M. Salmhofer's group

We work on the connections of statistical mechanics to quantum field theory, on the mathematical and physical aspects of renormalization group (RG) theory and on its applications to condensed matter physics, and on quantum kinetic theory. Our methods range from mathematical proofs to computational solution of large differential equations.

One of the central topics in our current research is an RG approach to many-fermion systems, which is used to investigate the properties of the Hubbard model in the parameter range relevant for high-temperature superconductivity. The RG method applied here is an exact functional transformation of the action of the system, which leads to an infinite hierarchy of equations for the Green functions. Truncations of this hierarchy are used in applications. In a number of nontrivial cases, this truncation can be justified rigorously, so that the method lends itself to mathematical studies. These mathematical aspects are also under investigation.

At present, we have collaborations with ETH Zurich, the Max-Planck Institute for Solid State Research in Stuttgart, the University of British Columbia, Vancouver, the University of Munich, and Harvard University.

Members

Prof. Dr. Manfred Salmhofer
Dr. Oliver Lauscher
Dr. Yohei Kashima
Christoph Husemann
Kay-Uwe Giering
Giulio Schober

Projects

Fermi surfaces with singularities

J. Feldman, M. Salmhofer

Fermi surfaces with van Hove singularities are interesting both from the point of view of applications (high-temperature superconductors) and from the theoretical point of view. The logarithmic singularity in the density of states caused by the zero of the bare Fermi velocity gives rise to new marginally relevant terms in the RG equation [FRS]. Work on the question of an inversion theorem generalizing that proven in [FST] for regular Fermi surfaces is in progress.

[FRS] N. Furukawa, T.M. Rice, M. Salmhofer, Phys. Rev. Lett. 81 (1998) 3195-3198

[FST] J. Feldman, M. Salmhofer, E. Trubowitz, Comm. Pure Appl. Math. 53 (2000) 1350-1384

[FS] J. Feldman, M. Salmhofer, Rev. Math. Phys. 20, No. 3 (2008) 233 - 274 and 275 - 334

Fermi Surface Flows

W. Pedra, M. Salmhofer

The method of posing counterterms in constructive field theoretic studies of two-dimensional fermion systems leads to the inversion problem which has been solved to all orders in perturbation theory [FST2] but not yet nonperturbatively. We introduce a new RG flow where the Fermi surface is adjusted dynamically in the flow. This allows us to give a nonperturbative construction of two-dimensional Fermi systems with a regular Fermi surface at the temperature above the critical temperature for superconductivity without using counterterms. In the proof we combine the tree expansion of [SW] with the arch expansion of Iagolnitzer and Magnen (see [DR]) to extract overlapping loops [FST1] which are crucial for the regularity properties of the selfenergy [PS].

[FST1] J. Feldman, M. Salmhofer, E. Trubowitz, J. Stat. Phys. 84 (1996) 1209-1336

[FST2] J. Feldman, M. Salmhofer, E. Trubowitz, Comm. Pure Appl. Math. 53 (2000) 1350-1384

[DR] M. Disertori, V. Rivasseau, Comm. Math. Phys. 215, 251,291 (2000)

PS] W. Pedra, M. Salmhofer, Fermi Systems in Two Dimensions and Fermi Surface Flows, to appear in the Proceedings of the ICMP 2003 and papers to appear

[SW] M. Salmhofer, C. Wieczerkowski, J. Stat. Phys. 99 (2000) 557-586

RG flows with symmetry breaking

C. Honerkamp, O. Lauscher, W. Metzner, M. Salmhofer

The flow to strong coupling observed in many RG studies of interacting fermion systems [HSFR,HM] indicates the occurrence of symmetry breaking. It is also a major technical problem for the attempt to give a more detailed description of the symmetry-broken phases of such models. Tendencies for a suppression of the quasiparticle weight are weaker than the drive towards symmetry breaking instabilities [HS]. We have developed techniques for flows in which symmetry breaking can occur and studied the flow of a BCS gap in detail [SHML]. We are also working on the question of how Ward identities that are broken by the RG flow are restored at the end of the flow.

[HM] C. Halboth, W. Metzner, Phys. Rev. B 61 (2000) 7364; Phys. Rev. Lett. 85 (2000) 5162;

[HSFR] C. Honerkamp, M. Salmhofer, N. Furukawa, T.M. Rice, Phys. Rev. B 63, 035109 (2001)

[HS] C. Honerkamp, M. Salmhofer, Phys. Rev. B 67, 174504 (2003)

[SHML] M. Salmhofer, C. Honerkamp, W. Metzner, O. Lauscher, Prog. Theor. Phys. 112 (2004) 943-970

Ferromagnetism and Superconductivity

C. Husemann, M. Salmhofer

We study the interplay of ferromagnetism and superconductivity in the two-dimensional Hubbard model with hopping amplitudes t between nearest neighbours and t' between next-to-nearest neighbours, in the regime 0.3 < - t'/t < 0.5 . In this regime, the temperature-flow RG predicts a zero-temperature transition between d-wave superconductivity and ferromagnetism if the Fermi surface has van Hove singularities [HS]. We have performed a mean-field analysis of possible coexistence (diploma thesis of C. Husemann). Work on an RG treatment that allows for an analysis of this transition and on possible other symmetry breaking effects is in progress.

[HS] C. Honerkamp, M. Salmhofer, Phys. Rev. Lett. 87 (2001) 187004, Phys. Rev. B 64 (2001) 184516

Quantum Boltzmann Equation

L. Erdös, M. Salmhofer, H.-T. Yau

We study the emergence of the quantum Boltzmann equation from the reversible dynamics given by the N-particle Schr" odinger equation for fermions, on the kinetic timescale t of order lambda^{-2}. We have shown that the problem can be reduced to showing restricted quasifreeness of the time-evolved state on this timescale [ESY]. The mathematical investigation of this property is work in progress.

[ESY] L. Erdös, M. Salmhofer, H.-T. Yau, On the Quantum Boltzmann Equation, J. Stat. Phys. 116 (2004) 367-380

Quantum Diffusion

L. Erdös, M. Salmhofer, H.-T. Yau

We study the long-time limit of the quantum Lorentz gas. We prove that, for a weakly coupled system with coupling strength lambda, the time evolution on timescale t of order lambda^{-2-eta}, eta > 0, is given by a diffusion equation. This is the first time that a proof of the behaviour of these systems on time scales bigger than O(lambda ^{-2}) is given. The essential complication is that the number of collisions that happen on such timescales diverges as an inverse power of lambda.

[ESY] L. Erdös, M. Salmhofer, H.-T. Yau:
Acta Mathematica 200 (2008) 211-277
Comm. Math. Phys. 271 (2007) 1-53
Annales Henri Poincaré 8 (2007) 621-685

Publications

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