Preformed Pairing Scenario in Condensed Matter

• Preformed pairing is defined as a regime where fermionic pairs form at a higher temperature than that required for global phase coherence, leading to a pseudogap state.
• Experimental diagnostics such as tunneling spectroscopy, shot-noise measurements, and transport studies reveal separate energy scales (E₍g₎ and Δ₍c₎) that distinguish local pairing from coherent superconductivity.
• Material realizations span disordered superconductors, cuprates, ultracold Fermi gases, and moiré systems, highlighting the significant role of phase fluctuations and dimensionality in controlling coherence.

The preformed pairing scenario denotes a regime in which fermionic pairing and macroscopic phase coherence are distinct phenomena rather than concomitant aspects of a single transition. In this framework, electrons, holes, or composite fermions form bound pairs at a temperature or energy scale higher than the onset of long-range coherent order, so that a pseudogap or related gap-like anomaly can appear in the normal state. The scenario is formulated across multiple settings: disordered superconductors, cuprates, iron-based superconductors, ultracold Fermi gases, semiconductor electron-hole systems, moiré materials, oxide nanostructures, quantum Hall bilayers, and excitonic charge-density-wave materials. Across these contexts, the recurring structure is a separation between a pair-formation scale and a lower condensation or coherence scale, often accompanied by strong phase fluctuations, Berezinskii-Kosterlitz-Thouless behavior, or transport anomalies (Dubouchet et al., 2018, Mishra et al., 2011, Wyk et al., 2015).

1. Core definition and separation of scales

In the conventional BCS limit, electron pairing and phase coherence emerge together at $T_c$. The preformed pairing scenario instead posits a two-step evolution: first, local or short-range pairs form without global phase locking; second, these pairs acquire long-range phase coherence at a lower temperature. In disordered amorphous indium oxide, this distinction is expressed through a pseudogap scale $E_g$ that remains large well above $T_c$, and a separate collective gap $\Delta_c$ tied to the superconducting transition itself (Dubouchet et al., 2018). In cuprates, the same logic appears as a distinction between the pseudogap energy $A_{\mathrm{PG}}$, identified with pairing onset, and the lower scale associated with coherent superconductivity (Niu et al., 2024).

A compact way to write the separation below $T_c$, used in a preformed-pair pseudogap formulation, is

$$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$

with $\Delta_{\rm sc}$ the condensate contribution and $\Delta_{\rm pg}$ the non-condensed-pair contribution; above $T_c$, $E_g$0 while $E_g$1 may remain finite (Ghosh, 8 Jul 2026). Related temperature hierarchies appear in several theories. In a two-dimensional imbalanced Fermi gas, the mean-field pair-formation scale $E_g$2, the pseudogap temperature $E_g$3, and the BKT temperature $E_g$4 are distinct, with fluctuations suppressing $E_g$5 relative to $E_g$6 (Klimin et al., 2012). In a recent phase-fluctuation framework for cuprates, the gap-closing temperature $E_g$7 exceeds the phase-ordering temperature $E_g$8, and the interval $E_g$9 is explicitly identified as an incoherent preformed Cooper pairing regime (Yang et al., 25 Sep 2025).

The same separation is not restricted to electron-electron superconductivity. In ZnO, the regime of preformed electron-hole Cooper pairs is placed between an electron-hole plasma and a coherent electron-hole BCS state, as part of an electron-hole analogue of the BCS-BEC crossover (Versteegh et al., 2011). In 2H-TaSe$T_c$0, the ordered charge-density-wave state is interpreted as a Bose condensation of preformed excitons that already exist in the correlated normal state (Taraphder et al., 2010). In $T_c$1 quantum Hall bilayers, composite electron-composite hole pairing is treated as a precursor to exciton condensation (Rüegg et al., 2023). These formulations differ microscopically, but they all assign physical meaning to a paired, non-condensed regime.

2. Spectroscopic and thermodynamic signatures

The preformed pairing scenario is usually identified by observables that distinguish local pairing from coherent order. In strongly disordered a:InO films, single-particle tunneling measures a gap $T_c$2 that can satisfy $T_c$3, far above the BCS value $T_c$4, while Andreev spectroscopy reveals an intra-gap collective scale $T_c$5 (Dubouchet et al., 2018). The Andreev conductance exhibits two peaks at

$T_c$6

for example at $T_c$7 in a representative sample, and these peaks appear only below $T_c$8, shift toward zero bias with increasing temperature, merge near $T_c$9, and disappear above $\Delta_c$0. The single-particle gap $\Delta_c$1, by contrast, remains essentially unchanged at $\Delta_c$2 and collapses only at substantially higher temperature. This directly separates pair formation from collective condensation (Dubouchet et al., 2018).

In cuprates, local shot-noise spectroscopy provides a different pairing diagnostic. The low-temperature relation

$\Delta_c$3

defines an effective tunneling charge $\Delta_c$4, and the finite-temperature form

$\Delta_c$5

permits extraction of the bias-dependent pairing onset. In Bi$\Delta_c$6Sr$\Delta_c$7CaCu$\Delta_c$8O$\Delta_c$9, $A_{\mathrm{PG}}$0 at $A_{\mathrm{PG}}$1, then shows a step-like rise when $A_{\mathrm{PG}}$2, implying that pairing starts at the pseudogap energy rather than at the smaller kink feature. The pairing energy is spatially heterogeneous, can exceed $A_{\mathrm{PG}}$3 meV, and tracks $A_{\mathrm{PG}}$4 with slope $A_{\mathrm{PG}}$5 in local scatter plots (Niu et al., 2024).

Thermodynamic signatures are especially developed in ultracold Fermi gases. Within the Nozières-Schmitt-Rink treatment above $A_{\mathrm{PG}}$6, the specific heat

$A_{\mathrm{PG}}$7

is enhanced near $A_{\mathrm{PG}}$8 in the unitary regime by metastable preformed Cooper pairs, whereas deep on the BEC side it approaches the behavior of an ideal Bose gas of stable molecules (Wyk et al., 2015). The decomposition

$A_{\mathrm{PG}}$9

separates free fermions, stable molecules, and scattering contributions from metastable pairs. In the unitary regime, the enhancement of $T_c$0 occurs with $T_c$1, so it is attributed to fluctuating preformed pairs rather than stable bosons. In the strong-coupling BEC regime, $T_c$2, and the internal energy reduces to that of an ideal gas of bosonic dimers plus a binding-energy offset. This leads to a phase diagram with NF, PG, NB, and superfluid regions, using two temperatures extracted from minima in $T_c$3: $T_c$4 for the normal-Fermi-gas to pseudogap crossover and $T_c$5 for the pseudogap to nearly ideal Bose gas crossover (Wyk et al., 2015).

Transport also provides indirect thermodynamic evidence. In a three-dimensional preformed-pair framework for underdoped cuprates, the pseudogap $T_c$6 transfers spectral weight to higher frequencies, suppresses low-frequency $T_c$7, gives a large diamagnetic susceptibility above $T_c$8, but yields a $T_c$9 that is suppressed to zero much closer to $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$0. The contrast is framed through gauge-invariant response functions and the transverse $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$1-sum rule (Mishra et al., 2011). This behavior is used to reconcile large diamagnetism with only a narrow temperature window of substantial superconducting-like inductive response.

3. Phase fluctuations, BKT physics, and coherence establishment

A central mechanism for separating pairing from superconductivity is the suppression of phase stiffness. In layered FeSe-based superconductors, the preformed-pair interpretation is built from $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$2Se Knight shift, $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$3, weak diamagnetism, and a remarkable Nernst effect. Pairing onset is placed at $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$4 K, while zero resistance occurs below $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$5 K in $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$6, and BKT analysis gives $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$7 K (Kang et al., 2019). The transport fits use

$$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$8

with the BKT condition identified from the current-voltage exponent.

A closely related structure is reported in the triclinic iron pnictide $$\Delta^2(T)=\Delta_{\rm sc}^2(T)+\Delta_{\rm pg}^2(T),$$9, where $\Delta_{\rm sc}$0 K and a higher pairing-related scale $\Delta_{\rm sc}$1 K are inferred from inelastic neutron scattering, Nernst, NMR, and field response (Li et al., 15 Jan 2025). The material is explicitly quasi-two-dimensional, with anisotropy

$\Delta_{\rm sc}$2

and BKT analysis gives $\Delta_{\rm sc}$3 at about $\Delta_{\rm sc}$4 K and $\Delta_{\rm sc}$5 K from the Halperin-Nelson form. The spin resonance mode at $\Delta_{\rm sc}$6 meV evolves continuously across $\Delta_{\rm sc}$7, and a $\Delta_{\rm sc}$8 T in-plane field suppresses the resonance intensity below $\Delta_{\rm sc}$9, not only below $\Delta_{\rm pg}$0. The authors interpret the interval $\Delta_{\rm pg}$1 as a regime of incoherent Cooper pairs driven by phase fluctuations (Li et al., 15 Jan 2025).

Phase-fluctuation theory has also been formulated microscopically for cuprates. In a recent self-consistent framework, the order parameter is written as

$\Delta_{\rm pg}$2

and the bare stiffness $\Delta_{\rm pg}$3 is renormalized by vortex-antivortex fluctuations through BKT flow equations

$\Delta_{\rm pg}$4

The renormalized stiffness $\Delta_{\rm pg}$5 vanishes at $\Delta_{\rm pg}$6, while the gap amplitude survives to $\Delta_{\rm pg}$7, producing a finite interval of incoherent pairing (Yang et al., 25 Sep 2025). The same framework predicts a finite uncondensed normal component even at $\Delta_{\rm pg}$8,

$\Delta_{\rm pg}$9

and a two-fluid optical response

$T_c$0

This formulation treats smooth Nambu-Goldstone fluctuations and topological BKT-type vortices as distinct renormalization channels (Yang et al., 25 Sep 2025).

The general principle is therefore not merely that pairing precedes coherence, but that the lower transition is controlled by a separate stiffness problem. This is explicit in disordered films, quasi-two-dimensional iron-based superconductors, layered FeSe systems, and cuprate phase-fluctuation models (Dubouchet et al., 2018, Li et al., 15 Jan 2025, Kang et al., 2019, Yang et al., 25 Sep 2025).

4. Material realizations across condensed-matter platforms

The preformed pairing scenario has been invoked in materially diverse settings, but the phenomenology is not identical across them.

In disordered superconductors near the superconductor-insulator transition, amorphous indium oxide provides a prototypical case. Highly disordered films show a pseudogap above $T_c$1, a large $T_c$2, and Andreev peaks at $T_c$3, whereas a less disordered control film behaves as a conventional dirty BCS superconductor with monotonic in-gap Andreev enhancement and no intra-gap peaks (Dubouchet et al., 2018).

In cuprates, the local shot-noise study on Bi-2212 argues against a purely local-order origin of the pseudogap by showing that the shot-noise step marking paired transport appears at the pseudogap energy, not at the smaller kink energy; magnetic field drives the effective charge back toward $T_c$4, consistent with suppression of Andreev-mediated pairing (Niu et al., 2024). A recent phase-fluctuation theory further interprets the underdoped regime as an SC-pseudogap regime with $T_c$5 and a shoulder-like anomaly in the $T_c$6-$T_c$7 superconducting dome near $T_c$8 (Yang et al., 25 Sep 2025). A still more recent proposal attributes the thermal Hall effect above $T_c$9 to the parity anomaly of a chiral preformed-pair pseudogap mass, with onset at $E_g$00 rather than $E_g$01 (Ghosh, 8 Jul 2026).

Iron-based systems supply multiple realizations. In Ca$E_g$02(Pt$E_g$03As$E_g$04)(Fe$E_g$05As$E_g$06)$E_g$07, optical conductivity shows a pseudogap above $E_g$08 K; spectral-weight analysis indicates that the pseudogap and superconducting gap are formed from the same Drude-2 band, supporting continuous evolution of a superconducting gap precursor (Seo et al., 2018). In layered FeSe-based superconductors, quasi-two-dimensionality, enhanced resistivity anisotropy, and fluctuation signatures above $E_g$09 are taken to confirm strong phase fluctuation physics (Kang et al., 2019). In the triclinic pnictide CaLa-10-3-8, neutron, Nernst, NMR, and field data are interpreted in the same direction (Li et al., 15 Jan 2025).

Magic-angle twisted bilayer graphene has been modeled as a granular array of AA regions with localized nearly flat-band states and AB/BA regions with delocalized fermions. In the strong-coupling limit $E_g$10, local pairs form in the AA grains at

$E_g$11

while intergrain Andreev coupling sets a lower coherence temperature $E_g$12, so that $E_g$13 and an incoherent paired state intervenes (Islam et al., 2022). The same local-pairing/intergrain-coherence logic also appears in a flat-band semimetal model, where a nearly dispersionless band promotes local pairing and dispersive bands mediate phase locking (Zyuzin et al., 2021).

LAO/STO nanostructures realize a different manifestation: electron pairing without superconductivity. In quasi-1D ballistic Hall crosses, a conductance plateau at $E_g$14 is interpreted as paired transport, and the depairing field $E_g$15 coincides with the Hall transition field $E_g$16, for example $E_g$17 T and $E_g$18 T in one device. Angle-dependent Hall data define a nematicity parameter

$E_g$19

and the onset of nematicity is correlated with depairing (Nethwewala et al., 2022).

Semiconductor electron-hole systems supply a non-superconducting analogue. In highly excited ZnO, stimulated emission identifies a crossover from exciton gas or electron-hole plasma into a preformed electron-hole Cooper-pair regime. At fixed density $E_g$20, cooling from $E_g$21 K to $E_g$22 K produces a new strong peak interpreted as gain from preformed electron-hole Cooper pairs; at $E_g$23 K, increasing density yields a new peak with onset near $E_g$24, close to the calculated gain threshold $E_g$25 and above the Mott crossover near $E_g$26 (Versteegh et al., 2011).

Finally, 2H-TaSe$E_g$27 extends the scenario to electron-hole pairing in the form of a preformed excitonic liquid. The normal state is described as a bad metal with a pseudogap around $E_g$28 K, short mean free paths, and strong spectral-weight transfer, while the low-temperature CDW is interpreted as a coherence-restoring condensation of preformed excitons rather than a nesting instability (Taraphder et al., 2010).

5. Microscopic models and formal diagnostics

Preformed pairing has been formalized using several non-equivalent theoretical frameworks, each with its own operational criterion.

In the attractive Hubbard model, the response to an external pairing forcing field

$E_g$29

is used as a diagnostic (Tagliavini et al., 2016). The order parameter

$E_g$30

responds concavely in the attractive model with preformed pairs,

$E_g$31

whereas the repulsive comparison model shows low-field convexity,

$E_g$32

up to an inflection point. The paper treats low-field convexity as evidence against preformed pairs, while noting that concavity is necessary but not sufficient (Tagliavini et al., 2016).

In the NSR description of the BCS-BEC crossover above $E_g$33, the thermodynamic potential is

$E_g$34

with $E_g$35 fixed by the Thouless criterion. This framework distinguishes metastable preformed Cooper pairs from stable molecules using the pole structure of $E_g$36 and the number decomposition $E_g$37 (Wyk et al., 2015).

The two-dimensional imbalanced Fermi-gas treatment develops a Gaussian pair-fluctuation formalism that resolves the infrared divergence of standard NSR in $E_g$38D. The fluctuation contribution to the thermodynamic potential is written in terms of

$E_g$39

and the pseudogap temperature is defined by the condition

$E_g$40

where $E_g$41 is the pair-fluctuation spectral density. This yields a fluctuation-corrected hierarchy $E_g$42 in the generic case (Klimin et al., 2012).

Granular or emergent-granular models supply another route. In flat-band semimetals, the system is modeled as an array of grains with local pair amplitudes $E_g$43. Local pairing first occurs at

$E_g$44

while intergrain Andreev coupling from dispersive bands produces a lower phase-coherent transition (Zyuzin et al., 2021). The same structural idea underlies the TBG model, where delocalized AB fermions mediate an effective intergrain coupling

$E_g$45

which phase-locks locally preformed pairs (Islam et al., 2022).

Excitonic variants use analogous constructs. In 2H-TaSe$E_g$46, the local excitonic amplitude

$E_g$47

diagnoses an incoherent excitonic liquid that later condenses into CDW order (Taraphder et al., 2010). In quantum Hall bilayers, RPA gauge fluctuations produce an effective CE-CH attraction and an Eliashberg analysis yields a dominant $E_g$48-wave pairing channel, distinct from the $E_g$49-wave dominance of CE-CE pairing (Rüegg et al., 2023).

6. Extensions, controversies, and broader implications

One persistent controversy concerns the origin of the pseudogap. In cuprates, the local-order scenario attributes the pseudogap to CDW or related order and treats superconductivity as a separate lower-energy feature, whereas the preformed-pair scenario identifies the pseudogap itself with the onset of pairing. The shot-noise results in Bi-2212 are presented as excluding a pure local-order origin because the effective-charge step occurs at $E_g$50, not at the smaller kink scale, although the authors explicitly allow that local order could remain intertwined with pairing (Niu et al., 2024). Conversely, the pairing-field study of Hubbard models uses curvature diagnostics to argue that some pseudogap regimes need not be interpreted as well-defined preformed pairs, and may instead reflect antiferromagnetic correlations, Mott physics, or other nonlocal effects (Tagliavini et al., 2016).

A second controversy concerns the meaning of superconducting-like response above $E_g$51. In a light-driven setting, a preformed-pair interpretation has been proposed for photoinduced “superconductivity.” There, irradiation redistributes fermions to higher energies, effectively increasing the attraction and the density of nearly condensed bosonic pairs. The optical response is then computed with Aslamazov-Larkin fluctuation theory and gives

$E_g$52

over the experimental window, without requiring long-range phase coherence (Wang et al., 2024). The same work argues that the photoexcited state should be understood as a short-range-coherent preformed-pair state rather than a true equilibrium condensate. This directly contrasts with a photo-enhanced Drude interpretation.

A third development is topological. A recent theory argues that the pseudogap of non-condensed chiral pairs can enter the $E_g$53-dimensional parity anomaly exactly as a condensate mass, yielding the parameter-free relation

$E_g$54

On this basis, thermal Hall onset is predicted at $E_g$55, not at $E_g$56, and a logarithmic-derivative test against ARPES or STM pseudogap data is proposed (Ghosh, 8 Jul 2026). This suggests that preformed-pair physics may control transport coefficients usually associated with broken-symmetry phases, provided a gapped chiral Bogoliubov-de Gennes spectrum survives above $E_g$57.

The scenario also motivates quantum simulation. An extended Hubbard-Holstein model realized with $E_g$58 fermions and $E_g$59 bosons in painted optical lattices is proposed to access both a pair-formation scale $E_g$60 and a lower BKT temperature $E_g$61, thereby producing a tunable regime

$E_g$62

of preformed pairs without condensation (Hague et al., 2020). The simulator is intended to isolate whether boson-mediated pairing itself can generate a pseudogap-like regime in the presence of strong repulsion.

A recurrent misconception is that any gap above $E_g$63 automatically implies preformed pairs. The literature surveyed here does not support that equivalence in a universal form. Rather, the preformed pairing scenario requires some independent indicator that the gap or anomaly is pairing-related: Andreev reflection with an effective transfer of $E_g$64, shot-noise effective charge enhancement, a field-sensitive Nernst or diamagnetic response, a collective gap distinct from a single-particle pseudogap, a phase-stiffness scale separated from a gap-closing scale, or a microscopic diagnosis from fluctuation theory or forcing-field response (Dubouchet et al., 2018, Niu et al., 2024, Kang et al., 2019, Tagliavini et al., 2016). A plausible implication is that the scenario is best viewed not as a single model but as a class of mechanisms in which pairing amplitude survives outside the coherent ordered phase.

Across these realizations, the common conclusion is structural rather than material-specific: preformed pairing describes systems in which the binding of fermionic constituents is established before the emergence of long-range order, and the experimentally relevant problem becomes how phase stiffness, topology, dimensionality, disorder, or competing correlations convert a paired normal state into a coherent condensed one.