Institute for Theoretical Physics • University of Cologne

The origin and development of cracks in earth materials is a subject of general interest and wide application: from seismologists studying earthquakes to engineers studying the strength of materials. Therefore many laboratory experiments have been carried out to investigate the response of rocks to an applied differential stress, often using acoustic emissions (AE) to track intermittent crack growth inside the rock specimen prior to system-sized sample failure. Under a constant applied stress in double-torsion tensile tests with a guide groove and a single dominant crack, independent observations of the stress intensity factor and crack growth velocity imply the size of the largest (sub-) critical crack grows with time t according to an inverse power-law. A similar law holds for the mean crack length in a population of micro-cracks growing under compressional stress with no pre-defined fault plane, often treated using mean field models that ignore localisation of damage clearly seen in the experiments. Here I present a new hypothesis for the origin of this formula by combining expressions for crack population growth and localisation in a single model. The model is tested and some of its parameters inferred from analysis of the AE rate and the spatial clustering of its source locations from laboratory data.

Some f-electron systems such as URu2Si2 and PrFe4P12 show ordered phases where the order parameter is difficult to identify. These systems are characterized by the non-Kramers configuration where the number of f -electrons per site is even. We have studied generalized Kondo lattice models for such non-Kramers systems, and have found a new kind of orders by combining the dynamical mean-field theory and the continuous-time quantum Monte Carlo calculation. In the case of two-channel Kondo lattice with two conduction electrons per site, for example, a strange order emerges where the one-body quantities such as magnetization and charge density remain the same as in the high- temperature phase. The (hidden) order parameter Ψ at each site is given by Ψ = ⟨S · (s1 − s2)⟩ where S is the localized pseudo spin and si (i = 1, 2) is the pseudo spin of conduction channel i at the same site. This order can be interpreted as itinerant higher multipoles of electrons, and also as analogue of the composite superconducting order with odd frequency pairing amplitude. We also discuss other exotic orders and implication of our results in actual f-electron systems.

The development of thermodynamics in the second half of the 19th century has had a strong impact on both technology and natural philosophy. It is true that the steam engine for the conversion of heat into work existed before thermodynamics was developed as a branch of physics. However, the systematic theory improved the conversion process, and it succeeded in developing other processes essential to modern life, notably refrigeration and rectification. So, altogether thermodynamics has provided humanity with cheap energy, and cheap fuel, - consequently with cheap, and abundant, and unspoiled food. Thus thermodynamics has made populations grow, and life expectancy increase beyond anything people could possibly have imagined 200 years ago. At the same time thermodynamics has uncovered the precarious balance between determinism and stochasticity which is essential to processes on earth, including life. The competition of those intentions is described by the doctrine of energy and entropy in thermodynamics; energy tends to force a system into one single state, and entropy tends to spread the system evenly over all possible states. These competing tendencies are weighed by temperature such that minimal energy determines cold systems and maximal entropy determines hot systems.

The Hubbard model is one of the most studied microscopic models in condensed matter physics. It describes on a very simple level the interplay between the kinetics and the Coulomb interaction of electrons in solid state systems. Generically, one expects at half filling a metallic phase for large kinetics while a Mott insulator is present for large interactions. In most cases the Mott phase shows additionally long-range spin order, since the Hubbard model can be mapped to a Heisenberg model in the strong-coupling limit. A direct transition between the metal and the long-range ordered Mott insulator was considered to be the standard case for a long time. But in recent years there are more and more evidences that especially on frustrated lattices there is the possibility of exotic and insulating intermediate phases without long range order. It is therefore an obviously relevant question what kind of effective low-energy theory describes such Mott phases and how to derive them. This is particular complicated when the spin liquid is located close to the metal-insulator transition as for the recently discovered spin liquid of the Hubbard model on the honeycomb lattice. In this talk we discuss these issues for the Mott phase of the Hubbard model on the triangular and on the honeycomb lattice.

Understanding the mutual interdependence between the behavior of dynamical processes on networks and the underlying topologies promises new insight for a large class of empirical networks. We present a generic approach to investigate this relationship which is applicable to a wide class of dynamics, namely to evolve networks using a performance measure based on the whole spectrum of the dynamics' time evolution operator. As an example, we consider the graph Laplacian describing diffusion processes, and we evolve the network structure such that a given sub-diffusive behavior emerges.

The description of non-Fermi liquid metals is one of the central problems in the theory of correlated electron systems. I present a holographic theory which builds on general features of the thermal entropy density and the entanglement entropy. Remarkable connections emerge between the holographic approach, and the postulated strong-coupling behavior of the field theory.