# Multi-loop functional renormalization group

The many-body problem of nonrelativistic quantum-field theory is equipped with a well-known set of exact equations for its correlation functions. If these self-consistent many-body relations are expressed in their energy-momentum representation, they interrelate the different correlation functions between all energy scales, often involving integrations over all energy-momenta. However, a typical feature of interacting quantum many-body systems is that their relevant energy scales span several orders of magnitude. Conventional perturbative approaches or approaches that directly work with the self-consistent many-body relations treat all energy scales at once—they are therefore prone to inaccuracies and often plagued by infrared divergences. A very successful approach to such systems is instead given by the renormalization group (RG) technique which treats energy scales successively, starting from high ones and progressing towards lower ones.

We have recently formulated a multi-loop functional renormalization group (mfRG) approach which enables us to efficiently solve the many-body parquet equations in the parquet approximation [1-3]. The advantages of mfRG versus previous 1-loop fRG approaches have been impressively demonstrated in a benchmark study of the 2D Hubbard model [4], performed using the Matsubara formalism. Currently we are engaged in implementing mfRG in the Keldysh formalism [5,6]. This will allow the computation of real-frequency correlation functions for interacting many-body systems at arbitrary temperatures, yielding deep insights into a sytem's underlying excitations at arbitrary energy scales. The method is extremely flexible and can be applied to any interacting many-body system. Possible applications include:

- non-equilibrium transport through quantum dots [7];
- one-dimensional quantum liquids beyond the Luttinger liquid pardigm [8];
- the Fermi-polaron problem, involving a mobile impurity coupled to a Fermi sea [9];
- nonequilibrium transport through quantum point contacts [10,11];
- quantum magnets, treated using a pseudo-fermion representation of spin operators [12];
- many-body localization in disordered interacting electron systems [13].
- combining fRG with Dynamical Mean Field Theory (DMFT) to study the regime of strong interactions [14].

Each of these topics could be pursued in the context of a Master or PhD Thesis project.

Mentors: vondelft@lmu.de, fabian.kugler@physik.uni-muenchen.de

[0] A pedagogical introduction to traditional fRG may be found in the following book:

P. Kopietz, L. Bartosch and F. Schütz, *Introduction to the Functional Renormalization Group*, Springer (2010),
in particular Part II, chapters 6 to 8. Knowledge of the material presented there is presumed in all the fRG references
below.

[1] F. B. Kugler and J. von Delft. *Multiloop Functional Renormalization Group That Sums Up All Parquet Diagrams.* Phys. Rev. Lett. 120, 057403 (2018).

[2] F. B. Kugler and J. von Delft. *Multiloop functional renormalization group for general models.* Phys. Rev. B 97, 035162 (2018).

[3] F. B. Kugler and J. von Delft. *Derivation of exact flow equations from the self-consistent parquet relations.* New J. Phys. 20, 123029 (2018).

[4] A. Tagliavini, C. Hille, F. B. Kugler, S. Andergassen, A. Toschi and C. Honerkamp. *Multiloop functional renormalization group for the two-dimensional Hubbard model: Loop convergence of the response functions.* SciPost Phys. 6, 009 (2019).

[5] A. Kamenev. *Book: Field Theory of Non-Equilibrium Systems. * Cambridge University Press, 2013.

[6] S. G. Jakobs, M. Pletyukhov and H. Schoeller. *Properties of multi-particle Green and vertex functions within Keldysh formalism. * J. Phys. A: Math. Theor. 43 43, 103001 (2010).

[7] F. Schwarz, I. Weymann, J. von Delft and A. Weichselbaum. * Nonequilibrium Steady-State Transport in Quantum Impurity Models: A Thermofield and Quantum Quench Approach Using Matrix Product States.* Phys. Rev. Lett. 121, 137702 (2018).

[8] A. Imambekov, T. L. Schmidt and L. I. Glazman. *One-dimensional quantum liquids: Beyond the Luttinger liquid paradigm.* Rev. Mod. Phys. 84, 1253-1306 (2012).

[9] R. Schmidt, M. Knap, D. A. Ivanov, J.-S. You, M. Cetina, and E. Demler, *Universal many-body response of heavy impurities coupled to a Fermi sea.*, Rep. Prog. Phys. 81, 024401 (2018).

[10] F. Bauer, J. Heyder, E. Schubert, D. Borowsky, D. Taubert, B. Bruognolo, D. Schuh, W. Wegscheider, J. von Delft and S. Ludwig. *Microscopic origin of the 0.7-anomaly in quantum point contacts.* Nature 501, 73-78 (2013).

[11] D. H. Schimmel, B. Bruognolo and J. von Delft. *Spin Fluctuations in the 0.7 Anomaly in Quantum Point Contacts.* Phys. Rev. Lett. 119, 196401 (2017).

[12] Y. Iqbal, R. Thomale, F. Parisen Toldin, S. Rachel, and J. Reuther, *Functional renormalization group for three-dimensional quantum magnetism.* Phys. Rev. B 94, 140408 (2016).

[13] D. A. Abanin and Z. Papíc, *Recent progress in many-body localization.* Annalen der Physik 529, 1700169 (2017).

[14] D. Vilardi, C. Taranto and W. Metzner. *Antiferromagnetic and d-wave pairing correlations in the strongly interacting two-dimensional Hubbard model from the functional renormalization group.* Phys. Rev. B 99, 104501 (2019).