Bogoliubov Fermi Surfaces Research Articles

Lifting of Gap Nodes by Disorder in Tetragonal FeSe_{1-x}S_{x} Superconductors.

The observation of time-reversal symmetry breaking and large residual density of states in tetragonal FeSe_{1-x}S_{x} suggests a novel type of ultranodal superconducting state with Bogoliubov Fermi surfaces (BFSs). Although such BFSs in centrosymmetric superconductors are expected to be topologically protected, the impurity effect of this exotic superconducting state remains elusive experimentally. Here, we investigate the impact of controlled defects introduced by electron irradiation on the superconducting state of tetragonal FeSe_{1-x}S_{x} (0.18≤x≤0.25). The temperature dependence of magnetic penetration depth is initially consistent with a model with BFSs in the pristine sample. After irradiation, we observe a nonmonotonic evolution of low-energy excitations with impurity concentrations. This nonmonotonic change indicates a transition from nodal to nodeless, culminating in gapless with Andreev bound states, reminiscent of the nodal s_{±} case. This points to the accidental nature of the possible BFSs in tetragonal FeSe_{1-x}S_{x}, which are susceptible to disruption by the disorder.

Transport signatures of Bogoliubov Fermi surfaces in normal metal/time-reversal symmetry broken d-wave superconductor junctions

In recent times, Bogoliubov Fermi surfaces (BFSs) in superconductors (SCs) have drawn significant attention due to a substantial population of Bogoliubov quasiparticles (BQPs) together with Cooper pairs (CPs) in them. The BQPs as zero energy excitations give rise to captivating and intricate charge dynamics within the BFSs. In this theoretical study, we propose to reveal the unique signatures of the topologically protected BFSs in bulk d-wave SCs using normal metal/time-reversal symmetry (TRS) broken d-wave SC hybrid setup, in terms of the differential conductance and Fano factor (FF). Orientation of crystal a axis with respect to junction normal, quantified by the parameter α, is crucial for transport properties in these hybrid devices. For α = 0, an enhancement in zero-bias conductance (ZBC) can be identified as a key signature of BFSs. However, for α≠0 , this feature does not replicate due to the presence of the localized Andreev bound state (ABS) at the interface. The interplay of ABS and BFSs gives rise to an anomalous behavior in ZBC compared to the α = 0 case. This behavior remains qualitatively similar even at finite temperatures. Finally, we explain this anomalous behavior by analyzing the effective charge of the carriers in terms of the FF. The simplicity of our setup based on d-wave SC makes our proposal persuasive.

Proximity-induced gapless superconductivity in two-dimensional Rashba semiconductor in magnetic field

Two-dimensional semiconductor-superconductor heterostructures form the foundation of numerous nanoscale physical systems. However, measuring the properties of such heterostructures, and characterizing the semiconductor in-situ is challenging. A recent experimental study by [Phys. Rev. Lett. 128, 107701 (2022)] was able to probe the semiconductor within the heterostructure using microwave measurements of the superfluid density. This work revealed a rapid depletion of superfluid density in semiconductor, caused by the in-plane magnetic field which in presence of spin-orbit coupling creates so-called Bogoliubov Fermi surfaces. The experimental work used a simplified theoretical model that neglected the presence of non-magnetic disorder in the semiconductor, hence describing the data only qualitatively. Motivated by experiments, we introduce a theoretical model describing a disordered semiconductor with strong spin-orbit coupling that is proximitized by a superconductor. Our model provides specific predictions for the density of states and superfluid density. Presence of disorder leads to the emergence of a gapless superconducting phase, that may be viewed as a manifestation of Bogoliubov Fermi surface. When applied to real experimental data, our model showcases excellent quantitative agreement, enabling the extraction of material parameters such as mean free path and mobility, and estimating gg-tensor after taking into account the orbital contribution of magnetic field. Our model can be used to probe in-situ parameters of other superconductor-semiconductor heterostructures and can be further extended to give access to transport properties.

Impurity effect on Bogoliubov Fermi surfaces: Analysis based on iron-based superconductors

Impurity effect on Bogoliubov Fermi surfaces: Analysis based on iron-based superconductors

Nodal band-off-diagonal superconductivity in twisted graphene superlattices

The superconducting state and mechanism are among the least understood phenomena in twisted graphene systems. Recent tunneling experiments indicate a transition between nodal and gapped pairing with electron filling, which is not naturally understood within current theory. We demonstrate that the coexistence of superconductivity and flavor polarization leads to pairing channels that are guaranteed by symmetry to be entirely band-off-diagonal, with a variety of consequences: most notably, the pairing invariant under all symmetries can have Bogoliubov Fermi surfaces in the superconducting state with protected nodal lines, or may be fully gapped, depending on parameters, and the band-off-diagonal chiral p-wave state exhibits transitions between gapped and nodal regions upon varying the doping. We demonstrate that band-off-diagonal pairing can be the leading state when only phonons are considered, and is also uniquely favored by fluctuations of a time-reversal-symmetric intervalley coherent order motivated by recent experiments. Consequently, band-off-diagonal superconductivity allows for the reconciliation of several key experimental observations in graphene moiré systems.

Spin fluctuations from Bogoliubov Fermi surfaces in the superconducting state of S-substituted FeSe

The study of the iron-based superconductor, FeSe, has resulted in various topics, such as the interplay among superconductivity, nematicity, and magnetism, Bardeen-Cooper-Schrieffer Bose-Einstein-condensation (BCS-BEC) crossover, and Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity. Recently, topologically protected nodal Fermi surfaces, referred to as Bogoliubov Fermi surfaces (BFSs), have garnered much attention. A theoretical model for the S-substituted FeSe system demonstrated that BFSs can manifest under the conditions of spin-orbit coupling, multi-band systems, and superconductivity with time-reversal symmetry breaking. Here we report the observation of spin fluctuations originating from BFSs in the superconducting (SC) state via 77Se-nuclear magnetic resonance measurements to 100 mK. In a heavily S-substituted FeSe, we found an anomalous enhancement of low-energy spin fluctuations deep in the SC state, which cannot be explained by an impurity effect. Such unusual behavior implies the presence of significant spin fluctuations of Bogoliubov quasiparticles, which are associated with possible nesting properties between BFSs.

Quantum geometry induced anapole superconductivity

Anapole superconductivity recently proposed for multiband superconductors [S. Kanasugi and Y. Yanase, Commun. Phys. 5, 39 (2022)] is a key feature of time-reversal ($\mathcal{T}$) symmetry broken polar superconductors. The anapole moment was shown to arise from the asymmetric Bogoliubov spectrum, which induces finite center of mass momenta of Cooper pairs at the zero magnetic field. In this paper, we show an alternative mechanism of anapole superconductivity: the quantum geometry induces the anapole moment when the interband pairing and Berry connection are finite. Thus, the anapole superconductivity is a ubiquitous feature of $\mathcal{T}$-broken multiband polar superconductors. Applying the theory to a minimal model of ${\mathrm{UTe}}_{2}$, we demonstrate the quantum geometry induced anapole superconductivity. Furthermore, we show the Bogoliubov Fermi surfaces (BFS) in an anapole superconducting state and predict an unusual temperature dependence of BFS due to the quantum geometry. Experimental verification of these phenomena may clarify the superconducting state in ${\mathrm{UTe}}_{2}$ and reveal the ubiquitous importance of quantum geometry in exotic superconductors.

Stability of Bogoliubov Fermi surfaces within BCS theory

It has recently been realized that the gap nodes of multiband superconductors that break time-reversal symmetry generically take the form of Fermi surfaces of Bogoliubov quasiparticles. However, these Fermi surfaces lead to a nonzero density of states (DOS) at the Fermi energy, which typically disfavors such superconducting states. It has thus not been clear whether they can be stable for reasonable pairing interactions or are in practice preempted by time-reversal-symmetric states with vanishing DOS. In this Letter, we show within BCS theory applied to a paradigmatic model that the time-reversal-symmetry-breaking states are indeed stabilized over broad parameter ranges at weak coupling. Moreover, we introduce a fast method that involves solving the inverse BCS gap equation, does not require iteration, does not suffer from convergence problems, and can handle metastable solutions.

Two superconducting states with broken time-reversal symmetry in FeSe1-xSx.

Iron-chalcogenide superconductors FeSe1-xSx possess unique electronic properties such as nonmagnetic nematic order and its quantum critical point. The nature of superconductivity with such nematicity is important for understanding the mechanism of unconventional superconductivity. A recent theory suggested the possible emergence of a fundamentally new class of superconductivity with the so-called Bogoliubov Fermi surfaces (BFSs) in this system. However, such an ultranodal pair state requires broken time-reversal symmetry (TRS) in the superconducting state, which has not been observed experimentally. Here, we report muon spin relaxation (μSR) measurements in FeSe1-xSx superconductors for 0 ≤ x ≤ 0.22 covering both orthorhombic (nematic) and tetragonal phases. We find that the zero-field muon relaxation rate is enhanced below the superconducting transition temperature Tc for all compositions, indicating that the superconducting state breaks TRS both in the nematic and tetragonal phases. Moreover, the transverse-field μSR measurements reveal that the superfluid density shows an unexpected and substantial reduction in the tetragonal phase (x > 0.17). This implies that a significant fraction of electrons remain unpaired in the zero-temperature limit, which cannot be explained by the known unconventional superconducting states with point or line nodes. The TRS breaking and the suppressed superfluid density in the tetragonal phase, together with the reported enhanced zero-energy excitations, are consistent with the ultranodal pair state with BFSs. The present results reveal two different superconducting states with broken TRS separated by the nematic critical point in FeSe1-xSx, which calls for the theory of microscopic origins that account for the relation between nematicity and superconductivity.

Majorana-Weyl cones in ferroelectric superconductors

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 ultraclean ${\mathrm{SrTiO}}_{3}$ opens new opportunities. Indeed, a promising route to engineer topological superconductivity is the combination of strong spin-orbit coupling and inversion-symmetry breaking. Here we study a three-dimensional parabolic band minimum with Rashba spin-orbit coupling, whose axis is aligned by the direction of a ferroelectric moment. We show that all of the aforementioned phenomena naturally emerge in this model when a magnetic field is applied. Above a critical Zeeman field, Majorana-Weyl cones emerge regardless of the electronic density. These cones manifest themselves as Majorana arc states appearing on surfaces and tetragonal domain walls. Rotating the magnetic field with respect to the direction of the ferroelectric moment tilts the Majorana-Weyl cones, eventually driving them into the type-II state with Bogoliubov Fermi surfaces. We then consider the consequences of the orbital magnetic field. First, the single vortex is found to be surrounded by a topological halo and is characterized by two Majorana zero modes: one localized in the vortex core and the other on the boundary of the topological halo. Based on a semiclassical argument we show that upon increasing the field above a critical value the halos overlap and eventually percolate through the system, causing a bulk topological transition that always precedes the normal state. Finally, we propose concrete experiments to test our predictions.

Anomalous Fano factor as a signature of Bogoliubov Fermi surfaces

Bogoliubov Fermi surfaces consist of an extensive number of Bogoliubov quasiparticles and can be created, for example, in semiconductor/superconductor hybrid devices. Noise spectroscopy is proposed to detect the presence of Bogoliubov Fermi surfaces in experiments.

Topological triplet-superconductivity in spin-1 semimetal

Superconductivity in topological semimetals gives a new paradigm of unconventional superconductors. Their exotic gap structures and topological properties have fascinated searching for material realizations and applications. In this work, we focus on a triple point semimetal where quasiparticle excitations, triple point fermions, carry the effective integer spin-1 in two distinct valleys. Our work demonstrates that the triple point fermion stabilizes inter-valley s-wave spin-triplet pairing. This is due to Fermi statistics, which strictly forbids the formation of inter-valley s-wave spin-singlet pairings. This feature is clearly distinct from the BCS and other multi-band superconductors. We find that two distinct inter-valley s-wave spin-triplet superconductors are allowed which in principle can be controlled by tuning the chemical potential: time-reversal symmetric (sz) state with topologically protected nodal lines and time-reversal broken (sx + isy) state with topologically protected Bogoliubov Fermi surfaces. Our study provides guidance in searching for spin-triplet superconductivity.

Impurity effects on Bogoliubov Fermi surfaces: A comparison with conventional BCS pairing

The superconductors can have a Fermi surface even below the transition temperature, which is known as the Bogoliubov Fermi surface. Focusing on the degrees of freedom at low energies, which are described by the spinless Bogoliubov particles for the case without time-reversal symmetry, we consider the self-energy from the impurity scattering on the Bogoliubov Fermi surfaces. The physical quantities such as the density of states are evaluated based on the single-particle Green’s functions. We systematically compare these results with those of the conventional s-wave spin-singlet pairing state, to illuminate the characteristic features of the Bogoliubov particles located at the Fermi level.

Bogoliubov Fermi surfaces from pairing of emergent j=32 fermions on the pyrochlore lattice

We examine the appearance of superconductivity in the strong-coupling limit of the Hubbard model on the pyrochlore lattice. We focus upon the limit of half filling, where the normal-state band structure realizes a $j=3/2$ semimetal. Introducing doping, we show that the pairing is favored in a $J=2$ quintet $E_g$ state. The attractive interaction in this channel relies on the fact that $E_g$ pairing on the pyrochlore lattice avoids the detrimental on-site repulsion. Our calculations show that a time-reversal symmetry-breaking superconducting phase is favored, which displays Bogoliubov Fermi surfaces.

Detecting Induced p±ip Pairing at the Al-InAs Interface with a Quantum Microwave Circuit.

Superconductor-semiconductor hybrid devices are at the heart of several proposed approaches to quantum information processing, but their basic properties remain to be understood. We embed a two-dimensional Al-InAs hybrid system in a resonant microwave circuit, probing the breakdown of superconductivity due to an applied magnetic field. We find a fingerprint from the two-component nature of the hybrid system, and quantitatively compare with a theory that includes the contribution of intraband p±ip pairing in the InAs, as well as the emergence of Bogoliubov-Fermi surfaces due to magnetic field. Separately resolving the Al and InAs contributions allows us to determine the carrier density and mobility in the InAs.

Theory of Hofstadter superconductors

We study mean-field states resulting from the pairing of electrons in time-reversal broken fractal Hofstadter bands, which arise in two-dimensional lattices where the unit cell traps magnetic flux $\Phi = (p/q)\Phi_0$ comparable to the flux quantum $\Phi_0 = h/e$. It is established that the dimension and degeneracy of the irreducible representations of the magnetic translation group (MTG) furnished by the charge 2e pairing fields have different properties from those furnished by single particle Bloch states, and in particular are shown to depend on the parity of the denominator $q$. We explore this symmetry analysis to formulate a Ginzburg-Landau theory describing the thermodynamic properties of Hofstadter superconductors at arbitrary rational flux $\Phi = (p/q)\Phi_0$ in terms of a multicomponent order parameter that describes the finite momentum pairing of electrons across different Fermi surface patches. This phenomenological theory leads to a rich phase diagram characterized by different symmetry breaking patterns of the MTG, which can be interpreted as distinct classes of vortex lattices. A class of $\mathbb{Z}_q$-symmetric Hofstadter SCs is identified, in which the MTG breaks down to a $\mathbb{Z}_q$ subgroup. We study the topological properties of such $\mathbb{Z}_q$-symmetric Hofstadter SCs and show that the parity of the Chern numbers is fixed by the parity of $q$. We identify the conditions for the realization of Bogoliubov Fermi surfaces in the presence of parity and MTG symmetries, establishing a novel topological invariant capturing the existence of such charge-neutral gapless excitations. Our findings, which could bear relevance to the description of re-entrant superconductivity in moir\'e systems in the Hofstadter regime, establish Hofstadter SC as a fertile setting to explore symmetry broken and topological orders.

Time-reversal symmetry breaking and d -wave superconductivity of triple-point fermions

We study the possibility of complex tensor ($d$-wave) superconducting order in three-dimensional semimetals with chiral spin-1/2 triple-point fermions, which have an effective orbital angular momentum of $L=1$ arising from a crossing of three bands. Retaining the first three lowest order terms in momentum and assuming rotational symmetry we show that the resulting mean-field $d$-wave ground state breaks time reversal symmetry, and depends crucially on the coefficients of the two quadratic terms in the Hamiltonian. The phase diagram at a finite chemical potential displays both the "cyclic" and the "ferromagnetic" states, distinguished by the average value of the magnetization; in the former state it is minimal (zero), whereas in the latter it is maximal (two). In both states we find mini Bogoliubov-Fermi surfaces in the quasiparticle spectrum, conforming to recent general arguments.

Symmetry, nodal structure, and Bogoliubov Fermi surfaces for nonlocal pairing

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.

Odd-frequency pairing inherent in a Bogoliubov Fermi liquid

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 with sign function of frequency) and the other is proportional to the inverse of the frequency, which exist in both the normal and anomalous parts of the self-energy. These characteristic self-energy structures are clearly reflected in the single-particle spectrum. Since the bogolons are originally composed of electrons, the connection between the two is also sought using the concrete $j=3/2$ fermion model, which reveals that the odd-frequency pairing of bogolons is mainly made of the electrons' odd-frequency pairing.

Odd-frequency pair density wave in the Kitaev-Kondo lattice model

We investigate the properties of the Kitaev-Kondo lattice model defined on a bilayer honeycomb lattice by means of the SO(3) Majorana representation for spin-$1/2$ moments. We first consider the pairing of neighboring sites for the parent Kitaev spin liquid (KSL) Hamiltonian to render the Majorana and the spin-$1/2$ Hilbert spaces perfectly equivalent to each other. As a consequence, we demonstrate that this decoupling of the Kitaev interaction in terms of the SO(3) Majorana fermions reproduces exactly the spectrum of the KSL model alone. Then, by considering the effect of a local Kondo coupling $J_K$ in the model and decoupling it in terms of an order parameter that physically must have a finite staggering phase, we obtain that the system undergoes a quantum phase transition from a fractionalized Fermi liquid to a nematic triplet superconducting (SC) phase as $J_K$ is increased. Depending on the model parameters, this SC phase can exhibit either Dirac points, Bogoliubov-Fermi lines, or Bogoliubov-Fermi surfaces as nodal bulk manifolds. The surface states in this latter case are also characterized by topologically protected antichiral edge modes. The SC phase breaks time-reversal symmetry and exhibits a coexistence of a dominant odd-frequency pairing with a small even-frequency component for electronic excitations localized on sites of the same sublattice of the system. Finally, we show that this SC phase is in fact a pair-density-wave state, with Cooper pairs possessing a finite center-of-mass momentum in zero magnetic field.