Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models

We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis.We first give two analytically accessible examples. Then we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches such as the functional renormalization group.Comment: revised version: 38 pages, 4 figure

Multiorbital effects in the functional renormalization group: A weak-coupling study of the Emery model

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, multi-orbital effects have a significant impact on a quantitative level.Comment: revised version: 22 pages, 11 figure

Functional renormalization and mean-field approach to multiband systems with spin-orbit coupling: Application to the Rashba model with attractive interaction

The functional renormalization group (RG) in combination with Fermi surface patching is a well-established method for studying Fermi liquid instabilities of correlated electron systems. In this article, we further develop this method and combine it with mean-field theory to approach multiband systems with spin-orbit coupling, and we apply this to a tight-binding Rashba model with an attractive, local interaction. The spin dependence of the interaction vertex is fully implemented in a RG flow without SU(2) symmetry, and its momentum dependence is approximated in a refined projection scheme. In particular, we discuss the necessity of including in the RG flow contributions from both bands of the model, even if they are not intersected by the Fermi level. As the leading instability of the Rashba model, we find a superconducting phase with a singlet-type interaction between electrons with opposite momenta. While the gap function has a singlet spin structure, the order parameter indicates an unconventional superconducting phase, with the ratio between singlet and triplet amplitudes being plus or minus one on the Fermi lines of the upper or lower band, respectively. We expect our combined functional RG and mean-field approach to be useful for an unbiased theoretical description of the low-temperature properties of spin-based materials.Comment: consistent with published version in Physical Review B (2016

Quantized shift response in multi-gap topological phases

We show that certain 3D multi-gap topological insulators can host quantized shift photoconductivities due to bulk invariants that are defined under reality conditions imposed by additional symmetries. We recast the quantization in terms of the integrated torsion tensor and the non-Abelian Berry connection constituting Chern-Simons forms. Physically, we recognize that the topological quantization emerges purely from virtual transitions contributing to the optical response. Our findings provide another quantized electromagnetic DC response due to the non-trivial band topology, beyond the quantum anomalous Hall effect of Chern insulators and quantized circular photogalvanic effect found in Weyl semimetals.Comment: 7+7 pages; 3+1 figure

Quantum geometry in superfluidity and superconductivity

We review the theoretical description of the role of quantum geometry in superfluidity and superconductivity of multiband systems, with focus on flat bands where quantum geometry is wholly responsible for supercurrents. This review differs from previous ones in that it is based on the most recent understanding of the theory: the dependence of the self-consistent order parameter on the supercurrent is properly taken into account, and the superfluid weight in a flat band becomes proportional to the minimal quantum metric. We provide a recap of basic quantum geometric quantities and the concept of superfluid density. The geometric contribution of superconductivity is introduced via considering the two-body problem. The superfluid weight of a multiband system is derived within mean-field theory, leading to a topological bound of flat band superconductivity. The physical interpretation of the flat band supercurrent in terms of Wannier function overlaps is discussed.Comment: 32 pages, 1 figure, Lecture notes for the Proceedings of the International School of Physics "Enrico Fermi" Course 211 "Quantum Mixtures with Ultra-Cold Atoms" (Varenna, Italy, 2022

Universal higher-order bulk-boundary correspondence of triple nodal points

Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. For spinless particles, the stability of a triple nodal point requires two ingredients: rotational symmetry of order three, four, or six; combined with mirror or space-time-inversion symmetry. However, despite ample studies of their classification, robust boundary signatures of triple nodal points have until now remained elusive. In this work, we first show that pairs of triple nodal points in semimetals and metals can be characterized by Stiefel-Whitney and Euler monopole invariants, of which the first one is known to facilitate higher-order topology. Motivated by this observation, we then combine symmetry indicators for corner charges and for the Stiefel-Whitney invariant in two dimensions with the classification of triple nodal points for spinless systems in three dimensions. The result is a complete higher-order bulk-boundary correspondence, where pairs of triple nodal points are characterized by fractional jumps of the hinge charge. We present minimal models of the various species of triple nodal points carrying higher-order topology, and illustrate the derived correspondence on Sc3AlC which becomes a higher-order triple-point metal in applied strain. The generalization to spinful systems, in particular to the WC-type triple-point material class, is briefly outlined

Revealing hidden topologies in photonic crystals

This thesis is part of an effort to bring together two very active fields of physics: band topology and photonics. The field of band topology has revealed exotic phenomena such as robust, unidirectional edge states that occur at the interfaces between materials that belong to different topological phases. The Nobel Prize in Physics 2016 was awarded to Thouless, Haldane, and Kosterlitz, for predicting such phases in electronic systems where the topological edge states may revolutionise electronics and quantum computing. There is now great interest in reproducing such topological phases in photonics using photonic crystals: periodic nanostructures with tunable photonic bands. Realising such topological edge states in photonic devices could revolutionise optical data transport and optical quantum computing. In this thesis, we focus on two symmetry-protected topological phases that have been difficult to realise in photonics: the quantum spin-Hall effect (QSHE, protected by the fermionic time-reversal symmetry of electrons) and square-root topological semimetals (protected by chiral symmetry, also known as sublattice symmetry). We introduce a new topological index for C2T symmetric crystals that emulate the QSHE using the angular momentum of light to mimic the spin of electrons. For example, in 2015 Wu & Hu proposed a photonic analogue of the QSHE where the crystalline symmetries and bosonic time-reversal symmetries of the photons generated a pseudo-fermionic time-reversal symmetry. Subsequent works suggested that this crystal was a trivial phase rather than a non-trivial QSHE phase. However, we believe that our new topological index demonstrates the non-trivial QSHE-like nature of the photonic crystal introduced by Wu & Hu while accounting for all of the valence bands determined from full-wave calculations. We then study the topology of networks of voids and narrow connecting channels that are formed by the space between closely spaced perfect conductors. In photonics, chiral symmetry is often broken by long-range interactions, but Vanel et al 2017 showed that such void-channel networks can be mapped to analagous mass-spring systems in an asymptotically rigorous manner and therefore have only short-range interactions. We demonstrate that topological tight-binding models, such as square-root semimetals, can be reproduced in these void-channel networks with appropriate boundary conditions. Finally, we discuss an interesting application of closely spaced nanoscopic metallic particles in the mid-to-far infrared and larger wavelengths. We show that despite being composed of highly dispersive and lossy metals, the effective dielectrics are virtually dispersion-free throughout the infrared spectrum and can be even more transparent than natural dielectrics such as germanium in the far-infrared. The effective index can be tuned locally, allowing us to design gradient-index lenses where light is guided by a continuously varied local refractive index. We propose a novel gradient-index lens that exploits the simultaneous transparency and high metallic filling fraction of the effective dielectrics to create intense ‘doubly-enhanced’ hotspots where light is focused on the microscale and the electric field ‘squeezed’ between the metallic particles on the nanoscale.Open Acces