Gapless Superconductivity

• Gapless superconductivity is a regime where the order parameter remains finite, yet the quasiparticle spectrum does not exhibit a full gap and shows a finite low-energy density of states.
• It arises from mechanisms such as magnetic disorder, anisotropic spin splitting, and interface-induced odd-frequency pairing, bridging conventional superconductivity with metallic transport properties.
• Experimental signatures include non-activated temperature behavior, finite zero-bias conductance, and distinctive vortex or defect-bound states despite sustained superconducting order.

Searching arXiv for recent and relevant papers on gapless superconducting states across different mechanisms and platforms. A gapless superconducting state is a superconducting regime in which the pairing amplitude or phase-coherent condensate remains nonzero, yet the quasiparticle excitation spectrum lacks a full gap. In a conventional fully gapped BCS superconductor,

$E(\mathbf{k})=\sqrt{\xi_{\mathbf{k}}^2+\Delta^2},$

so the minimum excitation energy is $|E|_{\min}=\Delta$, the density of states vanishes for $|E|<\Delta$, and low-temperature thermodynamics are exponentially activated. In a gapless superconductor, by contrast, $E(\mathbf{k})=0$ for some $\mathbf{k}$, or localized defect-bound or vortex-bound branches cross zero energy, or low-energy spectral weight fills the gap through disorder, topology, or strong fluctuations. The result is finite low-energy density of states, metallic-looking low-bias transport, and power-law rather than activated low-temperature behavior, without necessarily destroying superconducting order (Wei et al., 2023, Yerin et al., 2021, Zhang et al., 2021).

1. Definition and conceptual scope

The basic distinction is between the superconducting order parameter and the spectral gap. In Abrikosov–Gor’kov theory, the order parameter $\Delta$ and the spectral gap $\Delta_g$ separate: one may have $\Delta\neq 0$ while $\Delta_g=0$, with $N_s(0)>0$ and superconductivity still present (Yerin et al., 2021). A similar distinction is explicit in current-carrying holographic nanowires, where a superfluid component and a normal, momentum-carrying component coexist at $|E|_{\min}=\Delta$0, and in granular or strongly fluctuating systems where superfluid stiffness remains finite while low-energy dissipation saturates (Khlebnikov, 2013, Zhang et al., 2024).

A second distinction concerns where gaplessness resides. In some works it is a bulk property of the Bogoliubov spectrum, as in altermagnets, magnetic superconductors, quasicrystals, or disorder-broadened Shiba bands (Wei et al., 2023, Karmakar et al., 2016, Saito et al., 2024, Yang et al., 7 Jul 2025). In others, the bulk remains fully gapped while gapless excitations are confined to defects or interfaces, most notably vortex lines in superconducting topological semimetals and odd-frequency states at superconductor/ferromagnet interfaces (Zhang et al., 2021, Bernardo et al., 2016). A third, distinct usage appears in Fulde–Ferrell–Larkin–Ovchinnikov physics, where the collective amplitude mode rather than the quasiparticle spectrum becomes gapless (Huang et al., 2021).

Setting	What is gapless	Representative work
Bulk superconducting spectrum	Bogoliubov quasiparticles in the bulk	(Wei et al., 2023)
Defect-bound sector	Vortex or interface modes while the bulk is gapped	(Zhang et al., 2021)
Collective sector	Higgs amplitude mode in a modulated condensate	(Huang et al., 2021)

A recurrent misconception is therefore that “gapless superconducting state” always means a bulk nodal superconductor. The literature instead supports several inequivalent meanings: bulk gapless quasiparticles, defect-localized gapless channels, or gapless collective modes.

2. Pair breaking, magnetic disorder, and the gap–gapless transition

The canonical bulk mechanism is magnetic pair breaking. In Abrikosov–Gor’kov theory for an isotropic $|E|_{\min}=\Delta$1-wave superconductor with paramagnetic impurities, the pair-breaking parameter is

$|E|_{\min}=\Delta$2

At $|E|_{\min}=\Delta$3, the system is gapped for $|E|_{\min}=\Delta$4 and gapless for $|E|_{\min}=\Delta$5. On the gapped side,

$|E|_{\min}=\Delta$6

so the spectral gap collapses continuously at $|E|_{\min}=\Delta$7 while $|E|_{\min}=\Delta$8 remains finite (Yerin et al., 2021). This transition has been identified as a Lifshitz-type, $|E|_{\min}=\Delta$9-order phase transition: the free energy and its first two derivatives are continuous, but the third derivative is singular on the gapless side, and the density-of-states surface develops a cuspidal edge (Yerin et al., 2021). A complementary topological formulation assigns the transition to a change in Euler characteristic of the density-of-states surface, $|E|<\Delta$0, again emphasizing that the gap collapse is topological in the Lifshitz sense rather than a destruction of order (Yerin et al., 2022).

A modern realization of this logic appears in magnetic-impurity $|E|<\Delta$1-wave superconductors with Yu–Shiba–Rusinov bands. In a fully self-consistent treatment, a small amount of magnetic disorder broadens Shiba levels into impurity bands. For impurity concentration around $|E|<\Delta$2, the impurity-band half-width

$|E|<\Delta$3

exceeds the band-center energy $|E|<\Delta$4, the single-particle excitation gap closes, and a gapless superconducting phase emerges while $|E|<\Delta$5 remains finite. Superconductivity itself is lost only near a quantum critical point around $|E|<\Delta$6 (Yang et al., 7 Jul 2025). In that phase, linear THz conductivity acquires finite low-frequency absorption even for photon energies below the conventional threshold $|E|<\Delta$7, whereas the nonlinear response remains dominated by a coherent Higgs mode at $|E|<\Delta$8 (Yang et al., 7 Jul 2025).

Material-specific band structure can dramatically lower the magnetic-disorder scale required for gaplessness. In $|E|<\Delta$9-NbSe$E(\mathbf{k})=0$0S$E(\mathbf{k})=0$1, scanning tunnelling microscopy and self-consistent Bogoliubov–de Gennes calculations found gapless superconductivity at remarkably low magnetic impurity concentrations. The Se–S substitution significantly modifies the band structure, favors nesting, and dictates the in-gap scattering for $E(\mathbf{k})=0$2, in contrast to the dominant charge-density-wave interactions in pure $E(\mathbf{k})=0$3-NbSe$E(\mathbf{k})=0$4 (Moreno et al., 1 Aug 2025). The operational signature is a finite low-energy density of states at the Fermi level together with coherence peaks, vortex lattice formation, and a BCS-like gap-versus-$E(\mathbf{k})=0$5 evolution.

3. Bulk gaplessness from magnetic textures, spin splitting, and finite-momentum structure

A separate class of mechanisms does not rely on impurity pair breaking, but on band reconstruction or momentum-selective suppression of pairing. In altermagnets, the normal-state Hamiltonian

$E(\mathbf{k})=0$6

contains intrinsic anisotropic spin splitting even without spin–orbit coupling or external field. When such a 2D altermagnet is proximitized by a uniform $E(\mathbf{k})=0$7-wave gap $E(\mathbf{k})=0$8, the local excitation gap on the Fermi surface becomes

$E(\mathbf{k})=0$9

and the main gap closes once

$\mathbf{k}$0

The resulting state has a segmented Bogoliubov Fermi surface, coexisting spin-singlet and spin-triplet pairings with effective $\mathbf{k}$1-wave angular structure, and a pair of finite-energy “mirage gaps” centered at $\mathbf{k}$2 with width $\mathbf{k}$3 (Wei et al., 2023). Because $\mathbf{k}$4 vanishes on the zone diagonals, the gapless phase is intrinsically angle-selective rather than uniformly metallic.

Magnetic reconstruction of an otherwise uniform $\mathbf{k}$5-wave condensate yields analogous gaplessness in the Hubbard–Kondo model for magnetic superconductors. There the pairing amplitude remains spatially uniform and $\mathbf{k}$6-wave-like, but the magnetic order evolves from ferromagnet through non-collinear spiral states to Néel order with increasing density and exchange coupling. At intermediate magnetic coupling the antiferromagnetic–superconducting state is gapless, except in the Néel regime near half filling, because band folding by non-collinear order leaves an effective Fermi surface of Bogoliubov quasiparticles (Karmakar et al., 2016).

A still more explicit finite-momentum route appears in magnetically polarized media supporting triplet pair-density-wave order. In a fully spin-polarized system, a staggered $\mathbf{k}$7-wave $\mathbf{k}$8-triplet state,

$\mathbf{k}$9

can appear as the only nonzero order parameter. In that regime the Bogoliubov spectrum retains an extended Fermi surface of zero-energy excitations, the zero-energy density of states is finite, and the low-temperature specific heat is linear in $\Delta$0, even though superconducting phase coherence survives (Georgiou et al., 2018). The same work stresses that such gapless triplet superconductivity can obstruct the realization of localized Majorana modes in nominally spinless platforms.

A related but distinct notion occurs in FFLO states. There, the order parameter is spatially modulated, and the Goldstone mode associated with broken translation symmetry becomes an amplitude fluctuation of the condensate. In the LO state this yields a gapless Higgs mode with

$\Delta$1

a collective, rather than quasiparticle, realization of gaplessness (Huang et al., 2021).

4. Topology-bound gaplessness: vortices, interfaces, and Majorana transport

Gaplessness can also be topologically confined to defects. In superconducting topological semimetals, the bulk is assumed to be a conventional, fully gapped $\Delta$2-wave superconductor, yet vortex lines host dispersive bound-state bands $\Delta$3 that cross zero energy. This occurs in Dirac and Weyl semimetals, spin-1 fermion systems, double Weyl systems, and spin-$\Delta$4 Rarita–Schwinger–Weyl semimetals, and is tied to the Chern numbers of $\Delta$5-resolved normal-state slices. The number of chiral vortex modes obeys

$\Delta$6

with vorticity $\Delta$7 and slice Chern number $\Delta$8 (Zhang et al., 2021). In this sense, the superconducting state is gapless only in the one-dimensional vortex sector.

At superconductor/ferromagnet interfaces, inhomogeneous magnetization can generate odd-frequency spin-triplet pairing. Scanning tunnelling spectroscopy on Au/Nb/Ho structures showed that driving Ho from a helical antiferromagnetic state to a homogeneous ferromagnet changes the Nb subgap density of states from pronounced double-peak or zero-bias-peak structures back to a BCS-like gap. The double-peak structure corresponds to dominant spin-zero odd-frequency triplets, whereas a zero-bias peak signals strong spin-one odd-frequency triplet correlations. Because the gap is locally filled by these interface-induced states, the superconducting density of states becomes gapless even though Nb remains superconducting (Bernardo et al., 2016).

A different topological consequence of an extended gapless superconducting region was identified in a $\Delta$9-wave chain with complex hopping. Breaking effective time-reversal symmetry by a hopping phase $\Delta_g$0 produces an extended gapless region between two topological phases. For a finite open chain, a slow quench of the superconducting term can transport an edge Majorana from one topological phase to the other with nonzero probability, provided the transit time through the gapless region is tuned. The optimal transit time is proportional to system size and diverges in the thermodynamic limit, reflecting that the edge Majorana mixes mainly with low-lying inverted bulk states inside the gapless region (Rajak et al., 2014).

5. Inhomogeneity, quantum geometry, quasiperiodicity, and strong fluctuations

Disorder can create gaplessness even when the clean system is fully gapped and nodeless. In a two-orbital model with sign-changing but nodeless pairing, random nearest-neighbor attraction generates $\Delta_g$1-junction networks and Andreev bound states. Their localization length is controlled not only by the BCS coherence scale $\Delta_g$2 but by the quantum metric through

$\Delta_g$3

As the Fubini–Study metric parameter $\Delta_g$4 increases, Andreev states at distinct $\Delta_g$5-junctions hybridize into impurity-like bands, the inverse participation ratio drops, the disorder-averaged zero-energy spectral function develops a Bogoliubov “Fermi surface,” and the superfluid stiffness acquires a power-law temperature dependence $\Delta_g$6 with $\Delta_g$7 between about $\Delta_g$8 and $\Delta_g$9 in the calculations shown (Lesser et al., 21 May 2025).

Quasicrystals provide a clean, aperiodic route. In Ammann–Beenker quasicrystals with intrinsic $\Delta\neq 0$0-wave pairing, self-consistent Bogoliubov–de Gennes calculations found gapless superconductivity at and near half filling under magnetic field. The order parameter $\Delta\neq 0$1 remains finite everywhere in real space, yet the lowest quasiparticle excitation energy tends to zero because confined normal-state states and broken translational symmetry smear the coherence gap. With Rashba spin–orbit coupling, the gapless phase can be topological, characterized by a nonzero pseudospectrum invariant computed from a spectral localizer,

$\Delta\neq 0$2

and accompanied by near-zero-energy edge states (Saito et al., 2024).

Strong phase fluctuations produce a further route in quasi-one-dimensional wires. In thin superconducting wires where quantum phase slips proliferate, long-range phase coherence is destroyed, yet the pairing amplitude remains nonzero. The paper identifies a broad temperature window $\Delta\neq 0$3 in which the tunnelling density of states is finite at low energy and the conductivity falls off at most linearly with temperature, rather than exponentially, because electron–electron interaction induced dephasing acts as a pair-breaking mechanism (Meidan et al., 2011).

Low-frequency electrodynamics reveals analogous behavior in two-dimensional granular In/InO$\Delta\neq 0$4. Mutual-inductance measurements show a $\Delta\neq 0$5 saturating dissipative response in finite magnetic field coexisting with a robust superfluid density, and power-law spectra

$\Delta\neq 0$6

$\Delta\neq 0$7

These features were interpreted as signatures of gapless superconductivity and quantum critical behavior proximate to a quantum superconductor-to-anomalous-metal transition (Zhang et al., 2024).

6. Experimental signatures, misconceptions, and broader significance

Across these settings, the spectroscopic hallmark is finite low-energy spectral weight. Depending on context, this appears as finite $\Delta\neq 0$8, segmented Bogoliubov Fermi surfaces, zero-bias peaks, or continuous low-energy absorption. In altermagnet–superconductor junctions, angle-resolved Andreev reflection and conductance directly track gapless directions, spin-polarized segmented Fermi surfaces, and finite-energy mirage gaps whose centers follow $\Delta\neq 0$9 while the width remains $\Delta_g=0$0 (Wei et al., 2023). In magnetic-disorder $\Delta_g=0$1-wave superconductors, THz spectroscopy distinguishes the gapless phase by finite absorption below $\Delta_g=0$2, while the nonlinear Higgs response remains coherent (Yang et al., 7 Jul 2025). In Nb/Ho bilayers, scanning tunnelling spectra distinguish spin-zero and spin-one odd-frequency gapless states through double-peak and zero-bias-peak subgap structures, respectively (Bernardo et al., 2016). In $\Delta_g=0$3-NbSe$\Delta_g=0$4S$\Delta_g=0$5, finite zero-bias conductance far from individual impurities coexists with coherence peaks and vortex lattices, demonstrating that gaplessness need not eliminate global superconductivity (Moreno et al., 1 Aug 2025).

Thermodynamic and transport responses are equally diagnostic. Gapless states replace activated low-$\Delta_g=0$6 behavior by power laws or finite residual responses: linear specific heat in the staggered $\Delta_g=0$7-wave $\Delta_g=0$8-triplet state, power-law superfluid stiffness in quantum-geometric impurity bands, finite low-frequency dissipation in granular films, and enhanced heat or charge transport along topological vortex lines (Georgiou et al., 2018, Lesser et al., 21 May 2025, Zhang et al., 2024, Zhang et al., 2021). In the Abrikosov–Gor’kov setting, the gap–gapless transition is additionally predicted to generate giant thermoelectric anomalies near $\Delta_g=0$9 and a cuspidal change in the density-of-states surface (Yerin et al., 2021).

Several misconceptions recur. First, gaplessness does not imply the disappearance of superconducting order; many of the works explicitly retain finite $N_s(0)>0$0, finite superfluid stiffness, or robust Meissner response while $N_s(0)>0$1 (Yerin et al., 2021, Zhang et al., 2024). Second, gapless superconductivity is not restricted to intrinsic nodal order parameters: it can emerge from anisotropic spin splitting, magnetic band folding, odd-frequency conversion, Shiba-band overlap, quantum geometry, quasiperiodicity, or phase-slip proliferation (Wei et al., 2023, Karmakar et al., 2016, Bernardo et al., 2016, Lesser et al., 21 May 2025, Meidan et al., 2011). Third, the relevant “gapless” object need not be the bulk quasiparticle spectrum; it may be a vortex-line channel or a collective Higgs mode (Zhang et al., 2021, Huang et al., 2021).

Taken together, these studies broaden the concept of gapless superconductivity from a narrow Abrikosov–Gor’kov phenomenon to a family of superconducting states in which order persists while low-energy excitations survive. The resulting landscape includes field-free gapless superconductivity in altermagnets, topologically protected gapless vortex bands in semimetals, odd-frequency interface states, disorder- and metric-driven impurity bands, quasicrystalline and granular gapless phases, and collective-mode gaplessness in modulated condensates. The common thread is not the absence of superconductivity, but the decoupling of superconducting order from a hard single-particle gap.