Pair-density waves: finite momentum, harmonics and experimental evidence

Scope

A pair-density wave (PDW) concerns finite-wavevector components of a superconducting pair field. A charge-density wave (CDW), a modulated tunneling spectrum and a modulated pair field are different observables. Specify the underlying lattice or superlattice, the electromagnetic gauge convention and whether a uniform condensate coexists. A single plane-wave phase has constant magnitude; periodic magnitude alone also need not establish a new spontaneous broken translation symmetry. R1 R10

Fulde and Ferrell's 1964 publisher abstract specifies a spatially uniform spin-exchange field with no scattering. Their finite-momentum state has a compensating contribution from unpaired electrons, so its equilibrium total current is zero despite the spatial phase variation. This statement concerns their idealized model; the original derivation is not reproduced here. Larkin and Ovchinnikov's paper appeared in Russian in 1964 and in English translation in 1965. The author institute's bibliography confirms that identity, but the original full text remains an access gap and its derivation has not been checked. These historical assumptions do not by themselves validate a modern orbital-FFLO interpretation. R1 R2

Mathematical assumptions

For a scalar illustration in a fixed gauge, write $\Delta(x)=d_0+d_+e^{iQx}+d_-e^{-iQx}$. The Fourier coefficients of $|\Delta|^2$ are $C_0=|d_0|^2+|d_+|^2+|d_-|^2$, $C_Q=d_+d_0^*+d_0d_-^*$ and $C_{2Q}=d_+d_-^*$, with negative-wavevector coefficients given by conjugation. Four independent discrete Fourier examples reproduce these identities. They are algebraic intensity identities, not a derivation of electronic charge density or a tunneling response. R4 R6

For $d_0=d_-=0$, the Fulde–Ferrell-type plane wave has constant $|\Delta|$. Nonzero equal real $d_+=d_-=d/2$ instead give a Larkin–Ovchinnikov-type standing wave $d\cos Qx$. In this fixed gauge and for $Q\ne0$, the pair field has period $2\pi/|Q|$, while $|\Delta|^2$ has period $\pi/|Q|$. A nonzero real uniform component $u$ adds a Q harmonic, but a quadrature component does not: $\Delta=iu+d\cos Qx$ gives $|\Delta|^2=u^2+d^2\cos^2Qx$ for real $u,d$. Thus a missing Q intensity harmonic need not exclude even coexistence with a uniform component. These are imposed scalar-field identities, not calculated electronic charge harmonics: a measured 2Q density signal also requires a response model and does not uniquely identify a PDW. No free energies are compared. R1 R4 R6

Under an electron gauge phase $\alpha$, both pair components acquire $e^{2i\alpha}$. The product $d_+d_-^*$ is neutral, whereas $d_+d_-$ transforms with $e^{4i\alpha}$ and has zero net wavevector. Three numerical phase checks confirm the distinction. A composite with charge 4e is not by itself evidence for a thermodynamically coherent charge-4e phase. Berg, Fradkin and Kivelson's 2009 melting theory proposes such a phase and fractional flux as a test; that proposal is distinct from the 2007 layer-decoupling paper. R8

Berg and colleagues' 2007 stripe model suppresses leading interplane Josephson coupling through a particular antiphase pattern and stacking. An independent overlap calculation gives $\langle\cos Qx\cos Qy\rangle=0$ over complete periods for perpendicular stripes, but parallel stripes give $1/2$. Adding a common uniform amplitude $u$ gives $u^2$; a cropped window need not cancel. The quartic overlap remains $\langle\cos^2Qx\cos^2Qy\rangle=1/4$. This illustrates why cancellation of the lowest coupling does not mean all coupling vanishes. The paper retains higher-order/further-neighbor terms and discusses defects that spoil exact cancellation. R4

An illustrative vortex-halo envelope $A(r)=e^{-r/\xi}$ produces a Q amplitude proportional to $A\Delta_0^*$ and a 2Q amplitude proportional to $A^2$. If $\Delta_0$ is approximately constant, the latter decays twice as fast. Four independent logarithmic-slope checks verify this and show failure of that ratio when the uniform component recovers from zero. The relation is conditional, especially near the core; it is not a universal PDW diagnostic. R6

Experimental criteria

Tranquada and colleagues' 1995 neutron study concerns static spin/charge stripes in Nd-substituted La–Sr–Cu–O at $x=0.12$. That diffraction evidence does not directly measure a superconducting pair field. Layer decoupling and charge harmonics are useful cross-checks only when competing charge, magnetic and structural explanations are addressed. R3 R4

Hamidian and colleagues' 2016 experiment used a superconducting Bi2212 nanoflake tip for scanned Josephson tunneling microscopy below 50 mK. It reported modulations near $Q_P=(0.25\pm0.02)\,2\pi/a_0$, superposed on a stronger uniform component. It is not the later field-induced vortex-halo spectroscopy experiment. Their phase-diffusion model uses $I(V)=I_J^2 ZV/[2(V^2+V_c^2)]$, with $V_c=2eZk_BT^*/\hbar$. The measured peak $I_c=\hbar I_J^2/(8ek_BT^*)$ differs from the underlying Josephson critical current $I_J$; three independent numerical peak searches verify this relation. The effective noise temperature $T^*$ must not simply be replaced by refrigerator temperature. R5

Edkins and colleagues' 2019 study used single-electron STM, explicitly without scanned Josephson tunneling microscopy. At 2 K it compared 0 and 8.25 T in the same Bi2212 field of view using the same tip. Approximately eight-cell and four-cell electronic modulations support a PDW interpretation within the authors' model. The fitted Fourier widths are not exactly in a two-to-one ratio. Sublattice form factors belong to the measured electronic response and must not be copied directly onto the pair field. The core region is excluded from the simple envelope argument. Raw-data fitting and the complete supplement are not reproduced here. R6

Chen and colleagues' 2021 CsV3Sb5 work reports spatial gap/conductance modulations and Josephson spectroscopy on prepared surfaces. Bulk and surface temperature scales differ, and the author manuscript explicitly says roton excitations were not directly observed. Its proposed complex PDW interpretation is not a direct phase measurement or a universal statement about kagome superconductors. The selected manuscript and methods check is inherited from the chiral-pairing entry. R7

In May 2026, Cao and colleagues reported a hysteretic tunneling-gap transition in multilayer NbSe2 under an in-plane field, consistent with a possible orbital FFLO state. Their reported angle, disorder and thickness dependences matter; the selected theoretical discussion neglects vortices in its energy comparison. This is evidence for a transition with a proposed interpretation, not a direct image of finite pair momentum. Only the publisher abstract, introduction and selected model passage have been checked here. R9

Limits and disputes

Keep four distinctions explicit: pair tunneling versus single-electron tunneling; induced modulation versus a primary instability; field-induced versus spontaneous behavior; and a surface observation versus a bulk thermodynamic phase. A particular gap map cannot alone determine the pair phase. A nonzero composite amplitude, a half-vortex proposal or an unusual transport curve likewise needs a separate coherence and alternative-explanation analysis. R5 R6 R8

The 2026 report by Wang and colleagues deliberately calls its moiré result Cooper-pair density modulation (CPDM): the authors describe periodic order-parameter variation without additional translation-symmetry breaking. That distinction prevents treating every periodic pairing texture as a spontaneous PDW. This entry has checked the institutional abstract and publication identity, not the complete measurement or microscopic model. R10

Discovery implications

Store the probe, temperature, magnetic-field magnitude and orientation, surface preparation, Fourier convention, uniform component and normalization with every candidate. Compare observations with an explicit response model before assigning an order parameter. Our local algebra, peak and overlap checks test stated examples; they do not reproduce the experiments, establish a material ground state, or close full scientific acceptance. R4 R5 R6 R9

Related topics

Continue with charge-fluctuation pairing, d-wave pairing, chiral pairing and topological superconductivity.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.