Bogoliubov Fermi Surfaces (BFSs)

• Bogoliubov Fermi Surfaces (BFSs) are two-dimensional loci in momentum space where zero-energy Bogoliubov quasiparticles reside in unconventional, TRSB superconductors.
• They emerge from nonunitary, multiband pairing that inflates point or line nodes into extended surfaces, and are topologically protected by a ℤ₂ Pfaffian invariant.
• BFSs yield distinct experimental signatures including finite zero-energy DOS, characteristic ARPES patterns, and anomalous thermodynamic and transport properties.

Bogoliubov Fermi Surfaces (BFSs) are two-dimensional momentum-space loci of zero-energy Bogoliubov quasiparticles in superconductors, arising generically in multiband systems with nonunitary, time-reversal-symmetry-breaking (TRSB) pairing. Unlike conventional nodal gaps (points or lines), BFSs manifest as “inflated” nodes—closed surfaces of gapless excitations—when internally anisotropic degrees of freedom (such as orbital, sublattice, or higher-spin manifolds) permit nontrivial interband pairing. BFSs are topologically protected by a ℤ₂ Pfaffian invariant and can coexist with superconductivity, leading to ultranodal states characterized by finite zero-energy density of states (DOS), unique thermodynamic signatures, and magnetic or topological phenomena not seen in conventional superconductors (Brydon et al., 2018, Setty et al., 2019, Agterberg et al., 2016). BFSs have been both theoretically predicted and experimentally observed, notably in iron-based superconductors such as FeSe₁₋ₓSₓ, noncentrosymmetric and Rashba systems, and superconductors proximitized to altermagnetic materials (Yu et al., 2023, Fu et al., 23 Dec 2025).

1. General Theoretical Framework

The formation of BFSs requires superconductors with internal degrees of freedom beyond spin, allowing for pairing channels that are not purely intraband. The general Bogoliubov–de Gennes (BdG) Hamiltonian in a four-component basis (e.g., two orbitals × spin-½ or j = 3/2) takes the form:

$$H_k = \begin{pmatrix} H_0(k) & \Delta(k) \ \Delta^\dagger(k) & -H_0^T(-k) \ \end{pmatrix},$$

where $H_0(k)$ is the normal-state Hamiltonian and $\Delta(k)$ encodes the superconducting gap structure. The most general even-parity gap

$$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$

combines a singlet ($\eta_0$) and internally anisotropic “quintet” ($\vec{\eta}$) components (Brydon et al., 2018). Nonunitary pairing arises when the gap product $\Delta\Delta^\dagger$ contains terms not proportional to the identity, typically requiring $\eta_0$ and at least two anisotropic $\eta_n$ components with nontrivial relative phases, hence breaking T. The emergence of BFSs is closely tied to a nonvanishing time-reversal-odd (T-odd) part of the gap product, which acts as a “pseudo-magnetic” field inflating conventional nodes into extended zero-energy surfaces.

2. Topological Protection and Pfaffian Invariant

The existence and robustness of BFSs are protected by a ℤ₂ invariant defined via the Pfaffian of an antisymmetrized BdG Hamiltonian. For systems with particle-hole (C) and inversion (P) symmetry, one can construct an antisymmetric form $\tilde{H}_k$ such that

$H_0(k)$0

on the BFS. The sign-change of the Pfaffian across the momentum space defines the ℤ₂ invariant:

$H_0(k)$1

where $H_0(k)$2 and $H_0(k)$3 bracket the BFS (Brydon et al., 2018, Agterberg et al., 2016). This invariant prohibits the removal of the BFS unless it annihilates with another surface with opposite topological charge. Spheroidal pockets resulting from inflated point nodes also carry an even Chern number, ensuring further topological stability against perturbations.

3. Microscopic Models and Formation Mechanism

Several microscopic models demonstrate the genericity of BFSs:

• Cubic $H_0(k)$4 systems: In models such as the Luttinger Hamiltonian (e.g., YPtBi, half-Heuslers), mixing of singlet and quintet pairing channels leads to BFSs upon TRSB. For instance, chiral $H_0(k)$5 or $H_0(k)$6 order parameters break time-reversal symmetry and inflate nodal points/lines into spheroidal or toroidal BFSs (Brydon et al., 2018, Menke et al., 2019, Mori et al., 2024).
• Hexagonal two-orbital models: Systems with strong spin-orbit coupling and multi-orbital structure support BFSs with chiral or TRSB gaps, with double-Weyl points or line nodes being inflated into large BFS pockets (Brydon et al., 2018).
• Iron-based superconductors: In FeSe₁₋ₓSₓ, the interplay of spin-orbit coupling, nonunitary interband pairing, and TRSB generates nematic BFSs in a structurally tetragonal lattice. Both static Néel magnetic order (magnetic toroidal order) and pair density wave states can drive the parity-odd TRSB required for BFSs, and the broken $H_0(k)$7 symmetry naturally produces the observed C₂-symmetric banana-shaped segments in ARPES (Wu et al., 2023, Cao et al., 2023, Yu et al., 27 Jul 2025).
• Altermagnetic and Rashba systems: Altermagnetic exchange fields with $H_0(k)$8-wave-like spin splitting generate BFSs in singlet channels, promoting chiral $H_0(k)$9-wave or FF states at larger field strength (Hong et al., 2024, Fu et al., 23 Dec 2025). Noncentrosymmetric superconductors with Rashba SOC and in-plane Zeeman fields further demonstrate BFS formation at topological Lifshitz transitions in helical superconducting phases (Zhuang et al., 21 Oct 2025, Banerjee et al., 2022).

The general condition for BFSs is the closure of the quasiparticle gap on a two-dimensional manifold in $\Delta(k)$0-space, i.e., for some branch $\Delta(k)$1,

$\Delta(k)$2

defines a codimension-one surface, replacing the conventional gap node structure (Setty et al., 2019).

4. Magnetic, Supercurrent, and Topological Properties

BFSs are invariably linked to a low-energy magnetization associated with the T-odd part of the nonunitary gap product:

$\Delta(k)$3

where $\Delta(k)$4 is the time-reversed gap matrix. In real space, BFSs induce characteristic octupolar or dipolar spin or angular-momentum textures on the zero-energy surface, depending on the internal structure of the pairing state. This emergent magnetization can further drive intertwined orders, e.g., static or current-induced toroidal (Néel) magnetism or superconducting diodic effects in noncentrosymmetric materials (Brydon et al., 2018, Wu et al., 2023, Zhuang et al., 21 Oct 2025).

Surface spectra in the presence of BFSs feature exotic Fermi-arc states connected to the projected BFS pockets. Specifically, spheroidal pockets with nonzero Chern number carry multiple Fermi arcs per surface; toroidal BFSs with zero Chern number lack Fermi arcs but remain protected by mirror-sector Pfaffians (Brydon et al., 2018, Menke et al., 2019).

5. Experimental Signatures and Material Realizations

The presence of BFSs leads to quantitatively unique and experimentally accessible signatures:

• Finite zero-bias DOS: BFSs generically yield a nonzero residual density of states $\Delta(k)$5, directly resulting in linear temperature dependence of low-$\Delta(k)$6 specific heat ($\Delta(k)$7), a field-independent residual term in the NMR spin-lattice relaxation $\Delta(k)$8, and a nonzero thermal conductivity at $\Delta(k)$9 (Lapp et al., 2019, Yu et al., 27 Jul 2025).
• Transport and thermoelectric signatures: BFSs enhance zero-bias conductance and Seebeck coefficient in normal–superconductor junctions, with $$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$0 values as high as $$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$1 observed in models of $$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$2-wave SCs under Zeeman fields. Noise spectroscopy (Fano factor) reveals a reduced effective charge and robust discontinuities at BFS-controlling Lifshitz transitions (Pal et al., 2024, Pal et al., 2023, Banerjee et al., 2022).
• Angle-resolved photoemission (ARPES): Direct visualization of banana- or segment-shaped zero-energy contours—distinct from point or line nodes—has been achieved in FeSe₁₋ₓSₓ (Wu et al., 2023, Yu et al., 2023, Yu et al., 27 Jul 2025).
• Ultranodal "gap-filling" in STM and tunneling: BFSs replace gap-closing with gap-filling in tunneling spectra; as a function of tuning parameter (e.g., doping, field), zero-bias conductance rises sharply in the BFS state (Setty et al., 2019, Yu et al., 2023).
• Magnetic and thermodynamic anomalies: μSR and Kerr rotation reveal TRSB concomitant with the BFS onset. Enhanced low-energy spin fluctuations and their pressure-dependence have been traced to BFS nesting in NMR (Yu et al., 27 Jul 2025).
• Josephson and superconducting diode effects: The existence of BFSs in noncentrosymmetric, Rashba systems can be detected by sharp anisotropic collapses in Josephson current and SDE/JDE efficiency at helical–ultranodal phase boundaries (Zhuang et al., 21 Oct 2025).

Table: Representative experimental and theoretical probes linked to BFSs.

Signature	Observable	Systems/Theoretical Context
Residual DOS at $$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$3	$$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$4, STM $$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$5, $$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$6	FeSe$$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$7S$$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$8, half-Heuslers
Low-$$\Delta(k) = \eta_0(k) U_T + \vec{\eta}(k) \cdot \vec{\gamma} U_T$$9 enhancement in $\eta_0$0	NMR, pressure dependence	FeSe$\eta_0$1S$\eta_0$2 (Yu et al., 27 Jul 2025)
Fermi-arc surface states	ARPES	$\eta_0$3 heavy-fermions, half-Heuslers
Seebeck, $\eta_0$4 enhancement	Thermoelectric response	$\eta_0$5-wave SC junctions (Pal et al., 2024)
Josephson/diode anisotropy, collapse	SDE/JDE experiments	Rashba NCS (Zhuang et al., 21 Oct 2025)
Magnetic order: octupolar/dipolar	$\eta_0$6SR, Kerr, neutron, STM	BFS with magnetic/orbital texture

6. Thermodynamic Stability and Phase Competition

Although BFSs introduce a finite zero-energy DOS, which typically costs condensation energy, self-consistent BCS calculations demonstrate energetic stability for TRSB states with BFSs over broad parameter regimes at moderate coupling and spin-orbit interaction. The stabilization is promoted when the parent node structure is point-like rather than line-like, and for intermediate ratios of intra- to interband pairing strength (Bhattacharya et al., 2023, Menke et al., 2019, Brydon et al., 2018). However, at strong coupling, there is often a first-order transition to symmetry-preserving (gap-nodal or fully gapped) states. Cubic anisotropy, pressure, and tuning of singlet/triplet interband terms can all act as control parameters driving transitions to and from ultranodal BFS states.

7. Future Directions and Broader Implications

The ubiquity of BFSs in multiband TRSB superconductors suggests they are relevant to a wide spectrum of materials, including heavy-fermion compounds (UPt₃, UBe₁₃, URu₂Si₂), half-Heusler and oxide heterostructures, and Fe-based and altermagnetic–topological hybrid systems (Fu et al., 23 Dec 2025, Hong et al., 2024). BFSs create new routes for realizing topological superconductivity, Majorana modes (facet- or vortex-bound, in the presence of altermagnetism), and tunable quantum devices via phase transitions associated with BFS onset. Moreover, understanding the microscopics of nonunitary, intertwined, and symmetry-broken pairing underlying BFSs is central to deciphering the rich phenomenology of unconventional superconductors and the emergence of novel ultranodal, gapless, yet distinctly nonmetallic quantum phases.

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Key references: (Brydon et al., 2018, Agterberg et al., 2016, Setty et al., 2019, Wu et al., 2023, Cao et al., 2023, Yu et al., 27 Jul 2025, Yu et al., 2023, Pal et al., 2024, Menke et al., 2019, Fu et al., 23 Dec 2025, Zhuang et al., 21 Oct 2025, Bhattacharya et al., 2023, Hong et al., 2024, Pal et al., 2023, Banerjee et al., 2022, Miki et al., 2023, Mori et al., 2024).