# The first dynamical quantum simulator

When Feynman proposed quantum computing in 1982, he suggested as one application the simulation of quantum systems that are too hard for analytical methods or classical computers. In this collaboration with the experimental group of Immanuel Bloch (LMU and MPQ) a simulator for the study of the relaxation of a non-equilibrium many-body quantum state towards thermal equilibrium was proposed. While theory allowed the validation of this dynamical quantum simulator without a single free fit parameter, the experiment itself could go way beyond the simulation times reached by the best classical methods, allowing the observation oft he build-up of non-trivial quantum correlations and the thermalization process. Ref.: Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert, Bloch, Nature Physics 8, 325 (2012)

One of the most timely topics in condensed matter theory is many-body localization, where researchers try to understand the intricate interplay of disorder and interactions. The many-body localized phase has many intriguing properties: it is a perfect finite temperature insulator, fails to thermalize and is robust against changes in the microscopic Hamiltonian. Moreover, interactions can lead to delocalization, stabilizing an ergodic phase and thus a novel transition from that ergodic phase into the many-body localized (MBL) phase. In our recent work, we introduce the analysis of one-particle density matrices as a new framework to understand many-body localization from a single-particle perspective. Our key results are that eigenstates of one-particle density matrices behave very analogous to the eigenstates of the Anderson problem: they are localized in the MBL and extended in the ergodic phase. Most importantly, the MBL phase is quite similar to a Fermi liquid: both states emerge adiabatically out of noninteracting reference states (the Anderson insulator versus the noninteracting Fermi gas) and possess quasi-particles. In the MBL phase, these quasi-particles are localized in real space, with the eigenstates of one-particle density matrices providing the optimized single-particle approximation. Moreover, the occupation spectrum, i.e., the eigenvalues of one-particle density matrices, feature a discontinuity, just like in a Fermi liquid, where the discontinuity is proportional to the quasi-particle weight. This discontinuity, finite in the MBL phase but zero in the ergodic phase, provides a useful means to compute phase diagrams, as the figure shows. For more details, see: Bera, Schomerus, Heidrich-Meisner, Bardarson, Phys. Rev. Lett. 115, 046603 (2015) more

# Topological order in the kagome model: beyond Landau's paradigm of phase transitions

For more than 20 years, the ground state of the kagome model, one of the best-known frustrated quantum magnets, has remained elusive. In the world's largest DMRG simulations to date, we have demonstrated using topological Renyi entanglement entropy that, among the many proposals made over decades, it is overwhelmingly likely that this magnet has as ground state a gapped topological quantum spin liquid of the Z2 variety. Topological ground states in real-world systems have been quite elusive so far, with the exception of the fractional quantum Hall effect. Excitations are anyonic and do not obey conventional Fermi or Bose statistics. At the same time, this magnet does not break locally any symmetry group of the Hamiltonian, as would be the case in conventional Landau theory of phase transitions, but its topological order can only be classified in a global picture of the entire magnet. Ref.: Depenbrock, McCulloch and Schollwöck (Phys. Rev. Lett. 109, 067201 (2012))