Prof. Martin R. Zirnbauer
Research

Research Interests

I am a theoretical physicist with research focus on the mathematical physics of mesoscopic systems. The adjective "mesoscopic" here refers to physical systems situated at the boundary between the classical and quantum worlds. To give an example, electronic devices with sizes in the sub-micrometer range typically qualify as mesoscopic for temperatures below one degree Kelvin. Under such conditions the particles (in this case, the electrons) behave as waves according to the quantum mechanical principle of wave-particle duality. In fact, a large fraction of the research on mesoscopic systems is concerned with wave interference phenomena and their sensitivity to the temperature and other parameters.

Since 2001, my personal development as a scientist has been strongly influenced by the transregional Collaborative Research Center 12, "Symmetries and Universality of Mesoscopic Systems", whose research agenda I helped shape as its founding director (2003-2006). Funded by the Deutsche Forschungsgemeinschaft (the German National Science Foundation), CRC/TR 12 is unique in Germany and perhaps the world, in that it is centered about the close collaboration between physics and mathematics in the whole range from informal discussions to the publication of joint papers.

After CRC/TR 12 had been launched in 2003, I got engaged in the task of initiating a number of collaborations with mathematicians inside and outside the CRC. The challenge here was to develop a common language and formulate some fundamental questions of mesoscopic physics in a mathematically concise way so as to render them accessible for mathematical research. As a consequence of these efforts, we succeeded in various instances in achieving new insights and results which could not have been reached without the collaboration between physics and mathematics.

My own research addresses especially the influence of disorder (and chaotic dynamics) on quantum interference effects in mesoscopic systems. The theoretical study of such effects leads to a class of field-theoretical models, the so-called "nonlinear sigma models", variants of which appear in all of theoretical physics – from condensed matter theory to string theory. In one of my most highly cited papers I introduced (in 1996) a large and complete class of mathematical objects which figure as target spaces for nonlinear sigma models of disordered quantum Hamiltonian systems and became known as "Riemannian symmetric superspaces". These nonlinear spaces derive their significance from the fact that they are in one-to-one correspondence with prototypical random matrix models for the symmetry classes of disordered metals and superconductors. The latter are often called "Altland-Zirnbauer classes".

A good fraction of my current research is concerned with the study of Riemannian symmetric superspaces and their properties. This research combines the theory of Lie groups and their representations with methods of differential geometry and the theory of supermanifolds. An ongoing project (jointly with A. Alldridge and J. Hilgert, see [AHZ 10]) is to extend the classical theory of "harmonic analysis" to Riemannian symmetric superspaces. A spin-off of this work is a recent paper with Gruzberg, Ludwig and Mirlin [GLM 11] on symmetry relations for multifractal wave function statistics.

Beyond these conceptual and foundational issues, in my Cologne research group we pursue the development of a variety of mathematical techniques for disordered quantum systems. Some key words in this context are: hyperbolic Hubbard-Stratonovich transformation, Laplace-Borel transform, superbosonization, and the connection between the Wegner-Efetov supersymmetry method and the R-transform of Voiculescu's free probability theory
(see some of my recent talks).

By continuation of a long standing research interest of mine, I am still fascinated by the prospects of developing a conformal field theory description of disordered electron systems at critical points in two space dimensions. Here a paradigm is the topological phase transition between Hall plateaus for disordered electron systems exhibiting the integer quantum Hall effect. Good progress with this problem was recently made in [BWZ 13], by identifying a lattice realization for the proposed algebra of primary fields. There exist strong indications that these fields are governed by a variant of the quantum Liouville theory known from quantum gravity.

Last but not least, over the last few years I have been drawn into the research area of topological phases of quantum matter [ESI-Vienna 2014]. There, my work on the symmetry classes of disordered fermions is often cited as one of the foundations of the field of symmetry-protected topological phases of insulators and superconductors; and the 'symplectic zero mode' (one of my early discoveries [PRL 1992], which remained somewhat of a puzzle at the time) acquired meaning as the symmetry-protected edge mode of the quantum spin Hall insulator of Kane & Mele. With R. Kennedy we recently developed [KZ 14] a homotopy-theoretic proof of the Bott-Kitaev Table for Z/2Z symmetric ground states of gapped free-fermion systems. My mantra in this context is that symmetries must be defined primarily as TRUE symmetries, i.e. as operations that commute with the Hamiltonian (as opposed to anti-commute, like in the case of so-called chiral 'symmetries'). One vindication of this stringent approach is that it paves the way to include interactions.

Research Interests

Colloquium Talks