The defining difference is the momentum carried by Cooper pairs. In conventional Bardeen–Cooper–Schrieffer (BCS) superconductivity, pairs have zero center-of-mass momentum and the superconducting order is uniform. In pair-density-wave (PDW) superconductivity, pairs have finite center-of-mass momentum, producing a superconducting order parameter that varies periodically through space. PDW is still superconductivity; its modulation is in the paired-electron condensate, not merely in the material’s electric charge.
What does finite-momentum pairing mean?
A Cooper pair is made of two electrons, and its center-of-mass momentum describes the motion of the pair as a whole. In the conventional BCS reference state, that momentum is zero. The pair condensate has no built-in spatial modulation: its superconducting order parameter is uniform.
In a PDW, Cooper pairs have finite center-of-mass momentum. Their collective superconducting order therefore changes periodically with position. For a simple unidirectional example, the order parameter can be represented as Δ(r) proportional to cos(Q·r), where Q is the modulation wavevector. The direction and period of the modulation depend on the state; this expression illustrates the idea rather than defining every PDW.
This distinction concerns the momentum and spatial structure of the pair condensate. It is separate from whether the superconducting gap has s-wave, d-wave, or another symmetry.
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How do conventional superconductivity and PDW compare?
| Feature | Conventional BCS reference | Pair-density wave |
|---|---|---|
| Cooper-pair center-of-mass momentum | Zero | Finite |
| Superconducting order in space | Uniform | Periodically modulated |
| What the definition requires | Uniform, zero-momentum pairing | Finite-momentum pairing and modulated superconducting order |
| Relationship to charge order | No charge modulation is required by the reference state | May coexist with or induce charge-density-wave order; charge modulation alone does not establish PDW |
This is a conceptual comparison, not a claim that every material in either category has the same microscopic mechanism or gap symmetry. PDW order can be discussed as correlation-driven or intertwined order, but its origin and relationship to other orders depend on the material and model. The 2020 review by Agterberg and colleagues surveys PDW phenomena and the open questions around their microscopic picture and role in cuprates: The Physics of Pair-Density Waves: Cuprate Superconductors and Beyond.
Is a pair-density wave the same as an FFLO state?
Not necessarily. Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states share a defining feature with PDW: finite-momentum pairing and a nonuniform superconducting order. Classic FFLO proposals involve conditions such as high magnetic field and low temperature. The Annual Review describes FFLO as the weak-coupling version of PDW order while also treating PDW physics more broadly. Terminology varies, so a specific state should be identified by its material, conditions, symmetry, and proposed mechanism rather than by treating the names as universal synonyms.
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A 2023 report on centrosymmetric bilayer MoS2 presented evidence for finite-momentum pairing under its experimental conditions. The authors reported that the state occurred below the Pauli limit, was driven by the orbital effect, and did not rely on Fermi-surface segmentation. This is a material-specific result, not evidence that all PDW or finite-momentum states share that explanation. See Zhao and colleagues’ Nature Physics report.
Does a charge-density wave prove PDW superconductivity?
No. A charge-density wave (CDW) is a periodic modulation of electronic charge. A PDW is a modulation of superconducting pair order. The two can coexist or be related: PDW order may induce or accompany charge order. But observing charge modulation by itself does not show that Cooper pairs have finite center-of-mass momentum. Evidence for PDW requires a signature tied to the superconducting pairing state, interpreted in the context of the particular material and experiment.
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What is established, and what remains material-specific?
The zero- versus finite-momentum distinction is the central theoretical contrast. Whether a particular material realizes PDW order, whether it is a primary or competing order, and how it relates to charge order remain questions that require material-specific evidence. The 2020 review discusses mounting evidence in cuprates alongside debate over the microscopic picture; that review is a useful account of the field at its publication date, not proof that the debate has been settled.
Predictions from models should also be kept separate from measured properties. A 2026 study in npj Quantum Materials examined superfluid density in a generic two-dimensional, unidirectional PDW model. It found a broad parameter region with negative calculated superfluid density and, in the model’s stable regime, predicted a small longitudinal response, strong anisotropy, and unusual temperature dependence, including a transverse T2 behavior at low temperature. These are model-dependent predictions and possible diagnostics, not established universal properties of PDW materials. The analysis highlights stability and experimental confirmation as important questions; see Wang and colleagues’ study.
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