Renormalization group theory for fermions and order parameter fluctuations in interacting Fermi systems Von der Fakultat¨ Mathematik und Physik der Universitat¨ Stuttgart zur Erlangung der Wurde¨ eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung vorgelegt von Philipp Strack aus Frankfurt am Main Hauptberichter: Prof. Dr. Walter Metzner Mitberichter: Prof. Dr. Alejandro Muramatsu Tag der mundlichen¨ Prufung:¨ 22. April 2009 Max-Planck-Institut fur¨ Festkorperforschung,¨ Stuttgart, 2009 Acknowledgments Walter Metzner’s supervision privileged my scientific efforts: timely availability for questions, significant calculational help, and extremely flexible work arrangements left hardly any external factors to blame. Though not always easy to digest, his sober and matter-of-factly Wittgensteinian approach to penetrate problems has sharpened my wit during the last three years for what I am deeply thankful. Alejandro Mura- matsu is thanked for co-refereeing this thesis. Pawel Jakubczyk, So Takei, Sebastian Diehl, and Johannes Bauer are thanked wholeheartedly for contributing important stimuli and corrections during the course of this work. During initial stages of my PhD time, Sabine Andergassen, Tilman Enss, and Carsten Honerkamp were always available for questions. Roland Gersch and Julius Reiss have additionally provided me with code examples and many use- ful hints on programming. Further interactions with Inga Fischer, Andrey Katanin, Manfred Salmhofer, and Roland Zeyher are gratefully acknowledged. Christof Wetterich is thanked for providing the opportunity to interact frequently with his group at the Institute for Theoretical Physics in Heidelberg. Intense discus- sions with Jan Pawlowski, Holger Gies, and Hans Christian Krahl have shaped some ideas of this thesis. Jens Braun, Stefan Fl¨orchinger, Jens M¨uller, Georg Robbers, and Michael Scherer are also acknowledged for useful conversations. Gil Lonzarich is gratefully acknowledged for being an inspirational host during the summer 2007 in Cambridge, UK. The Quantum Matter group at Cavendish Labo- ratory and especially Stephen Rowley, Leszek Spalek, Montu Saxena are thanked for insightful exchanges. Anne Gerrit Knepel makes my life better in any respect; Rolf Dieter Strack’s engi- neering skills made our apartment more livable and freed up valuable time; Irmgard and Georg Walter Strack’s liquidity injections eased costs associated with frequent travel and coexisting apartments; Elisabeth Strack’s and Uwe Gs¨anger’s policies in- sured me safely; Eva Maria and Helmut Knepel enabled a most luxurious lifestyle for our two cats Mia and Momo during much of this PhD time. Thank you. Abstract The physics of interacting Fermi systems is extremely sensitive to the energy scale. Of particular interest is the low energy regime where correlation induced collective behavior emerges. The theory of interacting Fermi systems is confronted with the oc- curence of very different phenomena along a continuum of scales calling for methods capable of computing physical observables as a function of energy scale. In this thesis, we perform a comprehensive renormalization group analysis of two- and three-dimensional Fermi systems at low and zero temperature. We examine sys- tems with spontaneous symmetry-breaking and quantum critical behavior by deriving and solving flow equations within the functional renormalization group framework. We extend the Hertz-Millis theory of quantum phase transitions in itinerant fermion systems to phases with discrete and continuous symmetry-breaking, and to quantum critical points where the zero temperature theory is associated with a non-Gaussian fixed point. The order parameter is implemented by a bosonic Hubbard-Stratonovich field, which –for continuous symmetry-breaking– splits into two components cor- responding to longitudinal and transversal Goldstone fluctuations. We compute the finite temperature phase boundary near the quantum critical point explicitly including non-Gaussian fluctuations. We then set up a coupled fermion-boson renormalization group theory that cap- tures the mutual interplay of gapless fermions with massless order parameter fluc- tuations when approaching a quantum critical point. As a first application, we com- pute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimen- sions. Both, the order parameter propagator and the fermion propagator become non- analytic functions of momenta destroying the Fermi liquid behavior. We finally compute the effects of quantum fluctuations in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The flow equations capture the influence of longitudinal and Goldstone order parameter fluctuations on non-universal quantities such as the fermionic gap and the fermion-boson vertex, as well as the exact universal infrared asymptotics present in every fermionic superfluid. Contents 1 Introduction ................................................... 1 1.1 Experiments ..................................... ............ 1 1.1.1 Phase boundary close to a quantum critical point . ........ 1 1.1.2 Phase diagram of an attractive two-component Fermi gas...... 3 1.2 Thesisoutline ................................... ............. 4 Part I Theoretical Framework 2 Underlying concepts ............................................. 9 2.1 Fermiliquidinstabilities ........................ ............... 9 2.2 Spontaneoussymmetry-breaking .................... ............ 12 2.3 Quantumcriticality .............................. ............. 13 2.4 Therenormalizationgroup ......................... ............ 15 3 Functional renormalization group ................................. 17 3.1 Functional integral for quantum many-particle systems .............. 17 3.1.1 Superfield formulation for fermionic and bosonic models ...... 18 3.1.2 One-particle irreducible generating functional . ............. 19 3.2 Flowequations ................................... ............ 19 3.2.1 Exactflowequation.............................. ........ 19 3.2.2 Spontaneous symmetry-breaking.................. ......... 22 4 Summary Part I ................................................. 25 Part II Applications 5 Hertz-Millis theory with discrete symmetry-breaking ................ 29 5.1 Introduction.................................... .............. 29 Contents VII 5.2 Bosonicaction ................................... ............ 30 5.3 Method .......................................... ........... 32 5.3.1 Truncation .................................... ......... 33 5.3.2 Flowequations................................. ......... 34 5.4 Zero-temperature solution at the quantum critical point ............. 35 5.4.1 z 2 .................................................. 36 5.4.2 z =≥ 1 .................................................. 37 5.5 Finitetemperatures.............................. .............. 38 5.5.1 z = 3 .................................................. 39 5.5.2 z = 2 .................................................. 44 5.5.3 z = 1 .................................................. 45 5.6 Conclusion ...................................... ............ 48 6 Quantum critical points with Goldstone modes ...................... 49 6.1 Introduction.................................... .............. 49 6.2 σ-Π Model for continuoussymmetry-breaking ................ .... 50 6.3 Method .......................................... ........... 53 6.4 Finite temperature phase boundary in three dimensions . ............ 54 6.4.1 Flowequations................................. ......... 54 6.4.2 Classicalfixedpoint ............................ ......... 56 6.4.3 Shift exponent ψ ........................................ 56 6.5 Infrared asymptotics in the symmetry-broken phase . ............ 57 6.5.1 Flowequations................................. ......... 57 6.5.2 Analyticalresults............................. ........... 58 6.6 Conclusion ...................................... ............ 60 7 Fermi-Bose renormalization group for quantum critical fermion systems 61 7.1 Introduction.................................... .............. 61 7.2 Diracconemodel.................................. ........... 62 7.2.1 Mean-fieldtheory ............................... ........ 63 7.3 Method .......................................... ........... 64 7.3.1 Truncation .................................... ......... 65 7.3.2 Flowequations................................. ......... 68 7.4 Solutionatthequantumcriticalpoint ............... ............. 71 7.4.1 Quantumcritical flows in two dimensions ............ ....... 72 7.4.2 Quantumcriticalexponents ...................... ......... 75 7.5 Conclusion ...................................... ............ 76 8 Fermionic superfluids at zero temperature .......................... 77 8.1 Introduction.................................... .............. 77 8.2 Bareaction ...................................... ............ 78 8.3 Mean-fieldtheory................................. ............ 80 VIII Contents 8.4 Truncation...................................... ............. 83 8.4.1 Symmetricregime ............................... ........ 83 8.4.2 Symmetry-brokenregime......................... ........ 85 8.4.3 Flowequations................................. ......... 86 8.4.4 Relationtomean-fieldtheory..................... ......... 92 8.5 Results........................................