This thesis is concerned with ground state properties of two-dimensional fermionic superfluids, in which fluctuation effects like the renormalization of the order parameter or infrared singularities are important. In the superfluid state, the fermionic two-particle vertex develops rich and singular dependences on momentum and frequency, for which an efficient parametrization in terms of boson-exchange interactions in the particle-hole and particle-particle channels is formulated. Based on this decomposition of the vertex, flow equations for the effective interactions are derived, extending existing channel-decomposition schemes to i) the description of symmetry breaking in the Cooper channel and ii) the inclusion of those two-loop renormalization contributions to the vertex that are neglected in the Katanin scheme. In the second part of this thesis, the superfluid ground state of various model systems is studied. For the attractive Hubbard model at weak coupling, the momentum and frequency dependence of the two-particle vertex and the frequency dependence of the self-energy are determined on one- and two-loop level. Results for the suppression of the superfluid gap by fluctuations are in good agreement with the literature. The two-loop approximation captures the singular infrared behaviour that is expected in a fermionic superfluid at zero temperature. For the repulsive Hubbard model at weak coupling, the momentum dependence of the two-particle vertex and the d-wave superfluid gap are determined on one-loop level as a function of the interaction, the next-nearest neighbour hopping and the fermionic density. The results for the critical scales and superfluid gaps are in qualitative agreement with the literature and suggest the existence of an optimal value of the next-nearest neighbour hopping for pairing.
This thesis is concerned with ground state properties of two-dimensional fermionic superfluids, in which fluctuation effects like the renormalization of the order parameter or infrared singularities are important. In the superfluid state, the fermionic two-particle vertex develops rich and singular dependences on momentum and frequency, for which an efficient parametrization in terms of boson-exchange interactions in the particle-hole and particle-particle channels is formulated. Based on this decomposition of the vertex, flow equations for the effective interactions are derived, extending existing channel-decomposition schemes to i) the description of symmetry breaking in the Cooper channel and ii) the inclusion of those two-loop renormalization contributions to the vertex that are neglected in the Katanin scheme. In the second part of this thesis, the superfluid ground state of various model systems is studied. For the attractive Hubbard model at weak coupling, the momentum and frequency dependence of the two-particle vertex and the frequency dependence of the self-energy are determined on one- and two-loop level. Results for the suppression of the superfluid gap by fluctuations are in good agreement with the literature. The two-loop approximation captures the singular infrared behaviour that is expected in a fermionic superfluid at zero temperature. For the repulsive Hubbard model at weak coupling, the momentum dependence of the two-particle vertex and the d-wave superfluid gap are determined on one-loop level as a function of the interaction, the next-nearest neighbour hopping and the fermionic density. The results for the critical scales and superfluid gaps are in qualitative agreement with the literature and suggest the existence of an optimal value of the next-nearest neighbour hopping for pairing.
This thesis is concerned with ground state properties of two-dimensional fermionic superfluids. In such systems, fluctuation effects are particularly strong and lead for example to a renormalization of the order parameter and to infrared singularities. In the first part of this thesis, the fermionic two-particle vertex is analysed and the fermionic renormalization group is used to derive flow equations for a decomposition of the vertex in charge, magnetic and pairing channels. In the second part, the channel-decomposition scheme is applied to various model systems. In the superfluid state, the fermionic two-particle vertex develops rich and singular dependences on momentum and frequency. After simplifying its structure by exploiting symmetries, a parametrization of the vertex in terms of boson-exchange interactions in the particle-hole and particle-particle channels is formulated, which provides an efficient description of the singular momentum and frequency dependences. Based on this decomposition of the vertex, flow equations for the effective interactions are derived on one- and two-loop level, extending existing channel-decomposition schemes to i) the description of symmetry breaking in the Cooper channel and ii) the inclusion of those two-loop renormalization contributions to the vertex that are neglected in the Katanin scheme. In the second part, the superfluid ground state of various model systems is studied using the channel-decomposition scheme for the vertex and the flow equations. A reduced model with interactions in the pairing and forward scattering channels is solved exactly, yielding insights into the singularity structure of the vertex. For the attractive Hubbard model at weak coupling, the momentum and frequency dependence of the two-particle vertex and the frequency dependence of the self-energy are determined on one- and two-loop level. Results for the suppression of the superfluid gap by fluctuations are in good agreement with the literature. The two-loop approximation captures the singular infrared behaviour that is expected in a fermionic superfluid at zero temperature. For the repulsive Hubbard model at weak coupling, the momentum dependence of the two-particle vertex and the d-wave superfluid gap are determined on one-loop level as a function of the interaction, the next-nearest neighbour hopping and the fermionic density. The results for the critical scales and superfluid gaps are in qualitative agreement with the literature and suggest the existence of an optimal value of the next-nearest neighbour hopping for pairing.
We derive an efficient method for treating renormalization contributions at two-loop level within the functional renormalization group in the one-particle irreducible formalism for fermions. It is based on a decomposition of the two-particle vertex in charge, magnetic and pairing channels. The method treats single-particle and collective excitations in all channels on equal footing, allows for the description of symmetry breaking and captures collective mode fluctuation physics in the infrared. As a first application, we study the superfluid ground state of the two-dimensional attractive Hubbard model. We obtain superfluid gaps that are reduced by fluctuations in comparison to the one-loop approximation and demonstrate that the method captures the renormalization of the amplitude mode by long-range phase fluctuations.
In this thesis, we perform a comprehensive renormalization group analysis of two- and three-dimensional Fermi systems at low and zero temperature. We examine systems 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. 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 captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. 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.
We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter and fluctuations thereof are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. Physical properties of the system are computed from a set of approximate flow equations obtained by truncating the exact functional renormalization group flow of the coupled fermion-boson action. The equations capture the influence of fluctuations on nonuniversal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. The fermionic gap $\ensuremath{\Delta}$ is reduced significantly compared to the mean-field gap, and the bosonic order parameter $\ensuremath{\alpha}$, which is equivalent to $\ensuremath{\Delta}$ in mean-field theory, is suppressed to values below $\ensuremath{\Delta}$ by fluctuations. The fermion-boson vertex is only slightly renormalized. In the infrared regime, transverse order-parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations: the longitudinal mass and the bosonic self-interaction vanish linearly as a function of the scale in two dimensions, and logarithmically in three dimensions, in agreement with the exact behavior of an interacting Bose gas.
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 occurrence 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 systems 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 quantumphase 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 corresponding 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 captures the mutual interplay of gapless fermions with massless order parameter fluctuations when approaching a quantum critical point. As a first application, we compute the complete set of quantum critical exponents at the semimetal-to-superfluid quantum phase transition of attractively interacting Dirac fermions in two dimensions. Both, the order parameter propagator and the fermion propagator become nonanalytic 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.
We derive and evaluate one-loop functional flow equations for the effective interactions, self-energy, and gap function in spin-singlet superfluids. The flow is generated by a fermionic frequency cutoff, which is supplemented by an external pairing field to treat divergencies associated with the Goldstone boson. To parametrize the singular momentum and frequency dependencies of the effective interactions, the Nambu interaction vertex is decomposed in charge, magnetic, and normal and anomalous pairing channels. The one-loop flow solves reduced (mean-field) models for superfluidity exactly, and captures also important fluctuation effects. The Ward identity from charge conservation is generally violated, but can be enforced by projecting the flow. Applying the general formalism to the two-dimensional attractive Hubbard model, we obtain detailed results on the momentum and frequency dependencies of the effective interactions for weak and moderate bare interactions. The gap is reduced by fluctuations, with a stronger reduction at weaker interactions, as expected.