TRR 12: Symmetries and Universality in Mesoscopic Systems
Mathematics
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.
Publications
- Semiclassical foundation of universality in quantum chaos, Phys. Rev. Lett. 93 (2004) 014103
S. Müller, S. Heusler, P. Braun, F. Haake, A. Altland
(See online at https://doi.org/10.1103/PhysRevLett.93.014103) - Exact results for one-dimensional disordered bosons with strong repulsion, Phys. Rev. Lett. 94 (2005) 060402
A. De Martino, M. Thorwart, R. Egger, R. Graham
(See online at https://doi.org/10.1103/PhysRevLett.94.060402) - Symmetry classes of disordered fermions, Commun. Math. Phys. 257 (2005) 725-771
P. Heinzner, A.H. Huckleberry, M.R. Zirnbauer
(See online at https://doi.org/10.1007/s00220-005-1330-9) - Energy correlations for a random matrix model of disordered bosons, J. Math. Phys. 47 (2006) 103304/1-24
T. Lueck, H.-J. Sommers, M.R. Zirnbauer
(See online at https://doi.org/10.1063/1.2356798) - Semiclassical theory of chaotic conductors, Phys. Rev. Lett. 96 (2006) 066804
S. Heusler, S. Müller, P. Braun, F. Haake
(See online at https://doi.org/10.1103/PhysRevLett.96.066804) - Magnetic confinement of massless Dirac fermions in graphene, Phys. Rev. Lett. 98 (2007) 066802
A. De Martino, L. Dell’Anna, R. Egger
(See online at https://doi.org/10.1103/PhysRevLett.98.066802) - On the geometry of orbits of Hermann actions, Geometriae Dedicata 129 (2007) 101-118
O. Goertsches, G. Thorbergsso
(See online at https://doi.org/10.1007/s10711-007-9198-9) - Periodic-orbit theory of level correlations, Phys. Rev. Lett. 98 (2007) 044103
S. Heusler, S. Müller, A. Altland, P. Braun, F. Haake
(See online at https://doi.org/10.1103/PhysRevLett.98.044103) - Quantum dots in graphene, Phys. Rev. Lett. 98 (2007) 016802
P.G. Silvestrov, K.B. Efetov
(See online at https://doi.org/10.1103/PhysRevLett.98.016802) - Superbosonization formula and its application to random matrix theory, J. Stat. Phys. 129 (2007) 809-832
J.E. Bunder, K.B. Efetov, V.E. Kravtsov, O.M. Yevtushenko, M.R. Zirnbauer
(See online at https://doi.org/10.1007/s10955-007-9405-y) - Universality for orthogonal and symplectic Laguerre-type ensembles, J. Stat. Phys. 129 (2007) 949-1053
P. Deift, D. Gioev, T. Kriecherbauer, M. Vanlessen
(See online at https://doi.org/10.1007/s10955-007-9325-x) - Level dynamics and the ten-fold way, J. Geom. Phys. 58 (2008) 1231-1240
A. Huckleberry, M. Kuś, P. Schützdeller
(See online at https://doi.org/10.1016/j.geomphys.2008.04.007) - Riemannian supergeometry, Math. Zeitschrift 260 (2008) 557-593
O. Goertsches
(See online at https://doi.org/10.1007/s00209-007-0288-z) - Superbosonization of invariant random matrix ensembles, Commun. Math. Phys. 283 (2008) 343-395
P. Littelmann, H.-J. Sommers, M.R. Zirnbauer
(See online at https://doi.org/10.1007/s00220-008-0535-0) - Nonadiabaticity and large fluctuations in a many-particle Landau-Zener problem, Phys. Rev. A 79 (2009) 042703/1-22
A. Altland, V. Gurarie, T. Kriecherbauer, A. Polkovnikov
(See online at https://doi.org/10.1103/PhysRevA.79.042703) - A pedestrian’s view on interacting particle systems, KPZ universality, and random matrices, J. Phys. A 43 (2010) 403001/1-41
T. Kriecherbauer, J. Krug
(See online at https://doi.org/10.1088/1751-8113/43/40/403001) - Characterization of geodesic flows on T 2 with and without positive topological entropy, Geom. Funct. Anal. 20 (2010) 1259-1277
E. Glasmachers, G. Knieper
(See online at https://doi.org/10.1007/s00039-010-0087-2) - Chevalley’s restriction theorem for reductive symmetric superpairs, J. Algebra 323 (2010) 1159-1185
A. Alldridge, J. Hilgert, M.R. Zirnbauer
(See online at https://doi.org/10.1016/j.jalgebra.2009.11.014) - Emergence of coherence in the Mott-insulator-superfluid quench of the Bose-Hubbard model, Phys. Rev. A 82 (2010) 063603/1-4
P. Navez, R. Schützhold
(See online at https://doi.org/10.1103/PhysRevA.82.063603) - Multiparticle equations for interacting Dirac fermions in magnetically confined graphene quantum dots, J. Phys. A 43 (2010) 215202/1-18
R. Egger, A. De Martino, H. Siedentop, E. Stockmeyer
(See online at https://doi.org/10.1088/1751-8113/43/21/215202) - A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch- Gordan coefficients, J. Math. Phys. 52 (2011) 023507/1-21
A. Alex, M. Kalus, A. Huckleberry, J. von Delft
(See online at https://doi.org/10.1063/1.3521562) - Large deviations for disordered bosons and multiple orthogonal polynomial ensembles, J. Math. Phys. 52 (2011) 073510/1-16
P. Eichelsbacher, J. Sommerauer, M. Stolz
(See online at https://doi.org/10.1063/1.3603994) - Rare events in population genetics: stochastic tunneling in a two-locus model with recombination, Phys. Rev. Lett. 106 (2011) 088101
A. Altland, A. Fischer, J. Krug, I.G. Szendro
(See online at https://doi.org/10.1103/PhysRevLett.106.088101) - Signatures of Wigner molecule formation in interacting Dirac fermion quantum dots, Phys. Rev. B 83 (2011) 085409/1-7
T. Paananen, H. Siedentop, R. Egger
(See online at https://doi.org/10.1103/PhysRevB.83.085409) - Symplectic geometry of entanglement, Commun. Math. Phys. 305 (2011) 441-468
A. Sawicki, A. Huckleberry, M. Kuś
(See online at https://doi.org/10.1007/s00220-011-1259-0) - Universality of random matrices and local relaxation flow, Invent. Math. 185 (2011) 75-119
L. Erdös, B. Schlein, H.-T. Yau
(See online at https://doi.org/10.1007/s00222-010-0302-7) - Class D spectral peak in Majorana quantum wires, Phys. Rev. Lett. 109 (2012) 227005
D. Bagrets, A. Altland
(See online at https://doi.org/10.1103/PhysRevLett.109.227005) - Fermionic transport and out-of-equilibrium dynamics in a homogeneous Hubbard model with ultracold atoms, Nature Physics 8 (2012) 213
U. Schneider, L. Hackermüller, J.P. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch, A. Rosch
(See online at https://doi.org/10.1038/nphys2205) - Fluctuation-induced magnetization dynamics and criticality at the interface of a topological insulator with a magnetically ordered layer, Phys. Rev. Lett. 109 (2012) 237203
F.S. Nogueira, I. Eremi
(See online at https://doi.org/10.1103/PhysRevLett.109.237203) - Induced Ginibre Ensemble of Random Matrices and Quantum Operations, J. Phys. A 45 (2012) 075203/1-31
J. Fischmann, W. Bruzda, B.A. Khoruzhenko, H.-J. Sommers, K. Życzkowski
(See online at https://doi.org/10.1088/1751-8113/45/7/075203) - Non-abelian symmetries in tensor networks: a quantum symmetry space approach, Ann. Phys. 327 (2012) 2972-3047
A. Weichselbaum
(See online at https://doi.org/10.1016/j.aop.2012.07.009) - Supersymmetry Approach to Wishart Correlation Matrices: Exact Results, J. Stat. Phys. 148 (2012) 981-998
C. Recher, M. Kieburg, T. Guhr, M.R. Zirnbauer
(See online at https://doi.org/10.1007/s10955-012-0567-x) - A link between quantum entanglement, secant varities and sphericity, J. Phys. A 46 (2013) 265301/1-20
A. Sawicki, V.V. Tsanov
(See online at https://doi.org/10.1088/1751-8113/46/26/265301) - Multiterminal Coulomb-Majorana junction, Phys. Rev. Lett. 110 (2013) 196401
A. Altland, R. Egger
(See online at https://doi.org/10.1103/PhysRevLett.110.196401) - Pseudogap state near a quantum critical point, Nature Phys. 9 (2013) 442
K.B. Efetov, H. Meier, C. Pepin
(See online at https://doi.org/10.1038/nphys2641) - A class of nonergodic interacting particle systems with unique invariant measure, Ann. Appl. Prob. 24 (2014) 2595-2643
B. Jahnel, C. Külske
(See online at https://doi.org/10.1214/13-AAP987) - Anderson’s orthogonality catastrophe, Commun. Math. Phys. 329 (2014) 979-998
M. Gebert, H. Küttler, P. Müller
(See online at https://doi.org/10.1007/s00220-014-1914-3) - Singular superspaces, Math. Zeitschrift 278 (2014) 441-492
A. Alldridge, J. Hilgert, T. Wurzbacher
(See online at https://dx.doi.org/10.1007/s00209-014-1323-5) - Superbosonization via Riesz superdistributions, , Forum Math. Sigma 2 (2014) e9, 64
A. Alldridge, Z. Shaikh
(See online at https://doi.org/10.1017/fms.2014.5)