Pair Density Wave (PDW) Superconductivity

• Pair density wave (PDW) is a superconducting state where Cooper pairs condense with finite momentum, resulting in a spatially modulated order parameter.
• It exhibits intertwined orders such as charge density waves, fractional vortices, and lattice dislocations, highlighting its complexity and impact on unconventional superconductivity.
• PDW order emerges through diverse microscopic routes, with evidence from experiments in cuprates, Fe(Te,Se), and graphene-based systems, offering insights into defect dynamics and vestigial phases.

Pair density wave (PDW) denotes a superconducting state in which the Cooper-pair condensate carries nonzero center-of-mass momentum, so the complex order parameter is spatially modulated rather than translationally invariant. In the minimal unidirectional case, the pair field is built from $\pm \mathbf{Q}$ components; in multi-$\mathbf{Q}$ states it can involve several symmetry-related wavevectors, intertwining superconducting phase order with charge order, lattice symmetry breaking, vestigial superconductivity, and unconventional topological defects. Depending on whether a uniform $\mathbf{Q}=0$ component is present, one distinguishes coexistence phases from a pure PDW, for which the spatial average of the superconducting order parameter vanishes (Agterberg et al., 2019, Lesser et al., 9 Jun 2025).

1. Order parameter, finite-momentum pairing, and basic classifications

A convenient parametrization of PDW order is

$$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$

where $\Delta_0$ is the uniform component and $\Delta_{\mathbf{Q}}$ encodes finite-momentum pairing. In momentum space, the defining anomalous amplitude is

$$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$

so that pairing occurs between $\mathbf{k}$ and $-\mathbf{k}+\mathbf{Q}$, not strictly $\mathbf{k}$ and $\mathbf{Q}$0 (Liu et al., 2022, Ticea et al., 2024).

The simplest single-$\mathbf{Q}$1 state has $\mathbf{Q}$2, while a time-reversal-symmetric LO-type state contains both $\mathbf{Q}$3 and yields a standing-wave profile,

$\mathbf{Q}$4

Single-$\mathbf{Q}$5 and multi-$\mathbf{Q}$6 states are both realized in the literature. Unidirectional “stripe” PDW, bidirectional PDW, tri-directional PDW, and checkerboard-like PDW are all explicitly realized in microscopic or Ginzburg–Landau treatments (Soto-Garrido et al., 2014, Agterberg et al., 2019).

A pure PDW is characterized by $\mathbf{Q}$7. This distinction is not merely terminological: it changes the defect content, the allowed couplings, and the transport phenomenology. The 2025 analysis of a pure two-dimensional tri-$\mathbf{Q}$8 PDW emphasizes that a system with no uniform superconducting component supports fractional vortices whose structure is inseparable from crystalline dislocations of the emergent PDW lattice (Lesser et al., 9 Jun 2025).

2. Broken symmetries, composite orders, and vestigial phases

Because a PDW breaks both global $\mathbf{Q}$9 gauge symmetry and translational symmetry, it generically induces composite orders. For a unidirectional PDW with fields $\mathbf{Q}=0$0, the standard bilinears are

$\mathbf{Q}=0$1

Thus the PDW naturally intertwines with a charge-density wave at $\mathbf{Q}=0$2 and with a uniform charge-$\mathbf{Q}=0$3 superconducting order parameter (Rosales et al., 2024, Agterberg et al., 2019).

This composite structure underlies the concept of vestigial order. Partial melting of the PDW can restore only one of its broken symmetries. In the review literature, proliferation of Abrikosov vortices can destroy PDW order while leaving a $\mathbf{Q}=0$4 CDW, whereas proliferation of PDW dislocations can destroy the modulated pair field while sustaining charge-$\mathbf{Q}=0$5 superconductivity (Agterberg et al., 2019). Commensurability changes this hierarchy in a decisive way. For two-dimensional commensurate PDW with $\mathbf{Q}=0$6-times expanded unit vectors, the low-energy phase Hamiltonian contains a clock-like anisotropy $\mathbf{Q}=0$7, with $\mathbf{Q}=0$8 physically distinct translational states after gauge equivalence is taken into account (Gao et al., 8 Dec 2025).

For $\mathbf{Q}=0$9, the commensurate theory supports vestigial charge-$$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$0 or charge-$$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$1 superconductivity together with a translational-symmetry-broken CDW vestigial phase. For $$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$2, the restoration of translational symmetry proceeds through two successive Berezinskii–Kosterlitz–Thouless transitions, producing two critical vestigial phases: critical-PDW and critical-CDW. In these phases the discrete translational symmetry is quasily broken, and density-density correlations decay as power laws even in two dimensions (Gao et al., 8 Dec 2025).

Point-group symmetry is often reduced as well. A unidirectional PDW breaks $$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$3 to $$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$4, giving a smectic superconducting state. In EuRbFe$$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$5As$$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$6, the observed zero-field PDW is explicitly smectic: it is unidirectional, breaks translations along one in-plane direction, and restores both translational and $$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$7 rotational symmetry above the magnetic ordering temperature $$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$8 (Zhao et al., 2023).

3. Microscopic routes to PDW order

The PDW literature does not present a single universal mechanism; instead, it contains several distinct microscopic routes. One weak-coupling route is realized in a two-dimensional spin-1/2 Fermi fluid embedded in a spin-triplet nematic phase. There the nematic distortion reshapes the Fermi surface and shifts the peak of the pairing susceptibility to finite $$\Delta(\mathbf{r})=\Delta_0+\sum_{\mathbf{Q}}\Delta_{\mathbf{Q}}e^{i\mathbf{Q}\cdot\mathbf{r}},$$9, producing unidirectional, bidirectional, and tridirectional PDWs, as well as a triple-helix FF-type state, without requiring a Zeeman field (Soto-Garrido et al., 2014).

A second route is geometric. In multi-orbital superconductors with orbital-dependent pairing, the band-projected geometric contribution to the superfluid stiffness tensor can become locally non-positive-definite. When such negative regions are large enough, or when they are amplified by nodes in the projected gap, the total stiffness becomes non-positive-definite and the uniform BCS state is destabilized in favor of finite-$\Delta_0$0 order. In the corresponding Ginzburg–Landau expansion,

$\Delta_0$1

a PDW appears when the stiffness tensor $\Delta_0$2 acquires a negative eigenvalue (Jiang et al., 2022).

A closely related multiband mechanism starts from density-density interactions rather than explicit pair-hopping terms. After projection to the band basis, the interaction inherits nontrivial momentum dependence from Bloch-wave form factors, and the pairing vertex

$\Delta_0$3

can become առավել attractive at finite $\Delta_0$4. Mean-field analysis of checkerboard and three-band Hubbard models then yields a leading $\Delta_0$5-wave PDW instability at $\Delta_0$6 when nearest-neighbor attraction is sufficiently strong and competing CDW tendencies are suppressed (Ticea et al., 2024).

Another explicit route uses repulsive but non-monotonic BCS interactions. In that setting, the Fourier transform $\Delta_0$7 becomes negative at finite momentum, so the Hubbard–Stratonovich propagator satisfies

$\Delta_0$8

and the PDW instability occurs when $\Delta_0$9. This produces finite-temperature transitions to PDW order and a $\Delta_{\mathbf{Q}}$0 PDW quantum critical point with non-Fermi-liquid behavior (Wu et al., 2022).

Strong-coupling mechanisms are equally diverse. In a two-orbital square-lattice model with quasi-one-dimensional Fermi surfaces, interorbital interband pairing stabilizes an incommensurate $\Delta_{\mathbf{Q}}$1 PDW at low filling, while the strong-coupling hard-core pair Hamiltonian favors a commensurate period-2 PDW at $\Delta_{\mathbf{Q}}$2 because the effective bosonic dispersion is minimized there (Vadnais et al., 8 Apr 2026). In the strong-coupling Holstein–Hubbard limit on the triangular lattice, spontaneous valley polarization plus intra-pocket pairing yields a finite COM momentum $\Delta_{\mathbf{Q}}$3, giving an $\Delta_{\mathbf{Q}}$4-wave PDW at low density and a nematic $\Delta_{\mathbf{Q}}$5-wave PDW at intermediate density (Wang et al., 2024). A related DMRG study of the effective triangular-lattice $\Delta_{\mathbf{Q}}$6-$\Delta_{\mathbf{Q}}$7-$\Delta_{\mathbf{Q}}$8 model finds quasi-long-range PDW order with an incommensurate wavevector, spontaneous breaking of time-reversal and inversion symmetries, and an unusual Luther–Emery liquid on two-leg cylinders (Huang et al., 2021).

A further multiband route emphasizes opposite-dispersion interband pairing. Uniform interband pairing between electron-like and hole-like bands can exhibit negative superfluid weight,

$\Delta_{\mathbf{Q}}$9

which implies that the $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$0 condensate is unstable and that the true ground state is a PDW at finite momentum. Large-scale DMRG on ladder geometries then finds power-law PDW correlations with incommensurate modulations, and an exponent $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$1 that decreases systematically with increasing ladder width (Wang et al., 5 Dec 2025).

4. Topological defects, fractional vorticity, and quasiparticle structure

The topological defect content of PDW order is richer than that of a uniform superconductor because phase winding and translational dislocations are locked. For a unidirectional PDW with $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$2, one may define

$$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$3

and classify defects by vorticity $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$4 and dislocation charge $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$5. The three fundamental defects are the Abrikosov vortex $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$6, the half-vortex $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$7, and the double dislocation $$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$8 (Rosales et al., 2024).

In the pure two-dimensional tri-$$\Delta_{\mathbf{Q}}(\mathbf{k})=\left\langle c_{\mathbf{k}+\mathbf{Q}/2,\uparrow}\,c_{-\mathbf{k}+\mathbf{Q}/2,\downarrow}\right\rangle,$$9 PDW proposed for the quarter-metal phase of rhombohedral tetralayer graphene, the elementary topological defect has a dual identity. It is simultaneously a superconducting vortex, in which one of the three phases winds by $\mathbf{k}$0, and a crystalline dislocation of the emergent triangular lattice of PDW amplitude maxima, represented by a $\mathbf{k}$1–$\mathbf{k}$2 dislocation pair with Burgers vector $\mathbf{k}$3. The defect carries fractional flux

$\mathbf{k}$4

so the minimal excitation is a $\mathbf{k}$5 vortex (Lesser et al., 9 Jun 2025).

This duality has direct transport consequences. Mobile $\mathbf{k}$6 defects feel a Lorentz force $\mathbf{k}$7, drift preferentially along their glide direction, and generate dissipation with a strongly anisotropic resistivity. In the same framework, the Hall angle is purely geometric: $\mathbf{k}$8 where $\mathbf{k}$9 is the angle between the applied current and the Burgers vector. A small perpendicular magnetic field introduces full $-\mathbf{k}+\mathbf{Q}$0 vortices that create vacancies in the emergent PDW lattice and block defect glide, thereby restoring zero resistance (Lesser et al., 9 Jun 2025).

Spectroscopically, PDW defects also have distinctive electronic signatures. In the Bogoliubov–de Gennes analysis of half-vortices and dislocations, the half-vortex core is locally FF-like, inversion symmetry is violated because $-\mathbf{k}+\mathbf{Q}$1, and the Fourier-transformed LDOS develops split peaks around $-\mathbf{k}+\mathbf{Q}$2 with zero intensity exactly at $-\mathbf{k}+\mathbf{Q}$3. By contrast, Abrikosov vortices do not produce split $-\mathbf{k}+\mathbf{Q}$4 peaks, and double dislocations produce a stronger split associated with a $-\mathbf{k}+\mathbf{Q}$5 phase jump (Rosales et al., 2024).

The quasiparticle spectrum of a PDW need not resemble that of a conventional nodal superconductor. Quasi-one-dimensional PDW states can exhibit reconstructed Bogoliubov Fermi pockets rather than a fully gapped spectrum (Soto-Garrido et al., 2015). Conversely, the self-consistent $-\mathbf{k}+\mathbf{Q}$6 kagome $-\mathbf{k}+\mathbf{Q}$7 PDW is fully gapped, and its chiral realization carries Chern number $-\mathbf{k}+\mathbf{Q}$8, implying two co-propagating edge modes and a low-temperature thermal Hall conductance

$-\mathbf{k}+\mathbf{Q}$9

The corresponding real $\mathbf{k}$0 PDW is topologically trivial with $\mathbf{k}$1 (Yao et al., 2024).

5. Experimental realizations and diagnostic signatures

Several material platforms now exhibit evidence for PDW order or closely related phenomenology.

System or platform	Hallmark observation	Citation
Monolayer Fe(Te,Se)/SrTiO$\mathbf{k}$2	Domain-wall PDW with $\mathbf{k}$3, non-dispersive in-gap modulation, and $\mathbf{k}$4-phase boundaries near $\mathbf{k}$5 CDW dislocations	(Liu et al., 2022)
EuRbFe$\mathbf{k}$6As$\mathbf{k}$7	Zero-field smectic PDW with period $\mathbf{k}$8 unit cells, disappears at $\mathbf{k}$9 while uniform gap survives to $\mathbf{Q}$00	(Zhao et al., 2023)
Underdoped cuprates	Vortex-halo PDW signatures, $\mathbf{Q}$01, induced $\mathbf{Q}$02 charge modulations, and Josephson anomalies	(Agterberg et al., 2019)
Rhombohedral tetralayer graphene quarter-metal state	First experimental system where a pure PDW without uniform superconductivity is suspected	(Lesser et al., 9 Jun 2025)

The Fe(Te,Se) result is notable because the PDW is directly visualized by STM/STS at domain walls. Three mutually reinforcing signatures are reported there: in-gap LDOS modulation, coherence-peak and gap-depth modulation, and direct gap-energy modulation $\mathbf{Q}$03, all at the same wavevector $\mathbf{Q}$04–$\mathbf{Q}$05. The associated $\mathbf{Q}$06 modulation appears only at low energies within the superconducting gap, and $\mathbf{Q}$07-phase shift boundaries in the PDW phase are observed near vortices of the induced $\mathbf{Q}$08 CDW, which the authors identify as a topological “smoking gun” for a primary multi-$\mathbf{Q}$09 PDW (Liu et al., 2022).

EuRbFe$\mathbf{Q}$10As$\mathbf{Q}$11 provides a distinct case: a primary, zero-field, smectic PDW that coexists with a uniform superconducting gap. SI-STM resolves unidirectional modulations $\mathbf{Q}$12 with $\mathbf{Q}$13 mV, $\mathbf{Q}$14 mV, and $\mathbf{Q}$15. The modulation disappears at $\mathbf{Q}$16, not at $\mathbf{Q}$17, and it is suppressed inside vortex halos while remaining robust immediately outside them. Topography, high-energy tunneling maps, Raman spectroscopy, and the absence of elastic-scattering reports of density-wave order are used to exclude a primary CDW or SDW (Zhao et al., 2023).

Cuprates remain the most extensively debated PDW setting. In Bi$\mathbf{Q}$18Sr$\mathbf{Q}$19CaCu$\mathbf{Q}$20O$\mathbf{Q}$21, vortex halos exhibit field-induced modulations at $\mathbf{Q}$22 and $\mathbf{Q}$23, while La$\mathbf{Q}$24Ba$\mathbf{Q}$25CuO$\mathbf{Q}$26 near $\mathbf{Q}$27 displays layer decoupling, strong in-plane superconducting correlations, and Josephson signatures consistent with antiphase pairing between stripes (Agterberg et al., 2019).

Across platforms, the recurring diagnostics are technically specific. They include non-dispersive gap modulations in STM, induced $\mathbf{Q}$28 charge order, Josephson interference sensitive to finite pair momentum, anomalous Fraunhofer patterns, vortex-halo suppression or enhancement of PDW signatures, reconstructed Bogoliubov spectra in ARPES or QPI, and in pure PDW settings the possibility of fractional flux and transport signatures tied to defect motion (Liu et al., 2022, Zhao et al., 2023, Lesser et al., 9 Jun 2025).

6. Distinctions, debates, and open problems

PDW must be distinguished from several nearby concepts. It is not synonymous with FFLO. In FFLO, finite-$\mathbf{Q}$29 pairing is driven by Zeeman-split Fermi surfaces and typically breaks time-reversal symmetry; by contrast, many PDW constructions are zero-field instabilities that can preserve time-reversal symmetry, and their ordering vectors are selected by lattice symmetry, multiband geometry, or strong-correlation physics rather than by spin imbalance (Agterberg et al., 2019, Ticea et al., 2024). It is also not reducible to a conventional CDW or SDW: CDW and SDW dislocations do not carry fractional superconducting flux, whereas PDW defects can bind both phase winding and lattice dislocation in a single object (Lesser et al., 9 Jun 2025).

In cuprates, one major unresolved issue is whether PDW is a “mother order” or another competing order. The review literature presents both views. In one interpretation, PDW is the primary organizing tendency whose vestigial descendants account for charge order, nematicity, and pseudogap phenomenology. In another, PDW is nearly degenerate with uniform $\mathbf{Q}$30-wave superconductivity and charge order, with the actual state selected by crystal structure, magnetic field, or disorder (Agterberg et al., 2019).

Several technical open questions recur across the recent literature. The first is dimensionality: many controlled numerical realizations remain quasi-one-dimensional, while experiments motivate genuinely two-dimensional or layered PDWs (Jiang et al., 2023, Jiang, 2022). The second is disorder and pinning. Charge disorder can nucleate PDW defects, but strong pinning can immobilize them and obscure transport fingerprints; even weak disorder can destroy true long-range incommensurate PDW order in two dimensions (Lesser et al., 9 Jun 2025, Rosales et al., 2024). The third is the microscopic structure of defect cores, including bound states in fractional vortices and their interactions with ordinary $\mathbf{Q}$31 vortices (Lesser et al., 9 Jun 2025).

Materials-specific questions also remain unsettled. In monolayer Fe(Te,Se), the PDW has so far been observed only at domain walls, and its evolution under magnetic field, in pristine domains, and as a function of temperature remains to be mapped. Proposed follow-ups include phase-resolved STM in field, Josephson STM, ARPES, and QPI tomography (Liu et al., 2022). In EuRbFe$\mathbf{Q}$32As$\mathbf{Q}$33, the microscopic coupling between Eu magnetism and PDW order remains phenomenological, and more microscopic theory is explicitly called for (Zhao et al., 2023). In quarter-metal graphene, the definitive confirmation of a pure PDW would require the simultaneous observation of fractional-vorticity-controlled transport, anisotropy tied to the Burgers vector, and the geometric Hall response expected from mobile $\mathbf{Q}$34 vortices (Lesser et al., 9 Jun 2025).

Taken together, the modern PDW literature presents a coherent but still incomplete picture. The central idea—finite-momentum superconducting condensation with intertwined translational and gauge symmetry breaking—is now supported by direct spectroscopic visualization, microscopic model studies, and symmetry-based analyses across cuprates, iron-based superconductors, kagome systems, graphene-based platforms, and engineered multiband models. What remains unresolved is not the internal consistency of the concept, but the extent to which particular materials realize pure PDW, coexistence phases, or vestigial descendants, and which microscopic routes dominate in each case.