- 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

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

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

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

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]

Lukas Rammelmüller, Joaquín E. Drut, Jens Braun

The theoretical treatment of Fermi systems consisting of particles with unequal masses is challenging. Even in one spatial dimension analytic solutions are limited to special configurations and numerical progress with Monte Carlo simulations is hindered by the sign-problem. To circumvent this issue, we exploit the Complex Langevin approach and study one-dimensional mass-imbalanced two-component Fermi gases with attractive and repulsive interactions. We find perfect agreement with results obtained by other methods in a range of parameter space. Promisingly, our approach is not limited to the specific model presented here and can easily be extended to finite spin polarization and, most notably, can also be applied in higher dimensions. [arXiv:1710.11421]

A. C. Loheac, J. Braun, J. E. Drut

We calculate the pressure and density of polarized non-relativistic systems of two-component fermions coupled via a contact interaction at finite temperature. For the unpolarized one-dimensional system with an attractive interaction, we perform a third-order lattice perturbation theory calculation and assess its convergence by comparing with hybrid Monte Carlo. In that regime, we also demonstrate agreement with real Langevin. For the repulsive unpolarized one-dimensional system, where there is a so-called complex phase problem, we present lattice perturbation theory as well as complex Langevin calculations. For our studies, we employ a Hubbard-Stratonovich transformation to decouple the interaction and automate the application of Wick's theorem for perturbative calculations, which generates the diagrammatic expansion at any order. We find excellent agreement between the results from our perturbative calculations and stochastic studies in the weakly interacting regime. In addition, we show predictions for the strong coupling regime as well as for the polarized one-dimensional system. Finally, we show a first estimate for the equation of state in three dimensions where we focus on the polarized unitary Fermi gas. [arXiv:1710.05020]

Lukas Rammelmüller, William J. Porter, Joaquín E. Drut, Jens Braun

The calculation of the ground state and thermodynamics of mass-imbalanced Fermi systems is a challenging many-body problem. Even in one spatial dimension, analytic solutions are limited to special configurations and numerical progress with standard Monte Carlo approaches is hindered by the sign problem. The focus of the present work is on the further development of methods to study imbalanced systems in a fully non-perturbative fashion. We report our calculations of the ground-state energy of mass-imbalanced fermions using two different approaches which are also very popular in the context of the theory of the strong interaction (Quantum Chromodynamics, QCD): (a) the hybrid Monte Carlo algorithm with imaginary mass imbalance, followed by an analytic continuation to the real axis; and (b) the Complex Langevin algorithm. We cover a range of on-site interaction strengths that includes strongly attractive as well as strongly repulsive cases which we verify with non-perturbative renormalization group methods and perturbation theory. Our findings indicate that, for strong repulsive couplings, the energy starts to flatten out, implying interesting consequences for short-range and high-frequency correlation functions. Overall, our results clearly indicate that the Complex Langevin approach is very versatile and works very well for imbalanced Fermi gases with both attractive and repulsive interactions. [arXiv:1708.03149]

Lukas Rammelmüller, William J. Porter, Jens Braun, Joaquín E. Drut

Published in Phys. Rev. A 96, 033635 (2017)

We study the evolution from few- to many-body physics of fermionic systems in one spatial dimension with attractive pairwise interactions. We determine the detailed form of the momentum distribution, the structure of the one-body density matrix, and the pairing properties encoded in the two-body density matrix. From the low- and high-momentum scaling behavior of the single-particle momentum distribution we estimate the speed of sound and Tan's contact, respectively. Both quantities are found to be in agreement with previous calculations. Based on our calculations of the one-body density matrices, we also present results for the particle-partition entanglement entropy, for which we find a logarithmic dependence on the total particle number. [arXiv:1706.00031]

Jens Braun, Marc Leonhardt, Martin Pospiech

Published in Phys. Rev. D 96, 076003 (2017) - PRD Editors’ Suggestion

Nambu-Jona-Lasinio-type models are frequently employed as low-energy models in various research fields. With respect to the theory of the strong interaction, this class of models is indeed often used to analyze the structure of the phase diagram at finite temperature and quark chemical potential. The predictions from such models for the phase structure at finite quark chemical potential are of particular interest as this regime is difficult to access with lattice Monte Carlo approaches. In this work, we consider a Fierz-complete version of a Nambu-Jona-Lasinio model. By studying its renormalization group flow, we analyze in detail how Fierz-incomplete approximations affect the predictive power of such model studies. In particular, we investigate the curvature of the phase boundary at small chemical potential, the critical value of the chemical potential above which no spontaneous symmetry breaking occurs, and the possible interpretation of the underlying dynamics in terms of difermion-type degrees of freedom. We find that the inclusion of four-fermion channels other than the conventional scalar-pseudoscalar channel is not only important at large chemical potential but also leaves a significant imprint on the dynamics at small chemical potential as measured by the curvature of the finite-temperature phase boundary. [arXiv:1705.00074]

Paul Springer, Jens Braun, Stefan Rechenberger, Fabian Rennecke

Published in EPJ Web Conf. 137 (2017) 03022

The QCD phase diagram at finite temperature and density has attracted considerable interest over many decades now, not least because of its relevance for a better understanding of heavy-ion collision experiments. Models provide some insight into the QCD phase structure but usually rely on various parameters. Based on renormalization group arguments, we discuss how the parameters of QCD low-energy models can be determined from the fundamental theory of the strong interaction. We particularly fo- cus on a determination of the temperature dependence of these parameters in this work and comment on the effect of a finite quark chemical potential. We present first results and argue that our findings can be used to improve the predictive power of future model calculations. [arXiv:1611.06020]

Dietrich Roscher, Jens Braun

Published in
J. Phys. B50 (2017) 205301

The formation of bosonic bound states underlies the formation of a superfluid ground state in the many-body phase diagram of ultracold Fermi gases. We study bound-state formation in a spin- and mass-imbalanced ultracold Fermi gas confined in a box with hard-wall boundary conditions. Because of the presence of finite Fermi spheres, the center-of-mass momentum of the potentially formed bound states can be finite, depending on the parameters controlling mass and spin imbalance as well as the coupling strength. We exploit this observation to estimate the potential location of inhomogeneous phases in the many-body phase diagram as a function of spin- and mass imbalance as well as the box size. Our results suggest that a hard-wall box does not alter substantially the many-body phase diagram calculated in the thermodynamic limit. Therefore, such a box may serve as an ideal trap potential to bring experiment and theory closely together and facilitate the search for exotic inhomogeneous ground states. [arXiv:1611.02081]

Sandra Kemler, Martin Pospiech, Jens Braun

Published in Journal of Physics G: Nuclear and Particle Physics, Volume 44, Number 1

In nuclear physics, density functional theory (DFT) provides the basis for state-of-the art studies of ground-state properties of heavy nuclei. However, the direct relation of the density functional underlying these calculations and the microscopic nuclear forces is not yet fully understood. We present a combination of DFT and renormalization group (RG) techniques which allows to study selfbound many-body systems from microscopic interactions. We discuss its application with the aid of systems of identical fermions interacting via a long-range attractive and short-range repulsive two-body force in one dimension. We compute ground-state energies, intrinsic densities, and density correlation functions of these systems and compare our results to those obtained from other methods. In particular, we show how energies of excited states as well as the absolute square of the ground-state wave function can be extracted from the correlation functions within our approach. The relation between many-body perturbation theory and our DFT-RG approach is discussed and illustrated with the aid of the calculation of the second-order energy correction for a system of N identical fermions interacting via a general two-body interaction. Moreover, we discuss the control of spuriously emerging fermion self-interactions in DFT studies within our framework. In general, our approach may help to guide the development of energy functionals for future quantitative DFT studies of heavy nuclei from microscopic interactions. [arXiv:1606.04388]

Jens Braun, Felix Karbstein, Stefan Rechenberger, Dietrich Roscher

Published in Phys.Rev. D93 (2016) 1, 014032

Nambu-Jona-Lasinio-type models have been used extensively to study the dynamics of the theory of the strong interaction at finite temperature and quark chemical potential on a phenomenological level. In addition to these studies, which are often performed under the assumption that the ground state of the theory is homogeneous, searches for the existence of crystalline phases associated with inhomogeneous ground states have attracted a lot of interest in recent years. In this work, we study the Polyakov-loop extended Nambu--Jona-Lasinio model and find that the existence of a crystalline phase is stable against a variation of the parametrization of the underlying Polyakov loop potential. To this end, we adopt two prominent parametrizations. Moreover, we observe that the existence of a quarkyonic phase depends crucially on the parametrization, in particular in the regime of the phase diagram where inhomogeneous chiral condensation is favored. [arXiv:1510.04012]

A.C. Loheac, J. Braun, J.E. Drut, D. Roscher

Published in Phys.Rev. A92 (2015) 063609

We present a non-perturbative computation of the equation of state of polarized, attractively interacting, non-relativistic fermions in one spatial dimension. We show results for the density, spin magnetization, magnetic susceptibility, and Tan's contact. We compare with the second-order virial expansion and interpret our results in terms of pairing correlations. Our lattice Monte Carlo calculations implement an imaginary chemical potential difference to avoid the sign problem. The thermodynamic results on the imaginary side are analytically continued to obtain results on the real axis. We focus on an intermediate- to strong-coupling regime, and cover a wide range of temperatures and spin imbalances. [arXiv:1508.03314]

Dietrich Roscher, Jens Braun, Joaquín E. Drut

Published in Phys.Rev. A91 (2015) 5, 053611

We study the phase diagram of mass- and spin-imbalanced unitary Fermi gases, in search for the emergence of spatially inhomogeneous phases. To account for fluctuation effects beyond the mean-field approximation, we employ renormalization group techniques. We thus obtain estimates for critical values of the temperature, mass and spin imbalance, above which the system is in the normal phase. In the unpolarized, equal-mass limit, our result for the critical temperature is in accordance with state-of-the-art Monte Carlo calculations. In addition, we estimate the location of regions in the phase diagram where inhomogeneous phases are likely to exist. We show that an intriguing relation exists between the general structure of the many-body phase diagram and the binding energies of the underlying two-body bound-state problem, which further supports our findings. Our results suggest that inhomogeneous condensates form for mass ratios of the spin-down and spin-up fermions greater than three. The extent of the inhomogeneous phase in parameter space increases with increasing mass imbalance. [arXiv:1501.05544]

J. Braun, W. A. Mian, S. Rechenberger

We study the effect of an external magnetic field on the chiral phase transition in the theory of the strong interaction by means of a renormalization-group (RG) fixed-point analysis, relying on only one physical input parameter, the strong coupling at a given large momentum scale. To be specific, we consider the interplay of the RG flow of four-quark interactions and the running gauge coupling. Depending on the temperature and the strength of the magnetic field, the gauge coupling can drive the quark sector to criticality, resulting in chiral symmetry breaking. In accordance with lattice Monte-Carlo simulations, we find that the chiral phase transition temperature decreases for small values of the external magnetic field. For large magnetic field strengths, however, our fixed-point study predicts that the phase transition temperature increases monotonically. [arXiv:1412.6025]

Jens Braun, Leonard Fister, Jan M. Pawlowski, Fabian Rennecke

We present an analysis of the dynamics of two-flavour QCD in the vacuum. Special attention is payed to the transition from the high energy quark-gluon regime to the low energy regime governed by hadron dynamics. This is done within a functional renormalisation group approach to QCD amended by dynamical hadronisation techniques. The latter allow us to describe conveniently the transition from the perturbative high-energy regime to the nonperturbative low-energy limit without suffering from a fine-tuning of model parameters. In the present work, we apply these techniques to two-flavour QCD with physical quark masses and show how the dynamics of the dominant low-energy degrees of freedom emerge from the underlying quark-gluon dynamics. [arXiv:1412.1045]

Jens Braun, Stefan Finkbeiner, Felix Karbstein, Dietrich Roscher

Published in Phys.Rev. D91 (2015) 11, 116006

We revisit the Gross-Neveu model with N fermion flavors in 1+1 dimensions and compute its phase diagram at finite temperature and chemical potential in the large-N limit. To this end, we double the number of fermion degrees of freedom in a specific way which allows us to detect inhomogeneous phases in an efficient manner. We show analytically that this "fermion doubling trick" predicts correctly the position of the boundary between the chirally symmetric phase and the phase with broken chiral symmetry. Most importantly, we find that the emergence of an inhomogeneous ground state is predicted correctly. We critically analyze our approach based on this trick and discuss its applicability to other theories, such as fermionic models in higher dimensions, where it may be used to guide the search for inhomogeneous phases. [arXiv:1410.8181]

I. Boettcher, J. Braun, T. K. Herbst, J. M. Pawlowski, D. Roscher, C. Wetterich

Published in Phys. Rev. A 91, 013610

We investigate the phase structure of spin-imbalanced unitary Fermi gases beyond mean-field theory by means of the Functional Renormalization Group. In this approach, quantum and thermal fluctuations are resolved in a systematic manner. The discretization of the effective potential on a grid allows us to accurately account for both first- and second-order phase transitions that are present on the mean-field level. We compute the full phase diagram in the plane of temperature and spin-imbalance and discuss the existence of other conjectured phases such as the Sarma phase and a precondensation region. In addition, we explain on a qualitative level how we expect that in-situ density images are affected by our findings and which experimental signatures may potentially be used to probe the phase structure. [arXiv:1409.5070]

Jens Braun, Joaquín E. Drut, Dietrich Roscher

Published in Phys.Rev.Lett. 114 (2015) 5, 050404

We calculate the zero-temperature equation of state of mass-imbalanced resonant Fermi gases in an ab initio fashion, by implementing the recent proposal of imaginary-valued mass difference to bypass the sign problem in lattice Monte Carlo calculations. The fully non-perturbative results thus obtained are analytically continued to real mass imbalance to yield the physical equation of state, providing predictions for upcoming experiments with mass-imbalanced atomic Fermi gases. In addition, we present an exact relation for the rate of change of the equation of state at small mass imbalances, showing that it is fully determined by the energy of the mass-balanced system. [arXiv:1407.2924]

Jens Braun, Holger Gies, Lukas Janssen, Dietrich Roscher

Published in Phys. Rev. D 90, 036002

We analyze the many-flavor phase diagram of quantum electrodynamics (QED) in 2+1 (Euclidean) space-time dimensions. We compute the critical flavor number above which the theory is in the quasi-conformal massless phase. For this, we study the renormalization group fixed-point structure in the space of gauge interactions and pointlike fermionic self-interactions, the latter of which are induced dynamically by fermion-photon interactions. We find that a reliable estimate of the critical flavor number crucially relies on a careful treatment of the Fierz ambiguity in the fermionic sector. Using a Fierz-complete basis, our results indicate that the phase transition towards a chirally-broken phase occurring at small flavor numbers could be separated from the quasi-conformal phase at larger flavor numbers, allowing for an intermediate phase which is dominated by fluctuations in a vector channel. If these interactions approach criticality, the intermediate phase could be characterized by a Lorentz-breaking vector condensate. [arXiv:1404.1362]

Jens Braun, Joaquín E. Drut, Thomas Jahn, Martin Pospiech, Dietrich Roscher

Published in Phys. Rev. A 89, 053613

We analyze the phase structure of mass- and spin-imbalanced unitary Fermi gases in harmonic traps. To this end, we employ Density Functional Theory in the local density approximation. Depending on the values of the control parameters measuring mass and spin imbalance, we observe that three regions exist in the trap, namely: a superfluid region at the center, surrounded by a mixed region of resonantly interacting spin-up and spin-down fermions, and finally a fully polarized phase surrounding the previous two regions. We also find regimes in the phase diagram where the existence of a superfluid region at the center of the trap is not energetically favored. We point out the limitations of our approach at the present stage, and call for more detailed (ab initio) studies of the equation of state of uniform, mass-imbalanced unitary Fermi gases. [arXiv:1402.7042]