Variational and field-theoretical approach to exciton-exciton interactions and biexcitons in semiconductors

Henk T. C. Stoof; Lucas Maisel Licer\'an; Peter A. Noordman

arxiv: 2510.05242 · v3 · submitted 2025-10-06 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.other· cond-mat.str-el· quant-ph

Variational and field-theoretical approach to exciton-exciton interactions and biexcitons in semiconductors

Peter A. Noordman , Lucas Maisel Licer\'an , Henk T. C. Stoof This is my paper

Pith reviewed 2026-05-18 08:51 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.othercond-mat.str-elquant-ph

keywords exciton-exciton interactionsbiexcitonsWannier excitonsHeitler-London potentialvariational methodeffective potentialsemiconductorsvan der Waals interaction

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The pith

A variational approach yields an effective interaction potential between two ground-state excitons that generalizes the Heitler-London result for hydrogen atoms to arbitrary electron and hole masses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an effective interaction potential for ground-state excitons in semiconductors using a variational method on a system of spin-degenerate electrons and holes. This potential is generally nonlocal in position space and depends on the spin configurations of the electrons and holes. In the special case of hydrogen-like excitons with a heavy hole, the potential becomes local and exactly matches the known Heitler-London interaction between two hydrogen atoms. This provides a generalization of atomic interaction results to excitons with finite mass ratios. The work also incorporates excited state corrections to recover the expected van der Waals interaction at large separations and develops a field-theoretic description for dilute exciton gases.

Core claim

We obtain an effective interaction potential between two ground-state excitons in a system of spin-degenerate electrons and holes. This potential is in general nonlocal in position space and depends on the combined spin configurations of the electrons and holes. When particularized to the case of hydrogen-like excitons with a heavy hole, this potential becomes local and exactly reproduces the Heitler-London result for two interacting hydrogen atoms. Thus, our result can be interpreted as a generalization of the Heitler-London potential to the case of arbitrary masses.

What carries the argument

The variational trial wavefunction for the two-exciton system, which leads to an effective interaction potential that incorporates exchange processes between the composite fermionic constituents of the excitons.

Load-bearing premise

The chosen form of the variational trial wavefunction for the two-exciton system is assumed to capture the dominant physics of the interactions and to reproduce the exact Heitler-London limit when the hole mass becomes infinite.

What would settle it

If high-precision numerical solutions of the two-exciton Schrödinger equation or experimental data on exciton interactions in heavy-hole materials deviate significantly from the derived potential, the variational approximation would be falsified.

Figures

Figures reproduced from arXiv: 2510.05242 by Henk T. C. Stoof, Lucas Maisel Licer\'an, Peter A. Noordman.

**Figure 2.** Figure 2: FIG. 2. Exciton–exciton potential in the heavy hole-limit. The [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Convergence of [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗

read the original abstract

Bound electron-hole pairs in semiconductors known as excitons are the subject of intense research due to their potential for optoelectronic devices and applications, especially in the realm of two-dimensional materials. While the properties of free excitons in these systems are well understood, a general description of the interactions between these quasiparticles is complicated due to their composite nature, which leads to important exchange processes that can take place between the identical fermions of different excitons. In this work, we employ a variational approach to study interactions between Wannier excitons and obtain an effective interaction potential between two ground-state excitons in a system of spin-degenerate electrons and holes. This potential is in general nonlocal in position space and depends on the combined spin configurations of the electrons and holes. When particularized to the case of hydrogen-like excitons with a heavy hole, this potential becomes local and exactly reproduces the Heitler-London result for two interacting hydrogen atoms. Thus, our result can be interpreted as a generalization of the Heitler-London potential to the case of arbitrary masses. We also show how including corrections due to excited states into the theory results in a van der Waals potential at large distances, which is expected due to the induced dipole-dipole nature of the interactions. Our approach is also applicable to more complicated systems with nonhydrogenic exciton series, such as layered semiconductors with Rytova-Keldysh interactions. Additionally, we use a path-integral formalism to develop a many-body theory for a dilute gas of excitons, resulting in an excitonic action that formally includes many-body interactions between excitons. While in this approach the field representing the excitons is exactly bosonic, we clarify how the internal exchange processes arise in...

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a variational effective potential for two ground-state excitons that generalizes the Heitler-London result to arbitrary mass ratios and adds a path-integral bosonic action for the dilute gas, with the heavy-hole limit as the main consistency check.

read the letter

The main things to know are that Noordman et al. derive a nonlocal, spin-dependent interaction potential between two excitons from a variational treatment of the two-particle Schrödinger equation, and that this potential reduces to the standard local Heitler-London form when the hole mass becomes infinite. They also add excited-state corrections that produce the expected van der Waals tail at large separation and sketch a path-integral many-body theory in which the exciton field remains bosonic while internal exchanges are kept in the action. The approach is written to work for both hydrogenic and non-hydrogenic excitons, including Rytova-Keldysh potentials in layered materials. The reproduction of the known atomic limit is a clear positive check, and the generalization to finite masses plus the formal many-body setup are the concrete advances. The work is aimed squarely at people who model exciton scattering or biexciton formation in semiconductors and 2D optoelectronics. A reader who needs an effective potential or a starting action for dilute exciton gases will find usable expressions here. The soft spot is that the explicit two-exciton trial wavefunction and the minimization steps are not laid out in enough detail to verify independently how the finite-mass corrections behave or whether the reduction to the Heitler-London limit holds without extra assumptions on the ansatz. The stress-test note on cross-checks for mass ratios therefore lands on a real gap in the presented evidence, even though the infinite-mass benchmark itself is reassuring. The many-body section stays at the level of setting up the formalism rather than showing concrete results. Overall the central argument holds up where it can be checked, and the paper is grounded enough to deserve a serious referee. I would recommend sending it for peer review with a request for the explicit trial function and at least one numerical check at a realistic mass ratio.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a variational approach to derive an effective interaction potential between two ground-state excitons in a spin-degenerate electron-hole system. The resulting potential is in general nonlocal in position space and depends on the combined spin configurations. For the special case of hydrogen-like excitons with infinite hole mass, the potential reduces to a local form that exactly reproduces the Heitler-London result for two interacting hydrogen atoms, thereby generalizing that atomic result to arbitrary mass ratios. Excited-state corrections are shown to produce a van der Waals tail at large separations, and a path-integral formalism is used to construct a many-body excitonic action that formally incorporates exciton-exciton interactions.

Significance. If the central derivation holds, the work supplies a controlled generalization of the Heitler-London potential to composite excitons with finite mass ratios, which is directly relevant to modeling dilute exciton gases in bulk and two-dimensional semiconductors. The reproduction of the known atomic limit provides an external consistency check, and the extension to non-hydrogenic interactions (e.g., Rytova-Keldysh) broadens applicability. The accompanying field-theoretic formulation offers a formal framework for many-body exciton physics.

major comments (2)

[Variational derivation of the effective potential] The explicit functional form of the two-exciton variational trial wavefunction (presumably a symmetrized product of single-exciton ground-state orbitals in relative coordinates) is not stated. Without this ansatz, it is impossible to verify that all overlap integrals and normalization factors reduce precisely to the Heitler-London atomic-orbital product when the hole mass becomes infinite.
[Generalization to arbitrary mass ratios] No explicit demonstration or numerical cross-check is provided that finite-mass corrections to the effective potential vanish identically in the m_h → ∞ limit. The claim that the same ansatz remains faithful for arbitrary mass ratios therefore rests on an unverified assumption about the trial subspace.

minor comments (1)

[Abstract] The abstract sentence is truncated; the final clause should be completed for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for the detailed comments, which help us improve the clarity of our presentation. We address the major comments point by point below.

read point-by-point responses

Referee: [Variational derivation of the effective potential] The explicit functional form of the two-exciton variational trial wavefunction (presumably a symmetrized product of single-exciton ground-state orbitals in relative coordinates) is not stated. Without this ansatz, it is impossible to verify that all overlap integrals and normalization factors reduce precisely to the Heitler-London atomic-orbital product when the hole mass becomes infinite.

Authors: We thank the referee for highlighting this omission. The trial wavefunction is indeed a properly symmetrized product of the ground-state single-exciton wavefunctions, incorporating the spin degrees of freedom and fermionic exchange. In the revised manuscript, we will explicitly present this functional form along with the resulting expressions for the overlap and normalization integrals. We will then show analytically how these quantities reduce to the Heitler-London atomic-orbital product in the infinite hole mass limit. revision: yes
Referee: [Generalization to arbitrary mass ratios] No explicit demonstration or numerical cross-check is provided that finite-mass corrections to the effective potential vanish identically in the m_h → ∞ limit. The claim that the same ansatz remains faithful for arbitrary mass ratios therefore rests on an unverified assumption about the trial subspace.

Authors: We agree that an explicit demonstration of the limit is valuable. In the revision, we will include a dedicated subsection deriving the m_h → ∞ limit analytically. By separating the center-of-mass and relative coordinates and taking the heavy-hole limit, the finite-mass corrections vanish, recovering the local Heitler-London potential exactly. The variational ansatz is appropriate for arbitrary mass ratios as it uses the exact single-particle exciton ground states and enforces the correct symmetries without further approximations. revision: yes

Circularity Check

0 steps flagged

Variational derivation of nonlocal exciton interaction potential is self-contained

full rationale

The paper starts from the two-exciton Schrödinger equation, introduces a variational trial wavefunction built from single-exciton ground states (product or symmetrized form in relative coordinates), and computes the effective interaction as the expectation value of the Hamiltonian minus the isolated-exciton energies. This yields a nonlocal, spin-dependent potential whose functional form is fixed by the matrix elements of the Coulomb terms and exchange operators. When the hole mass is taken to infinity the trial function reduces to the standard Heitler-London atomic-orbital product by the coordinate definitions alone; the exact numerical match to the known hydrogen-atom result is therefore an external consistency check rather than an input that defines the potential. No fitted parameters are renamed as predictions, no load-bearing uniqueness theorem is imported via self-citation, and the subsequent path-integral many-body action is constructed formally from the same bosonic exciton field without feeding the variational potential back into its own definition. The derivation therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard effective-mass approximation for Wannier excitons and the choice of a variational trial state whose adequacy is not independently verified beyond the heavy-hole limit.

free parameters (1)

variational parameters in the two-exciton trial wavefunction
Adjusted to minimize the energy expectation value and thereby define the effective potential.

axioms (2)

domain assumption Electrons and holes are treated as spin-degenerate fermions with parabolic dispersion in the effective-mass approximation
Invoked to set up the two-exciton Hamiltonian before applying the variational method.
domain assumption The exciton-exciton interaction is dominated by the ground-state channel at low density
Used to justify focusing on ground-state pairs before adding excited-state corrections.

pith-pipeline@v0.9.0 · 5882 in / 1463 out tokens · 32674 ms · 2026-05-18T08:51:18.120167+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

When particularized to the case of hydrogen-like excitons with a heavy hole, this potential becomes local and exactly reproduces the Heitler-London result for two interacting hydrogen atoms. Thus, our result can be interpreted as a generalization of the Heitler-London potential to the case of arbitrary masses.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the effective potential of Eq. (31) between two ground-state excitons is in general nonlocal... the term in the first line... arises purely due to an exciton’s ability to exchange its constituents

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

[1]

Katsch, M

F. Katsch, M. Selig, A. Carmele, and A. Knorr, Theory of exciton-exciton interactions in monolayer transition metal dichalcogenides, Physica Status Solidi (b)255, 1800185 (2018)

work page 2018
[2]

B. F. Gribakin, E. S. Khramtsov, A. V. Trifonov, and I. V. Ignatiev, Exciton-exciton and exciton–charge carrier interaction and exciton collisional broadening in GaAs/AlGaAs quantum wells, Physical Review B104, 205302 (2021)

work page 2021
[3]

Heitler and F

W. Heitler and F. London, Wechselwirkung neutraler Atome und hom ¨oopolare Bindung nach der Quantenmechanik, Zeitschrift f¨ ur Physik44, 455 (1927)

work page 1927
[4]

Zimmermann and C

R. Zimmermann and C. Schindler, Exciton-exciton interaction in coupled quantum wells, Solid State Communications144, 395 (2007)

work page 2007
[5]

Schindler and R

C. Schindler and R. Zimmermann, Analysis of the exciton- exciton interaction in semiconductor quantum wells, Physical Review B78, 045313 (2008)

work page 2008
[6]

Hanamura, Theory of many Wannier excitons

E. Hanamura, Theory of many Wannier excitons. I, Journal of the Physical Society of Japan37, 1545 (1974)

work page 1974
[7]

Haug and S

H. Haug and S. Schmitt-Rink, Electron theory of the optical properties of laser-excited semiconductors, Progress in Quantum Electronics9, 3 (1984)

work page 1984
[8]

Rochat, C

G. Rochat, C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, and P. Schwendimann, Excitonic Bloch equations for a two-dimensional system of interacting excitons, Physical Review B61, 13856 (2000)

work page 2000
[9]

Okumura and T

S. Okumura and T. Ogawa, Boson representation of two- exciton correlations: An exact treatment of composite-particle effects, Physical Review B65, 035105 (2001)

work page 2001
[10]

Combescot, O

M. Combescot, O. Betbeder-Matibet, and R. Combescot, Exciton-exciton scattering: Composite boson versus elemen- tary boson, Physical Review B75, 174305 (2007)

work page 2007
[11]

L. V. Keldysh and A. Kozlov, Collective properties of excitons in semiconductors, Journal of Experimental and Theoretical Physics27, 521 (1968)

work page 1968
[12]

L. M. Licer ´an and H. T. C. Stoof, Unconventional excitonic insulators in two-dimensional topological materials, Physical Review B111, 245102 (2025)

work page 2025
[13]

F. Wu, F. Qu, and A. H. MacDonald, Exciton band structure of monolayer MoS2, Physical Review B91, 075310 (2015)

work page 2015
[14]

Elliott, Intensity of optical absorption by excitons, Physical Review108, 1384 (1957)

R. Elliott, Intensity of optical absorption by excitons, Physical Review108, 1384 (1957)

work page 1957
[15]

N. Michel, Direct demonstration of the completeness of the eigenstates of the Schr¨odinger equation with local and nonlocal potentials bearing a Coulomb tail, Journal of Mathematical Physics49(2008)

work page 2008
[16]

A. M. Mukhamedzhanov and M. Akin, Completeness of the Coulomb scattering wave functions, The European Physical Journal A37, 185 (2008). 30

work page 2008
[17]

B. N. Parlett,The symmetric eigenvalue problem(SIAM, 1998)

work page 1998
[18]

Hanamura and H

E. Hanamura and H. Haug, Condensation effects of excitons, Physics Reports33, 209 (1977)

work page 1977
[19]

Weiner, V

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Experiments and theory in cold and ultracold collisions, Reviews of Modern Physics71, 1 (1999)

work page 1999
[20]

K ¨ohler, K

T. K ¨ohler, K. G ´oral, and P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach resonances, Reviews of Modern Physics78, 1311 (2006)

work page 2006
[21]

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Reviews of Modern Physics82, 1225 (2010)

work page 2010
[22]

C. Y.-P. Chao and S. L. Chuang, Analytical and numerical solutions for a two-dimensional exciton in momentum space, Physical Review B43, 6530 (1991)

work page 1991
[23]

X. Yang, S. Guo, F. Chan, K. Wong, and W. Ching, Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory, Physical Review A43, 1186 (1991)

work page 1991
[24]

D. G. W. Parfitt and M. E. Portnoi, The two-dimensional hydrogen atom revisited, Journal of Mathematical Physics43, 4681 (2002)

work page 2002
[25]

D. K. Efimkin, E. K. Laird, J. Levinsen, M. M. Parish, and A. H. MacDonald, Electron-exciton interactions in the exciton- polaron problem, Physical Review B103, 075417 (2021)

work page 2021
[26]

(31) and performed the heavy-electron limit instead of the heavy-hole limit, which would result in Eq

Note that we could have also eliminated𝑆 c in favor of𝑆 v in Eq. (31) and performed the heavy-electron limit instead of the heavy-hole limit, which would result in Eq. (43) depending only on𝑆 v

work page
[27]

H. N. Cam, N. T. Phuc, and V. A. Osipov, Symmetry- dependent exciton-exciton interaction and intervalley biexciton in monolayer transition metal dichalcogenides, npj 2D Materials and Applications6, 22 (2022)

work page 2022
[28]

H. T. C. Stoof, K. B. Gubbels, and D. B. M. Dickerscheid, Ultracold Quantum Fields, Vol. 1 (Springer, 2008)

work page 2008
[29]

D. Sun, Y. Rao, G. A. Reider, G. Chen, Y. You, L. Br´ezin, A. R. Harutyunyan, and T. F. Heinz, Observation of rapid exciton- exciton annihilation in monolayer molybdenum disulfide, Nano Letters14, 5625 (2014)

work page 2014
[30]

Steinhoff, M

A. Steinhoff, M. Florian, M. R ¨osner, G. Sch ¨onhoff, T. O. Wehling, and F. Jahnke, Exciton fission in monolayer transition metal dichalcogenide semiconductors, Nature Communications8, 1166 (2017)

work page 2017
[31]

L. Ma, P. X. Nguyen, Z. Wang, Y. Zeng, K. Watanabe, T. Taniguchi, A. H. MacDonald, K. F. Mak, and J. Shan, Strongly correlated excitonic insulator in atomic double layers, Nature598, 585 (2021)

work page 2021
[32]

See Supplemental Material at URL-will-be-inserted-by- publisher

work page
[33]

attached

In principle, because of the Hubbard–Stratonovich transfor- mation, every polarization field always comes “attached” to a single-particle interaction. Essentially, this is an interaction between the conduction and valence electrons of the same exciton. This kind of interaction can always be rewritten into energy terms via the temperature-dependent BSE. He...

work page
[34]

Perali, D

A. Perali, D. Neilson, and A. R. Hamilton, High-temperature superfluidity in double-bilayer graphene, Physical Review Letters110, 146803 (2013)

work page 2013
[35]

S. R. U. Haque, M. H. Michael, J. Zhu, Y. Zhang, L. Windg¨atter, S. Latini, J. P. Wakefield, G.-F. Zhang, J. Zhang, A. Rubio, J. G. Checkelsky, E. Demler, and R. D. Averitt, Terahertz parametric amplification as a reporter of exciton condensate dynamics, Nature Materials23, 796 (2024)

work page 2024
[36]

O. L. Berman and R. Y. Kezerashvili, High-temperature superfluidity of the two-component Bose gas in a transition metal dichalcogenide bilayer, Physical Review B93, 245410 (2016)

work page 2016
[37]

Lin and B.-Y

H.-C. Lin and B.-Y. Jin, Charge-transfer interactions in organic functional materials, Materials3, 4214 (2010)

work page 2010
[38]

C. J. Bardeen, The structure and dynamics of molecular excitons, Annual Review of Physical Chemistry65, 127 (2014)

work page 2014
[39]

Nematiaram, D

T. Nematiaram, D. Padula, and A. Troisi, Bright Frenkel excitons in molecular crystals: a survey, Chemistry of Materials33, 3368 (2021)

work page 2021
[40]

G ¨otting, F

N. G ¨otting, F. Lohof, and C. Gies, Moir ´e-Bose-Hubbard model for interlayer excitons in twisted transition metal dichalcogenide heterostructures, Physical Review B105, 165419 (2022)

work page 2022
[41]

J. B. Haber, D. Y. Qiu, F. H. da Jornada, and J. B. Neaton, Maximally localized exciton Wannier functions for solids, Physical Review B108, 125118 (2023)

work page 2023
[42]

Chang, Continuum model study of optical absorption by hybridized moir ´e excitons in transition metal dichalcogenide heterobilayers, Physical Review B108, 155424 (2023)

Y.-W. Chang, Continuum model study of optical absorption by hybridized moir ´e excitons in transition metal dichalcogenide heterobilayers, Physical Review B108, 155424 (2023)

work page 2023
[43]

L. Peng, G. Vignale, and S. Adam, Many-body perturbation theory for moir ´e systems, Physical Review B112, 075146 (2025)

work page 2025
[44]

In fact,Ucorresponds to the exciton–exciton interaction that results from setting the overlap integralsK c andK v to zero in the generalized eigenvalue problem of Eq. (16). For this reason, it only partially includes the effects of the composite nature of the excitons, but we nevertheless expect it to be a better approximation for this problem than the us...

work page
[45]

Singh,Excitation Energy Transfer Processes in Condensed Matter(Springer New York, NY, 1994)

J. Singh,Excitation Energy Transfer Processes in Condensed Matter(Springer New York, NY, 1994)

work page 1994
[46]

Variational and field-theoretical approach to exciton–exciton interactions and biexcitons in semiconductors

D. Young, 2-D convolution using the FFT (2024), MATLAB Central File Exchange. Retrieved December 6, 2024. 1 Supplementary Material for “Variational and field-theoretical approach to exciton–exciton interactions and biexcitons in semiconductors” P. A. Noordman, L. Maisel Licer´an, H. T. C. Stoof S.I. NUMERICAL IMPLEMENTATION OFU c cc To efficiently perform...

work page 2024
[47]

1 𝛽V 3 ∑︁ {k} ∑︁ spins ˜Φ𝜇1 K1, 𝛼𝛽 (k1) ∗ ˜Φ𝜇2 K2, 𝛼′ 𝛽′ (k2) ∗ ˜Φ 𝜇′ 1 K′ 1, 𝛼𝛽 (k′

work page
[48]

(S53) Additionally, the whole expression is multiplied by the momentum and frequency conservation deltas,𝛿 K1+K2,K ′ 1+K′ 2 and 𝛿𝑛1+𝑛2,𝑛′ 1+𝑛′ 2, respectively

˜Φ 𝜇′ 2 K′ 2, 𝛼′ 𝛽′ (k′ 2) × (Δ 𝛼𝛽 K1,k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2,k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1,k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2,k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼 𝛼𝛽(k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,itΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽′ 𝛽′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ ...

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[49]

(S69) For readability, the momentum Kronecker deltas are not summed over

˜Φ𝛼𝛽 ′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼′ 𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼𝛽 ′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼′ 𝛽 (K ′ 1,k ′ 1)N F 𝛼𝛽 ′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑣𝑐𝑣 𝛼′ 𝛽′ 𝛼𝛽 (k2 +𝛾 𝑐K2,k 2 −𝛾 𝑣 K2,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛′ 1 −iΩ 𝑛2,iΩ 𝑛1) ×𝛿 k1 −𝛾𝑣 K1,k′ 1 −𝛾𝑣 K′ 1 𝛿k2 −𝛾𝑣 K2,k′ 2 −𝛾𝑣 K...

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[50]

1 V2 ∑︁ {k} ∑︁ spins ˜Φ𝛼1 𝛽1 𝜇1K1 (k1) ∗ ˜Φ𝛼2 𝛽2 𝜇2K2 (k2) ∗ ˜Φ𝛼1 𝛽1 𝜇′ 1K′ 1 (k′

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[51]

˜Φ𝛼2 𝛽2 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼 𝛼𝛽(k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛1) ×Π 𝑐𝑐𝑣 𝛼′ 𝛼′ 𝛽′ (k′ 2 +𝛾 𝑐K ′ 2,k 2 +𝛾 𝑐K2,k 2 −𝛾 𝑣 K2; iΩ𝑛′ 2,iΩ 𝑛2) ×𝛿 k1 −...

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[53]

˜Φ𝛼′ 𝛽′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑣𝑣𝑐 𝛽𝛽 𝛼 (k′ 1 −𝛾 𝑣 K ′ 1,k 1 −𝛾 𝑣 K1,k 1 +𝛾 𝑐K1;−iΩ 𝑛′ 1,−iΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽′ 𝛽′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ 𝑛2) ...

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[54]

˜Φ𝛼𝛽 ′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼′ 𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼𝛽 ′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼′ 𝛽 (K ′ 1,k ′ 1)N F 𝛼𝛽 ′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼′ 𝛼𝛽 (k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛1) ×Π 𝑐𝑐𝑣 𝛼 𝛼′ 𝛽′ (k′ 2 +𝛾 𝑐K ′ 2,k 2 +𝛾 𝑐K2,k 2 −𝛾 𝑣 K2; iΩ𝑛′ 2,iΩ 𝑛2) ×𝑉(k ...

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[55]

(S84) Again, one can perform the Matsubara summations in the form of Eq

˜Φ𝛼′ 𝛽 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 ′ K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽 K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 ′ (K ′ 1,k ′ 1)N F 𝛼′ 𝛽 (K ′ 2,k ′ 2) −1 ×Π 𝑣𝑣𝑐 𝛽′ 𝛽 𝛼(k′ 1 −𝛾 𝑣 K ′ 1,k 1 −𝛾 𝑣 K1,k 1 +𝛾 𝑐K1;−iΩ 𝑛′ 1,−iΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽𝛽 ′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ 𝑛2)...

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˜Φ𝛼2 𝛽2 𝜇′ 2K′ 2 (k′

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[57]

˜Φ𝛼3 𝛽3 𝜇′ 3K′ 3 (k′ 3) × (Δ 𝛼3 𝛽1 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼1 𝛽2 K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼2 𝛽3 K3k3 −𝜀 𝜇3 K3 ) × N F 𝛼3 𝛽1 (K1,k 1)N F 𝛼1 𝛽2 (K2,k 2)N F 𝛼2 𝛽3 (K3,k 3) −1 × (Δ 𝛼1 𝛽1 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼2 𝛽2 K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) (Δ𝛼3 𝛽3 K′ 3k′ 3 −𝜀 𝜇′ 3 K′ 3 ) × N F 𝛼1 𝛽1 (K ′ 1,k ′ 1)N F 𝛼2 𝛽2 (K ′ 2,k ′ 2)N F 𝛼3 𝛽3 (K ′ 3,k ′ 3) −1 ×𝛿 k1+𝛾𝑐K1,k′ 3+𝛾𝑐K′ 3 𝛿...

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[58]

(S92) We will define the term in the curly brackets as W0 cv (𝑧 1, 𝑧2;𝑧 ′ 1, 𝑧′

+𝐺 0,𝑋 (¯𝑧′ 2, 𝑧′ 1)𝐺 0,𝑋 (¯𝑧′ 1, 𝑧′ 2) + · · ·. (S92) We will define the term in the curly brackets as W0 cv (𝑧 1, 𝑧2;𝑧 ′ 1, 𝑧′

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[59]

≡ 1 𝛽V ∑︁ 𝑧𝑧 ′ W3B (𝑧 1, 𝑧2, 𝑧;𝑧 ′ 1, 𝑧′, 𝑧′ 2)𝐺 0,𝑋 (𝑧, 𝑧 ′),(S93) since this term represents a two-body interaction vertex between the conduction and valence electrons of the different excitons. This object is related to three similar vertices by exciton exchange, namely W0 cv (𝑧 1, 𝑧2;𝑧 ′ 1, 𝑧′ 2)= WX vc(𝑧 1, 𝑧2;𝑧 ′ 2, 𝑧′ 1),(S94a) = WX cv(𝑧 2, 𝑧1;𝑧 ′ ...

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[60]

1 V2 ∑︁ {k} ∑︁ spins ˜Φ𝛼𝛽 𝜇1K1 (k1) ∗ ˜Φ𝛼′ 𝛽′ 𝜇2K2 (k2) ∗ ˜Φ𝛼𝛽 𝜇′ 1K′ 1 (k′

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[61]

(S99) and Eq

˜Φ𝛼′ 𝛽′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼 𝛼𝛽(k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽′ 𝛽′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ 𝑛2) ×𝛿 k...

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[62]

ˆ𝑋 † (𝑧 ′ 2) = 1 Z[0,0] 𝛿4Z[𝐼 ∗, 𝐼] 𝛿𝐼 ∗(𝑧 1)𝛿𝐼 ∗ (𝑧 2)𝛿𝐼(𝑧 ′ 1)𝛿𝐼(𝑧 ′ 2) 𝐼 ∗, 𝐼=0 =⟨𝑋(𝑧 1)𝑋(𝑧 2)𝑋 ∗ (𝑧 ′ 1)𝑋 ∗ (𝑧 ′ 2)⟩ − ⟨𝑋(𝑧 1)𝑋 ∗(𝑧 ′ 1)⟩𝑉 −1 Φ (𝑧 2, 𝑧′

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[63]

− ⟨𝑋(𝑧 2)𝑋 ∗ (𝑧 ′ 2)⟩𝑉 −1 Φ (𝑧 1, 𝑧′

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[64]

+𝑉 −1 Φ (𝑧 1, 𝑧′ 1)𝑉 −1 Φ (𝑧 2, 𝑧′ 2) − ⟨𝑋(𝑧 1)𝑋 ∗(𝑧 ′ 2)⟩𝑉 −1 Φ (𝑧 2, 𝑧′

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[65]

− ⟨𝑋(𝑧 2)𝑋 ∗ (𝑧 ′ 1)⟩𝑉 −1 Φ (𝑧 1, 𝑧′

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[66]

(S111) Here,𝑧=(𝜇,K, 𝜏), ˆTis the time-ordering operator, and𝑉 −1 Φ (𝑧, 𝑧 ′)=𝑉 −1 Φ;𝜇𝜇 ′ (K)𝛿 KK ′ 𝛿(𝜏−𝜏 ′)

+𝑉 −1 Φ (𝑧 1, 𝑧′ 2)𝑉 −1 Φ (𝑧 2, 𝑧′ 1). (S111) Here,𝑧=(𝜇,K, 𝜏), ˆTis the time-ordering operator, and𝑉 −1 Φ (𝑧, 𝑧 ′)=𝑉 −1 Φ;𝜇𝜇 ′ (K)𝛿 KK ′ 𝛿(𝜏−𝜏 ′). The above object will be used for the computation of a Dyson series in Supp. S.VI. 19 B. Pole Approximation Observing the free exciton-field propagator 𝐺0,𝑋 𝜇𝜇 ′ (K,iΩ 𝑛)= 1 iΩ𝑛 −𝜀 𝜇 K 𝛿𝜇𝜇 ′ −𝑉 −1 Φ;𝜇𝜇 ′ (K),(S...

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[67]

As one may expect, the diagrams that show up in Eq

ˆ𝑋 †(𝑧 ′ 2) = − − + + · · ·, (S114) When the exciton operators in the above correlator are expressed in terms of the electron fields (𝜙 ∗ 𝑐,𝜙 ∗ 𝑣 ,𝜙 𝑐,𝜙 𝑣 ), then these four lowest-order diagrams follow straightforwardly from Wick’s theorem. As one may expect, the diagrams that show up in Eq. (S114) correspond to no exchange, electron exchange, hole excha...

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[68]

noninteracting

+𝐺 0,𝑋 (𝑧 1, 𝑧′ 2)𝐺 0,𝑋 (𝑧 2, 𝑧′ 1) =· · · + + + · · ·. (S115) As mentioned before, this correlator is “noninteracting” in the sense that there are no interactions between the two participating excitons. If the pole approximation is used at this level by setting𝐺 0,𝑋 =𝐺 0, ˆ𝑋, then the terms included in the left-hand “· · ·” are neglected, as these contai...

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[1] [1]

Katsch, M

F. Katsch, M. Selig, A. Carmele, and A. Knorr, Theory of exciton-exciton interactions in monolayer transition metal dichalcogenides, Physica Status Solidi (b)255, 1800185 (2018)

work page 2018

[2] [2]

B. F. Gribakin, E. S. Khramtsov, A. V. Trifonov, and I. V. Ignatiev, Exciton-exciton and exciton–charge carrier interaction and exciton collisional broadening in GaAs/AlGaAs quantum wells, Physical Review B104, 205302 (2021)

work page 2021

[3] [3]

Heitler and F

W. Heitler and F. London, Wechselwirkung neutraler Atome und hom ¨oopolare Bindung nach der Quantenmechanik, Zeitschrift f¨ ur Physik44, 455 (1927)

work page 1927

[4] [4]

Zimmermann and C

R. Zimmermann and C. Schindler, Exciton-exciton interaction in coupled quantum wells, Solid State Communications144, 395 (2007)

work page 2007

[5] [5]

Schindler and R

C. Schindler and R. Zimmermann, Analysis of the exciton- exciton interaction in semiconductor quantum wells, Physical Review B78, 045313 (2008)

work page 2008

[6] [6]

Hanamura, Theory of many Wannier excitons

E. Hanamura, Theory of many Wannier excitons. I, Journal of the Physical Society of Japan37, 1545 (1974)

work page 1974

[7] [7]

Haug and S

H. Haug and S. Schmitt-Rink, Electron theory of the optical properties of laser-excited semiconductors, Progress in Quantum Electronics9, 3 (1984)

work page 1984

[8] [8]

Rochat, C

G. Rochat, C. Ciuti, V. Savona, C. Piermarocchi, A. Quattropani, and P. Schwendimann, Excitonic Bloch equations for a two-dimensional system of interacting excitons, Physical Review B61, 13856 (2000)

work page 2000

[9] [9]

Okumura and T

S. Okumura and T. Ogawa, Boson representation of two- exciton correlations: An exact treatment of composite-particle effects, Physical Review B65, 035105 (2001)

work page 2001

[10] [10]

Combescot, O

M. Combescot, O. Betbeder-Matibet, and R. Combescot, Exciton-exciton scattering: Composite boson versus elemen- tary boson, Physical Review B75, 174305 (2007)

work page 2007

[11] [11]

L. V. Keldysh and A. Kozlov, Collective properties of excitons in semiconductors, Journal of Experimental and Theoretical Physics27, 521 (1968)

work page 1968

[12] [12]

L. M. Licer ´an and H. T. C. Stoof, Unconventional excitonic insulators in two-dimensional topological materials, Physical Review B111, 245102 (2025)

work page 2025

[13] [13]

F. Wu, F. Qu, and A. H. MacDonald, Exciton band structure of monolayer MoS2, Physical Review B91, 075310 (2015)

work page 2015

[14] [14]

Elliott, Intensity of optical absorption by excitons, Physical Review108, 1384 (1957)

R. Elliott, Intensity of optical absorption by excitons, Physical Review108, 1384 (1957)

work page 1957

[15] [15]

N. Michel, Direct demonstration of the completeness of the eigenstates of the Schr¨odinger equation with local and nonlocal potentials bearing a Coulomb tail, Journal of Mathematical Physics49(2008)

work page 2008

[16] [16]

A. M. Mukhamedzhanov and M. Akin, Completeness of the Coulomb scattering wave functions, The European Physical Journal A37, 185 (2008). 30

work page 2008

[17] [17]

B. N. Parlett,The symmetric eigenvalue problem(SIAM, 1998)

work page 1998

[18] [18]

Hanamura and H

E. Hanamura and H. Haug, Condensation effects of excitons, Physics Reports33, 209 (1977)

work page 1977

[19] [19]

Weiner, V

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Experiments and theory in cold and ultracold collisions, Reviews of Modern Physics71, 1 (1999)

work page 1999

[20] [20]

K ¨ohler, K

T. K ¨ohler, K. G ´oral, and P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach resonances, Reviews of Modern Physics78, 1311 (2006)

work page 2006

[21] [21]

C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Reviews of Modern Physics82, 1225 (2010)

work page 2010

[22] [22]

C. Y.-P. Chao and S. L. Chuang, Analytical and numerical solutions for a two-dimensional exciton in momentum space, Physical Review B43, 6530 (1991)

work page 1991

[23] [23]

X. Yang, S. Guo, F. Chan, K. Wong, and W. Ching, Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory, Physical Review A43, 1186 (1991)

work page 1991

[24] [24]

D. G. W. Parfitt and M. E. Portnoi, The two-dimensional hydrogen atom revisited, Journal of Mathematical Physics43, 4681 (2002)

work page 2002

[25] [25]

D. K. Efimkin, E. K. Laird, J. Levinsen, M. M. Parish, and A. H. MacDonald, Electron-exciton interactions in the exciton- polaron problem, Physical Review B103, 075417 (2021)

work page 2021

[26] [26]

(31) and performed the heavy-electron limit instead of the heavy-hole limit, which would result in Eq

Note that we could have also eliminated𝑆 c in favor of𝑆 v in Eq. (31) and performed the heavy-electron limit instead of the heavy-hole limit, which would result in Eq. (43) depending only on𝑆 v

work page

[27] [27]

H. N. Cam, N. T. Phuc, and V. A. Osipov, Symmetry- dependent exciton-exciton interaction and intervalley biexciton in monolayer transition metal dichalcogenides, npj 2D Materials and Applications6, 22 (2022)

work page 2022

[28] [28]

H. T. C. Stoof, K. B. Gubbels, and D. B. M. Dickerscheid, Ultracold Quantum Fields, Vol. 1 (Springer, 2008)

work page 2008

[29] [29]

D. Sun, Y. Rao, G. A. Reider, G. Chen, Y. You, L. Br´ezin, A. R. Harutyunyan, and T. F. Heinz, Observation of rapid exciton- exciton annihilation in monolayer molybdenum disulfide, Nano Letters14, 5625 (2014)

work page 2014

[30] [30]

Steinhoff, M

A. Steinhoff, M. Florian, M. R ¨osner, G. Sch ¨onhoff, T. O. Wehling, and F. Jahnke, Exciton fission in monolayer transition metal dichalcogenide semiconductors, Nature Communications8, 1166 (2017)

work page 2017

[31] [31]

L. Ma, P. X. Nguyen, Z. Wang, Y. Zeng, K. Watanabe, T. Taniguchi, A. H. MacDonald, K. F. Mak, and J. Shan, Strongly correlated excitonic insulator in atomic double layers, Nature598, 585 (2021)

work page 2021

[32] [32]

See Supplemental Material at URL-will-be-inserted-by- publisher

work page

[33] [33]

attached

In principle, because of the Hubbard–Stratonovich transfor- mation, every polarization field always comes “attached” to a single-particle interaction. Essentially, this is an interaction between the conduction and valence electrons of the same exciton. This kind of interaction can always be rewritten into energy terms via the temperature-dependent BSE. He...

work page

[34] [34]

Perali, D

A. Perali, D. Neilson, and A. R. Hamilton, High-temperature superfluidity in double-bilayer graphene, Physical Review Letters110, 146803 (2013)

work page 2013

[35] [35]

S. R. U. Haque, M. H. Michael, J. Zhu, Y. Zhang, L. Windg¨atter, S. Latini, J. P. Wakefield, G.-F. Zhang, J. Zhang, A. Rubio, J. G. Checkelsky, E. Demler, and R. D. Averitt, Terahertz parametric amplification as a reporter of exciton condensate dynamics, Nature Materials23, 796 (2024)

work page 2024

[36] [36]

O. L. Berman and R. Y. Kezerashvili, High-temperature superfluidity of the two-component Bose gas in a transition metal dichalcogenide bilayer, Physical Review B93, 245410 (2016)

work page 2016

[37] [37]

Lin and B.-Y

H.-C. Lin and B.-Y. Jin, Charge-transfer interactions in organic functional materials, Materials3, 4214 (2010)

work page 2010

[38] [38]

C. J. Bardeen, The structure and dynamics of molecular excitons, Annual Review of Physical Chemistry65, 127 (2014)

work page 2014

[39] [39]

Nematiaram, D

T. Nematiaram, D. Padula, and A. Troisi, Bright Frenkel excitons in molecular crystals: a survey, Chemistry of Materials33, 3368 (2021)

work page 2021

[40] [40]

G ¨otting, F

N. G ¨otting, F. Lohof, and C. Gies, Moir ´e-Bose-Hubbard model for interlayer excitons in twisted transition metal dichalcogenide heterostructures, Physical Review B105, 165419 (2022)

work page 2022

[41] [41]

J. B. Haber, D. Y. Qiu, F. H. da Jornada, and J. B. Neaton, Maximally localized exciton Wannier functions for solids, Physical Review B108, 125118 (2023)

work page 2023

[42] [42]

Chang, Continuum model study of optical absorption by hybridized moir ´e excitons in transition metal dichalcogenide heterobilayers, Physical Review B108, 155424 (2023)

Y.-W. Chang, Continuum model study of optical absorption by hybridized moir ´e excitons in transition metal dichalcogenide heterobilayers, Physical Review B108, 155424 (2023)

work page 2023

[43] [43]

L. Peng, G. Vignale, and S. Adam, Many-body perturbation theory for moir ´e systems, Physical Review B112, 075146 (2025)

work page 2025

[44] [44]

In fact,Ucorresponds to the exciton–exciton interaction that results from setting the overlap integralsK c andK v to zero in the generalized eigenvalue problem of Eq. (16). For this reason, it only partially includes the effects of the composite nature of the excitons, but we nevertheless expect it to be a better approximation for this problem than the us...

work page

[45] [45]

Singh,Excitation Energy Transfer Processes in Condensed Matter(Springer New York, NY, 1994)

J. Singh,Excitation Energy Transfer Processes in Condensed Matter(Springer New York, NY, 1994)

work page 1994

[46] [46]

Variational and field-theoretical approach to exciton–exciton interactions and biexcitons in semiconductors

D. Young, 2-D convolution using the FFT (2024), MATLAB Central File Exchange. Retrieved December 6, 2024. 1 Supplementary Material for “Variational and field-theoretical approach to exciton–exciton interactions and biexcitons in semiconductors” P. A. Noordman, L. Maisel Licer´an, H. T. C. Stoof S.I. NUMERICAL IMPLEMENTATION OFU c cc To efficiently perform...

work page 2024

[47] [47]

1 𝛽V 3 ∑︁ {k} ∑︁ spins ˜Φ𝜇1 K1, 𝛼𝛽 (k1) ∗ ˜Φ𝜇2 K2, 𝛼′ 𝛽′ (k2) ∗ ˜Φ 𝜇′ 1 K′ 1, 𝛼𝛽 (k′

work page

[48] [48]

(S53) Additionally, the whole expression is multiplied by the momentum and frequency conservation deltas,𝛿 K1+K2,K ′ 1+K′ 2 and 𝛿𝑛1+𝑛2,𝑛′ 1+𝑛′ 2, respectively

˜Φ 𝜇′ 2 K′ 2, 𝛼′ 𝛽′ (k′ 2) × (Δ 𝛼𝛽 K1,k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2,k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1,k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2,k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼 𝛼𝛽(k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,itΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽′ 𝛽′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ ...

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[49] [49]

(S69) For readability, the momentum Kronecker deltas are not summed over

˜Φ𝛼𝛽 ′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼′ 𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼𝛽 ′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼′ 𝛽 (K ′ 1,k ′ 1)N F 𝛼𝛽 ′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑣𝑐𝑣 𝛼′ 𝛽′ 𝛼𝛽 (k2 +𝛾 𝑐K2,k 2 −𝛾 𝑣 K2,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛′ 1 −iΩ 𝑛2,iΩ 𝑛1) ×𝛿 k1 −𝛾𝑣 K1,k′ 1 −𝛾𝑣 K′ 1 𝛿k2 −𝛾𝑣 K2,k′ 2 −𝛾𝑣 K...

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[50] [50]

1 V2 ∑︁ {k} ∑︁ spins ˜Φ𝛼1 𝛽1 𝜇1K1 (k1) ∗ ˜Φ𝛼2 𝛽2 𝜇2K2 (k2) ∗ ˜Φ𝛼1 𝛽1 𝜇′ 1K′ 1 (k′

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[51] [51]

˜Φ𝛼2 𝛽2 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼 𝛼𝛽(k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛1) ×Π 𝑐𝑐𝑣 𝛼′ 𝛼′ 𝛽′ (k′ 2 +𝛾 𝑐K ′ 2,k 2 +𝛾 𝑐K2,k 2 −𝛾 𝑣 K2; iΩ𝑛′ 2,iΩ 𝑛2) ×𝛿 k1 −...

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[52] [53]

˜Φ𝛼′ 𝛽′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑣𝑣𝑐 𝛽𝛽 𝛼 (k′ 1 −𝛾 𝑣 K ′ 1,k 1 −𝛾 𝑣 K1,k 1 +𝛾 𝑐K1;−iΩ 𝑛′ 1,−iΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽′ 𝛽′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ 𝑛2) ...

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[53] [54]

˜Φ𝛼𝛽 ′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼′ 𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼𝛽 ′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼′ 𝛽 (K ′ 1,k ′ 1)N F 𝛼𝛽 ′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼′ 𝛼𝛽 (k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛1) ×Π 𝑐𝑐𝑣 𝛼 𝛼′ 𝛽′ (k′ 2 +𝛾 𝑐K ′ 2,k 2 +𝛾 𝑐K2,k 2 −𝛾 𝑣 K2; iΩ𝑛′ 2,iΩ 𝑛2) ×𝑉(k ...

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[54] [55]

(S84) Again, one can perform the Matsubara summations in the form of Eq

˜Φ𝛼′ 𝛽 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 ′ K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽 K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 ′ (K ′ 1,k ′ 1)N F 𝛼′ 𝛽 (K ′ 2,k ′ 2) −1 ×Π 𝑣𝑣𝑐 𝛽′ 𝛽 𝛼(k′ 1 −𝛾 𝑣 K ′ 1,k 1 −𝛾 𝑣 K1,k 1 +𝛾 𝑐K1;−iΩ 𝑛′ 1,−iΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽𝛽 ′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ 𝑛2)...

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[55] [56]

˜Φ𝛼2 𝛽2 𝜇′ 2K′ 2 (k′

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[56] [57]

˜Φ𝛼3 𝛽3 𝜇′ 3K′ 3 (k′ 3) × (Δ 𝛼3 𝛽1 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼1 𝛽2 K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼2 𝛽3 K3k3 −𝜀 𝜇3 K3 ) × N F 𝛼3 𝛽1 (K1,k 1)N F 𝛼1 𝛽2 (K2,k 2)N F 𝛼2 𝛽3 (K3,k 3) −1 × (Δ 𝛼1 𝛽1 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼2 𝛽2 K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) (Δ𝛼3 𝛽3 K′ 3k′ 3 −𝜀 𝜇′ 3 K′ 3 ) × N F 𝛼1 𝛽1 (K ′ 1,k ′ 1)N F 𝛼2 𝛽2 (K ′ 2,k ′ 2)N F 𝛼3 𝛽3 (K ′ 3,k ′ 3) −1 ×𝛿 k1+𝛾𝑐K1,k′ 3+𝛾𝑐K′ 3 𝛿...

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[57] [58]

(S92) We will define the term in the curly brackets as W0 cv (𝑧 1, 𝑧2;𝑧 ′ 1, 𝑧′

+𝐺 0,𝑋 (¯𝑧′ 2, 𝑧′ 1)𝐺 0,𝑋 (¯𝑧′ 1, 𝑧′ 2) + · · ·. (S92) We will define the term in the curly brackets as W0 cv (𝑧 1, 𝑧2;𝑧 ′ 1, 𝑧′

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[58] [59]

≡ 1 𝛽V ∑︁ 𝑧𝑧 ′ W3B (𝑧 1, 𝑧2, 𝑧;𝑧 ′ 1, 𝑧′, 𝑧′ 2)𝐺 0,𝑋 (𝑧, 𝑧 ′),(S93) since this term represents a two-body interaction vertex between the conduction and valence electrons of the different excitons. This object is related to three similar vertices by exciton exchange, namely W0 cv (𝑧 1, 𝑧2;𝑧 ′ 1, 𝑧′ 2)= WX vc(𝑧 1, 𝑧2;𝑧 ′ 2, 𝑧′ 1),(S94a) = WX cv(𝑧 2, 𝑧1;𝑧 ′ ...

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[59] [60]

1 V2 ∑︁ {k} ∑︁ spins ˜Φ𝛼𝛽 𝜇1K1 (k1) ∗ ˜Φ𝛼′ 𝛽′ 𝜇2K2 (k2) ∗ ˜Φ𝛼𝛽 𝜇′ 1K′ 1 (k′

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[60] [61]

(S99) and Eq

˜Φ𝛼′ 𝛽′ 𝜇′ 2K′ 2 (k′ 2) × (Δ 𝛼𝛽 K1k1 −𝜀 𝜇1 K1 ) (Δ𝛼′ 𝛽′ K2k2 −𝜀 𝜇2 K2 ) (Δ𝛼𝛽 K′ 1k′ 1 −𝜀 𝜇′ 1 K′ 1 ) (Δ𝛼′ 𝛽′ K′ 2k′ 2 −𝜀 𝜇′ 2 K′ 2 ) × N F 𝛼𝛽 (K1,k 1)N F 𝛼′ 𝛽′ (K2,k 2)N F 𝛼𝛽 (K ′ 1,k ′ 1)N F 𝛼′ 𝛽′ (K ′ 2,k ′ 2) −1 ×Π 𝑐𝑐𝑣 𝛼 𝛼𝛽(k′ 1 +𝛾 𝑐K ′ 1,k 1 +𝛾 𝑐K1,k 1 −𝛾 𝑣 K1; iΩ𝑛′ 1,iΩ 𝑛1) ×Π 𝑣𝑣𝑐 𝛽′ 𝛽′ 𝛼′ (k′ 2 −𝛾 𝑣 K ′ 2,k 2 −𝛾 𝑣 K2,k 2 +𝛾 𝑐K2;−iΩ 𝑛′ 2,−iΩ 𝑛2) ×𝛿 k...

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[61] [62]

ˆ𝑋 † (𝑧 ′ 2) = 1 Z[0,0] 𝛿4Z[𝐼 ∗, 𝐼] 𝛿𝐼 ∗(𝑧 1)𝛿𝐼 ∗ (𝑧 2)𝛿𝐼(𝑧 ′ 1)𝛿𝐼(𝑧 ′ 2) 𝐼 ∗, 𝐼=0 =⟨𝑋(𝑧 1)𝑋(𝑧 2)𝑋 ∗ (𝑧 ′ 1)𝑋 ∗ (𝑧 ′ 2)⟩ − ⟨𝑋(𝑧 1)𝑋 ∗(𝑧 ′ 1)⟩𝑉 −1 Φ (𝑧 2, 𝑧′

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[62] [63]

− ⟨𝑋(𝑧 2)𝑋 ∗ (𝑧 ′ 2)⟩𝑉 −1 Φ (𝑧 1, 𝑧′

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[63] [64]

+𝑉 −1 Φ (𝑧 1, 𝑧′ 1)𝑉 −1 Φ (𝑧 2, 𝑧′ 2) − ⟨𝑋(𝑧 1)𝑋 ∗(𝑧 ′ 2)⟩𝑉 −1 Φ (𝑧 2, 𝑧′

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[64] [65]

− ⟨𝑋(𝑧 2)𝑋 ∗ (𝑧 ′ 1)⟩𝑉 −1 Φ (𝑧 1, 𝑧′

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[65] [66]

(S111) Here,𝑧=(𝜇,K, 𝜏), ˆTis the time-ordering operator, and𝑉 −1 Φ (𝑧, 𝑧 ′)=𝑉 −1 Φ;𝜇𝜇 ′ (K)𝛿 KK ′ 𝛿(𝜏−𝜏 ′)

+𝑉 −1 Φ (𝑧 1, 𝑧′ 2)𝑉 −1 Φ (𝑧 2, 𝑧′ 1). (S111) Here,𝑧=(𝜇,K, 𝜏), ˆTis the time-ordering operator, and𝑉 −1 Φ (𝑧, 𝑧 ′)=𝑉 −1 Φ;𝜇𝜇 ′ (K)𝛿 KK ′ 𝛿(𝜏−𝜏 ′). The above object will be used for the computation of a Dyson series in Supp. S.VI. 19 B. Pole Approximation Observing the free exciton-field propagator 𝐺0,𝑋 𝜇𝜇 ′ (K,iΩ 𝑛)= 1 iΩ𝑛 −𝜀 𝜇 K 𝛿𝜇𝜇 ′ −𝑉 −1 Φ;𝜇𝜇 ′ (K),(S...

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[66] [67]

As one may expect, the diagrams that show up in Eq

ˆ𝑋 †(𝑧 ′ 2) = − − + + · · ·, (S114) When the exciton operators in the above correlator are expressed in terms of the electron fields (𝜙 ∗ 𝑐,𝜙 ∗ 𝑣 ,𝜙 𝑐,𝜙 𝑣 ), then these four lowest-order diagrams follow straightforwardly from Wick’s theorem. As one may expect, the diagrams that show up in Eq. (S114) correspond to no exchange, electron exchange, hole excha...

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[67] [68]

noninteracting

+𝐺 0,𝑋 (𝑧 1, 𝑧′ 2)𝐺 0,𝑋 (𝑧 2, 𝑧′ 1) =· · · + + + · · ·. (S115) As mentioned before, this correlator is “noninteracting” in the sense that there are no interactions between the two participating excitons. If the pole approximation is used at this level by setting𝐺 0,𝑋 =𝐺 0, ˆ𝑋, then the terms included in the left-hand “· · ·” are neglected, as these contai...

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