User:Condmatstrel/Roth Approximation

Roth approximation is an approximate solution method for many-body problems in condensed-matter systems. It is based on the equation-of-motion method (or Liouvillian superoperator method ) and Green's function formalism. It is originally developed by Laura M. Roth for the Hubbard model. This method is applicable to systems with any electron filling, and any strength of disorder and interaction.

Very basics
We shall explain the basic idea of the Roth approximation in words first, and this will be followed by the rigorous derivation. Roth approximation includes two steps. First, one truncates the exponentially growing fermionic operator basis due to that the time evolution of the single fermionic creation/annihilation operators in the retarded Green's function lead to many fermionic operators in the presence of interactions. Second, one uses Roth's idea to approximately calculate some unknown correlations due to the truncation of the basis. Note that truncation of the basis is the same thing as the decoupling of the hierarchy of the Green's functions which occur when electrons interact.

Generalized Green's function
The rigorous derivation of the Roth approximation is as follows. Define the retarded Green's function (set $$ \hbar=1 $$):
 * $$ G_{ij} = -i\langle \{ \hat c_i(t_1), \hat c_j^\dagger (t_2) \} \rangle \theta(t_1 - t_2)$$.

Also define the Liouvillian superoperator via its acting on an operator
 * $$ \hat L \hat O \equiv [ \hat H, \hat O ] $$

which is nothing but the time evolution of the operator $$ \hat O $$. Rewrite the retarded Green's function using the Liouvillian superoperator:
 * $$ G_{ij} = -i\langle \{ \hat c_i(0), \hat c_j^\dagger(0) e^{-i\hat L t} \} \rangle \theta(t)$$,

and Fourier transfrom to the energy space:
 * $$ G_{ij}(\omega) = \langle \{ c_i(0), \frac{1}{\omega - \hat L} c_j^\dagger(0) \} \rangle $$

where $$ 1/(\omega - \hat L) $$ above should be understood as the inverse operator of $$ (\omega -\hat L) $$. Generalize this Green's function via including many-fermionic operators in the basis: define $$ \hat A $$ be an (zero time) operator in the generalized basis including many-fermionic operators, and explicitly write down the generalized Green's function:
 * $$ g_{ij}(\omega) = \langle \{ \hat A_i, \frac{1}{\omega - \hat L} \hat A_j^\dagger \} \rangle $$.

It is convenient to express the generalized Green's function in the matrix form. For this purpose define: the normalization matrix $$ \mathbf{\chi}_{ij} = \langle \{ \hat A, \hat A_j^\dagger \} \rangle $$, and the energy matrix $$ \mathbf{E}_{ij} = \langle \{ A_i, \hat L \hat A_j \} \rangle $$. Then the matrix form of the generalized Green's function reads:
 * $$ \mathbf{g} = \chi [\omega \mathbf{I} - \chi^{-1} \mathbf{E} ]^{-1} $$

where $$ \mathbf{I} $$ is the identity matrix, and $$ ^{-1} $$ is the matrix inverse.

Truncation of the basis
Include only the operators $$ \hat c_i^\dagger $$ and $$ \hat b_{i}^\dagger \equiv \hat c_i^\dagger ( \hat n_i - n_i )$$ where $$ n_i \equiv \langle \hat n_i \rangle $$. The idea is that the operators that is believed to be most relevant to the physics of the system that is studied is included in the basis. For example, one expects a gap in the excitation spectrum when the electron-electron repulsion is large. So, the operator which gives a gap in the excitation spectrum is included in the basis. Additionally, the clue of what operator to include comes from $$ \hat L \hat c^\dagger $$.

For this basis, $$ \mathbf{E} $$ matrix takes the following form for an ordered system for example:

Fixing the self consistency cycle
The solution method of that is used is a numerical procedure called "the self consistency cycle" or "iteration".

Notation

 * $$ i $$: complex constant
 * $$ \hat c_i(t_1) $$: a second quantized fermionic operator annihilating a given electron from site $$ i $$ at time $$ t_1$$
 * $$ \hat c_j^\dagger(t_2) $$: a second quantized operator creating an electron at site $$ j $$ at time $$ t_2 $$
 * $$ \{,\} $$: anticommutation
 * $$ \theta $$: the step or Heaviside function.