Pairing-Plus-Particle-Hole Model

• The pairing-plus-particle-hole model is a many-body framework that integrates superconductivity with density and excitonic orders through complementary pairing and particle-hole channels.
• It employs methods such as Hartree-Fock-Bogoliubov, RPA, and the Bethe-Salpeter equation to capture correlations via ladder and ring diagrams across nuclear and condensed matter systems.
• The model reveals that phenomena like pair-density-wave and charge-density-wave orders emerge as matrix components, guiding experimental probes like STM and Josephson STM.

The expression pairing-plus-particle-hole model is used for several closely related many-body constructions in which pairing correlations are not treated as an isolated particle-particle instability, but are formulated together with particle-hole structure, particle-hole symmetry, or particle-hole collective modes. In one quasi-one-dimensional realization of a pair-density-wave (PDW), superconducting and charge-density-wave order parameters are components of the same matrix field and are related by particle-hole symmetry (Tsvelik, 2023). In nuclear applications, the same general idea appears as a self-consistent treatment of the normal density and pairing tensor within Hartree-Fock-Bogoliubov theory, or as a number-conserving particle-hole RPA built on a particle-number-projected HFB reference state (Romero et al., 2017, Dukelsky et al., 2019). In layered Dirac materials, electron-hole pairing is treated as a momentum-space Coulomb problem whose solutions describe excitons and excitonic-insulator tendencies (Lokot, 2014). This suggests that the term is best understood as a structural motif of many-body theory rather than as a single universal Hamiltonian.

1. Conceptual scope and channel decomposition

A recurring theme is that the pairing channel and the particle-hole channel are complementary rather than mutually exclusive. In the density-functional and Green-function setting, standard adiabatic-connection fluctuation-dissipation theory uses dynamic density fluctuations in the particle-hole channel, whereas the particle-particle random phase approximation uses dynamic pairing-matrix fluctuations in the particle-particle and hole-hole channels. The pairing matrix is introduced as

$$\kappa_{ij}(t)=\langle \Psi_0^N|a_i a_j|\Psi_0^N\rangle,$$

and the corresponding response is encoded in the particle-particle Green function. The formal message is explicit: ph-RPA captures correlation through particle-hole excitations and ring diagrams, while pp-RPA captures correlation through pair addition-removal excitations and ladder diagrams. The same work also states that it does not build a combined pp+ph approximation, but instead frames the two channels as parallel and complementary (Aggelen et al., 2013).

A closely related reformulation appears in the particle-particle Bethe-Salpeter equation. There, the usual two-body Green’s function is re-read through a different time ordering so that it describes the propagation of two electrons or two holes rather than an electron-hole pair. The corresponding response is generated by a pairing field and anomalous propagators, and the pp-BSE kernel takes the form

$\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$

This is the direct particle-particle analogue of the familiar electron-hole BSE kernel. In that sense, a pairing-plus-particle-hole model can refer not only to a Hamiltonian with both sectors present, but also to a response framework that places pp and ph propagators on analogous formal footing (Marie et al., 2024).

2. Intertwined order in quasi-one-dimensional pair-density-wave theory

A particularly explicit use of the phrase occurs in a tractable quasi-one-dimensional model of PDW order. The physical setting is a set of weakly coupled chains with two partially filled, slightly warped Fermi sheets. Because the system is quasi-1D, the quasiparticle dispersion can be linearized around each Fermi surface, and the important low-energy processes are inter-sheet scattering processes connecting the two bands. The crucial nesting condition is

$\epsilon_a(k)=-\epsilon_b(k+Q),$

which produces singularities in both the particle-particle and particle-hole channels. The central claim is that the superconducting and charge-order channels are not separate or competing ingredients in the low-energy theory; instead, they emerge as two components of the same matrix order parameter,

$$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$

with the diagonal entries identified as CDW amplitudes and the off-diagonal entries as PDW amplitudes (Tsvelik, 2023).

The microscopic formulation uses right- and left-moving fermions on each chain and inter-chain couplings chosen so that the spin sector becomes gapped while the charge sector remains critical. In the Abelian bosonized description, the relevant $+$ sector operators are

$$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$

After gauge fixing, both operators share the same spin-sector dependence and differ only in the charge-sector phase, which makes their particle-hole relation explicit. For equal Fermi velocities, the spin sector is governed by a sine-Gordon Hamiltonian with a relevant cosine term for $g>0$, opening a spin gap $M_s$. Below that gap, the spin part acquires a nonzero expectation value, leaving an effective low-energy theory of coupled charge-like order parameters (Tsvelik, 2023).

Finite-momentum pairing is a direct consequence of pairing fermions from different Fermi sheets. The PDW and CDW wave vectors are

$$Q_{\rm pdw}=k_{F1}-k_{F2}, \qquad Q_{\rm cdw}=k_{F1}+k_{F2},$$

so the model naturally yields $Q_{\rm pdw}\neq Q_{\rm cdw}$ even though PDW and CDW are components of the same matrix field. The paper is explicit that this is not a Fulde-Ferrell-Larkin-Ovchinnikov state driven by Zeeman splitting; it arises from inter-sheet nesting and the structure of the quasi-1D bands. Depending on interchain couplings, the PDW can be unidirectional and may oscillate in sign between chains. For experimental signatures, ordinary STM is expected to mostly detect the CDW-induced modulation of the local density of states, whereas Josephson STM is identified as the more suitable probe because the local Josephson current oscillates with the PDW wave vector (Tsvelik, 2023).

3. Electron-hole realizations in layered Dirac materials and graphene

In layered Dirac materials, the pairing-plus-particle-hole idea is realized as an exactly solvable electron-hole problem. The model starts from low-energy Dirac-like band structures with a band gap $\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$0, electron-hole symmetry of reflection, and Coulomb attraction between an electron and a hole. Because center-of-mass and relative motion are not easily separable, the problem is formulated as an integral two-dimensional Schrödinger equation in momentum space,

$\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$1

and solved by projection of the momentum plane onto a three-dimensional sphere and expansion in spherical harmonics. The paper applies this construction to gapped graphene, monolayer MoS$\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$2, and bilayer graphene, reports quantized excitonic energies and analytic momentum-space wave functions, and states that electron-hole pairing leads to exciton-insulator states in single-layer graphene, monolayer MoS$\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$3, and bilayer graphene. For bilayer graphene it specifically reports an excitonic-insulator gap of about $\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$4 meV. The interpretive claim is that particle-hole symmetry of the Dirac equation produces identical electron and hole Fermi surfaces, enabling “perfect” pairing and driving a Cooper instability in the electron-hole sector (Lokot, 2014).

A graphene-specific superconducting variant replaces neutral excitons by a condensate built from conduction-band electrons and valence-band holes. In this relativistic ferromagnetic-superconductivity model, the exchange field $\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$5 can drive a crossover from ordinary BCS pairing to a Sarma regime and then to a p–n regime in which pairing occurs between electron-like carriers in the conduction band and hole-like carriers in the valence band. The mean-field order parameters may be on-site $\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$6-wave or nearest-neighbor $\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$7-wave, and the quasiparticle spectrum is

$\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$8

The crucial distinction is that the p–n state is unstable for $\Xi^{\pp}(44';33')=\left.\frac{\delta \Sigma^{\ee}(33')}{\delta G^{\ee}(44')}\right|_{U=0}.$9-wave symmetry, whereas the $\epsilon_a(k)=-\epsilon_b(k+Q),$0-wave p–n condensate is stable and can persist to much larger exchange fields, especially near the Dirac point. The same Hamiltonian is also stated to describe interlayer pairing in oppositely doped double-layer graphene, with exchange field and chemical-potential difference playing corresponding roles (Hosseini et al., 2012).

4. Nuclear many-body implementations

In nuclear theory, the pairing-plus-particle-hole model appears in a deliberately schematic but algebraically controlled form. One example is a single degenerate $\epsilon_a(k)=-\epsilon_b(k+Q),$1-shell with total degeneracy

$\epsilon_a(k)=-\epsilon_b(k+Q),$2

designed to isolate the competition between isovector and isoscalar pairing. The Hamiltonian is

$\epsilon_a(k)=-\epsilon_b(k+Q),$3

where $\epsilon_a(k)=-\epsilon_b(k+Q),$4 creates isovector pairs and $\epsilon_a(k)=-\epsilon_b(k+Q),$5 creates isoscalar pairs. The key mean-field statement is that the Hartree-Fock-Bogoliubov equations contain both the particle-hole field $\epsilon_a(k)=-\epsilon_b(k+Q),$6 and the pairing field $\epsilon_a(k)=-\epsilon_b(k+Q),$7,

$\epsilon_a(k)=-\epsilon_b(k+Q),$8

with $\epsilon_a(k)=-\epsilon_b(k+Q),$9 determined by the normal density $$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$0 and $$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$1 by the pairing tensor $$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$2. Neutron-proton mixing is introduced by isocranking in isospin space through a Routhian

$$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$3

The comparison to the exact $$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$4 solution shows that HFB reproduces the exact trends well and is closer to the exact solution than generalized BCS, specifically because BCS neglects the particle-hole mean field $$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$5 (Romero et al., 2017).

A second nuclear implementation concerns collective particle-hole excitations across the normal-to-superfluid transition. The benchmark is the original Agassi Hamiltonian, described as a superposition of the Lipkin model and the two-level pairing model, with pairing and monopole particle-hole terms. The formal innovation is a number-conserving particle-hole RPA built on a particle-number-projected HFB state, equivalently an antisymmetric geminal power state,

$$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$6

The NCphRPA phonon operator is constructed so that its backward term annihilates the PHFB reference state, and the resulting RPA eigenvalue problem is built from double commutators evaluated in that projected reference. In the Agassi benchmark at half filling and $$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$7, the excitation energy evolves smoothly across the normal-to-superfluid transition at $$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$8, in contrast to conventional phRPA and QRPA, which require a basis change at the broken-symmetry point. The formal point is that ph excitations can be described on the same footing on both sides of the phase boundary by keeping the same PHFB reference state (Dukelsky et al., 2019).

5. Response theory, density functionals, and particle-particle Bethe-Salpeter equations

The most systematic formal development of the pairing channel as a counterpart to the particle-hole channel occurs in the pairing-matrix formulation of exchange-correlation energy. Starting from the adiabatic connection Hamiltonian and the Hellmann-Feynman theorem, the correlation energy is written in terms of pairing transition amplitudes and the particle-particle Green function. At the pp-RPA level, the central Dyson-like equation is

$$\hat{\Delta}= \begin{pmatrix} \Delta_{\rm cdw} & \Delta_{\rm pdw}\ \Delta_{\rm pdw}^* & \Delta_{\rm cdw}^* \end{pmatrix},$$9

and the corresponding correlation energy is

$+$0

This is identified with the sum of all ladder diagrams, whereas the familiar ph-RPA expression sums ring diagrams. The same paper emphasizes several properties that are presented as distinctive for pp-RPA: nearly linear energy behavior for fractional charges, nearly constant behavior for fractional spin, correct derivative discontinuity at integer $+$1, good van der Waals descriptions for dimers such as Ar$+$2 and NeAr, and improved dissociation behavior for systems such as $+$3, $+$4, $+$5, $+$6, and $+$7 (Aggelen et al., 2013).

The particle-particle Bethe-Salpeter equation extends this channel decomposition from density functionals to a linear-response Green-function framework based on anomalous propagators. The starting point is a pairing perturbation that breaks particle-number conservation and generates nonzero $+$8 and $+$9. The pp propagator is defined as the response of $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$0 to the pairing field $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$1, and the final equation takes the form

$$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$2

with the kernel

$$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$3

This formulation supports kernels derived from first-order Coulomb interaction, second-Born, $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$4, and $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$5-matrix self-energies. The poles of $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$6 correspond to double ionization potentials and double electron affinities rather than neutral excitations. For a set of 23 small molecules, the reported qualitative hierarchy is that pp-RPA@HF is the least accurate, GF(2) starting points improve the DIPs, the static $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$7 kernel improves further, a perturbative dynamical correction on top of the TDA/static $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$8 kernel brings results very close to DIP-EOM-CCSD, and the pp $$\Delta^{(+)}_{\rm pdw} =R_{1\uparrow}L_{2\downarrow}-R_{1\downarrow}L_{2\uparrow}, \qquad \Delta^{(+)}_{\rm cdw} =R_{1\sigma}L_{2\sigma}.$$9-matrix kernel has the smallest MAE among the kernels tested. For double core-hole states, ppBSE@$g>0$0 improves over pp-RPA, but state-specific $g>0$1SCF/MOM remains preferable when feasible because orbital relaxation is important (Marie et al., 2024).

6. Symmetry breaking, duality, and unconventional variants

In the Dirac composite-fermion description of the half-filled Landau level, pairing-plus-particle-hole structure appears as a competition between BCS pairing and explicit particle-hole symmetry breaking. The massive DCF Lagrangian,

$g>0$2

uses the Dirac mass $g>0$3 as a proxy for Landau-level mixing and as an explicit PH-breaking term. The paper states that a PH Pfaffian-like topological state is impossible in the strictly massless PH-symmetric case, where the PH Pfaffian is critical rather than gapped. For small $g>0$4, the sign of $g>0$5 selects anti-Pfaffian-like or Pfaffian-like instability; for sufficiently large $g>0$6, the instability becomes PH Pfaffian-like irrespective of sign. A common misconception is therefore explicitly rejected: arbitrary weak PH breaking does not generically stabilize a PH Pfaffian; the strong-mixing regime is the one associated with PH Pfaffian-like pairing (Antonic et al., 2018).

An exact solvable realization of particle-hole structure appears in the $g>0$7 Hamiltonian revisited in the hole-pair representation. Under the particle-hole transformation $g>0$8, the attractive and repulsive systems are related, and a second Bethe-Ansatz solution is written directly in terms of hole pairs built on the filled vacuum. The central conclusion is a genuine asymmetry between particle-pair and hole-pair descriptions. In the attractive model’s strong pairing regime, states in the hole-pair picture take the form of a quasi-condensate involving two distinct hole-pair creation operators, one associated with zero-energy hole-pairs and one with infinite-energy hole-pairs. The paper states that no analogous feature is observed in the particle-pair picture (Links et al., 2015).

Other unconventional versions preserve the same organizing principle while changing the microscopic mechanism. In a two-band Hubbard model with overlapping bands, one- and two-particle on-site hybridization between bands produce an effective attraction in the $g>0$9-band, and the attraction is strongest when $M_s$0, favoring $M_s$1-pairing with total momentum $M_s$2; the one-particle hybridization $M_s$3 only shifts energies, whereas the two-particle hybridization $M_s$4 drives the pairing mechanism (Karnaukhov, 6 Jan 2026). In a gauge approach to the $M_s$5-$M_s$6 model for underdoped cuprates, the physical hole is a gauge-bound composite of a spinless holon and a bosonic spinon, opposite-chirality spin vortices on opposite Néel sublattices generate holon pairing, gauge interaction induces spin-singlet RVB spinon pairing, and superconductivity emerges only after coherent recombination of these ingredients. The paper explicitly describes this mechanism as non-BCS-like because the energy gain is associated with lowering the spinon gap and because the superconducting transition is “almost” of the classical 3D XY-type (Marchetti et al., 2011).

Taken together, these constructions show that the pairing-plus-particle-hole model is not a single canonical scheme. It is a family of formulations in which pairing is organized together with density order, electron-hole symmetry, collective ph excitations, anomalous response functions, or exact particle-hole dualities. The unifying content is the refusal to treat pairing as an isolated weak-coupling instability on a fixed Fermi surface; instead, the low-energy theory is built so that the particle-particle and particle-hole sectors constrain one another from the outset.