We show that the general low-energy Bogoliubov-de Genness Hamiltonian in a multiband superconductor with broken time reversal and preserved inversion symmetry is a generator of real four-dimensional representation of $SO(4)$. In the particular representation such an effective Hamiltonian is a purely imaginary matrix, and it is proportional to the antisymmetric tensor of a fictitious electromagnetic field which one can define in the momentum space. The quantum time evolution of the low-energy quasiparticle state becomes this way closely related to the classical relativistic motion of a charged particle in the presence of the Lorentz force that would be derived from such an electromagnetic field configuration. The condition for the emergence of a Bogoliubov-Fermi surface can then be understood as orthogonality of the fictitious electric and magnetic fields, which would allow zero Lorentz force. The corresponding zero-energy eigenstates are identified as the physical timelike and the unphysical spacelike solutions of the Lorentz force equation. We study the looming instability of the inversion-symmetric Bogoliubov-Fermi surface in presence of electron-electron interaction by formulating a concrete interacting model on the Lieb lattice that features the requisite $SO(4)$ kinetic energy term together with nearest-neighbor two-body repulsion. The latter is shown to favor dynamical breaking of the inversion symmetry. The inversion symmetry in our lattice model indeed becomes spontaneously broken at zero temperature at infinitesimal repulsion, with the original Bogoliubov-Fermi surface deformed and reduced in size. General features of this symmetry breaking phenomenon are discussed and a comparison with other works in literature is presented.
We show that when the time reversal symmetry is broken in a multicomponent superconducting condensate without inversion symmetry the resulting Bogoliubov quasiparticles generically exhibit mini-Bogoliubov-Fermi (BF) surfaces, for small superconducting order parameter. The absence of inversion symmetry makes the BF surfaces stable with respect to weak perturbations. With sufficient increase of the order parameter, however, the Bogoliubov-Fermi surface may disappear through a Lifshitz transition, and the spectrum this way become fully gapped. Our demonstration is based on the computation of the …
Quasiparticle interference (QPI) by means of scanning tunneling microscopy/spectroscopy (STM/STS), angle resolved photoemission spectroscopy (ARPES), and multi-orbital tight bind- ing calculations are used to investigate the band structure and superconducting order parameter of LiFeAs. Using this combination we identify intra- and interband scattering vectors between the hole (h) and electron (e) bands in the QPI maps. Discrepancies in the band dispersions inferred from previous ARPES and STM/STS are reconciled by recognizing a difference in the $k_z$ sensitivity for the two probes. The observ…
Exotic quantum phases including topological states and non-Fermi liquids may be realized by quantum states with total angular momentum $j=3/2$, as manifested in HgTe and pyrochlore iridates. Recently, an exotic superconducting state with non-zero density of states of zero energy Bogoliubov (BG) quasiparticles, Bogoliubov Fermi-surface (BG-FS), was also proposed in a centrosymmetric $j=3/2$ system, protected by a Z$_2$ topological invariant. Here, we consider interaction effects of a centrosymmetric BG-FS and demonstrate its instability by using mean-field and renormalization group analysis. Th…
Centrosymmetric multiband superconductors which break time-reversal symmetry generically have two-dimensional nodes, i.e., Fermi surfaces of Bogoliubov quasiparticles. We show that the coupling of the electrons to the lattice always leads to a weak-coupling instability of such a state towards spontaneous breaking of inversion symmetry at low temperatures. This instability is driven by a Cooper logarithm in the internal energy but the order parameter is not superconducting but distortional. We present a comprehensive symmetry analysis and introduce a measure that allows to compare the strengths…
The disorder and interaction effects on Bogoliubov-Fermi surfaces with preserved inversion symmetry are studied for a low-energy effective model coupled to bosonic degrees of freedom. It is shown that the non-ideal Bogoliubov quasiparticles (bogolons) generically induce the odd-frequency pair amplitude which reflects a Cooper pairing at different time. The self-energy of bogolons is mainly contributed by the disorder effects in the low frequency limit as in the usual electron liquid. Depending on the choice of the parameters, there are two kinds of solutions: one is frequency-independent (but …
Symmetry indicators have proven to be extremely helpful in identifying topologically non-trivial crystalline insulators using symmetry-group representations of their Bloch states. An extension of this approach to superconducting systems requires defining an appropriate atomic limit for Bogoliubov-de-Gennes Hamiltonians. Here, we introduce such a notion of atomic limit and derive a $\mathbb{Z}_{2^d}$-valued symmetry indicator for inversion-symmetric superconductors of $d$ dimensions. This indicator allows for a refined topological classification including higher-order phases for systems in the …
We uncover the properties of complex tensor (d-wave) superconducting order in three-dimensional Rarita-Schwinger-Weyl semimetals that host pseudospin-3/2 fermions at a fourfold linear band crossing point. Although the general theory of d-wave order was originally developed for materials displaying quadratic band touching, it directly applies to the case of semimetals with linear dispersion, several candidate compounds of which have been discovered experimentally very recently. The spin-3/2 nature of the fermions allows for the formation of spin-2 Cooper pairs which may be described by a comple…
Coexistence of Chiral Majorana Edge States and Bogoliubov Fermi Surfaces in Two-Dimensional Nonsymmorphic Dirac Semimetal/Superconductor Heterostructures
We present a comprehensive theory for Bogoliubov Fermi surfaces in inversion-symmetric superconductors which break time-reversal symmetry. A requirement for such a gap structure is that the electrons posses internal degrees of freedom apart from the spin (e.g., orbital or sublattice indices), which permits a nontrivial internal structure of the Cooper pairs. We develop a general theory for such a pairing state, which we show to be nonunitary. A time-reversal-odd component of the nonunitary gap product is found to be essential for the appearance of Bogoliubov Fermi surfaces. These Fermi surface…
In the presence of certain symmetries, three-dimensional Dirac semimetals can harbor not only surface Fermi arcs, but also surface Dirac cones. Motivated by the experimental observation of rotation-symmetry-protected Dirac semimetal states in iron-based superconductors, we investigate the potential intrinsic topological phases in a $C_{4z}$-rotational invariant superconducting Dirac semimetal with $s_{\pm}$-wave pairing. When the normal state harbors only surface Fermi arcs on the side surfaces, we find that an interesting gapped superconducting state with a quartet of Bogoliubov-Dirac cones o…
Electromagnetic responses in superconductors provide valuable information on the pairing symmetry as well as physical quantities such as the superfluid density. However, at the superconducting gap energy scale, optical excitations of the Bogoliugov quasiparticles are forbidden in conventional Bardeen-Cooper-Schrieffer superconductors when momentum is conserved. Accordingly, far-infrared optical responses have been understood in the framework of a dirty-limit theory by Mattis and Bardeen for over 60 years. Here we show, by investigating the selection rules imposed by particle-hole symmetry and …
We develop an analytical tool to extract bulk electronic properties of unconventional superconductors through inelastic neutron scattering (INS) spectra. Since the spin excitation spectrum in the superconducting (SC) state originates from Bogoliubov quasiparticle scattering associated with Fermi surface nesting, its energy-momentum relation--the so called `hour-glass' feature--can be inverted to reveal the Fermi momentum dispersion of the single-particle spectrum as well as the corresponding SC gap function. The inversion procedure is analogous to the quasiparticle interference (QPI) effect in…
We study the charge character of the Andreev bound states (ABSs) in one-dimensional topological superconductors with spatial inversion symmetry (SIS) breaking. Despite the absence of the SIS, we show a hidden symmetry for the Bogoliubov de Gennes equations around Fermi points in addition to the particle-hole symmetry. This hidden symmetry protects that the charge of the ABSs is solely dependent on the corresponding Fermi velocities. On the other hand, if the SIS is present, the ABSs are charge neutral, similar to Majorana fermions. We demonstrate that the charge of the ABSs can be experimental…
We show that spontaneous time-reversal-symmetry (TRS) breaking can naturally arise from the interplay between pair density wave (PDW) ordering at multiple momenta and nesting of Fermi surfaces (FS). Concretely, we consider pair-density-wave superconductivity on a hexagonal lattice with nested FS at $3/4$ electron filling, which is related to a recently discovered superconductor CsV$_3$Sb$_5$. Because of nesting of the FS, each momentum $\bk$ on the FS has at least two counterparts $-\bk\pm\mathbf{Q}_α$ ($α=1,2,3$) on the FS to form finite momentum ($\pm\mathbf{Q}_α$) Cooper pairs, resulting in…
Topological superconductors are predicted to exhibit outstanding phenomena, including non-abelian anyon excitations, heat-carrying edge states, and topological nodes in the Bogoliubov spectra. Nonetheless, and despite major experimental efforts, we are still lacking unambiguous signatures of such exotic phenomena. In this context, the recent discovery of coexisting superconductivity and ferroelectricity in lightly doped and ultra clean SrTiO$_3$ opens new opportunities. Indeed, a promising route to engineer topological superconductivity is the combination of strong spin-orbit coupling and inve…
Two topics of high current interest in the field of unconventional superconductivity are non-centrosymmetric superconductors and multiband superconductivity. Half-Heusler superconductors such as YPtBi exemplify both. In this paper, we study bulk and surface states in nodal superconducting phases of the half-Heusler compounds, belonging to the $A_1$ ($s+p$-like) and $T_2$ ($k_zk_x+ik_zk_y$-like) irreducible representations of the point group. These two phases preserve and break time-reversal symmetry, respectively. For the $A_1$ case, we find that flat surface bands persist in the multiband sys…
