Quantum Computers will constrain the Equation of State of Neutron Stars

Adri\'an Casta\~no-Garc\'ia; Felipe J. Llanes-Estrada; J. J. G\'alvez-Viruet; Mar\'ia G\'omez-Rocha; Mar\'io Logros\'an-\'Alvarez; Nahia J. Dios-Bilbao; Nicol\'as M. Arenaza

REVIEW 4 major objections 4 minor 58 references

Quantum Computers will constrain the Equation of State of Neutron Stars

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A future quantum computer could compute the neutron-star equation of state directly from QCD, avoiding the sign problem, and this paper demonstrates the full pipeline on small simulated systems.

desk verdict A solid, honest methods proof-of-concept for quantum-computing QCD; the operator expansions are the substance, the Gauss-law residual is the flaw, and the physics claim runs ahead of the evidence. read the letter →

arxiv 2608.06515 v1 pith:EOHCUCWV submitted 2026-08-06 hep-ph nucl-thquant-ph

Adrián Castaño-García , Nahia J. Dios-Bilbao , J. J. Gálvez-Viruet , Felipe J. Llanes-Estrada , Marío Logrosán-Álvarez , Nicolás M. Arenaza , María Gómez-Rocha This is my paper · ORCID

classification hep-phnucl-thquant-ph MSC 81P6881V05 PACS 12.38.-t03.67.Ac97.60.Jd

keywords quantumcomputingneutronstarequationofstatedenseQCDsignproblemWeylgaugeparticle-registerencodingGauss'slawvariationaleigensolver

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to show that a quantum computer can extract the equation of state of neutron-star matter directly from Quantum Chromodynamics, bypassing the sign problem that blocks lattice QCD at finite chemical potential. It sets a concrete goal: obtaining one energy-density--pressure pair at intermediate densities with uncertainty three times smaller than the current microscopic band. The authors develop a canonical Hamiltonian in Weyl (time-axial) gauge, enforce Gauss's law through a squared Gauss operator, and demonstrate the method end-to-end on classical simulators for systems of three to four particles. If the approach scales as the analysis suggests, it would give first-principles predictions for neutron-star radii and tidal deformability to accompany third-generation gravitational-wave detectors.

What carries the argument

The key device is the squared Gauss operator $\mathcal{G}^2 = \sum_a \int d^3x\, G_a(x)G_a(x)$, which is nonnegative because each $G_a$ is Hermitian; minimizing its expectation value to zero enforces Gauss's law on the variational state. This is combined with the Weyl-gauge Hamiltonian expanded in normal modes and the particle-register encoding, which maps creation and annihilation operators to operations on qubit registers and allows the Hamiltonian and $\mathcal{G}^2$ to be decomposed into Pauli strings. The polynomial scaling analysis of these Pauli strings, supported by a kinematic cutoff on small matrix elements, is what makes the end-to-end cost plausible on future hardware.

What would settle it

Run the same constrained minimization with progressively larger Fock spaces at fixed chemical potential: if the minimized $\langle \mathcal{G}^2 \rangle$ does not tend toward zero, or if the resulting pressure changes by more than the stated target when going from the 3q1g to the 1q2g ansatz, the extracted states are not physical QCD states and the equation of state is not trustworthy.

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Extended reading notes

Core claim

The central claim is that the neutron-star equation of state can be obtained beginning-to-end from QCD by combining Weyl-gauge canonical quantization, a particle-register encoding of quark and gluon fields, and a variational eigensolver with a quadratic penalty enforcing Gauss's law. The authors show that minimizing the expectation value of the Hamiltonian together with the squared Gauss operator yields energy and pressure values, and that the resulting sound speed is compatible with causality and stability constraints. They further show that the number of Pauli strings needed to represent the Hamiltonian and Gauss operator grows polynomially with the number of momentum modes, especially after discarding kinematically suppressed vertices. The demonstration is explicitly a proof of concept: current classical hardware limits the Fock space to at most three quarks plus one gluon or one quark plus two gluons, and the authors state that residual violation of Gauss's law may or may not be a truncation artifact.

Load-bearing premise

The load-bearing premise is that the heavily truncated Fock space, with at most three quarks plus one gluon or one quark plus two gluons and only two momentum modes, is a sufficient stand-in for the dense-QCD ground state, so that residual violation of Gauss's law is only a truncation artifact.

Editorial extensions

If this is right

A quantum computer with thousands of qubits could compute $\rho$ and $P$ points in the intermediate-density regime where lattice QCD is blocked by the sign problem, narrowing the equation-of-state band.
If a single point is computed near $\rho \sim 400\,\mathrm{MeV/fm^3}$, the paper finds that the predicted radius of a $1.4\,M_\odot$ neutron star narrows from a 36% relative uncertainty to about 15%.
The same constrained EoS would sharpen predictions for tidal deformability, making it easier to exclude $\Lambda = 0$ in gravitational-wave observations.
The Pauli-string count for $H+\mathcal{G}^2$ remains polynomial in the number of momentum modes after the proposed cutoff, suggesting that the resource cost does not grow exponentially as the momentum grid is refined.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same penalty-based enforcement of Gauss's law could be applied to other gauge theories in canonical quantization, not just QCD, wherever a sign problem afflicts Monte Carlo methods.
Editorial inference: the residual violation of Gauss's law is a natural convergence diagnostic; if enlarging the Fock space does not drive $\langle \mathcal{G}^2 \rangle$ toward zero, the time-axial approach may need to be replaced by a formulation in Coulomb gauge, where the constraint is solved explicitly.
Editorial inference: the paper's target of a 95%-confidence point with three-times-smaller uncertainty gives a concrete benchmark for early quantum hardware, and one could test the method on a lower-dimensional toy model where exact or lattice results are available.
Editorial inference: the strong sensitivity of the results to the choice of variational wavefunction (3q1g versus 1q2g) suggests that the same pipeline, run with more particles, could be used to search for signs of a phase transition between hadronic and quark matter.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: this is a carefully built proof-of-concept, not a physics result. The authors map QCD in Weyl gauge onto a particle-register encoding, write out the full normal-mode expansions of the Hamiltonian and the squared Gauss operator, and run VQE on a few particles with a classical simulator. They also quantify what a single quantum-computed (rho,P) point would do to radius and tidal-deformability bands. That TOV forecast is a concrete target for the field.

The paper earns credit for being upfront. Section V.B says plainly that the residual Gauss-law violation may or may not be a truncation artifact, that the subtraction is 'naive', and that the EoS depends on the trial wavefunction. The subtraction constants are calibrated to pQCD at a matching scale; that's a boundary condition, not circular reasoning. The appendices contain a complete tabulation of H and G^2 terms in momentum space, which is the kind of formal result that stays useful after the numerics are forgotten.

The soft spots, in proportion. The main one is the one they flag: Gauss's law is enforced by a penalty term, and no final residual is reported. In the truncated Fock space (3q+1g or 1q+2g), G acting on the ansatz produces components outside the variational space, so G^2_min is likely strictly positive. The u -> infinity limit then minimizes G^2 rather than enforcing G^a|Psi>=0. The extracted EoS is therefore not guaranteed to come from physical QCD states. The authors know this, and it does not kill the paper as a methods demonstration, but it does undercut the 'beginning-to-end' claim if read literally.

Eq. (2) writes [A_i, A_j]=delta_ij delta^3, which is not the canonical equal-time commutator; the fields commute, and the canonical commutator is [A_i, Pi_j]=i delta_ij delta^3. The mode algebra in Eqs. (12-13) is correct, so this is likely a typo, but it should be fixed. Also: no code, no specified quark masses in the numerics, and the EoS depends on the fitting function (Exp vs Fraction). Those are minor-to-moderate for a proof-of-concept, but they limit reproducibility.

Who this is for: people working on Hamiltonian QCD for quantum computing, and nuclear astrophysicists who want to know what quantum computers could deliver. It deserves a serious referee. I would send it to peer review with a request to report the G^2 residual, fix the commutator, and either tighten the truncation argument or qualify the central claim. I would not cite it for a physics number, but the operator expansion is worth pointing to.

Referee Report

4 major / 4 minor

Summary. The paper proposes a quantum-computing pipeline for computing the equation of state (EoS) of neutron-star matter directly from QCD at finite chemical potential, a regime where lattice QCD suffers from the sign problem. It first quantifies the astrophysical payoff by showing how a single improved (rho,P) point would narrow the mass-radius and tidal-deformability bands. It then formulates QCD in Weyl (time-axial) gauge, expands the Hamiltonian and the squared Gauss operator in momentum modes, and describes a particle-register encoding of quark and gluon Fock states. As a proof of concept, the authors run classical VQE simulations with very small truncated Fock spaces: up to three quarks plus one gluon, or one quark plus two gluons, with momenta along a single axis. They extract a subtracted energy, convert it to pressure and sound speed, and give a scaling analysis of the number of Pauli strings as a function of momentum-grid size. The paper is explicitly framed as a demonstration rather than a realistic calculation, and several important limitations are acknowledged in Sec. V B.

Significance. If the proposed method could be made to work, it would address a genuinely important problem: first-principles QCD constraints on the neutron-star EoS at intermediate densities. The paper contains substantial constructive material: the complete normal-mode expansions of the Hamiltonian and squared Gauss operator in Appendices A and B, a concrete particle-register encoding with explicit qubit counts, an end-to-end classical VQE demonstration, and a polynomial-scaling analysis of the operator encodings. The authors are also commendably explicit about the proof-of-concept nature of the numerical demonstration. I do not regard the pQCD matching in Eq. (67) as circular; it is calibration against an external benchmark. However, the claim that the calculation extracts the EoS "directly from Quantum Chromodynamics" is not yet supported, because the variational states are not demonstrated to lie in the physical, Gauss-law-constrained Hilbert space.

major comments (4)

[Sec. II, Eq. (2)] The central physical-subspace enforcement is unverified. The penalty term in Eq. (40) minimizes (G^2 - G^2_min) as the penalty strength grows, but the paper never reports the residual value of G^2 at the final variational states. Section V B explicitly leaves open "whether the residual violation of Gauss' law observed in the present calculation is merely an artifact of the truncation." This is not a cosmetic gap: in the 3q+1g sector, acting with G on the 1g component generates two-gluon and quark-antiquark components that are absent from the ansatz, and similarly the 1q+2g sector lacks three-gluon and q-qbar components. Generically, then, G^2_min is strictly positive in the truncated space, and the u -> infinity limit of Eq. (40) minimizes G^2 rather than enforcing G^a|Psi>=0. The EoS extracted via Eq. (70) is therefore computed from states with residual gauge violation. The manuscript should report the final G^2 expectation values, demonstrate convergence with penalty strength and basis size, or clearly restrict the conclusions to a method demonstration rather than an EoS extraction.
[Eq. (2)] Equation (2) states [A_i(x), A_j(y)] = delta_ij delta^{(3)}(x-y), which is not the canonical equal-time commutation relation. The correct relation is [A_i(x), Pi_j(y)] = i delta_ij delta^{(3)}(x-y) with [A_i, A_j] = 0, where Pi_i is the conjugate momentum. The later mode algebra in Eqs. (12)-(13) appears to be the one actually used, so this is probably a typo, but as written Eq. (2) would invalidate the canonical quantization and should be corrected.
[Eq. (67)] The subtraction in Eq. (67) is, as the paper itself says, a "naive subtraction" rather than a proper renormalization. The constants H0, h0, and h2 are fixed by requiring the subtracted Hamiltonian to reproduce the pQCD pressure and energy density at a matching scale, together with a vacuum normalization condition. This means the EoS at the matching point is imposed by construction, and the resulting curve is a calibration of the truncated Hamiltonian to pQCD rather than a fully independent QCD prediction. The claim in the Conclusions that the EoS is extracted "directly from Quantum Chromodynamics" should be softened, and the sensitivity of the final EoS to the matching scheme and scale should be estimated.
[Fig. 4] The final EoS and sound speed depend substantially on both the fitting function and the trial wavefunction. Figure 4 shows visible differences between the "Exp" and "Fraction" fits, particularly for the 1q2g wavefunction, and Tables X and XI give fit parameters that differ at the few-percent level in the exponents but lead to different c_s^2 curves. Since pressure in Eq. (70) and c_s^2 in Eq. (71) are derivatives of the fitted H'(Lambda), the uncertainty in the derivative is not controlled by the small fit errors quoted in Tables X and XI. The paper correctly notes the strong sensitivity to the trial wavefunction, but the quantitative EoS results should be presented as an illustration of the method, with explicit error bars that include the fitting ambiguity, rather than as constraints on the neutron-star EoS.

minor comments (4)

[Abstract] There are several typographical issues: "resistsab initio" is missing a space, and the Conclusions contain the awkward phrase "Neutron Star Matter relevant for neutron–star." Please proofread these passages.
[Sec. II B] The symbol Lambda is used both for the ultraviolet cutoff and for the tidal deformability in Eq. (1). The authors note this, but the double use is genuinely confusing in a paper about both quantities; a different symbol for the cutoff would improve readability.
[Eq. (51)] Immediately before Eq. (51) the text states that h_g11(p,i,j) with i != j vanishes for momenta along one direction, but the displayed equation appears to retain terms such as h_g11(1,1,2) and h_g11(2,1,2). Please clarify whether these terms are zero or whether the statement about i != j applies only to a particular subset of the matrix elements.
[Eq. (74)] The matrix-element cutoff eta = 10% in Eq. (74) is introduced for computational convenience, but no physical or convergence justification is given for this particular value. A short discussion of how the results and scaling exponents change with eta would strengthen the cost analysis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EoS extraction is calibrated to pQCD at one explicit matching point, and the residual Gauss-law violation is an acknowledged limitation rather than a circular reduction.

full rationale

The derivation chain is self-contained in the sense that no predicted quantity is, by construction, equal to an input. The subtraction constants in Eq. (67) are explicitly fit to reproduce the pQCD pressure and energy density at one matching scale, as stated in Sec. V A: "To set the correct energy and pressure at one point, the matching scale to pQCD, we apply the following subtraction to the Hamiltonian." This is transparent calibration, not a hidden input renamed as a prediction. The EoS at other densities is obtained from the derivative of the minimized, subtracted Hamiltonian with respect to volume, Eq. (70), so those points are not forced to equal the pQCD input. The variational wavefunctions are constrained by symmetry to resemble a color-singlet, mixed-symmetry neutron-like state, but the paper does not claim that this ansatz is derived from first principles; it is a proof-of-concept truncation. The particle-register encoding is imported from the authors' prior work [30], and the paper relies on that reference for the canonical (anti)commutation relations "up to a boundary term." That is a self-citation, but the cited construction is a published, externally checkable formalism rather than a uniqueness theorem invoked to forbid alternatives, and the paper's central EoS result does not reduce to that encoding by definition. The residual violation of Gauss's law is explicitly left open in Sec. V B: "whether the residual violation of Gauss' law observed in the present calculation is merely an artifact of the truncation." That is an unverified assumption or correctness risk, not a circular step, because the paper does not define the physical subspace to be whatever the truncated variational minimum satisfies. Equation (2) appears to be a typo in the canonical commutator, but the mode algebra subsequently used is Eqs. (12)-(13), so this does not create a circular dependency. Overall, the paper is honest about its calibrations and limitations, and no load-bearing step reduces to its own inputs.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or dimensions are postulated; the particle-register encoding is a computational representation, not a physical entity. The main additions are the explicit expansions of H and G² in momentum space for this encoding, plus the numerical demonstration and scaling analysis.

free parameters (5)

Subtraction constants H0, h0, h2 = H0=0.84/1.33 GeV; h0=11.79/11.76; h2=1.55/1.59 (Table IX)
Fitted so that the subtracted Hamiltonian (Eq. 67) matches pQCD pressure and energy density at the matching scale [44], and the free vacuum expectation vanishes (Eq. 69). These constants shift the energy and hence the EoS.
Energy-fit parameters a1,b1,a2,b2,c2,a3,b3,c3 = See Tables X-XII
The minimized energies H'(Λ) are fit to three functions (Eqs. 68, 72, 73). Pressure (Eq. 70) and sound speed (Eq. 71) are derivatives of the fit, so the reported EoS depends on the fit form; two different forms give visibly different c_s².
Chemical-potential interpolation endpoints = µ_B = 1.4, 2.0, 2.6 GeV for Core, Middle, Match (Table VIII)
Eq. (66) tunes µ_B(Λ) linearly between chosen endpoints; this sets the physical density scale and is not derived from the dynamics.
Quark masses m_f = not stated
The spinor basis (Appendix C) and Hamiltonian require quark masses, but the numerical values used in the simulation are not given in the text.
Truncation cutoff η = η = 0.1 (Eq. 74)
Matrix elements with |c_mom| < ε are discarded to reduce Pauli-string counts; this affects the scaling analysis rather than the EoS demo.

assumptions (6)

domain assumption Time-axial gauge A0=0 with Gauss law as a subsidiary condition (Eq. 6) yields a valid canonical quantization of QCD.
Standard gauge-fixing; used to derive the Hamiltonian in Eq. (4).
domain assumption Minimizing ⟨H⟩ + ũ/Λ² ⟨G²⟩ (Eq. 40) with increasing ũ converges to a state satisfying Gauss law.
The penalty method is standard for constrained optimization, but in a truncated Fock space it is not proven to reach the physical subspace; residual violation is acknowledged in Sec. V B.
domain assumption The box volume V = (2π)^3(1+2N)^3/Λ^3 with fixed particle number gives density n=N/V, and µ_B is a linear function of Λ (Eq. 66).
This discretization replaces the thermodynamic limit and sets the density/chemical-potential correspondence.
ad hoc to paper A Fock space truncated to at most 3 quarks + 1 gluon or 1 quark + 2 gluons captures the relevant physics.
The authors state this is a proof of concept and that the consequences of truncation are not fully understood (Sec. V B).
ad hoc to paper The linear counterterm subtraction H' = H - H0 - (h0 + h2 g²) Λ (Eq. 67) is a sufficient renormalization.
Called 'naïve' by the authors in Sec. V B; a proper renormalization is left for future work.
domain assumption Free-field spinor solutions (Appendix C) can be used as the basis for interacting fields in the variational ansatz.
The spinor basis is defined from the free Dirac equation; the variational wavefunction is a superposition of these free-field states.

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Pith. "Pith review of Quantum Computers will constrain the Equation of State of Neutron Stars." pith.science (2026). https://pith.science/paper/EOHCUCWV

@misc{pith2026260806515,
  author       = {Pith},
  title        = {Pith review of: Quantum Computers will constrain the Equation of State of Neutron Stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EOHCUCWV}},
  note         = {Machine review of arXiv:2608.06515}
}

abstract

The Equation of State (EoS) of Nuclear Matter at high densities, and particularly that of neutron stars, resists $\mathit{ab}$ $\mathit{initio}$ Quantum Chromodynamics (QCD) computations due to the notorious sign problem of Lattice Gauge Theory at finite chemical potential. A quantum computer deploying QCD in canonical quantization should be able to make substantial progress. We set some basic goals for a future quantum computer to predict the EoS, and thus the basic static observables of the star (mass, radius and Tidal deformability, for example). We then develop the basic theory to address the canonical Hamiltonian in Weyl (time-axial) gauge expressed in normal modes, together with the squared Gauss operator $\mathcal{G}^2$ necessary to execute energy minimization algorithms restricted to the physical Fock subspace. Finally, we deploy our particle-quantum register encoding of a generic field theory to demonstrate QCD at finite chemical potential for a few (three-four) particles with a modest number of momentum modes, by simulating the quantum computer on a classical cluster. This opens the possibility for effective quantum computers to constrain the microscopic physics of neutron stars simultaneously to the operation of third--generation gravitational wave detectors such as the Einstein Telescope, providing more detailed predictions than has been possible until now.

Figures

Figures reproduced from arXiv: 2608.06515 by the authors.

**Figure 2.** FIG. 2. Top panel: Mass (in solar masses) against neutron [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Energies obtained after applying Eq. (67) to the mini [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. Pressure and sound velocity squared as functions of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. Left: number of non-vanishing matrix elements for the combined operator [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6. Number of Pauli strings in the combined operator [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7. Percentage of Pauli strings remaining after cutting off irrelevantly small matrix element, as a funnction of the momentum [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: FIG. 8. Number of Pauli strings for individual, selected matrix elements of the operator [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

**Figure 9.** Figure 9: FIG. 9. Quantum circuit implementing the Hadamard test [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

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Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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3 Quarks + 0, 1 Gluon. The first wavefunction is a superposition of states with three quarks and three quarks and a gluon: |Ψ1(θ)⟩= cos(θ 1) Ψ3q + sin(θ1)e iθ2 Ψ3q1g ,(41) whereθ 1, θ2 are the first two parameters. Note that the only phase introduced via parameters is that of Eq. (41), all other amplitudes are written in terms of real trigono- metric func...

work page

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1 Quark + 0, 1 or 2 Gluons Restricting the Fock space to at most one gluon closes the exploration of the non-Abelian interaction terms ap- pearing in both the Hamiltonian and the squared Gauss’ law operator. To probe the effect of these interaction terms, we considered states with one quark and up to two gluons: |Ψ2(θ)⟩= cos(θ 1) Ψ1q + sin(θ1) cos(θ3)e iθ...

work page

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Note that, as a consequence of the gluon polarization ba- sis Eq

Kinetic energy terms The simplest operator in the momentum basis is the kinetic energy Hg11 = Z d3p 2Ep hg11(p;i, j)a† piaapja ,(47) whereh g11(p;i, j) can be found in the supplementary ma- terial along with an explanation of the subindex notation. Note that, as a consequence of the gluon polarization ba- sis Eq. (10), which differs from the usual decompo...

work page

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Anomalous gluon-pair term The Hamiltonian operator and squared Gauss’ law con- straint have anomalous terms that create and annihilate particles from the vacuum, for example Hg20 = Z d3p 4Ep hg20(p;i, j) (a† piaa† −pja + h.c.).(52) Such operators do not conserve particle number and their eigenstates are therefore superpositions of states contain- ing arbi...

work page

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Exp” denoting an exponential function fit, see Eq. (72) and “Fraction

Gluon-pair interaction The last term we consider is (we omit the Dirac delta function of momentum conservation) Hg22 = g2 2 Z [pqkl] f abcf adeδi i′δj j′ +f abef adc δi i′δj j′ −δ i jδj′ i′ a† piba† kjc aqi′dalj′e,(58) where [pqkl] is defined in the appendices. This interaction term also involves two registers, but in contrast to the previous case, they b...

2000

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