Self-energy functional theory

The self-energy functional theory is a variational calculation method of the quantum mechanical many-body theory , which describes systems with a large number of interacting particles. The method is advantageously used in the case of strongly correlated (coupled) electron systems, such as those found in magnets and superconductors .

issue

The computing time for the computation of multi-particle systems increases strongly with the number of particles and at least with it . If, for example, the accuracy is to be improved by doubling the number of observed particles, the waiting time in front of the computer increases eight times. A popular approach to a solution combines groups of particles, i.e. approximates with clusters while simplifying the interactions of the particles in a cluster. ${\displaystyle N}$ ${\displaystyle N^{3}}$

Another approach consists in calculating similar comparison systems as precisely as possible and "converting" these systems into the given system. Self-energy functional theory (SFT) pursues this approach. In the language of the SFT this means: The self-energy functional theory (SFT) provides an approach with which the self-energy of large systems can be calculated very precisely using small reference systems that can be solved as precisely as possible with relatively little computing effort. With this method, the computing effort is largely decoupled from the size of the system.

System description

The SFT, first published as a self-energy functional approach (SFA) by M. Potthoff in 2003, represents a variation principle without approximation of the functional dependency with self-energy as the dynamic basic variable. The grand canonical potential known from thermodynamics is constructed as a functional of self-energy ( ), which becomes stationary at the exact self-energy of the system . For the test self-energies, reference systems that can be solved as precisely as possible with the same interaction as the original system are required. Here, or denote the so-called hopping parameter of the non-interacting part of the respective Hamilton operator and the strength of interaction. ${\displaystyle \,\Sigma }$ ${\displaystyle \,\Omega [\Sigma ]}$ ${\displaystyle \,H=H_{0}(t)+H_{1}(U)}$ ${\displaystyle H^{\prime }=H_{0}(t^{\prime })+H_{1}(U)}$ ${\displaystyle \,t}$ ${\displaystyle t^{\prime }}$ ${\displaystyle \,U}$

More precise representation

(The following development uses the Green Functionalism of many-particle theory (see web links)).

The grand- canonical potential is given as a functional of the temperature-dependent Greens function (Matsubara function) by: ${\displaystyle \,\Omega }$ ${\displaystyle G}$

${\displaystyle \Omega [G]=\Phi [G]+{\text{Tr}}\ln(-G)-{\text{Tr}}((G_{0}^{-1}-G^{-1})G),}$

where the track is defined as ${\displaystyle {\text{Tr}}}$

${\displaystyle {\text{Tr}}A=\beta ^{-1}\sum _{\omega }e^{\mathrm {i} \omega 0^{+}}{\text{tr}}A}$

and is the inverse temperature. The Luttinger-Ward functional occurring in the grand canonical potential is defined as the sum of the contributions of all closed, connected, drawn skeleton diagrams: ${\displaystyle \,\beta }$ ${\displaystyle \,\Phi [G]}$

${\displaystyle \Phi =\sum _{n=1}^{\infty }{\frac {1}{2n\beta }}\sum _{\omega }{\text{tr}}(\Sigma ^{(n)}G)}$

Since i. A. The exact functional relationship is unknown, the transition to the functional of self-energy is a good idea : ${\displaystyle \Omega }$ ${\displaystyle \Sigma }$

${\displaystyle \Sigma [G]\rightarrow G[\Sigma ]}$

It must be assumed that it is locally invertible, i.e. that the system is not at a critical point for a phase transition. The grand-canonical potential as a functional of self-energy is then: ${\displaystyle \,\Sigma [G]}$

${\displaystyle \Omega [\Sigma ]={\text{Tr}}\,\ln(-(G_{0}^{-1}-\Sigma )^{-1})+F[\Sigma ]}$

where is the Legendre transform of the Luttinger-Ward potential . ${\displaystyle \,F[\Sigma ]=\Phi [G[\Sigma ]]-{\text{Tr}}(\Sigma G[\Sigma ])}$ ${\displaystyle \,\Phi [G]}$

Since the Luttinger-Ward functional can be formally constructed as the sum of the contributions of all closed, connected, drawn skeleton diagrams, it is only dependent on the interaction. So the Legendre transform is only dependent on the interaction. If the reference systems now have the same interaction as the original system, then is universal and can be eliminated from the grand canonical potential by expressing through the reference system: ${\displaystyle \,\Phi [G]}$ ${\displaystyle \,F[\Sigma ]}$ ${\displaystyle \,F}$ ${\displaystyle \,F[\Sigma ]}$

${\displaystyle F[\Sigma ]=\Omega '[\Sigma ]-{\text{Tr}}\ln(-(G_{0}^{'-1}-\Sigma )^{-1})}$

So the grand canonical potential of the original system is:

${\displaystyle \Omega [\Sigma ]=\Omega '[\Sigma ]+{\text{Tr}}\ln(-(G_{0}^{-1}-\Sigma )^{-1})-{\text{Tr}}\ln(-(G_{0}^{'-1}-\Sigma )^{-1}))}$

Steadiness condition

The exact self-energy fulfills the following condition of steadiness: ${\displaystyle \,\Sigma }$

${\displaystyle {\frac {\delta \Omega [\Sigma ]}{\delta \Sigma }}=0,}$

d. H. for the exact self-energy there is an extremum or a saddle point for the grand-canonical potential. This condition of steadiness only needs to be tested with the help of the self-energies of the reference systems, calculated as precisely as possible using other means. If a self-energy is found which approximately fulfills this condition, then one has an approximate solution for the self-energy and the grand-canonical potential of the original system.

literature

• S. Bäse, Self-energy functional theory with stochastic cluster solver, University of Hamburg (2008)
• M. Potthoff, Eur. Phys. J. B 32, 429 (2003)
• M. Potthoff, Phys. Rev. Lett. 91, 206402 (2003)
• M. Potthoff, Eur. Phys. J. B 36, 335 (2003)
• C. Dahnken et al., Phys. Rev. B 70, 245110 (2004)

Web links

• Information page about the SFT on the homepage of M. Potthoff ( Memento from June 5, 2008 in the Internet Archive )