** Highlights **

** origin of orbital order**

The origin of orbital order has been debated for decades. Two mechanism have been proposed, the purely electronic super-exchange interaction of Kugel and Khomskii and the classical (perhaps Coulomb enhanced) Jahn-Teller coupling. We have solved this problem by developing a new approach which we combined with LDA+DMFT. In a series of papers we have shown that, in the text-book examples of orbitally ordered materials, KCuF

_{3}and LaMnO

_{3}, (i) the Kugel-Khomskii mechanism yields large orbital-ordering temperature but (ii) unfortunately too low to explain the existence of orbital order in real materials at very high temperature. To explain the latter, a static crystal-field, as the one due to the Jahn-Teller effect, is necessary.

Main papers:

**101**, 266405 (2008)

**104**, 086402 (2010)

**85**, 035124 (2012)

**87**, 195141 (2013)

**89**, 155109 (2014)

For a pedagogical introduction to orbital orderng you might see the lecture notes:

** Mott transition and Fermi surface of correlated ruthenates **

We have explained the nature of the metal-insulator transition in Ca

_{2-x}Sr

_{x}RuO

_{4}. We have shown that the L-Pbca --> S-Pbca structural change plays a key role. We have shown that, contrarily to what has been suggested, the spin-orbit interaction is not crucial. We have shown that Coulomb-enhanced spin-orbit and low-symmetry Coulomb terms are instead both essential to explain the Fermi surface of Sr

_{2}RuO

_{4}. Our studies are the

**first LDA+DMFT calculations**including at the same time the actual crystal structure, the crystal-field splitting and the

**spin-orbit interaction, and the full Coulomb vertex**.

Main papers:

**104**, 226401 (2010)

**116**, 106402 (2016)

**95**, 075145 (2017)

**97**, 085141 (2018)

** generalized CT-HYB and CT-INT QMC solvers for DMFT **

We have developed generalized CT-INT (continuous-time interaction-expansion) and CT-HYB (continuous-time hybridization-expansion) DMFT quantum Montecarlo solvers. These allow us to study Hamiltonians and Coulomb matrices of any symmetry, with and without spin-orbit interaction.

**We have found that the sign problem can be minimized by optimal basis choices, e.g., rotating to a crystal-field basis, which diagonalizes the one-electron part of the on-site Hamiltonian**[Flesch et al, 2013]. This idea has been later exploited also by other groups to reduce the sign problem in the presence of spin orbit coupling.

We have found a scheme for accounting for the double-counting correction in the presence of low-symmetry Coulomb terms.

Main papers:

**104**, 226401 (2010)

**87**, 195141 (2013)

**116**, 106402 (2016)

** building many-body models for molecular systems **

We have developed a general and efficient method to construct ab-initio many-body models for strongly-correlated molecular nano-magnets.

Main theory papers:

Phys. Rev. Lett.

**110**, 157204 (2013)

Phys. Rev. B

**94**, 224422 (2016)

For applications see publication list.

** the spin-state transitions in cobaltates **

We have explained the spin-state transitions in cobaltates as the result of a delicate balance between super-exhange energy, multiplets and crystal-field. We have mapped this competition into a

**Ising-like model with parameters calculated ab-initio**. We have shown that commonly used approximations to the Coulomb tensor might lead to completely incorrect results.

Main papers:

**86**, 184413 (2012)