Superconductivity from Chaos: How Defects in Graphene Give Birth to Quantum Order ✛ ✦
Context
Superconductivity is the brightest example of a macroscopic quantum phenomenon, where electrons form a Bose-Einstein condensate of Cooper pairs, obeying Bose-Einstein statistics. The classic BCS theory, created by John Bardeen and colleagues, describes this phase transition in pure metals. However, in two-dimensional systems like graphene, the situation is complicated: at the charge neutrality point, the density of states tends to zero, and fantastically large attractive forces are needed for pairing. Electrons in graphene behave as massless fermions, obeying a relativistic equation, which was realized early on by Paul Dirac and later developed in the Fermi liquid theory of Lev Landau. Recent breakthroughs in twisted multilayer graphene have shown that creating flat bands with high density of states is key. This work proposes an alternative path: disorder engineering. Random vacancies or chemisorbed hydrogen create a peak in the density of states near the Fermi energy, much like a flat band but born from chaos. The question is whether such an inhomogeneous medium can support global phase coherence—and the answer, obtained via numerical simulation, was affirmative.
Methods
To describe the electronic structure of disordered graphene, a tight-binding model on a lattice of 5000 carbon atoms with periodic boundary conditions was used. Vacancies were modeled by removing sites, and hydrogenation by adding orbitals with finite energy and hybridization. Superconducting pairing was investigated in two channels: conventional s-wave (on-site attraction) and non-trivial d-wave (on nearest-neighbor bonds). To find the order parameter, the Bogoliubov–de Gennes equations were solved in a mean-field formalism, using a quantum field approach, and the critical temperature was determined from the linearized gap equation using a rational pole expansion. The quantum geometry of the normal state was computed via the quantum metric, and the superfluid weight and BKT temperature from the linear response and the Nelson–Kosterlitz criterion. An important role was played by the superposition character of defect-state wavefunctions, which exhibit a multifractal structure, analyzed through the singularity spectrum. All calculations were performed on graphics processors, ensuring interferometric precision in diagonalizing large matrices.
Results
It turned out that even small defect concentrations (1–10%) radically reduce the threshold interaction for superconductivity. If in pure graphene the critical coupling constant $U_c$≈2.3t, then with 4% vacancies, a superconducting order parameter emerges at U/$U_c$∼0.2 and grows linearly with U, characteristic of flat-band superconductivity. The spatial distribution of the gap is highly inhomogeneous: it is concentrated around defects, but due to the algebraic decay of multifractal states, it permeates the entire sample. Global phase coherence is established thanks to a finite superfluid weight $D_s$, which, as shown, has a purely geometric origin—the conventional (dispersive) contribution is absent. The value of $D_s$ correlates with the quantum metric of the normal state and reaches a maximum at a vacancy concentration of about 12%. The BKT temperature $T_{BKT}$ computed from the Nelson–Kosterlitz criterion reaches tens of kelvin already at U/$U_c$∼0.2 and grows with defect concentration. For hydrogenation, the results are qualitatively similar, but with smaller gap and $T_{BKT}$ values due to partial breaking of chiral symmetry and a shift of the density-of-states peak. In the case of d-wave pairing, quantum-critical behavior is observed: superconductivity arises only at finite interaction, but the threshold rapidly decreases with increasing defect concentration. The pairing symmetry is a mixture of $d_{x2-y2}$ and $d_{xy}$ components with a suppressed relative phase, indicating nematicity induced by disorder. In all cases, a crucial role is played by the quantum entanglement of electrons in Cooper pairs, obeying Fermi-Dirac statistics in the normal phase and bosonic in the condensate, and the BKT transition is governed by the balance of energy and entropy of vortices in accordance with the second law of thermodynamics.
Implications
The discovery demonstrates that disorder is not an enemy, but a creator of quantum correlated phases. It lays the foundations for a new paradigm of 'superconductivity from chaos,' applicable to a broad class of two-dimensional materials. In particular, it becomes clear that the traditional requirement of high crystallinity can be relaxed, and controlled defect creation can become a tool for designing superconductors with desired properties. The concept of geometric superfluid weight, previously considered exotic for flat bands in moiré systems, now extends to disordered media. Furthermore, the observed multifractality of the order parameter indicates a deep connection between Anderson critical states and superconductivity, calling for further theoretical understanding.
Future development
In the future, this direction could lead to the creation of superconducting devices based on monolayer graphene with defects, using electrostatic doping for precise tuning. Further research may include realistic modeling accounting for lattice reconstruction and multi-orbital effects, as well as the search for optimal types of disorder. It is also important to study the competition with magnetic phases that may arise from the same defect states. Experimental verification of the predictions will require samples with controlled concentrations of vacancies or hydrogen adatoms, measurements of the tunneling gap and transport properties. Another promising path is the application of quantum-thermodynamic approaches to optimize the transition temperature.
Impact
The field of condensed matter physics and quantum materials will gain a new class of disordered superconductors, and technology will potentially have a simple path to integrating superconducting elements into flexible electronics and sensors based on graphene.
Next steps
Immediate next steps include taking into account structural relaxation of vacancies, multi-orbital corrections, and competition with magnetism, as well as experimental confirmation of the predicted effects in tunneling spectroscopy and superfluid response measurements.
Key open problems
The work is directly connected to unsolved questions in physics: how can disorder generate, rather than destroy, quantum coherence? Where is the boundary between an Anderson localized state and a superconducting condensate? And what are the fundamental limits on the superconducting transition temperature in disordered systems? Answers to these are important for understanding high-temperature superconductivity and quantum phase transitions.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This work numerically investigates superconductivity in disordered graphene with vacancies and hydrogenation. The authors show that even small defect concentrations (1–10%) sharply reduce the threshold interaction for pairing, and the quantum metric associated with the multifractal structure of defect states acts as a "glue" for Cooper pairs. Phase diagrams are obtained and the Berezinskii–Kosterlitz–Thouless temperature is estimated.
Where its strength is
The work demonstrates that disorder can not only fail to destroy superconductivity but actually generate it, with the geometry of wavefunctions playing a key role, not just the density of states. This opens the way to using cheap and abundant materials without fine-tuning.
Where it could go next
- It might be interesting to compare the obtained results with the behavior of superconductivity in twisted layers, where flat bands also arise but due to the moiré potential rather than disorder.based on: Invisible Superconductivity in Twisted Layers arXiv:2605.03766
- Nearby is a work on switching superconductivity by a magnetic field in graphene—perhaps the combined action of field and disorder will yield new regimes.based on: Magnetic field switches on superconductivity in graphene arXiv:2607.06520
- It is interesting to see how disordered states with multifractal structure affect nonequilibrium transport, for example, the diode effect discussed in one of the neighboring works.based on: Graphene Makes Current Flow One Way arXiv:2606.17673
Worth trying
- Test whether superconductivity persists when a magnetic field is added, using the model from work 2607.06520.
- Estimate the magnitude of the diode effect in the superconducting state with defects, following the approach from 2606.17673.
- Compare the quantum geometry in disordered graphene and in twisted bilayers to identify universal features.
Why this may matter
If disorder can indeed serve as a tool for creating superconductivity, this will simplify technology and expand the class of materials for quantum devices.
Nearby works in our archive
- Graphene Makes Current Flow One Waysimilarity 0.67
- The Quantum 'Cook': Superfluidity of the Quantum Hall Effectsimilarity 0.65
- Invisible Superconductivity in Twisted Layerssimilarity 0.65
- Electronic dance on the edge: how emptiness gives rise to superconductivitysimilarity 0.65
- Chaos in a Crystal: How Disorder Awakens Quantum Unpredictabilitysimilarity 0.63
- Superconductivity Under Voltage: Current and Resistance Togethersimilarity 0.62
- Magnetic field switches on superconductivity in graphenesimilarity 0.62
The closest work in the archive sits at 0.67 by meaning — the usual level for our corpus.
Original abstract
as written by the authors arXiv:2607.02267
Correlated phases of matter are typically investigated in crystalline systems, where disorder is considered to be detrimental. However, intriguing exceptions exist, such as superconductivity being enhanced in amorphous realizations of Al and Bi. Here, we demonstrate that superconductivity can even emerge entirely from disorder, using monolayer graphene as an example. In the clean limit, the semi-metallic nature of graphene requires prohibitively strong electronic interactions to achieve superconductivity. Despite the inherently random nature of disorder, we show that introducing low concentrations of vacancies or hydrogenation in graphene provides a large density of low-energy states that easily induce superconductivity. For conventional $$s$$-wave pairing, disorder induces a finite superconducting order parameter for arbitrarily weak attractive interactions. Rather than forming isolated superconducting puddles, global phase coherence is established through a finite superfluid weight of purely geometrical origin. Away from the chiral limit of vacancies, hydrogenation similarly yields a finite transition temperature and nonzero superfluid weight for weak interactions. For unconventional nearest-neighbor pairing, typically more disrupted by disorder, superconductivity exhibits quantum-critical-like behavior, yet phase coherence persists at low interaction strengths, with mixed $$d$$-wave symmetries. Our work demonstrates the robust emergence of macroscopic superconducting phase coherence engineered entirely from microscopic disorder.
Interestingly, superconductivity in disordered graphene is literally 'visible at your fingertips': a scanning tunneling microscope can detect the inhomogeneous superconducting gap near defects, and applying a weak magnetic field will generate vortices 'pinned' to vacancies — a kind of quantum disorder that can be mapped out.
The idea of chaos as a source of order echoes Frank Herbert's 'Dune,' where navigators through folded space used prescience born of melange — here, crystal lattice defects open 'folds' in the electronic spectrum, giving birth to a superconducting state.
Key numbers
- vacancy concentration for maximum D_s: 12%
- BKT temperature: up to ~200 K
- threshold interaction for pure graphene: U_c ≈ 2.23 t (≈6 eV)
- threshold reduction at 4% vacancies: to U ≈ 0.2 U_c
- defect concentration in study: 1–25%