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Register a new userSymmetry, nodal structure, and Bogoliubov Fermi surfaces for nonlocal pairing
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Multiband effects can lead to fundamentally different electronic behavior of solids, as exemplified by the possible emergence of Fermi surfaces of Bogoliubov quasiparticles in centrosymmetric superconductors which break time-reversal symmetry. We extend the analysis of possible pairing symmetries, the corresponding nodal structure, and the Bogoliubov Fermi surfaces in two directions: We include nonlocal pairing and we consider internal degrees of freedom other than the effective angular momentum of length $j=3/2$ examined so far. Since our main focus is on the Bogoliubov Fermi surfaces we concentrate on even-parity pairing. The required symmetry analysis is illustrated for several examples, as a guide for the reader. We find that the inclusion of nonlocal pairing leads to a much larger range of possible pairing symmetries. For infinitesimal pairing strength, we find a simple yet powerful criterion for nodes in terms of a scalar product of form factors.
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Superconductors involving electrons with internal degrees of freedom beyond spin can have internally anisotropic pairing states that are impossible in single-band superconductors. As a case in point, in even-parity multiband superconductors that break time-reversal symmetry, nodes of the superconducting gap are generically inflated into two-dimensional Bogoliubov Fermi surfaces. The detection and characterization of these quasiparticle Fermi surfaces requires the understanding of their experimental consequences. In this paper, we derive the low-energy density of states for a broad range of possible nodal structures. Based on this, we calculate the low-temperature form of observables that are commonly employed for the characterization of nodal superconductors, i.e., the single-particle tunneling rate, the electronic specific heat and Sommerfeld coefficient, the thermal conductivity, the magnetic penetration depth, and the NMR spin-lattice relaxation rate, in the clean limit. We also address the question whether the topological invariant of the Bogoliubov Fermi surfaces is associated with topologically protected surface states, with negative results. This work is meant to serve as a guide for experimental searches for Bogoliubov Fermi surfaces in time-reversal-symmetry-breaking superconductors.
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 of competing distortional instabilities. Moreover, we discuss the instability using an effective single-band model. This framework reveals a duality mapping of the effective model which maps the distortional order parameter onto a superconducting one, providing a natural explanation for the Cooper logarithm and the weak-coupling nature of the instability. Finally, we consider the possibility of a pair-density wave state when inversion symmetry is broken. We find that it can indeed exist but does not affect the instability itself.
Recent development in exact classification of a superconducting gap has elucidated various unconventional gap structures, which have not been predicted by the classification of order parameter based on the point group. One of the important previous results is that all symmetry-protected line nodes are characterized by nontrivial topological numbers. Another intriguing discovery is the gap structures depending on the angular momentum $j_z$ of normal Bloch states on threefold and sixfold rotational-symmetric lines in the Brillouin zone. Stimulated by these findings, we classify irreducible representations of the Bogoliubov-de Gennes Hamiltonian at each $boldsymbol{k}$ point on a high-symmetry $n$-fold ($n = 2$, $3$, $4$, and $6$) axis for centrosymmetric and paramagnetic superconductors, by using the combination of group theory and $K$ theory. This leads to the classification of all crystal symmetry-protected nodes (including $j_z$-dependent nodes) on the axis that crosses a normal-state Fermi surface. As a result, it is shown that the classification by group theory completely corresponds with the topological classification. Based on the obtained results, we discuss superconducting gap structures in SrPtAs, CeCoIn$_5$, UPt$_3$, and UCoGe.
It has recently been pointed out that Fermi surfaces can remain even in the superconductors under the symmetric spin-orbit interaction and broken time-reversal symmetry. Using the linear response theory, we study the instability of such systems toward ordering, which is an intrinsic property of the Fermi surfaces. The ordered states are classified into diagonal and offdiagonal ones, each of which respectively indicates the Pomeranchuk instability and Cooper pairing not of original electron but of Bogoliubov particles (bogolons). The corresponding order parameters are expanded by multipole moments (diagonal order parameter) and multiplet pair amplitudes (offdiagonal order parameter) of original electrons, which are induced by the internal fields arising from bogolons ordering. While the bogolons order parameters partially inherit the characters of the original electrons, many order parameter components mix with similar magnitude. Hence there is no clear-cut distinction whether the phase transition is diagonal or offdiagonal ordering in terms of the original electrons. These ordering instabilities inside the superconducting states provide insights into the superconductors which have the second phase transition below the first transition temperature.
In order to incorporate spatial inhomogeneity due to nonmagnetic impurities, Anderson [1] proposed a BCS-type theory in which single-particle states in such an inhomogeneous system are used. We examine Andersons proposal, in comparison with the Bogoliubov-de Gennes equations, for the attractive Hubbard model on a system with surfaces and impurities. [1] P. W. Anderson, J. Phys. Chem. Solids {bf 11}, 26 (1959).
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