We study the quantum many-body instabilities of the
t−JK−JH Kitaev-Heisenberg Hamiltonian on the honeycomb lattice as a minimal model for a doped spin-orbit Mott insulator. This spin-
1/2 model is believed to describe the magnetic properties of the layered transition-metal oxide Na
2IrO
3. We determine the ground-state of the system with finite charge-carrier density from the functional renormalization group (fRG) for correlated fermionic systems. To this end, we derive fRG flow-equations adapted to the lack of full spin-rotational invariance in the fermionic interactions, here represented by the highly frustrated and anisotropic Kitaev exchange term. Additionally employing a set of Ward identities for the Kitaev-Heisenberg model, the numerical solution of the flow equations suggests a rich phase diagram emerging upon doping charge carriers into the ground-state manifold (
Z2 quantum spin liquids and magnetically ordered phases). We corroborate superconducting triplet
p-wave instabilities driven by ferromagnetic exchange and various singlet pairing phases. For filling
δ>1/4, the
p-wave pairing gives rise to a topological state with protected Majorana edge-modes. For antiferromagnetic Kitaev and ferromagnetic Heisenberg exchange we obtain bond-order instabilities at van Hove filling supported by nesting and density-of-states enhancement, yielding dimerization patterns of the electronic degrees of freedom on the honeycomb lattice. Further, our flow equations are applicable to a wider class of model Hamiltonians.
The functional renormalization group (FRG) approach for spin models relying on a pseudofermionic description has proven to be a powerful technique in simulating ground-state properties of strongly frustrated magnetic lattices. A drawback of the FRG framework is that it is formulated in the imaginary-time Matsubara formalism and thus only able to access static correlations, a limitation shared with most other many-body approaches. A description of the dynamical properties of magnetic systems is the key to bridging the gap between theory and neutron-scattering spectra. We take the decisive step of expanding the scope of pseudofermion FRG to the Keldysh formalism, which, while originally developed to address nonequilibrium phenomena, enables a direct calculation of the equilibrium dynamical spin structure factors on generic lattices in arbitrary dimensions. We identify the principal features characterizing the low-energy spectra of exemplary zero-, one- and two-dimensional spin-
1/2 Heisenberg models as well as the Kitaev honeycomb model while identifying current limitations of the method that have to be improved upon.
In this dissertation, which is situated in the field of theoretical condensed matter physics, we investigate many-body effects in altermagnets and electron–phonon systems. Specifically, we study a Heisenberg model for two-dimensional altermagnets, a dissipative variant of the Yukawa–Sachdev–Ye–Kitaev (YSYK) model, and the Holstein model. In all of these systems, the presence or absence of quasiparticles plays a central role. Consequently, the description of these quasiparticles and their influence on the state of the system constitute the primary focus of this work. Since the Heisenberg model can be bosonized using the Holstein–Primakoff transformation, all systems considered in this dissertation can be analyzed using many-body methods, in particular within the framework of Green's function techniques. Altermagnets constitute a recently identified class of magnetic materials characterized by their distinct symmetry properties. They combine properties of both ferromagnets and antiferromagnets, thereby offering promising opportunities for future applications, for example in spintronics. Consequently, they have become the subject of intense current research, and new candidate altermagnetic materials are continuously being proposed. The central question addressed in this part of the dissertation is, using nonlinear spin-wave theory, whether the collective magnetic excitations, namely magnons, are stable at zero temperature or undergo spontaneous decay driven exclusively by quantum-mechanical effects. The dissipative YSYK model belongs to the class of SYK models, which describe strongly coupled chaotic systems with Gaussian-distributed interactions. In the limit of an infinite number of particles, these models can be solved exactly at the level of the self-energy. They do not possess quasiparticles, exhibit a non-Fermi-liquid (NFL) phase associated with strange-metal behavior and high-temperature superconductivity, and display several additional remarkable properties that have also attracted considerable interest in gravitational physics. The primary objective of this part of the dissertation is to explain the self-tuning criticality of the NFL phase in the dissipative YSYK model. Although this phenomenon had previously been found by solving the exact Dyson–Schwinger equations (DSEs) in the infinite-particle limit, its origin has remained unexplained. To this end, we develop a functional renormalization group (FRG) approach for the dissipative YSYK model. Although the influence of lattice vibrations on electronic systems has been investigated for decades using well-established many-body and kinetic-theory methods, the behavior of strongly coupled electron–phonon systems remains incompletely understood. To address this problem, we study the microscopic Holstein model with dispersionless Einstein phonons at zero temperature. By combining the Dyson–Schwinger equations of the Holstein model with the exact FRG flow equations formulated within the chemical-potential cutoff scheme, we derive a Ward identity relating the electronic compressibility to the renormalized phonon frequency. This identity implies that the compressibility diverges when the renormalized phonon frequency vanishes, leading to a Pomeranchuk instability in the density channel with zero angular momentum. Using the nonperturbative FRG, we investigate the presence of this instability and propose a phase diagram of the Holstein model in arbitrary dimensions.