Research
Dynamical mean-field theory & beyond
Hansmann et al. (2012)
In the last decades, dynamical mean-field theory (DMFT) has established itself as the state-of-the-art for many-body calculations. DMFT is able to capture quantum fluctuations, i.e., the local dynamics of a many-body system. and hence to describe both coherent (quasi-particle) and incoherent excitations, at the expense of momentum resolution. Moreover, in combination with ab-initio density functional theory (i.e., DFT+DMFT) it allows bridging the gap between model Hamiltonians and realistic materials, opening the path towards quantitative theoretical predictions and direct comparison to experiments.
However, momentum resolution becomes of importance for the description of, e.g., dispersive excitations, critical behavior, and in general for low-dimensional systems, as the local approximation behind DMFT is no longer adequate. Extensions of DMFT can be classified under two categories, i.e., cluster or diagrammatic extensions, which follow quite different philosophies.
Within quantum cluster theories, the auxiliary many-body problem is no longer an individual atom but a system with a finite spatial extension, and thus includes short-range correlations within the cluster exactly, while longer-range correlations are retained only at a mean-field level. In contrast, within diagrammatic extensions, while the auxiliary problem remains local spatial correlations on all length scales can be taken into account on equal footing, exploiting the knowledge of local vertex functions to obtain a momentum-resolved self-energy. There have also been attempts to combine the two above philosophies within multi-scale approaches.
The parquet formalism is the natural framework to describe fluctuations in all possible scattering channels, as well as their interplay, which is believed to lie at the heart of the complex phase diagram of strongly correlated materials.
The standard parquet decomposition of the two-particle vertex is based on the concept of irreducibility of Feynman diagrams with respect to a pair of fermionic lines. A pioneering analysis of local two-particle vertex functions yields a physical interpretation of the characteristic frequency structure of reducible and irreducible vertex functions, which paved the path for advances of diagrammatic theories of strongly correlated electron systems [1].
A self-consistent numerical solution of the parquet equations within the dynamical vertex approximation has been demonstrated for finite systems with periodic boundary conditions., e.g., benzene and organic molecules [2,3] as well as for lattice models.
[1] Rohringer et al., Phys. Rev. B 86, 125114 (2012)
[2] Valli et al., Phys. Rev. B 91, 115115 (2015)
[3] Pudleiner et al., Phys. Rev. B 99, 125111 (2019)
Related readings:
Rohringer et al., Rev. Mod. Phys. 90, 025003 (2018)
Kauch et al., Phys. Rev. Lett. 124, 047401 (2020)
An alternative to the standard decomposition of the full vertex function can be introduced considering irreducibility with respect to bare interaction lines. In this framework, all reducible contributions can be represented in terms of fermion-boson vertices and screened interactions and can be interpreted in terms of single-boson exchange (SBE) processes, while irreducible contributions are instead connected to multi-boson scattering processes [4].
It is possible to reformulate the parquet equations following the SBE decomposition. Since all diagrammatic blocks of the theory are physical response functions, the novel parquet theory avoids local vertex divergences and, at the same time, allows for a significant reduction of the numerical workload with respect to standard techniques [5].
The long- and short-range vertex corrections to susceptibilities are identified with single-boson (Maki-Thompson) and multi-boson (Aslamazov-Larkin) processes, respectively. While the diagrammatic resummation of single-boson contributions can be performed efficiently, the multi-boson contributions are harder to evaluate. A rapid converge both in frequency and in a momentum form-factor expansion allows achieving high momentum resolution while still retaining one- and two-particle spectral information [6].
[4] Krien et al., Phys. Rev. B 100, 155149 (2019)
[5] Krien & Valli, Phys. Rev. B 100, 245147 (2019)
[6] Krien et al., Phys. Rev. B 102, 195131 (2020)
Related readings:
Krien et al., Phys. Rev. Research 3, 013149 (2021)