We perform an instability analysis of the Emery three-band model at hole doping and weak coupling within a channel-decomposed functional renormalization group flow proposed in Phys. Rev. B 79, 195125 (2009). In our approach, momentum dependences are taken into account with improved precision compared to previous studies of related models. Around a generic parameter set, we find a strong competition of antiferromagnetic and $d$-wave Cooper instabilities with a smooth behavior under a variation of doping and additional hopping parameters. For increasingly incommensurate ordering tendencies in the magnetic channel, the $d$-wave pairing gap is deformed at its maxima. Comparing our results for the Emery model to those obtained for the two-dimensional one-band Hubbard model with effective parameters, we find that, despite considerable qualitative agreement, multiorbital effects have a significant impact on a quantitative level.
We perform an instability analysis of the Emery three-band model at hole doping and weak coupling within a channel-decomposed functional renormalization group flow proposed in Phys. Rev. B 79, 195125 (2009). In our approach, momentum dependences are taken into account with improved precision compared to previous studies of related models. Around a generic parameter set, we find a strong competition of antiferromagnetic and d-wave Cooper instabilities with a smooth behavior under a variation of doping and additional hopping parameters. For increasingly incommensurate ordering tendencies in the magnetic channel, the d-wave pairing gap is deformed at its maxima. Comparing our results for the Emery model to those obtained for the two-dimensional one-band Hubbard model with effective parameters, we find that, despite considerable qualitative agreement, multiorbital effects have a significant impact on a quantitative level.
This thesis is devoted to purely fermionic functional renormalization-group (fRG) studies of multiband effects and of antiferromagnetically ordered phases. The formal framework of the one-particle irreducible fRG scheme is recapitulated and extended for charge-conserving theories with a collinear spin symmetry. For multiband models, point-group symmetries and their interplay with the phases of Bloch basis states are discussed. Further, an improved truncation of the fRG flow equations is proposed that keeps track of three-particle interactions generated by virtual excitations of bands away from the Fermi surface. The feedback of this three-particle term on the two-particle interaction is first investigated in a g-ology approach to a simple two-band model for cuprate superconductors at weak coupling. The results suggest a potentially strong impact of the three-particle term. Subsequently, the three-band Emery model is analyzed within a channel-decomposition, again at weak coupling. In contrast to earlier work, the approach pursued here does not rely on a form-factor expansion, which allows for the observation of a deformed d-wave pairing gap for increasingly incommensurate ordering tendencies in the magnetic channel. In this more refined study, the three-particle feedback plays a minor role. Despite considerable qualitative agreement with results for the one-band Hubbard model with effective parameters, multi-orbital effects are found to play an important role on a quantitative level. Phases of antiferromagnetic ordering are first studied for a reduced mean-field model. In order to go beyond the mean-field picture, a hierarchy of approximations is devised, which starts from the corresponding channel-decomposed one-loop flow equations. Interaction terms breaking discrete symmetries are thereby successively neglected. In the course of these approximations, also an exchange parametrization is employed. For the spin-density wave phase of a two-pocket model initially proposed by Chubukov et al. in Phys. Rev. B 78:134512, the flow equations are then integrated numerically at the most approximate level of the hierarchy. This yields a gap that is significantly reduced compared to mean-field theory. The resulting violation of the SU(2) Ward identity appears acceptable on a qualitative level, suggesting that the approximations causing this violation are physically meaningful.
The Bethe-Salpeter equation is combined with the temperature-cutoff functional renormalization group approach to analyze the order parameter structure for the leading instabilities of the 2D $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model. We find significant deviations from the conventional s-, p-, or d-wave forms, which is due to the frustration of antiferromagnetism at small and intermediate ${t}^{\ensuremath{'}}$. With adding a direct antiferromagnetic spin-exchange coupling the eigenfunctions in the particle-hole channel have extended s-wave form, while in the particle-particle singlet pairing channel a higher angular momentum component arises besides the standard ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ wave component, which flattens the angular dependence of the gap. For ${t}^{\ensuremath{'}}$ closer to t∕2 we find a delicate competition of ferromagnetism and triplet pairing with a nontrivial pair-wavefunction.
In the present thesis, we consider interaction induced instabilities in different electronic models consisting of multiple orbitals. We employ the functional renormalization group method, to tackle the many-particle problem. The main goal of this work is the identification of leading instabilities of different interacting electron systems and the relevant energy scales for the onset of correlations. First, we recapitulate the functional renormalization group formalism and discuss details of its application. We proceed with the investigation of two models, a continuum model and a model on a checkerboard lattice, which exhibit a single quadratic band crossing point at the Fermi level. Here, we find that electron-electron interactions can lead to ordering with non-trivial topological properties, i.e. the existence of topologically protected edge states. We further discuss the emergence of superconductivity, when the band crossing point is shifted away from the Fermi surface. Then, we turn to more realistic models featuring quadratic and cubic band crossing points, namely the honeycomb bi- and trilayer, which are used to describe the graphene layered systems. We obtain rich phase diagrams, with antiferromagnetic spin-density, two types of charge-density waves and a quantum spin Hall phase, which features in the case of the trilayer topologically protected edge states. With interaction parameters extracted from ab initio calculations, we observe a close competition between the antiferromagnetic and quantum spin Hall state. Here, the precise shape of the interaction decides which instability is the leading one. A comparison of the energy scales of the ordering phenomena in tri-, bi- and single layer reveals that when the interaction parameters from ab initio are appropriately rescaled, the functional renormalization group can qualitatively reproduce experimental gap sizes in the respective materials consistently. Afterwards, we study a multiband model of the copper-based unconventional superconductors. This is motivated by a puzzling material trend in copper-based high-Tc superconductors, pointed out by Pavarini et al. [PRL. 87, 047003 (2001)]: Compounds with larger next-nearest-neighbor hopping generally exhibit larger critical temperatures, which cannot be easily understood within spin-fluctuation theories of the one-band Hubbard model. Therefore, we consider the extension of the original model by including additional orbitals. We investigate a two-band model with an additional 4s orbital within a two-patch approximation and the N-patch-scheme. By this, a general understanding of the influence of the additional orbitals is established. Afterwards, we additionally include the 3dz2 orbital and show that the material trend, found in experiments, can, at least, qualitatively be reproduced. Lastly, we go back to the single band Hubbard model and examine the consequences of the widely used approximation of neglecting the self-energy and the frequency dependence of the interaction vertex. We establish that the phase diagram obtained from the functional renormalization group is only quantitatively affected. From this study, we conclude that neglecting self-energy and frequency dependence, is a sensible approach, in order to obtain a reliable qualitative picture of the low energy behavior of a weakly or moderately coupled system at reasonable expense.
The treatment of strongly interacting two-dimensional Fermi systems constitutes one of the most challenging problems in the field of condensed matter physics. Many theoretical works focused on the strongly coupled Hubbard model, since it is expected to capture the most important physics of cuprate superconductors. Due to our methodological improvements and understanding of the frequency dependence of the two-particle vertex function, the functional renormalization group combined with the dynamical mean field theory can now be used to study competing correlations in the strongly interacting regime. While limited to moderate interaction strengths, the functional renormalization group describes efficiently systems with a hierarchy of different energy scales and competing correlations. For instance, it provides definite evidence for d-wave superconductivity in the two-dimensional Hubbard model at moderate coupling. In a first project, we study the frequency dependence of the one-particle irreducible vertex function generated by the functional renormalization group flow. The frequency dependence, which becomes singular for strong interactions, appears to be important already for moderate couplings, and it cannot be represented by separate channels each depending on a single linear combination of frequencies. For strongly interacting systems, the dynamical mean field theory captures strong local correlation effects nonperturbatively. With this approximation, we study the impact of local correlations on the magnetic susceptibility. The local dynamics strongly affects the spin response function and its momentum dependence. In contrast to the widely used random-phase approximation with self-energy corrections that predicts Néel antiferromagnetic order, the local vertex corrections favor an incommensurate order similar to the ordering instability predicted by the Fermi surface geometry, as for weakly interacting systems. The dynamical mean field theory is also used as starting point for the functional renormalization group flow. We demonstrate that, due to our improvements for the parametrization of the vertex function, this approach is actually able to access the strong coupling physics. Moreover, we derive a flow scheme that conserves the local contributions and reduces the truncation error of the flow equations. In the strongly interacting regime, we capture strong d-wave pairing correlations driven by magnetic fluctuations with a mechanism similar to the one observed in the weakly interacting system.
We derive a differential equation for the one-particle-irreducible vertex functions of interacting fermions as a function of the temperature. Formally, these equations correspond to a Wilsonian renormalization group (RG) scheme that uses the temperature as an explicit scale parameter. Our method allows us to analyze the competition between superconducting and various magnetic Fermi surface instabilities in the one-loop approximation. In particular, this includes ferromagnetic fluctuations, which are difficult to treat on an equal footing in conventional Wilsonian momentum space techniques. Applying the scheme to the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model we investigate the RG flow of the interactions at the van Hove filling with varying next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ Starting at ${t}^{\ensuremath{'}}=0$ we describe the evolution of the flow to strong coupling from an antiferromagnetic nesting regime over a d-wave regime at moderate ${t}^{\ensuremath{'}}$ to a ferromagnetic region at larger absolute values of ${t}^{\ensuremath{'}}.$ Upon increasing the particle density in the latter regime the ferromagnetic tendencies are cut off and the leading instability occurs in the triplet superconducting pairing channel.
We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping p and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the (p,T) phase diagram centered around 15% hole doping.