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TRR 12:  Symmetries and Universality in Mesoscopic Systems

Subject Area Physics
Mathematics
Term from 2003 to 2015
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 5485756
 
Final Report Year 2015

Final Report Abstract

The novelty and challenge of SFB/TR 12 was to create – arguably for the first time in Germany if not the world, outside of string theory – an interdisciplinary research platform for the synergetic and productive collaboration between theoretical physics and pure mathematics (as opposed to applied mathematics or computer science). A golden opportunity for such a platform – this was our vision of fifteen years ago – was offered by the research field of mesoscopic systems (located at the boundary between the microscopic/quantum and macroscopic/classical worlds), a research field still vibrant and full of promise as an area at the cutting edge of physics, yet mature enough in order for sound mathematical research to be feasible. In twelve years of interdisciplinary effort, driven by a regular series of week-long workshops and short retreats, we established a common language and culture of communication and cooperation between the two disciplines, spanning the range from foundational lecture series to joint publications at the frontier of research. In doing so, we trained a generation of young scientists to be fluent in both disciplines; many of them have now been appointed to tenured positions or are well on their way to academic careers. On the physics side of the SFB, we have done major work impacting the research areas of ultracold atoms, graphene, and topological insulators. A particular highlight among our results is a comprehensive theory of universality of the energy level statistics of quantum chaotic systems, developed on the basis of a semiclassical expansion that sums over the periodic orbits of the chaotic classical system. On the mathematics side, we have analyzed semiclassical limit phenomena for Lie group actions and representations. Motivated by the supersymmetry methods of mesoscopic physics, we have established a vigorous activity, leading to numerous new results, in the theory of supermanifolds. In the interdisciplinary realm, our main achievements are the following. (i) The scheme for symmetry classification of disordered electron systems (the “Tenfold Way”) was put in its definitive form and mathematically proved. Reinterpreted in an surprising way, this scheme now makes its imprint on the flourishing field of topological insulators. (ii) Using tools from symplectic geometry, we have introduced a new measure of entanglement for use in quantum information theory. (iii) We have invented, applied, and given a rigorous proof of a new random-matrix method (called the “superbosonization formula”) which significantly enhances the tractable range of probabilistic models in mesoscopic physics and beyond. (iv) A very general numerical code library (“QSpace”) was developed, optimized, and put to good use in applications. QSpace allows a computationally efficient treatment of quantum systems with non-commuting symmetries, especially of two-dimensional tensor networks.

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