For the sake of simplicity of the derivation, I will use a simple example. So let me first introduce the Ising model [62,63]. Let there be a number of independent variablesσi ={−1,1}. One can assume these variables to be spins on a lattice.
The Hamiltonian for this model reads H= 1
2 X
i,j
Ji,jσiσj−hX
i
σi.
The constants Ji,j represent a coupling between the spins, and h is a homo-geneous magnetic field acting on all spins. The property, which completely describes the physics and behavior of the system in equilibrium for a given temperatureT, is the free energyF or the partition function Z, respectively,
Z(β, Ji,j, h) = Tr
{σ}exp (−βH) F(β, Ji,j, h) = −1
βlogZ,
whereβ= 1/(kBT). Expectation values and correlation functions can be easily calculated by derivatives of the free energy. For example the magnetization, given by M =P
iTrσ(σiexp (−βH))/Z, can be written as M =−dF(β, Ji,j, h)
dh .
The magnetization represents an order parameter for the ferromagnetic phase transition occurring in this model [62, 63]. The Ising model at T = 0 with ferromagnetic coupling Ji,j < 0 between nearest neighbors only is clearly fer-romagnetic. The lowest energy configuration is such, that all spins are aligned in the same direction. At this point the system is Z2 symmetric, meaning that the ground state is twofold degenerate. The value of all spins is simultaneously either σ = 1 or σ =−1. At very high temperatures, Ji,j ≪ kBT, the system will be in the paramagnetic state, where all spins are in principle decoupled and thus M = 0. The existence of a finite temperature, T > 0, below which the spins begin to order, critically depends on the dimension of the system. For dimension d = 1, the spins align only for T = 0 [62, 63], while for d = 2 the system exhibits a finite temperature phase transition, as found by Onsager [64].
Already for the three dimensional system there is no analytical solution to this problem.
For strongly interacting electrons on a lattice, as introduced in chapter 1.3, the situation is even more difficult. Nevertheless, there are numerical and analytical methods to study strongly correlated electron systems. A numerical method that is able to directly simulate a lattice model is Quantum Monte Carlo [65–
67]. But Quantum Monte Carlo is limited to small systems and simulations of rather high temperatures due to the computational effort. Besides this, there are parameter regions where the simulations will fail because of the sign-problem [68, 69]. There are also other approaches like exact diagonalization, which is limited to even smaller clusters. These examples are by no means all in the zoo of possible methods for such systems. I wanted to illustrate that there are different methods all having advantages and disadvantages. There is no best method for the model I am intending to study. An overview about analytical and numerical methods for interacting quantum systems focusing on metal insulator transitions can be found in M. Imada [14].
In this work I use the dynamical mean field theory. Using this approach it is possible to scan through the whole parameter region, temperature and interac-tion parameters. I am able to identify and analyze different phases. Of course, this method has also some drawbacks. Besides the approximations, which I will state below, there are also sometimes problems stabilizing ordered phases. I will come back to this point later.
The aim of any mean field theory is to relate the lattice problem to a pure local problem, called impurity problem, by partially tracing out all degrees of freedom but one site, here denoted as “site 0”. This is called a cavity construction [70].
For this purpose, one splits the trace of the partition function into a part containing only degrees of freedom of “site 0”, Tr0, and one part containing the rest of the system, TrC. The Hamiltonian is split into three parts: H0 containing only parts of the single site 0, H0C parts which connect the single
2.2 Cavity Construction 27
site to the rest of the system and HC the rest of the system, which contains no degrees of freedom from the single site. The partition function for the Ising model can then be written as
Z = Tr exp(−βH) since it denotes a classical model. Expanding exp(−βH0C), one finds
Z = Tr exp(−βH)
If one now performs the trace over the system without site 0, one can perform a cumulant expansion and obtains
Z = Tr
Here h·iCcum denotes the cumulant within the cavity system. Until now every-thing is exact. But noevery-thing was gained, as the new action contains all powers ofσ0 and infinite many cumulant expectation values have to be calculated.
The first approximation is to assume that the expectation value in the system without site 0 is the same as in the system with site 0, soh·iCcum =h·icum. The next step is to neglect all terms in the sum for n > 1. Both approximations can be justified by an infinite coordination numberz and an infinite extended lattice. One ends up with the following formula
Z = Tr
wherehσiicum =hσiiholds. Analyzing a homogeneous system with interactions betweenz nearest neighbors only
J0i =Ji0 =
J i is nearest neighbor of 0
0 else ,
the partition function for the Ising model in the mean field approximation reads Z =Tr [exp(−βσ(−h+zJhσi))],
where hσi has to be calculated self-consistently. It can be shown that this formula is exact in the limit of an infinite dimensional lattice, in the sense of an infinite coordination numberz. There one has to scalezJ →J⋆ =constfor z→ ∞[70]. Calculating the expectation valuehσifinally yields the familiar self-consistency equation in the Weiss molecular field theory of the ferromagnet [71]
m= tanh (−β(−h+J⋆m)).
A spontaneous magnetization,h= 0 butm6= 0, can occur forβ|J⋆|>1 leading to a finite temperature phase transition at T =|J⋆|/kB.