We discuss different approximations for effective low-energy interactions in multiband models for weakly correlated electrons. In the study of Fermi-surface instabilities of the conduction band(s), the standard approximation consists only keeping those terms in the bare interactions that couple only to the conduction band(s), while corrections due to virtual excitations into bands away from the Fermi surface are typically neglected. Here, using a functional renormalization group approach, we present an improved truncation for the treatment of the effective interactions in the conduction band that keeps track of the generated three-particle interactions (six-point term) and hence allows one to include important aspects of these virtual interband excitations. Within a simplified two-patch treatment of the conduction band, we demonstrate that these corrections can have a rather strong effect in parts of the phase diagram by changing the critical scales for various orderings and the phase boundaries.
We discuss different approximations for effective low-energy interactions in multiband models for weakly correlated electrons. In the study of Fermi-surface instabilities of the conduction band(s), the standard approximation consists only keeping those terms in the bare interactions that couple only to the conduction band(s), while corrections due to virtual excitations into bands away from the Fermi surface are typically neglected. Here, using a functional renormalization group approach, we present an improved truncation for the treatment of the effective interactions in the conduction band that keeps track of the generated three-particle interactions (six-point term) and hence allows one to include important aspects of these virtual interband excitations. Within a simplified two-patch treatment of the conduction band, we demonstrate that these corrections can have a rather strong effect in parts of the phase diagram by changing the critical scales for various orderings and the phase boundaries.
In correlated electron materials, the application of many-body techniques for the study of interaction effects or unconventional superconductivity often requires the formulation of an effective low-energy model that contains only the relevant bands near the Fermi level. However, the bands away from the Fermi level are known to renormalize the low-energy interactions substantially. Here we compare different schemes to derive low-energy effective theories for interacting electrons in solids. The frequently used constrained random phase approximation (cRPA) is identified as a particular resummation of higher-order interaction terms that includes important virtual corrections. We then propose an adapted functional renormalization group scheme that includes the cPRA, but also allows one to go beyond the cRPA approximation. We study a simple two-band model in order to explore the differences between the approximations.
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.
We describe an efficient approximation for the electron-electron interaction in the determination of the low-energy effective interaction in multiband lattice systems. By using ideas for channel decomposition, form-factor expansion, and the truncated-unity technique we describe the interaction as arising from the nonlocal and orbital-dependent coupling of particle-hole and particle-particle bilinears formed by fields residing in the same one or two orbitals. This allows us to employ the constrained functional renormalization group (cfRG) with a suitable momentum and frequency discretization. The approach gives insights into the nonlocal screening of spin and charge interactions when bands away from the Fermi level are integrated out. Specifically, we compute the effective low-energy interactions in the low-energy target band of a three-band model with on-site and nonlocal bare interactions. We show that the cfRG adds important features to the effective target-band interaction that cannot be found using the constrained random phase approximation (cRPA).
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.
In the derivation of low-energy effective models for solids targeting the bands near the Fermi level, the constrained random-phase approximation (cRPA) has become an appreciated tool to compute the effective interactions. The Wick-ordered constrained functional renormalization group (cfRG) generalizes the cRPA approach by including all interaction channels in an unbiased way. Here we present applications of the cfRG to two simple multiband systems and compare the resulting effective interactions to the cRPA. First, we consider a multiband model for monolayer graphene, where we integrate out the $\ensuremath{\sigma}$ bands to get an effective theory for $\ensuremath{\pi}$ bands. It turns out that terms beyond cRPA are strongly suppressed by the different xy-plane reflection symmetry of the bands. In our model the cfRG corrections to cRPA become visible when one disturbs this symmetry difference slightly, however, without qualitative changes. This study shows that the embedding or layering of two-dimensional electronic systems can alter the effective interaction parameters beyond what is expected from screening considerations. The second example is a one-dimensional model for a diatomic system reminiscent of a CuO chain, where we consider an effective theory for Cu 3d-like orbitals. Here the fRG data shows relevant and qualitative corrections compared to the cRPA results. We argue that the new interaction terms affect the magnetic properties of the low-energy model.
We consider a one-dimensional model of interacting electrons. The Fermi surface crosses two bands, with light and heavy masses, respectively. Both intraband and interband interactions are assumed. The earlier renormalization-group treatment of the model by Varma and Zawadowski is extended to check the validity of the scaling assumption and to calculate the next corrections to the scaling equations. These corrections indicate a Fermi-velocity renormalization. Depending on the initial parameters, two kinds of behavior can appear. Either the interband charge fluctuations are negligible and then the dominant instabilities are determined by the interband exchange and the intraband backscattering with the two velocities remaining different, or the interband charge fluctuations become dominant and the velocities scale to a common value. The phase diagram of the model is also determined.