- Renormalization Group (RG), flow equations
- Gauge Theories
- Strongly-coupled fermionic systems
- QCD phase diagram
- Finite-Volume Effects in Quantum Field Theories
- Cold atomic gases
- Many-body physics
- RG approaches to density functional theories

Jens Braun, Timon Dörnfeld, Benedikt Schallmo, Sebastian Töpfel

Dense relativistic matter has attracted a lot of attention over many decades now, with a focus on an understanding of the phase structure and thermodynamics of dense strong-interaction matter. The analysis of dense strong-interaction matter is complicated by the fact that the system is expected to undergo a transition from a regime governed by spontaneous chiral symmetry breaking at low densities to a regime governed by the presence of a Cooper instability at intermediate and high densities. Renormalization group (RG) approaches have played and still play a prominent role in studies of dense matter in general. In the present work, we study RG flows of dense relativistic systems in the presence of a Cooper instability and analyze the role of the Silver-Blaze property. In particular, we critically assess how to apply the derivative expansion to study dense-matter systems in a systematic fashion. This also involves a detailed discussion of regularization schemes. Guided by these formal developments, we introduce a new class of regulator functions for functional RG studies which is suitable to deal with the presence of a Cooper instability in relativistic theories. We close by demonstrating its application with the aid of a simple quark-diquark model. [arXiv:2008.05978]

Jens Braun, Wei-jie Fu, Jan M. Pawlowski, Fabian Rennecke, Daniel Rosenblüh, Shi Yin

We calculate chiral susceptibilities in (2+1)-flavour QCD for different masses of the light quarks using the functional renormalisation group (fRG) approach to first-principles QCD. We follow the evolution of the chiral susceptibilities with decreasing masses as obtained from both the light-quark and the reduced quark condensate. The latter compares very well with recent results from the HotQCD collaboration for pion masses ≳100MeV. For smaller pion masses, the fRG and lattice results are still consistent. In particular, the estimates for the chiral critical temperature are in very good agreement. We close by discussing different extrapolations to the chiral limit. [arXiv:2003.13112]

Lukas Rammelmüller, Joaquín E. Drut, Jens Braun

We study spin- and mass-imbalanced mixtures of spin-1/2 fermions interacting via an attractive contact potential in one spatial dimension. Specifically, we address the influence of unequal particle masses on the pair formation by means of the complex Langevin method. By computing the pair-correlation function and the associated pair-momentum distribution we find that inhomogeneous pairing is present for all studied spin polarizations and mass imbalances. To further characterize the pairing behavior, we analyze the density-density correlations in momentum space, the so-called shot noise, which is experimentally accessible through time-of-flight imaging. At finite spin polarization, the latter is known to show distinct maxima at momentum configurations associated with the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) instability. Besides those maxima, we find that additional features emerge in the noise correlations when mass imbalance is increased, revealing the stability of FFLO-type correlations against mass imbalance and furnishing an experimentally accessible signature to probe this type of pairing. [arXiv:2003.06853]

Jens Braun, Marc Leonhardt, Martin Pospiech

Our understanding of the dynamics and the phase structure of dense strong-interaction matter is to a large extent still built on the analysis of low-energy models, such as those of the Nambu-Jona-Lasinio-type. In this work, we analyze the emergence of the latter class of models at intermediate and low energy scales from fundamental quark-gluon interactions. To this end, we study the renormalization group flow of a Fierz-complete set of four-quark interactions and monitor their strength at finite temperature and quark chemical potential. At small quark chemical potential, we find that the scalar-pseudoscalar interaction channel is dynamically rendered most dominant by the gauge degrees of freedom, indicating the formation of a chiral condensate. Moreover, the inclusion of quark-gluon interactions leaves a significant imprint on the dynamics as measured by the curvature of the finite-temperature phase boundary which we find to be in accordance with lattice QCD results. At large quark chemical potential, we then observe that the dominance pattern of the four-quark couplings is changed by the underlying quark-gluon dynamics, without any fine-tuning of the four-quark couplings. In this regime, the scalar-pseudoscalar interaction channel becomes subleading and the dominance pattern suggests the formation of a chirally symmetric diquark condensate. In particular, our study confirms the importance of explicit UA(1) breaking for the formation of this type of condensate at high densities. [arXiv:1909.06298]

Casey E. Berger, Lukas Rammelmüller, Andrew C. Loheac, Florian Ehmann, Jens Braun, Joaquín E. Drut

We review the theory and applications of complex stochastic quantization to the quantum many-body problem. Along the way, we present a brief overview of a number of ideas that either ameliorate or in some cases altogether solve the sign problem, including the classic reweighting method, alternative Hubbard-Stratonovich transformations, dual variables (for bosons and fermions), Majorana fermions, density-of-states methods, imaginary asymmetry approaches, and Lefschetz thimbles. We discuss some aspects of the mathematical underpinnings of conventional stochastic quantization, provide a few pedagogical examples, and summarize open challenges and practical solutions for the complex case. Finally, we review the recent applications of complex Langevin to quantum field theory in relativistic and nonrelativistic quantum matter, with an emphasis on the nonrelativistic case. [arXiv:1907.10183]

Marc Leonhardt, Martin Pospiech, Benedikt Schallmo, Jens Braun, Christian Drischler, Kai Hebeler, Achim Schwenk

We study the equation of state of symmetric nuclear matter at zero temperature over a wide range of densities using two complementary theoretical approaches. At low densities up to twice nuclear saturation density, we compute the energy per particle based on modern nucleon-nucleon and three-nucleon interactions derived within chiral effective field theory. For higher densities we derive for the first time constraints in a Fierz-complete setting directly based on quantum chromodynamics using functional renormalization group techniques. We find remarkable consistency of the results obtained from both approaches as they come together in density and the natural emergence of a maximum in the speed of sound cS at supranuclear densities with a value beyond the asymptotic c2S=1/3. The presence of a maximum appears tightly connected to the formation of a diquark gap. [arXiv:1907.05814]

Lukas Rammelmüller, Andrew C. Loheac, Joaquín E. Drut, Jens Braun

Published in Phys. Rev. Lett. 121 (2018) no.17, 173001

We study in a nonperturbative fashion the thermodynamics of a unitary Fermi gas over a wide range of temperatures and spin polarizations. To this end, we use the complex Langevin method, a first principles approach for strongly coupled systems. Specifically, we show results for the density equation of state, the magnetization, and the magnetic susceptibility. At zero polarization, our results agree well with state-of-the art results for the density equation of state and with experimental data. At finite polarization and low fugacity, our results are in excellent agreement with the third-order virial expansion. Our results suggest that the curvature of the finite-temperature phase boundary is very small close the spin-balanced case. [arXiv:1807.04664]

Jens Braun, Marc Leonhardt, Jan M. Pawlowski

Published in SciPost Phys. 6 (2019) no.5, 056

Low-energy effective theories have been used very successfully to study the low-energy limit of QCD, providing us with results for a plethora of phenomena, ranging from bound-state formation to phase transitions in QCD. These theories are consistent quantum field theories by themselves and can be embedded in QCD, but typically have a physical ultraviolet cutoff that restricts their range of validity. Here, we provide a discussion of the concept of renormalization group consistency, aiming at an analysis of cutoff effects and regularization-scheme dependences in general studies of low-energy effective theories. For illustration, our findings are applied to low-energy effective models of QCD in different approximations including the mean-field approximation. More specifically, we consider hot and dense as well as finite systems and demonstrate that violations of renormalization group consistency affect significantly the predictive power of the corresponding model calculations. [arXiv:1806.04432]

Andrew C. Loheac, Jens Braun, Joaquín E. Drut

Published in Phys. Rev. D 98 (2018) no.5, 054507

We calculate the finite-temperature density and polarization equations of state of one-dimensional fermions with a zero-range interaction, considering both attractive and repulsive regimes. In the path-integral formulation of the grand-canonical ensemble, a finite chemical potential asymmetry makes these systems intractable for standard Monte Carlo approaches due to the sign problem. Although the latter can be removed in one spatial dimension, we consider the one-dimensional situation in the present work to provide an efficient test for studies of the higher-dimensional counterparts. To overcome the sign problem, we use the complex Langevin approach, which we compare here with other approaches: imaginary-polarization studies, third-order perturbation theory, and the third-order virial expansion. We find very good qualitative and quantitative agreement across all methods in the regimes studied, which supports their validity. [arXiv:1804.10257]

Jens Braun, Marc Leonhardt, Martin Pospiech

Published in Phys. Rev. D 97, 076010 (2018)

Nambu-Jona-Lasinio-type models are often employed as low-energy models for the theory of the strong interaction to analyze its phase structure at finite temperature and quark chemical potential. In particular at low temperature and large chemical potential, where the application of fully first-principles approaches is currently difficult at best, this class of models still plays a prominent role to guide our understanding of the dynamics of dense strong-interaction matter. In this work, we consider a Fierz-complete version of the Nambu-Jona-Lasinio model with two massless quark flavors and study its renormalization group flow and fixed-point structure at leading order of the derivative expansion of the effective action. Sum rules for the various four-quark couplings then allow us to monitor the strength of the breaking of the axial UA(1) symmetry close to and above the phase boundary. We find that the dynamics in the ten-dimensional Fierz-complete space of four-quark couplings can only be reduced to a one-dimensional space associated with the scalar-pseudoscalar coupling in the strict large-Nc limit. Still, the interacting fixed point associated with this one-dimensional subspace appears to govern the dynamics at small quark chemical potential even beyond the large-Nc limit. At large chemical potential, corrections beyond the large-Nc limit become important and the dynamics is dominated by diquarks, favoring the formation of a chirally symmetric diquark condensate. In this regime, our study suggests that the phase boundary is shifted to higher temperatures when a Fierz-complete set of four-quark interactions is considered. [arXiv:1801.08338]