Recent Thesis Projects
# Recent Thesis Projects

**December 2019**
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Scaling Behavior in Dense Matter

**December 2019**
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Exploring Imbalanced Fermi Gases with Stochastic Quantization

**November 2019**
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From Hot to Cold, from Dense to Dilute – Renormalization Group Studies of Strongly-Interacting Matter

**Oktober 2019**
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Phase Structure and Equation of State of Dense Strong-Interaction Matter

**August 2019**
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Ground-State Equation of State of Spin-Polarized Unitary Fermi Gases

**Januar 2019**
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Stochastic Calculation of Self-Bound Quantum Mechanical States

**August 2018**
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Langevin-Approach to Quantum Mechanics

The following thesis projects have recently been completed in this group. if
you are interested in writing a thesis here, **do not hesitate to contact
us**

Sebastian Töpfel, M. Sc.

We apply functional renormalization group methods to scalar field theories and low-energy models of quantum chromodynamics. First we present a comprehensive derivation of the Wetterich equation in order to generate flow equations in O(n) symmetric scalar field theories. The system of flow equations is numerically solved for the case of n = 2 to illustrate temperature-dependent phase transitions from broken to symmetric regimes. It is found that the presence of a heat bath further drives the bosonic system into the symmetric regime and can even restore a symmetry that is spontaneously broken in the infrared for T = 0. We also study the scaling behavior of the critical scale for the quark-meson model and a simple diquark model at finite chemical potential and zero temperature. For the quark-meson model we find the critical exponent of the critical scale to be 1/2. Our results suggest that the μ-dependence is mostly governed by the Silver-Blaze property of the system, i.e. the observables do not exhibit any dependence on μ until a certain value is exceeded. For the diquark model we encounter severe obstacles in the process of computing the threshold function associated with the diquark mass parameter. It turns out that the usage of Silver-Blaze symmetric evaluation points leads to incorrect predictions in this setting. We present a regularization scheme that allows to obtain the BCS scaling behavior as expected in phases of color superconductivity.

Lukas Rammelmüller, Dr. rer. nat.

Strongly coupled quantum matter displays a rich phenomenology including phase transitions and often unexpected collective behavior. Remarkable advances in experiments with ultracold Fermi gases allow us to gain deep insight into these intriguing systems. Their theoretical description, however, is often challenging as exact analytic solutions are available only in a few special cases, and approximate techniques such as mean-field or perturbation theory are of limited use. Numerical treatment with Monte Carlo (MC) methods has led to profound success in this regard. Unfortunately, for many systems - and especially for asymmetric quantum gases - the infamous sign problem slows progress due to an exponentially scaling of the computational effort with inceasing system size. In this thesis, we set out to explore the rich physics of two-component Fermi gases in the presence of finite spin polarization and/or mass imbalance. To surmount an arising sign problem, we learn from methodological advances made in the field of quantum chromodynamics and further develop these lattice approaches in the context of nonrelativistic Fermi gases. An extensive overview of the numerical methods is presented, including several toy problems to detail the capabilities and shortcomings of the developed approaches. With these tools in hand, we perform extensive benchmarks of the hybrid Monte Carlo method with imaginary asymmetries (iHMC) and the complex Langevin (CL) method, which is based on a complex version of stochastic quantization. Both approaches are shown to yield excellent results for the ground-state energy equation of state of mass-imbalanced Fermi gases in one spatial dimension. Due to its great versatility, the CL method is subsequently employed to study pairing in one-dimensional Fermi gases, for which suitable two-body correlations are computed, revealing unexpected pairing patterns for spin- and mass-imbalanced systems. Another major system of interest in this thesis is the paradigmatic unitary Fermi gas which is investigated at finite temperature and spin polarization. A precise determination of the density equation of state in the normal phase enables us to explore a broad range of thermodynamic properties. We infer valuable information on the finite-temperature phase diagram, such as a flat phase boundary of the normal-to-superfluid transition near the balanced limit and indications for the absence of an extensive pseudogap phase above this transition. The presented results provide experimentally testable ab initio predictions for a range of previously inaccessible thermodynamic quantities.

Martin Pospiech, Dr. rer. nat.

We study the nature of strongly-interacting fermion matter by employing functional Renormalization Group (RG) techniques. In the first part of this thesis, we examine relativistic hot and dense quark matter focusing on the mechanism of spontaneous symmetry breaking in Quantum Chromodynamics (QCD) with two massless quark flavors. To this end, we consider Nambu–Jona-Lasinio-type (NJL) models serving as effective low-energy descriptions of QCD. We highlight the significance of Fierz completeness in such studies, analyze the fixed-point structure, study the RG flows of the four-fermion couplings, and explore the phase structure at finite temperature and quark chemical potential where we investigate the influence of different truncations. Using a Fierz-complete four-quark basis, we then study the impact of gauge degrees of freedom on the thermal phase boundary and explore the phase structure of chiral two-flavor QCD. We find that the phase boundary is significantly altered when Fierz-incomplete ansätze are considered. Moreover, our Fierz-complete studies suggest that the dynamics at low quark chemical potential is predominantly controlled by a non-Gaussian fixed point, ensuring that the low-energy physics is governed by chiral degrees of freedom. For the regime at large quark chemical potential, we find strong indications for the formation of a chiral diquark condensate. In the second part, we study bound-state properties of non-relativistic few-fermion systems at zero temperature using a functional Renormalization Group approach to Density Functional Theory (DFT-RG). We give a short introduction to DFT and the famous Kohn-Sham (KS) equations, discuss the derivation of the DFT-RG flow equation, and study a one-dimensional nuclear model as an introductory example. To improve the precision of the truncated DFT-RG equations, we propose an improvement based on the KS equations optimizing the starting point of the RG flow. As a feasibility study for this new development, we consider a system of quasi-one-dimensional dipolar fermions confined in a harmonic trap. For up to N = 5 particles, we compute ground-state energies for various interaction strengths and let different truncations compete against each other. Within our approximation, our KS-optimized DFT-RG method performs best for attractive interaction strengths but appears to be less suited in the repulsive regime of our benchmark system. Compared to exact results, we observe that the relative deviation decreases for higher particle numbers.

Marc Leonhardt, Dr. rer. nat.

The understanding of matter at extreme temperatures or densities is of great importance since it is essential to various fundamental phenomena and processes, such as the evolution of the early universe or the description of astrophysical objects. Under such conditions, the governing interaction is the strong force between the elementary constituents of matter, i.e., quarks and gluons, which is described by quantum chromodynamics (QCD). In this work, we study the phase structure of dense strong-interaction matter with two massless quark flavors at finite temperature and the equation of state in the zero-temperature limit employing functional renormalization group techniques. Four-quark self-interactions, which play an essential role in the description of the strongly correlated low-energy dynamics, are fully incorporated in the sense of Fierz-complete interactions only constrained by symmetries. In order to analyze the importance of Fierz completeness and how incomplete approximations affect the predictive power, we study different versions of the Nambu–Jona-Lasinio model. The predictions from such low-energy effective models for dense QCD matter are of great interest as this regime is at least difficult to access with fully first-principles approaches such as lattice Monte Carlo techniques. We analyze the fixed-point and phase structure at finite temperature and quark chemical potential based on the RG flow of the four-quark interactions at leading order of the derivative expansion. By studying the relative strengths of the various four-quark couplings, we obtain insights into condensate formation in phases governed by spontaneous symmetry breaking. We find that Fierz completeness is particularly important at large quark chemical potentials and leads to a shift of the phase boundary to higher temperatures. The incorporation of dynamical gauge fields allows us to adopt an approach directly based on quark-gluon dynamics. Without any fine-tuning, we observe a natural emergence of dominances among the four-quark couplings indicating spontaneous chiral symmetry breaking at small chemical potentials and a color superconducting phase at high chemical potentials. These dominances are found to be very robust against details of the approximations in the gauge sector, indicating that the dynamics within the quark sector are crucial in this respect. Toward lower energy scales, we recast the RG flow in the form of a quark-meson-diquark-model truncation in order to access the regime governed by spontaneously broken symmetries. This allows us to derive for the first time constraints on the equation of state of cold isospin-symmetric QCD matter at high densities in a Fierz-complete setting directly anchored in the fundamental gauge theory. Our results are found to be remarkably consistent with chiral effective field theory approaches applicable at smaller densities and with perturbative QCD approaches at very high densities. At supranuclear densities, we observe that condensation effects are essential and give rise to a maximum in the speed of sound which exceeds the asymptotic non-interacting limit, with potential implications for astrophysical applications.

Florian Ehmann, M. Sc.

This work applies recent progress on the *Complex Langevin* method to
perform Monte-Carlo simulations of a spin-polarized *Unitary Fermi Gas*
which are normally plagued by the *sign-problem*. In this work an
equation of state for the systems ground-state is obtained.

Felix Hermsen, B. Sc.

In this thesis *stochastic quantization* is applied to self-bound
quantum mechanical systems with different potentials. Beyond the harmonic
oscillator, systems with Gaussian and delta potentials are studied and their
ground-state energies are calulated. Furthermore a two-body system is
quantized using relative coordinates and a three-body system is studied
through the use of *Jacobi coordinates* and the quantization of
multiple coordinate fields.

Fabian Brauneis, B. Sc.

This work explores the application of *stochasting quantization* to
systems from simple toy problems to quantum mechanical systems. In
particular the *Langevin approach* is applied to the harmonic and
anharmonic oscillator and observables like energies and correlation
functions are calculated.