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## Perspectives and Challenges of Many-Particle Methods: Efficient Strategies and Tools for the Description of Complex Systems

#### University of Bremen, Germany

#### Organisers

The field of computational material science made recently a tremendous step toward a first-principle description of correlated electronic systems including transition metal and rare-earth elements. A great impact is expected in the new area of artificially constructed magnetic nano-systems and at the interface of traditional inorganic chemistry, physics, biology and materials science in general. Of prime importance is the development of next-generation realistic many-body computational tools which are fast, reliable and are able to describe non-trivial quantum dynamics of complex systems. In order to address these problems, new integrated software tools for realistic quantum simulations of correlated systems need to be developed for a broad scientific community [1].

Recently a new generation of continuous-time Quantum Monte Carlo (CT-QMC) methods for numerically exact calculation of complicated fermionic path integrals have been proposed for interacting electrons based on the weak-coupling [2] and strong-coupling [3] perturbation expansion. This methodological breakthrough in the quantum many-body theory stimulate a great progress in the electronic calculations of realistic strongly correlated systems within the dynamical mean-field theory (DMFT) where the solution of effective multiband impurity problem is the main point [4-8]. The most important achievement of the CT-QMC scheme is related with an opportunity to simulate real systems with arbitrary complex interaction vertex including spin-flip terms and retardation effects. Such progress was impossible with the conventional Hirsch-Fye QMC algorithm based on a time discretization approach.

New QMC technique stimulate the fast and important developments in the field of strongly correlated materials related with the realistic LDA+DMFT calculations for f-orbital case of plutonium [4,5], heavy-fermion systems [6], strongly correlated thermoelectric compounds [7] and novel high-temperature superconducting pnictides [8]. The first attempt to combine the new computational scheme with analytical perturbation approach in correlated fermion models has been presented [9].

Over the past decades, density functional theory (DFT) has become the effective single-particle theory of choice both for physicists from the solid state community as well as for quantum chemists. However, this tremendous success is also accompanied by an ever growing list of documented failures, especially for strongly correlated systems. Modern approaches to improve the description by better exchange-correlation functionals are based on an adapted treatment for short and long-range electron-electron interaction (so called range-seperated functionals) [10] or the seperate treatment of exchange and correlation functionals (exact exchange + random phase approximation) [11].

Beyond DFT, the GW approximation based on MBPT has found widespread use in the calculation of single-particle spectra. Originally used only to compute band structures in solids, GW has recently also been applied to study surfaces and molecules [12]. A detailed comparison of the self energy in the GW context with the exchange-correlation functional in DFT, is expected to pave the road for further improvement of functionals. This interplay of different approaches to correlated systems was already exploited in the realm of time dependent DFT [13].

Yet other approaches for interacting many-electron systems were put forward in the quantum chemistry community (see e.g. Refs. [14-16]). Wavefunction-based schemes allow to systematically approach the exact solution of the electronic Schroedinger equation and in this way offer a hierarchy useful for estimating error bars of simpler approximations. The methods range from efficient MBPT methods mostly employing Gauss-type basis functions to coupled-cluster theory and also multi-reference approaches. To deal with the cusp problem and basis deficiencies, r12 and more recently F12 methods have been brought forward [14]. The steep increase of the computational effort with molecular size has been circumvented by introducing linear-scaling methods for many quantum-chemical methods and for computing various molecular properties. They exploit the local electronic structure and open the way to treat large molecular systems with 1000 atoms and more at the HF, DFT, and MP2 levels [15]. Also the possibilities in performing highly accurate CC calculations has been dramatically increased. In addition, a variety of approximations were introduced for reducing prefactors by, e.g., auxiliary basis set expansions or Cholesky decompositions. Finally, also partially periodic boundary conditions have been accounted for [16]. The relation of all of these schemes to the correlated methods in the physics community has not yet been fully investigated and hence not been exploited.

An interesting first-principle alternative to LDA+U scheme for correlated materials is related with developments of Reduced-density-matrix-functional theory (RDMFT) [17, 18]. It is based on old Gilbert’s theorem, which shows that the expectation value of any observable in the ground state can be expressed as a functional of the one-body reduced density matrix. The advantage of RDMFT approach, compared to DFT, is that the exact many-body kinetic energy is easily expressed in terms of reduced-density-matrix. The RDMFT calculations of transition metal oxides gives correct insulating states of these correlated compounds.

Recently a new generation of continuous-time Quantum Monte Carlo (CT-QMC) methods for numerically exact calculation of complicated fermionic path integrals have been proposed for interacting electrons based on the weak-coupling [2] and strong-coupling [3] perturbation expansion. This methodological breakthrough in the quantum many-body theory stimulate a great progress in the electronic calculations of realistic strongly correlated systems within the dynamical mean-field theory (DMFT) where the solution of effective multiband impurity problem is the main point [4-8]. The most important achievement of the CT-QMC scheme is related with an opportunity to simulate real systems with arbitrary complex interaction vertex including spin-flip terms and retardation effects. Such progress was impossible with the conventional Hirsch-Fye QMC algorithm based on a time discretization approach.

New QMC technique stimulate the fast and important developments in the field of strongly correlated materials related with the realistic LDA+DMFT calculations for f-orbital case of plutonium [4,5], heavy-fermion systems [6], strongly correlated thermoelectric compounds [7] and novel high-temperature superconducting pnictides [8]. The first attempt to combine the new computational scheme with analytical perturbation approach in correlated fermion models has been presented [9].

Over the past decades, density functional theory (DFT) has become the effective single-particle theory of choice both for physicists from the solid state community as well as for quantum chemists. However, this tremendous success is also accompanied by an ever growing list of documented failures, especially for strongly correlated systems. Modern approaches to improve the description by better exchange-correlation functionals are based on an adapted treatment for short and long-range electron-electron interaction (so called range-seperated functionals) [10] or the seperate treatment of exchange and correlation functionals (exact exchange + random phase approximation) [11].

Beyond DFT, the GW approximation based on MBPT has found widespread use in the calculation of single-particle spectra. Originally used only to compute band structures in solids, GW has recently also been applied to study surfaces and molecules [12]. A detailed comparison of the self energy in the GW context with the exchange-correlation functional in DFT, is expected to pave the road for further improvement of functionals. This interplay of different approaches to correlated systems was already exploited in the realm of time dependent DFT [13].

Yet other approaches for interacting many-electron systems were put forward in the quantum chemistry community (see e.g. Refs. [14-16]). Wavefunction-based schemes allow to systematically approach the exact solution of the electronic Schroedinger equation and in this way offer a hierarchy useful for estimating error bars of simpler approximations. The methods range from efficient MBPT methods mostly employing Gauss-type basis functions to coupled-cluster theory and also multi-reference approaches. To deal with the cusp problem and basis deficiencies, r12 and more recently F12 methods have been brought forward [14]. The steep increase of the computational effort with molecular size has been circumvented by introducing linear-scaling methods for many quantum-chemical methods and for computing various molecular properties. They exploit the local electronic structure and open the way to treat large molecular systems with 1000 atoms and more at the HF, DFT, and MP2 levels [15]. Also the possibilities in performing highly accurate CC calculations has been dramatically increased. In addition, a variety of approximations were introduced for reducing prefactors by, e.g., auxiliary basis set expansions or Cholesky decompositions. Finally, also partially periodic boundary conditions have been accounted for [16]. The relation of all of these schemes to the correlated methods in the physics community has not yet been fully investigated and hence not been exploited.

An interesting first-principle alternative to LDA+U scheme for correlated materials is related with developments of Reduced-density-matrix-functional theory (RDMFT) [17, 18]. It is based on old Gilbert’s theorem, which shows that the expectation value of any observable in the ground state can be expressed as a functional of the one-body reduced density matrix. The advantage of RDMFT approach, compared to DFT, is that the exact many-body kinetic energy is easily expressed in terms of reduced-density-matrix. The RDMFT calculations of transition metal oxides gives correct insulating states of these correlated compounds.

## References

**France**

Andreas Savin ( CNRS and Sorbonne University ) - Organiser

**Germany**

Thomas Frauenheim ( University of Bremen ) - Organiser

Alexander I. Lichtenstein ( University of Hamburg ) - Organiser

Christian Ochsenfeld ( University of Munich / LMU ) - Organiser