We explore multiband effects on $d$-wave superconducting instabilities driven by electron-electron interactions. Our models on the two-dimensional square lattice consist of a main band with an extended Fermi surface and predominant weight from ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ orbitals, whose orbital character is influenced by the admixture of other energetically neighbored orbitals. Using a functional renormalization group description of the superconducting instabilities of the system and different levels of approximations, we study how the energy scale for pairing and hence the critical temperature is affected by the band structure. We find that a reduction of orbital admixture as a function of the orbital energies can cause a ${T}_{c}$ enhancement although the Fermi surface becomes more curved and hence less favorable for antiferromagnetic spin fluctuations. While our study does not allow a quantitative understanding of the ${T}_{c}$ differences in realistic high-${T}_{c}$ cuprate systems, it may reveal an underlying mechanism contributing to the actual material trends.
We explore multiband effects on d-wave superconducting instabilities driven by electron-electron interactions. Our models on the two-dimensional square lattice consist of a main band with an extended Fermi surface and predominant weight from dx2\ensuremath−y2 orbitals, whose orbital character is influenced by the admixture of other energetically neighbored orbitals. Using a functional renormalization group description of the superconducting instabilities of the system and different levels of approximations, we study how the energy scale for pairing and hence the critical temperature is affected by the band structure. We find that a reduction of orbital admixture as a function of the orbital energies can cause a Tc enhancement although the Fermi surface becomes more curved and hence less favorable for antiferromagnetic spin fluctuations. While our study does not allow a quantitative understanding of the Tc differences in realistic high-Tc cuprate systems, it may reveal an underlying mechanism contributing to the actual material trends.
Motivated by exploring superconductivity in multi-orbital systems, we study two orbital models of spinful fermions representing (px,py) or (dxz,dyz) orbitals on the square lattice. For minimal interorbital t-J or t-V on-site interactions, a random phase approximation uncovers regimes of instability towards incommensurate dxy pair density wave (d-PDW) superconductivity with driven by interband pairing. We study the competition of PDW order with uniform nodal dxy pairing states and magnetic and charge density wave (CDW) instabilities. At strong coupling, we derive an effective hard-core Cooper pair Hamiltonian which we study using a bosonic Gutzwiller ansatz to reveal a period-2 PDW over a wide range of fillings as well as a checkerboard CDW at quarter-filling. Our results apply to correlated multi-orbital materials with quasi-1D bands, Hubbard models on the square-octagon lattice, and atomic fermions in p-orbitals. Our work highlights the role of the orbital content and multiband Fermi surfaces in stabilizing interband PDW states.
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.
Our previous mean-field study concerning superconducting pairings in the extended Hubbard model with on-site repulsive and intersite attractive interaction and arbitrary electron density is extended to include the effects of nearest-neighbor exchange (J) and correlated hopping (K) interaction. Detailed numerical analysis of s-, p-, and d-wave pairing solutions is performed for the two-dimensional square lattice with nearest- (t) and next-nearest- (t2) neighbor hopping, and the resulting mutual stability phase diagrams are given as a function of band filling and the interaction parameters. Antiferromagnetic exchange can enhance both the d-wave and the extended s-wave pairing. If it prevails over the repulsive intersite density-density interaction, such a term can be a leading mechanism for superconductivity. However, its effects strongly depend on the band filling and the tt2 ratio, giving in particular for t2=0s-wave (d-wave) pairing as most favorable in the low (high) density limit, but increasing t2 can even reverse this tendency. The correlated hopping term which breaks the electron-hole symmetry affects s-wave pairing mostly. A condensation transition (phase separation), possible in the presence of the attractive density-density interaction, is also analyzed within the random-phase approximation, and stability conditions of such an electron droplet phase with respect to other types of ordering are determined as a function of the band filling for the two-dimensional square lattice.
We analyze the influence of phonons on the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-pairing instability in the Hubbard model on the two-dimensional square lattice at weak to moderate interaction U, using a functional renormalization group scheme with frequency-dependent interaction vertices. As measured by the pairing scale, the B1g buckling mode enhances the pairing, while other phonon modes decrease the pairing. When various phonon modes are included together, the net effect on the scale is small. However, in situations where d-wave superconductivity and other tendencies, e.g., antiferromagnetism, are closely competing, the combined effect of different phonons may be able to tip the balance towards pairing.
In the present work the Roth s two-pole approximation (Phys. Rev. 184, 451 (1969)) has been used to investigate the role of d− p hybridization in the normal and superconducting states of an extended d− p Hubbard model. Superconductivity with singlet dx2−y2 -wave pairing is treated by following Beenen and Edwards formalism (Phys. Rev. B 52, 13636 (1995)). In the first part of this work, the effects of the hybridization on the superconductivity, in the hole-doped regime, have been studied treating Roth s band shift within two different approximations. In the first one, the band shift has been calculated in the limit U →¥ (U is the Coulomb interaction), with zero temperature and without consider the superconducting effects. These regards, are restrict to the band shift. In the other parts of the problem, U, the temperature and the superconducting effects have been considered finites. In the second approximation, the Coulomb interaction, the temperature and the superconductivity have been considered in the calculation of some relevant correlation functions present in the Roth s band shift. The obtained results show that the hybridization acts in the sense of to suppresses the superconductivity. Also, it has been verified that the first approximation overstimates the effects of the hybridization on the superconductivity. In the second part of these work, hoppings to second-nearest-neighbors have been included in the model with the purpose of reproduces adequately the asymetries (mainly those related with the Fermi surface, band structure and phase diagram) between the hole- and electron-doped systems. Particularly, it is shown that the crossover from hole-like to electronlike Fermi surface is deeply affected by the d − p hybridization in the hole doping case. It has been verified that the effect of the hybridization is most pronounced around the saddle-points, where the superconducting gap is maximum in the particular case of dx2−y2 -wave symmetry. As a consequence, the critical temperature Tc is directly affected by the hibridization. Moreover, the obtained results suggest that in the hole doped regime, the hybridization may act on the transport properties of the system due to the sign changes of the Hall coefficient when the crossover of the Fermi surface occurs. In the electron doped case, the crossover in the Fermi surface is not verified. Nevertheless, as the hybridization suppresses the density of states near the Fermi level, the superconductivity is affected. The topology of the Fermi surface in the hole and electron doping regime agree with recent experimental ARPES results for La2−xSrxCuO4 (hole doping) and Nd2−xCexCuO4 (electron doping).
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.