occupation of the site i (0 or 1) and U the potential. The Hubbard model therefore describes a lattice occupied by electrons of spin up and down. They have a (complex) amplitude t to jump from one site to the next. If that site is occupied by an electron of the other spin-family an extra energy U has to be paid. The rst part of the Hamiltonian describes the kinetic energy, the second one the coulomb interaction of electrons. The Falicov-Kimball model is very similar to the Hubbard model, only in the Falicov-Kimball model only one spin channel has a hopping amplitude i.e. in (3.4) hi;ji; ! i;j and the other one acts as a static background potential via the n i ni; »Pi;# term.
Anderson impurity model
The Anderson impurity model (AIM) is used to describe an electron embedded in an environment (which is usually referred to as bath and which can be both metallic and insulating) [11, 7, 1]. Its Hamiltonian yields
X ay X (cy a + ay c
H = a k + V k ) + U n n # (n » + n ) (3.5)
k k k k » #
k; k;
where ayk and ak are creation and annihilation operator of an electron in the bath with spin and energy k, cy and ck are creation and annihilation operator of an elec-tron at the impurity site, Vk de nes the hybridization (coupling) strength of bath and impurity, n » and n# are the occupation operator at the impurity site, U is the local potential at the impurity site i.e. the coulomb repulsion and the chemical potential. Under certain circumstances the Hubbard model can be mapped to the Anderson impu-rity model. This is part of the next chapter. More information on the Anderson model can be found in [15].
Dynamical Mean Field Theory
When working on a solid state physics problem the Hamiltonian of the system grows exponentially with the size of the system. Therefore the Hilbertspace gets extremely large and it becomes impossible to give an exact solution of the full problem. The aim of Dynamical Mean Field Theory (DMFT) is to reduce the complexity of the problem such that it can be treated at least numerically. In DMFT each site in the Hubbard model is mapped to a separate AIM, which means that the spacial correlations are treated in a mean eld way. The AIM is then in general solved numerically, in our case using a DMRG impurity solver, to produce a local self-energy.
To see how the Hubbard model can be mapped to the AIM it is useful to look at the kinetic term of (3.4). If z the number of neighbours at every site then the hopping probability to any neighbour gets proportional to P / zt2. Furthermore it can be shown that in order to have non-trivial physics in the limit z ! 1 the hopping amplitude t has to scale like t / 1 i.e. the kinetic energy stays nite. Then the probability for an electron to visit the same site twice goes to 0 i.e. the system can described by a one-particle Green function and (k; !) ! (!). The diagrammatic content of the AIM is exactly the same, since it is a purely local model. Therefore treating the Hubbard model with DMFT corresponds to taking into account all purely local one-particle irreducible diagrams as the electronic self-energy (without any perturbative restriction). More about the mapping to impurity models can be found in [4, 9].
Now in the DMFT self consistency cycle [7] rst the lattice Green function G(!+; k) is calculated from (!+; k) and the lattice G0(!+; k). Then this lattice Green function is projected to a local one Gloc(!+). Taking [G0 1(!+)] = (!+) + Gloc1(!+) as the non-interacting impurity Green function de nes an AIM with a hybridization function (!+) = !+ G 0 1(!+). The impurity solver then produces a new impurity Green function G(!+) from which a new self energy can be obtained via (!+) = G0 1(!+) G 1(!+). Now the self consistency is iterated with the new . More in DMFT can be found in [17].
When mapping the Hubbard model to the AIM the e ect of the bath on the impurity is fully described by the hybridization function (!+ = ! + i ) = Xk j Vk j 2 (3.6) ! + i k
Solving the DMFT self consistency cycle then yields the local part of the Green func-tion of the Hubbard model. It gets exact when assuming a Hubbard model on a Bethe-lattice (A lattice on which every site as in nitely many neighbours, sometimes also called Bethe-graph).
Going back to the di erence of the Falicov-Kimball and Hubbard model they show dif-ferent properties, at least when solved via a DMFT calculation. E.g. the Hubbard model has a metallic, thus conducting, a Mott-isolating (a phase which using simple band theories should be conducting but is not when measured) and an isolating phase, whereas the Falicov-Kimball model does only have the latter two. Nevertheless in our case we used the paramagnetic half lled one-band Hubbard model which does not show the band insulating phase but only the Mott-insulating phase when solved via DMFT. Another one (see section 3.5) is the behaviour of the divergence lines of the irreducible vertex around zero temperature.
Divergence lines of the irreducible vertex r
The irreducible vertex 0 is connected to the general susceptibility of DMFT [12]
ph; 0 0 =Z0 d 1d 2d 3 e i 1 ei( + ) 2 e i( 0 +!) 3 (3.7)
[hT cy ( 1)c ( 2)cy 0 ( 3)c 0 (0)i h T cy ( 1)c ( 2)ihT cy 0 ( 3)c 0 (0)i]
where T denotes the time ordering symbol, c and cy the annihilation and creation operator of an electron with spin , bosonic and fermionic Matsubara frequencies, = 1=T and ph stands for particle-hole. It is related to a particle-hole scattering event with an energy transfer . The irreducible vertex then describes the e ective interaction between electrons in a given scattering channel. For = 0 it can be expressed in terms of functional derivatives [12]
A divergence of implies that G( ) has a zero derivative there i.e. G= = Therefore there is a connection between G, and In [12] the results of calculating divergence lines of in Falicov-Kimball and Hubbard model are published. Plotting them in the T =U (the same U as in (3.4)) plane shows that near T = 0 they behave di erent, see Figure 3.3. In the Falicov-Kimball model all p lines meet at one value U = 1= 2, whereas for the Hubbard model this is not the case. Tracking down this di erence in the transition from the Falicov-Kimball to the Hubbard model was the o set of this project.
Density Matrix Renormalization Group
Density Matrix Renormalization Group (DMRG) is a numerical variational technique which is very e cient for 1-dimensional systems. Most commonly it is used to calculate the ground state of a given system. As already mentioned one of the main problems when doing many body physics is that the Hilbert space of the system grows exponentially with the size (number of sites) of the system. DMRG tackles this problem by performing a truncation i.e. a reduction of the dimension of the Hilbert space if the dimension grows beyond a prede ned limit. It provides very accurate results and can work on the real energy axis, both properties necessary for the calculations performed throughout this interaction in (3.4) Picture taken from [12] . project.
Following will be a brief overview about the basic concepts of DMRG. The reader is highly encouraged to read more about this in [5, 6, 14, 16, 13].
In general their are two types of DMRG. In nite size DMRG and nite size DMRG. The impurity solver used in this project uses the latter one. Nevertheless for pedagogical reasons it is easier to rst understand the rst one.
DMRG as it will be presented now can deal with Hamiltonians of the form H= OO (3.9) where O can be any operator. This is a one dimensional Hamiltonian with chain structure. This means that in order to perform DMRG it is necessary to transform the System such that it has a chain geometry.
Table of contents :
1 Abstract
2 Introduction
3 Theoretical background
3.1 Greens functions and self energy
3.2 Hubbard and Falicov-Kimball model
3.3 Anderson impurity model
3.4 Dynamical Mean Field Theory
3.5 Divergence lines of the irreducible vertex r
3.6 Density Matrix Renormalization Group
3.6.1 Innite size DMRG
3.6.2 nite size DMRG
3.7 The Program
3.7.1 Chebyshev expansion
3.7.2 Linear prediction
4 Calculations
4.1 Dening the upper boundary of the local potential
4.2 Results
4.2.1 U=1.00
4.2.2 U=1.50
4.2.3 U=1.80
4.3 Table of kinks in the self energy
5 Conclusion
Bibliography
