Topological Bogoliubov Fermi Surfaces

• TBFSs are codimension-1 manifolds of zero-energy Bogoliubov quasiparticles that result from inflated nodal structures in superconductors.
• Their formation is enabled by mechanisms such as multiband pairing, inversion and particle-hole symmetries, and altermagnetic spin-splitting, with protection diagnosed through Pfaffian sign changes or Wilson-loop invariants.
• TBFSs yield unique experimental signatures including altered thermal transport, modified spectroscopy, and persistent Majorana edge states despite a gapless bulk.

Topological Bogoliubov Fermi surfaces (TBFSs) are codimension-1 manifolds of zero-energy Bogoliubov quasiparticles in superconducting Bogoliubov–de Gennes (BdG) spectra that are stabilized by symmetry and topology rather than by accidental fine tuning. In three dimensions they are genuine two-dimensional zero-energy surfaces, whereas in two dimensions the same terminology is conventionally used for closed zero-energy contours. Relative to ordinary point or line nodes, TBFSs imply a finite zero-energy quasiparticle phase space within a superconducting state, and modern work identifies several routes to them, including even-parity multiband time-reversal-symmetry-breaking superconductors, inversion- and particle-hole-symmetric heterostructures, field-driven nodal inflation, and altermagnetic spin-splitting mechanisms (Agterberg et al., 2016, Mo et al., 2024).

1. Definition, geometric character, and place within nodal superconductivity

TBFSs generalize the nodal structures of unconventional superconductors. Instead of isolated point nodes or one-dimensional line nodes, the zero-energy condition $E(\mathbf{k})=0$ is satisfied on a codimension-1 manifold. In the foundational multiband even-parity time-reversal-symmetry-breaking setting, the relevant picture is that point nodes inflate into small spheroidal surfaces and line nodes inflate into toroidal surfaces; the zero-energy manifold is therefore finite-area in momentum space rather than singular only on lower-dimensional sets (Agterberg et al., 2016).

Within crystalline classification, Bogoliubov Fermi surfaces are not treated as an unrelated anomaly. A symmetry analysis on high-symmetry $n$-fold axes in centrosymmetric paramagnetic superconductors yields four local possibilities on the axis—full gap, point nodes, line nodes, and surface nodes—and the surface-node case is identified with Bogoliubov Fermi surfaces. In that framework, even-parity and time-reversal-symmetry-breaking order parameters are the cases that produce “a part of surface nodes (Bogoliubov FSs)” on the axis, while the global three-dimensional object is the full BFS (Sumita et al., 2018).

In two-dimensional systems, the same codimension-1 logic yields closed zero-energy contours rather than literal surfaces. Several papers therefore speak of Bogoliubov Fermi surfaces while explicitly noting that, in 2D, these are contours or loops in the Brillouin zone. This convention is used in nonsymmorphic Dirac-semimetal heterostructures, field-driven $d$-wave models, and altermagnetic 2D superfluids (Mo et al., 2024, Pal et al., 2024, Liu et al., 11 Aug 2025).

2. Symmetry structure and topological protection

A standard protection mechanism is based on the coexistence of inversion symmetry and BdG particle-hole symmetry. When these symmetries are present, the BdG Hamiltonian can be unitarily transformed into an antisymmetric matrix,

$$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$

so that one can define a Pfaffian

$$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$

Because

$\det \mathcal H(\mathbf k)=P^2(\mathbf k),$

zeros of the Pfaffian coincide with zero-energy degeneracies, and a sign change of $P(\mathbf k)$ across momentum space provides a $\mathbb Z_2$-type protection of the TBFS (Agterberg et al., 2016, Mo et al., 2024).

This protection is often local in momentum space rather than a single global invariant over the entire Brillouin zone. In the 2D nonsymmorphic Dirac-semimetal/superconductor heterostructure, the protected BFS contours are precisely the loci $P(\mathbf k)=0$, and their stability is diagnosed by

$$\mathrm{sgn}[P(\mathbf k)] : +1 \leftrightarrow -1.$$

That sign reversal prevents removal of the contour unless two contours annihilate or the protecting symmetry is broken. The same work emphasizes that, once the BFSs appear, the usual Brillouin-zone Chern number of occupied BdG bands becomes ill-defined because the bulk is no longer globally gapped (Mo et al., 2024).

Crystalline classifications refine this picture. On high-symmetry axes, the local node is classified by a $n$0D $n$1 or $n$2 index within rotation-eigenvalue sectors, and the full BFS is then understood as the three-dimensional extension of that local topological crystalline node. In this sense, the high-symmetry-line classification and the Pfaffian description are complementary rather than competing (Sumita et al., 2018).

Not every TBFS is organized by a full two-dimensional bulk invariant. In a proximitized Rashba electron gas under an in-plane Zeeman field, the gapless phase with a single pair of BFSs is topological as a family of 1D class-D superconductors parameterized by transverse momentum $n$3. There the relevant indicator is a Wilson-loop/Berry-Zak-type invariant $n$4, which is $n$5 in the nontrivial interval of $n$6, while the bulk remains gapless because a $n$7-odd tilt term pushes the low-energy cones through zero energy (Ruiz et al., 12 Jun 2025).

3. Microscopic mechanisms and representative model classes

Several distinct microscopic routes to TBFSs are now established. The earliest modern route is the multiband even-parity time-reversal-symmetry-breaking mechanism: interband pairing in a multiband superconductor generates a pseudomagnetic splitting in band space, so the zero-energy condition becomes the equation of a finite surface rather than a point or line (Agterberg et al., 2016). Iron-based “ultranodal” models recast the same logic for dominant spin-singlet intraband pairing supplemented by spin-orbit-coupling-enabled interband pairing and type-2 time-reversal-symmetry breaking (Setty et al., 2019). Spin-$n$8 and $n$9 systems provide a natural microscopic environment because quintet pairing in a multicomponent basis directly supplies the necessary internal structure (Ohashi et al., 2023, Kobayashi et al., 2021).

Later work broadened the landscape substantially. A two-dimensional nonsymmorphic Dirac semimetal on a checkerboard lattice, proximitized by a conventional $d$0-wave superconductor and subjected to an out-of-plane Zeeman field, realizes Pfaffian-protected TBFSs even though the same ingredients are more commonly associated with a fully gapped chiral topological superconductor. In that model, the essential TBFS ingredients are the multiband Dirac structure and the coexistence of inversion symmetry with particle-hole symmetry, while the nonsymmorphic symmetries chiefly supply the parent Dirac-semimetal band structure (Mo et al., 2024).

A distinct weak-topology route appears in a 2D Rashba electron gas with proximity-induced $d$1-wave pairing and an in-plane Zeeman field. There the $d$2 slice is the familiar 1D topological-wire problem, a finite interval of $d$3 inherits that topology, and a momentum-dependent $d$4 tilt then creates a single pair of zero-energy BFSs (Ruiz et al., 12 Jun 2025). Field-induced $d$5-wave heterostructures realize a simpler spin-$d$6 route, where an in-plane Zeeman field inflates nodal points into closed Bogoliubov contours protected by a $d$7-based Pfaffian sign change (Pal et al., 2024, Pal et al., 2023).

Altermagnetism introduced another family of TBFS mechanisms. In a 2D single-band altermagnetic Fermi gas with $d$8-wave spin splitting and ordinary $d$9-wave pairing, the quasiparticle energies

$$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$0

yield a TBFS when

$$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$1

and the topology is again diagnosed by a Pfaffian sign reversal. That model is notable because it uses only one band, zero net magnetization, and conventional $$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$2-wave pairing (Liu et al., 11 Aug 2025). In a separate three-dimensional altermagnetic topological-insulator platform, facet-dependent surface BFSs arise under conventional $$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$3-wave proximity because the altermagnetic term anisotropically shifts surface Dirac cones and frustrates ordinary singlet pairing on selected crystal faces (Fu et al., 23 Dec 2025).

Setting	Key ingredients	Topological diagnostic
Even-parity multiband TRSB superconductor	Interband pairing, inversion, broken TRS	Pfaffian $$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$4 sign change
Nonsymmorphic DSM heterostructure	Dirac bands, $$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$5-wave pairing, out-of-plane Zeeman, inversion + PHS	Pfaffian sign change
Rashba 2DEG with in-plane field	$$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$6-wave pairing, in-plane Zeeman, $$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$7-resolved 1D class D	Wilson loop $$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$8
2D $$\tilde{\mathcal H}(\mathbf k)=U\mathcal H(\mathbf k)U^\dagger,\qquad \tilde{\mathcal H}(\mathbf k)=-\tilde{\mathcal H}^{\mathbb T}(\mathbf k),$$9-wave heterostructure	In-plane Zeeman, nodal pairing, $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$0 structure	Pfaffian sign change
2D altermagnet	$$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$1-wave spin splitting, $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$2-wave pairing, parity + PHS	Pfaffian sign change
AMTI surface state	Facet-shifted Dirac cones, $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$3-wave proximity, crystal anisotropy	Class-D nanowire Pfaffian after confinement

4. Boundary physics, coexistence phases, and Majorana structures

A major development is that TBFSs need not eliminate boundary topology. In the nonsymmorphic Dirac-semimetal heterostructure, chiral Majorana edge states can persist even after BFSs appear and the usual bulk Chern number becomes ill-defined. The underlying bulk-boundary argument is highly model-specific: the inter-sublattice coupling that creates the BFSs vanishes at special edge momenta such as $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$4 or $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$5, so edge crossings pinned there are not perturbed even though the bulk becomes gapless (Mo et al., 2024).

In the in-plane-field Rashba system, the gapless topological phase supports antichiral edge states: two co-propagating Majorana modes localized on opposite edges perpendicular to the magnetic field. The low-energy bulk Hamiltonian consists of a chiral-symmetric Dirac part plus a $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$6-odd tilt term. The former supplies the 1D class-D topology of fixed-$$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$7 slices, while the latter simultaneously creates the BFSs and shifts the edge modes to finite energy, producing the antichiral dispersion $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$8 (Ruiz et al., 12 Jun 2025).

Boundary-selective BFSs can themselves become a resource for Majorana engineering. In an altermagnetic topological insulator proximitized by an $$P(\mathbf k)=\mathrm{Pf}[\tilde{\mathcal H}(\mathbf k)].$$9-wave superconductor, some facets are fully gapped while others host anisotropic surface BFSs. Confinement of those facet-selective BFSs in a quasi-1D prism nanowire produces class-D topological phase transitions, diagnosed by

$\det \mathcal H(\mathbf k)=P^2(\mathbf k),$0

and yields Majorana zero modes at the wire ends. In vortex geometries, the same platform supports a switch between vortex-line Majorana zero modes and boundary Majorana zero modes, depending on the altermagnetic strength and chemical potential (Fu et al., 23 Dec 2025).

These coexistence phenomena show that TBFS topology is not exhausted by the statement “the bulk is gapless.” A plausible implication is that gapless superconducting phases may retain sharply structured boundary signatures even when the global invariant of a fully gapped phase is unavailable, provided a lower-dimensional or local topological characterization remains meaningful (Mo et al., 2024, Ruiz et al., 12 Jun 2025).

5. Experimental and spectroscopic signatures

Because TBFSs leave a finite zero-energy quasiparticle phase space, they imply a nonzero residual density of states, linear-in-$\det \mathcal H(\mathbf k)=P^2(\mathbf k),$1 specific heat at low temperature, and anomalous heat transport while the system remains superconducting (Agterberg et al., 2016). Concrete model studies sharpen these expectations.

Thermal transport gives one direct bulk signature. In a fully gapped chiral topological superconductor, the low-temperature thermal Hall conductivity is quantized as $\det \mathcal H(\mathbf k)=P^2(\mathbf k),$2, but in the nonsymmorphic Dirac-semimetal heterostructure this quantization is lost once BFSs emerge because zero-energy Bogoliubov quasiparticles contribute with nonzero Berry curvature. The deviation is strongest when BFS formation also destroys the remnant chiral edge mode (Mo et al., 2024).

Junction spectroscopy has become a particularly detailed probe. In a normal-metal/$\det \mathcal H(\mathbf k)=P^2(\mathbf k),$3-wave-superconductor heterostructure with an in-plane Zeeman field, the field-generated BFSs produce strong thermoelectric signatures, and in the presence of interface Andreev bound states the Seebeck coefficient reaches $\det \mathcal H(\mathbf k)=P^2(\mathbf k),$4V/K while the figure of merit reaches $\det \mathcal H(\mathbf k)=P^2(\mathbf k),$5 (Pal et al., 2024). In related normal-metal/TRS-broken-$\det \mathcal H(\mathbf k)=P^2(\mathbf k),$6-wave junctions, the zero-bias conductance and Fano factor depend sharply on the crystal orientation $\det \mathcal H(\mathbf k)=P^2(\mathbf k),$7: for $\det \mathcal H(\mathbf k)=P^2(\mathbf k),$8, enhanced zero-bias conductance in the tunneling limit is the cleanest BFS signature, whereas for $\det \mathcal H(\mathbf k)=P^2(\mathbf k),$9 the interplay of BFSs with interface Andreev bound states produces anomalous nonmonotonic behavior (Pal et al., 2023).

Surface density of states and tunneling conductance also change systematically when parent point or line nodes inflate into BFSs. In spin-$P(\mathbf k)$0 quintet superconductors, zero-bias conductance peaks associated with surface Andreev bound states are blunted or split after BFS formation; when the parent state has no surface bound state on the chosen surface, the BFS instead produces a small but finite zero-bias conductance through bulk zero-energy states (Ohashi et al., 2023).

NMR offers a complementary dynamical probe. In heavily S-substituted FeSe, $P(\mathbf k)$1Se NMR finds an anomalous enhancement of low-energy spin fluctuations deep in the superconducting state, which is interpreted as arising from Bogoliubov quasiparticles on expanded BFSs and possible nesting between them. At lower sulfur content the low-$P(\mathbf k)$2 $P(\mathbf k)$3 saturates instead of turning upward, consistent with smaller residual zero-energy manifolds (Yu et al., 2023).

A conceptually distinct but related case is the polar phase of superfluid $P(\mathbf k)$4He. There the equilibrium state at rest has a Dirac nodal line, and superflow Doppler-shifts the spectrum into two touching Bogoliubov pockets bounded by flow-induced BFSs. The touching points carry $P(\mathbf k)$5, and the resulting “super-Landau” state remains stable up to a finite observed critical velocity $P(\mathbf k)$6 even though the Landau critical velocity is formally zero (Autti et al., 2020).

6. Materials platforms, conceptual distinctions, and open problems

The materials and platform landscape is now broad. Proposed or analyzed hosts include multiband even-parity TRSB superconductors of the Agterberg–Brydon–Timm type, iron-based “ultranodal” systems such as FeSe$P(\mathbf k)$7S$P(\mathbf k)$8, $P(\mathbf k)$9 and spin-$\mathbb Z_2$0 superconductors, pyrochlore-lattice systems with emergent $\mathbb Z_2$1 fermions, checkerboard nonsymmorphic Dirac semimetal/$\mathbb Z_2$2-wave-superconductor heterostructures, in-plane-field Rashba 2DEGs, altermagnetic Fermi gases, and altermagnetic topological-insulator/superconductor structures potentially relevant to $\mathbb Z_2$3 (Setty et al., 2019, Kobayashi et al., 2021, Mo et al., 2024, Ruiz et al., 12 Jun 2025, Liu et al., 11 Aug 2025, Fu et al., 23 Dec 2025).

At the same time, the literature distinguishes sharply between BFSs and TBFSs. Some papers explicitly provide a topological protection theory—typically a Pfaffian sign criterion or a slice-wise invariant—whereas others study ordinary BFS phenomenology without establishing such protection. A 2D $\mathbb Z_2$4-wave-magnet model, for example, analyzes BFSs in gapless FF and LO states but does not provide a Pfaffian invariant, $\mathbb Z_2$5 charge, enclosed Chern number, or analogous protection theory for the BFSs themselves (Pal et al., 3 Mar 2026). Likewise, rhombohedral tetralayer graphene supports a Bogoliubov Fermi surface near mean-field $\mathbb Z_2$6, but the paper’s explicit topological analysis concerns the gapped chiral superconducting phase rather than the BFS itself (Yang et al., 2024).

Another open issue concerns the stability of TBFS states once interactions between Bogoliubov quasiparticles are included. In a centrosymmetric $\mathbb Z_2$7 BG-FS system protected at the noninteracting BdG level by a $\mathbb Z_2$8 invariant, attractive interactions generate a BCS-type logarithmic instability in an inversion-symmetry channel. The striking conclusion is not that the Fermi surface must disappear, but that a centrosymmetric BG-FS is generically unstable toward inversion breaking and may evolve into a non-centrosymmetric BG-FS rather than a fully gapped state (Oh et al., 2019).

Several structural limitations recur across recent TBFS papers. Some lattice studies do not provide an explicit low-energy continuum Hamiltonian near the BFS, an explicit formula for the unitary transformation used in the Pfaffian construction, or a global invariant for gapless coexistence phases (Mo et al., 2024). Other works establish rich boundary phenomenology while leaving disorder response and beyond-mean-field fluctuation effects open (Ruiz et al., 12 Jun 2025, Fu et al., 23 Dec 2025). A plausible synthesis is that TBFS research is now divided between three complementary agendas: exact symmetry/topology diagnostics, microscopic mechanism design, and experimental identification through transport, spectroscopy, NMR, and flow phenomena.

TBFSs therefore occupy a distinctive position in superconductivity theory. They preserve superconducting order while supporting a codimension-1 zero-energy manifold, they often require genuinely multicomponent internal structure or momentum-dependent band asymmetry, and they force a departure from the usual binary distinction between fully gapped topological superconductors and ordinary nodal phases. Across current work, the most robust conceptual statement is that TBFSs are neither accidental metallic remnants nor merely enlarged nodes: they are symmetry-constrained, topologically organized zero-energy Bogoliubov manifolds whose bulk, boundary, and experimental consequences are now documented across several independent model classes (Agterberg et al., 2016, Sumita et al., 2018).