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REVIEW 4 major objections 6 minor 98 references

Density-Functional Green Function Theory: Dynamical exchange-correlation field in lieu of self-energy

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that the one-particle Green function can be obtained from a local dynamical exchange-correlation field instead of the nonlocal self-energy, with simple approximations already matching benchmark spectra for strongly…

desk verdict A well-written consolidation of the author's own Vxc program; the formal rewriting is sound and the benchmarks are honestly presented, but the practical transferability claim rests on uncontrolled cluster extrapolations. read the letter →

arxiv 2505.19852 v1 pith:HOGYUB55 submitted 2025-05-26 cond-mat.str-el cond-mat.mtrl-sci

F. Aryasetiawan This is my paper · ORCID
classification cond-mat.str-elcond-mat.mtrl-sci
keywords Greenfunctiondynamicalexchange-correlationfieldxcholeself-energystronglycorrelatedelectronsHubbardchainAndersonimpuritymodelhomogeneouselectrongas
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

The paper argues that the one-particle Green function of an interacting electron system can be found from a local-in-time equation of motion that contains a dynamical exchange-correlation field, $V_{\rm xc}(r,r';t)$, instead of the traditional nonlocal self-energy. $V_{\rm xc}$ is the Coulomb potential of a dynamical exchange-correlation hole $\rho_{\rm xc}$, which obeys a sum rule and an exact on-top constraint inherited from density-functional theory. The paper presents this replacement as exact and shows that simple approximations for $\rho_{\rm xc}$ -- extrapolated from small clusters or parametrized from the homogeneous electron gas -- reproduce benchmark spectra for the one-dimensional Hubbard and Heisenberg chains, the single-impurity Anderson model, and sodium photoemission. If the claim holds, spectral calculations that currently require expensive self-energy construction could be replaced by a local-in-time propagation problem.

What carries the argument

The central object is the dynamical exchange-correlation hole $\rho_{\rm xc}(r,r',r'';t)$, defined by writing the two-particle Green function as $G^{(2)}(r,r',r'';t)=[\rho(r'')+\rho_{\rm xc}(r,r',r'';t)]G(r,r';t)$. Its Coulomb potential is $V_{\rm xc}$; the sum rule and on-top constraint make $V_{\rm xc}$ act like a local density-dependent field, and only the spherical average of $\rho_{\rm xc}$ and its first radial moment enter $V_{\rm xc}$. The paper also uses a quasiparticle-effective-field decomposition $\Xi_q(r,t)\approx\Xi^S_q(r)+\Xi^D_q(r)e^{i\Omega t}$ to turn the equation of motion into a quasiparticle picture of a static energy shift plus a dynamic satellite-generating term.

What would settle it

Calculate the exact $V_{\rm xc}$ for the one-dimensional Hubbard chain at $U=4$ using a large reference calculation and compare with the dimer- and six-site-extrapolated forms: if the difference at small $k$ and finite $t$ does not shrink as the cluster grows, the extrapolation premise fails.

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Extended reading notes

Core claim

The central claim is that there exists an exact local-in-time equation of motion for the time-ordered Green function, $[i\partial_t - h(r) - V_{\rm xc}(r,r';t)]G(r,r';t)=\delta(r-r')\delta(t)$, where $V_{\rm xc}$ is generated by the dynamical exchange-correlation hole through $V_{\rm xc}(r,r';t)=\int dr''\, v(r-r'')\rho_{\rm xc}(r,r',r'';t)$. The hole satisfies $\int d^3r''\, \rho_{\rm xc}(r,r',r'';t)=-\delta_{\sigma\sigma''}\theta(-t)$ and $\rho_{\rm xc}(r,r',r;t)=-\rho(r)$, the same normalization and on-top conditions as the static exchange-correlation hole used in ground-state density-functional theory. The paper reports that approximate $V_{\rm xc}$ built from the Hubbard dimer and a six-site cluster produces the spinon-holon structure and the $\alpha U$ gap of the half-filled Hubbard chain; a four-site spin cluster extrapolated with a two-spinon energy reproduces the Heisenberg dynamic structure factor; an ansatz matched to a 50-site Anderson cluster reproduces the Hubbard side bands and Kondo resonance; and an electron-gas parametrization reproduces sodium photoemission with plasmon satellites.

Load-bearing premise

The practical results stand on the assumption that a small cluster's exchange-correlation field, with a few fitted parameters, accurately represents the infinite system's field.

Editorial extensions

If this is right

  • Spectra for photoemission and inverse photoemission could be computed by propagating the Green function pointwise in time, because $V_{\rm xc}$ is local in time and no self-energy convolution is needed.
  • Density-functional-style approximations such as a local-density approximation built from the homogeneous electron gas become available for one-particle excitation spectra, not just ground-state energies.
  • For strongly correlated models, a $V_{\rm xc}$ parametrized from small clusters can capture correlation gaps, spinon continua, and Kondo features at much lower cost than exact diagonalization or impurity solvers.
  • The static-plus-dynamic form of the effective quasiparticle field gives a systematic language for describing satellites: the dynamic term couples electrons to the main collective mode and transfers spectral weight to a replica peak.
  • Because hole and electron see different $V_{\rm xc}$, the gap underestimation of ground-state density-functional theory is explained without invoking a derivative discontinuity.

Reading between the lines

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

  • An extension the paper leaves implicit is that diagrammatic intuition could be redirected from self-energy expansions to constructing $\rho_{\rm xc}$ directly, for example from density-response data.
  • The cluster-transfer success suggests a nearsightedness test: computing $V_{\rm xc}$ for systematically larger clusters and checking convergence would turn the practical claim into a quantitative statement about correlation length.
  • The electron-gas local-density model could be probed on a second simple metal such as aluminum; the paper shows the sodium spectrum but does not test whether the same parametrization transfers.
  • Since only the spherical average and first radial moment of the xc hole determine $V_{\rm xc}$, approximate holes with incorrect higher moments may still yield accurate spectra, which could be exploited in data-driven constructions of $V_{\rm xc}$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. This manuscript develops "density-functional Green function theory", in which the one-particle Green function obeys the local-in-time equation of motion [i∂t − h(r) − Vxc(r,r′;t)]G(r,r′;t)=δ(r−r′)δ(t) (Eq. 44), with Vxc defined as the instantaneous Coulomb potential of a dynamical exchange-correlation hole ρxc (Eq. 46). The hole is shown to satisfy the sum rule (Eq. 40) and the on-top constraint (Eq. 43), both derived from exact operator identities. The paper presents analytic and numerical Vxc for the hydrogen atom, the Holstein model, the Hubbard dimer, and the homogeneous electron gas within RPA/plasmon-pole approximations; derives quasiparticle equations, a total-energy formula, and thermal/nonequilibrium extensions; and applies cluster-extrapolated Vxc to the 1D Hubbard chain, the 1D antiferromagnetic Heisenberg chain, and the single-impurity Anderson model, comparing against Bethe ansatz, DMRG, NRG, and KCuF3/Na data. The formal part is internally consistent, but the load-bearing practical step is the assumption that a small-cluster Vxc can be extrapolated to infinite systems; this assumption is stated in Sec. XVIII but never validated by a systematic convergence study.

Significance. If established, the formalism would be conceptually attractive and computationally useful: local-in-time propagation avoids the convolution in the Dyson equation, and the xc-hole picture provides a natural link between Green function theory and DFT. The formal identities—Eqs. (40), (43), (44), and (46)—are correct and are derived cleanly. The Hubbard gap αU is a genuine parameter-free prediction and agrees remarkably well with the Bethe ansatz over a wide range of U (Fig. 23); the DMRG, NRG, KCuF3, and Na comparisons are real and nontrivial. However, the applications rest on uncontrolled cluster-to-lattice extrapolations and on several fitted parameters (e.g., γ, Z, η1 in Sec. XVII E; A, B, ωsp in Sec. XVIII B; λ, ω1, C in Sec. XVIII C). The central practical claim—that a short-range, cluster-derived Vxc can replace the self-energy without significant loss—is therefore not yet established. The exactness of the rewriting does not by itself validate any approximation scheme.

major comments (4)
  1. [XVIII A, Eqs. (361)-(370)] The central practical claim that a cluster-computed Vxc transfers to the infinite lattice is not supported by any convergence test. The dimer Vxc is truncated at nearest-neighbour level; the paper explicitly states it "clearly neglects components of Vxc beyond nearest neighbours" and attributes the small-k disagreement with DMRG to this neglect. The six-site calculation (Fig. 24) improves agreement, but no N=2,4,6 convergence sequence, no data on the distance decay of Vxc matrix elements, and no error estimate are provided. Since all applications in this section inherit this transferability assumption, the benchmark agreement, including the parameter-free gap αU, cannot be taken as evidence that the infinite-system Vxc is sufficiently short-ranged.
  2. [XVIII B, Eqs. (424)-(428)] The Heisenberg ansatz VD(k,t)=A(k)e^{-iωsp(k)t}+B(k) is not derived from the cluster calculation: ωsp(k) is fitted to the exact two-spinon boundary (Eq. 428) and B(k) is adjusted by hand to remove finite-size effects. This puts part of the target spectrum into the input, so the agreement with DMRG and KCuF3 is partly a consistency check of the ansatz rather than a predictive test of Vxc transferability. A protocol that determines A, B, and ωsp from cluster data alone, together with a sensitivity analysis, is needed before the comparison can be interpreted as support for the extrapolation premise.
  3. [XVIII C, Eq. (464)] The SIAM ansatz Vxc(t)=[λ(ω1+C)+(1−λ)Ce^{iω1t}]/[λ+(1−λ)e^{iω1t}] has its parameters determined from the same 50-site cluster whose physics it is then used to represent; benchmarking against NRG validates the functional form in the wide-band limit but does not test whether a Vxc computed on a small cluster converges to the infinite-system Vxc. No cluster-size study is reported for this model, and the wide-band-limit parameters are effectively fitted, so Fig. 31 does not isolate the transferability assumption.
  4. [XVII C-E, Eqs. (302)-(348)] The HEG-based LDA is not parameter-free: the unoccupied-momentum cutoff is set to about 1.5kF to reproduce the static correlation hole (Sec. XVII C), and γ(rs), Z(rs), and η1(rs) are extracted from GW calculations or treated as fitting parameters (Eqs. 339, 346, 347). The Na photoemission comparison therefore demonstrates that a GW-calibrated model can reproduce known spectra, but it does not establish a density-functional approximation for inhomogeneous systems; no inhomogeneous test of the LDA prescription in Eq. (100) is reported.
minor comments (6)
  1. [V B, Eq. (56)] The bosonic spectral representation of the dynamic part of Vxc is introduced as a conjecture; since it may be used in later constructions, it should be labelled as an assumption and its range of validity stated.
  2. [XVIII A, Eqs. (371)-(377)] The renormalized weights Ae0(q) and Ah0(q) are allowed to deviate from unity, but the paper does not state whether the resulting spectral functions still satisfy the relevant sum rules; a brief check would be helpful.
  3. [IV, Eqs. (30)-(31)] The notation g(r,r′,r′′;t) is introduced as a correlator, but its physical meaning and the regime in which it can be complex are not discussed until much later; a short comment near Eq. (30) would improve readability.
  4. [XVII C, Eqs. (302)-(303)] The validity condition for the q-independent plasmon-pole approximation (q≤qc with qc values given for rs=3,4,5) is stated without derivation or reference; a citation or one-line justification is needed.
  5. [XVIII B, Eq. (416)] The definition of Vxc(k,t) as a two-index object is stated without explaining how the four-index object of Eq. (49) reduces to it in the translationally invariant chain; an explicit reduction would prevent confusion.
  6. [Various] The manuscript is a synthesis of the author's previous publications (Refs. [35,44,45,58,66,84,86]); a short paragraph at the start of Sec. XIX summarizing what is new relative to those works would help readers place the contributions.

Circularity Check

2 steps flagged · score 6.0 of 10

The exact Vxc equation is a definitional rewrite of the interaction term, and the Heisenberg-chain spectrum fits the spinon boundary it then reproduces.

  1. self definitional [Secs. IV-V, Eqs. (30)-(32), (44), (46)]
    "G^{(2)} can be rewritten as G^{(2)}(r,r',r'';t)=[\rho(r'')+\rho_{xc}(r,r',r'';t)]G(r,r';t) (Eq. 31); V_{xc}(r,r';t)=\int dr'' v(r-r'')\rho_{xc}(r,r',r'';t) (Eq. 46); "Using the expression for G^{(2)} in Eq. (31), the equation of motion ... becomes [i\partial_t-h(r)-V_{xc}(r,r';t)]G(r,r';t)=\delta(r-r')\delta(t)" (Eq. 44)."

    V_{xc} is not an independent physical quantity: \rho_{xc} is defined by Eq. (31) so that, after substitution into the exact equation of motion, Eq. (44) holds identically. The sum rule (40) and on-top constraint (43) then follow from the same definition and elementary operator identities. Thus the central statement that an exact local V_{xc} exists and obeys a local equation of motion is a rewriting of the definition of V_{xc} in terms of G^{(2)}, not a derived prediction. The exactness is true by construction; all predictive content is delegated to subsequent approximations for \rho_{xc}.

  2. fitted input called prediction [Sec. XVIII B, Eqs. (424), (428), Fig. 27]
    ""The spinon excitation energy is estimated by fitting the cluster \omega_{sp} to the two-spinon spectrum boundary, \omega_{sp}\to(-J)\pi(\sin k/2-\frac{1}{2}|\sin k|)" (Eq. 428); this is inserted into the ansatz V_D(k,t)=A(k)e^{-i\omega_{sp}(k)t}+B(k) (Eq. 424), and the resulting spectrum is compared with the KCuF_3 structure factor "in both the peak locations and the relative weights"."

    The predicted peak positions are constructed from the exact two-spinon boundaries. The static part V_S(k) is chosen to reproduce the lower boundary \Omega_L(k), and \omega_{sp} in the dynamical part is fitted to \Omega_U(k)-\Omega_L(k). Hence the main peak near \Omega_L+B and the satellite near \Omega_U+B are inputs, not outputs. The agreement with DMRG and KCuF_3 in peak locations is therefore partly by construction; only the weights A/\omega_{sp} and the finite-size shift B carry independent cluster information.

full rationale

The central formal object is constructed rather than derived: substituting Eq. (31) into the equation of motion and defining V_{xc} by Eq. (46) makes Eq. (44) an identity, and the accompanying sum rule and on-top constraint are immediate consequences of the same definitions. This is a legitimate exact rewriting, but it is not an independent falsifiable prediction, so it counts as a self-definitional step rather than an external theorem. The Hubbard-dimer application is not circular: \alpha U is computed from the dimer and compared with the Bethe-ansatz gap without fitting to that gap. The Anderson application fits an ansatz to a 50-site cluster and benchmarks against NRG, which is an independent transferability test. The electron-gas/Na model takes \gamma, Z, and \eta from GW calculations and then compares with photoemission, again not fitted to the target spectrum. The one clear case of reduction is the Heisenberg chain: the spinon excitation frequency is fitted to the two-spinon spectral boundary, so the peak positions compared with DMRG and KCuF_3 are partly inputs. Overall, the strong exactness claim is definitional and one benchmark is partly a fit, but several results retain independent anchors, giving partial rather than complete circularity.

Assumptions & free parameters 6 free parameters · 7 assumptions · 4 invented entities

The exact formalism pulls no free parameters, but it also contains no physics beyond the defining identities: g is introduced so that G^(2) = G g rho, and Vxc is the remainder that makes Eq. (44) exact. All physical content enters through axioms and fits: the cluster extrapolation of Vxc (Sec. XVIII), the RPA/plasmon-pole correlation hole with a fitted momentum cutoff (Sec. XVII), the static-plus-one-mode decomposition of the effective field (Sec. XIV B), and the ansatz forms for the Heisenberg and Anderson models. Free parameters: 1.5kF cutoff, gamma, Z, eta1, A, B, omega_sp, lambda, omega1, C. The independent benchmarks (Bethe ansatz, DMRG, NRG, KCuF3, Na photoemission) are external and not used to set the parameters, except in the Heisenberg section where omega_sp and B encode the known two-spinon boundary, and in the HEG section where GW supplies the constants.

free parameters (6)
  • HEG unoccupied-momentum cutoff (k' upper limit) = 1.5 kF for rs = 4
    Sec. XVII C: the cutoff is treated as a parameter and set to approximately 1.5 kF to reproduce approximately the static correlation hole [72]; the reported Vxc and the smoothness claim depend on this fit.
  • gamma(rs), momentum-broadening factor = from one-shot GW, Fig. 16
    Sec. XVII E: XiS_q = (1 - gamma Z)(EF - epsilon_q); gamma is extracted from GW calculations, so the proposed LDA inherits its constants from the method it aims to replace.
  • Z(rs), quasiparticle renormalization factor = average of q=0 and q=kF GW values, Fig. 16
    Sec. XVII E: fixes lambda = 1 - sqrt(2Z - 1) and the satellite weights A0, A1, A2; no uncertainty is reported.
  • eta1(rs), lifetime broadening parameter = from one-shot GW, Fig. 17
    Sec. XVII E: Eq. (347) models the lifetime broadening; eta0 is set finite by hand to broaden the delta-function peaks.
  • A(k), B(k), omega_sp(k) for the Heisenberg chain = linear interpolation from a 12-site cluster; omega_sp fitted to the two-spinon boundary
    Sec. XVIII B 2: A, B, G(k,0+) estimated by interpolation; omega_sp fitted to Eq. (428), B adjusted downward for finite-size effects; the spectral positions therefore encode the known answer.
  • lambda, omega1, C for the SIAM ansatz = determined from a 50-site cluster calculation
    Sec. XVIII C 3: Eq. (464) is fitted to cluster Vxc; the agreement with NRG validates the functional form, not a parameter-free prediction.
assumptions (7)
  • domain assumption The correlator g is well-defined by the division G^(2) = G g rho (Eq. 30), requiring G(r,r';t)rho(r'') != 0 on the domain of interest.
    Zeros of the exact time-dependent Green function would make rho_xc and hence Vxc singular; the paper does not address this technical condition.
  • ad hoc to paper Vxc has a bosonic spectral representation (Eq. 56) with a static and a dynamic part.
    Sec. V B states 'it can be conjectured that analogous to W, Vxc should be bosonic.' The satellite analysis and the decomposition used throughout rely on this unproven form.
  • ad hoc to paper Cluster-to-lattice extrapolation: Vxc from a dimer, 4-site, or 6-site cluster represents the infinite chain.
    Sec. XVIII A extrapolates the dimer Vxc and admits it 'clearly neglects components of Vxc beyond nearest neighbours'; the small-k Hubbard spectra deviate from DMRG for this reason.
  • ad hoc to paper RPA plus a q-independent plasmon-pole approximation, with the fitted 1.5kF cutoff, gives a reliable HEG correlation hole.
    Sec. XVII C: the approximation is valid for q <= qc at rs = 3, 4, 5; the cutoff is fitted to the static correlation hole, so the HEG Vxc inherits a fitted input.
  • domain assumption For a renormalized G, exchange plus density-response contributions preserve the xc-hole sum rule without vertex corrections.
    Sec. VIII explicitly doubts this: 'it is not impossible that the sum of these two terms would fulfill the sum rule, but this seems unlikely.' The RPA correlation-hole construction rests on this unresolved point.
  • ad hoc to paper The effective field decomposes as a static term plus a single dynamic mode, Xi = XiS + XiD e^{iOmega t} (Eq. 172).
    Sec. XIV B: 'physically suggestive' decomposition underpinning the HEG model and the Heisenberg ansatz; the paper notes the q-averaged version fails for the homogeneous system.
  • domain assumption The Hubbard dimer ground-state solution transfers to the infinite chain, including the gap alpha U.
    Sec. XVIII A: the comparison with the Bethe ansatz gap (Fig. 23) is good at large U but overestimates at small U, quantified by the paper as missing long-range correlations.
invented entities (4)
  • Dynamical exchange-correlation hole rho_xc(r,r',r'';t) independent evidence
    purpose: Defined by Eq. (32); its Coulomb potential generates Vxc and it carries the sum rule (Eq. 40) and on-top constraint (Eq. 43).
    It is defined exactly from G^(2), and its constraints are checkable; the spectra generated by its Coulomb potential are compared with Bethe ansatz, DMRG, NRG, neutron, and photoemission data, giving falsifiable handles.
  • Temporal density rho(r,t) and temporal current density j(r,r';t)
    purpose: Diagonal/adjacent components of G used to build a continuity-like equation (Eq. 122) for the total spectral function.
    Sec. XII A: 'the temporal current density should be regarded as a construct and does not necessarily correspond to a physical current density.' No experimental handle is proposed.
  • Kinetic potential VK(r,t)
    purpose: Absorbs the divergence term so that i partial_t ln rho = VMF + Vxc + VK (Eq. 129).
    Sec. XII B concedes 'sufficiently accurate approximations for the kinetic energy as an explicit functional of the electron density are not yet available,' so the practical scheme is left incomplete.
  • Quasiparticle wave function psi_k(r,t) and effective field Xi_q(r,t) independent evidence
    purpose: Landau-like picture where each quasiparticle feels its own field; psi_k defined from G via Eq. (149).
    The norm bound sum_k' |G_kk'(t)|^2 <= 1 (Eq. 162) is a provable constraint, and the Xi_q model is tested against Na photoemission and cumulant results (Figs. 19, 20).

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Pith. "Pith review of Density-Functional Green Function Theory: Dynamical exchange-correlation field in lieu of self-energy." pith.science (2026). https://pith.science/paper/HOGYUB55

@misc{pith2026250519852,
  author       = {Pith},
  title        = {Pith review of: Density-Functional Green Function Theory: Dynamical exchange-correlation field in lieu of self-energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HOGYUB55}},
  note         = {Machine review of arXiv:2505.19852}
}
read the original abstract

The one-particle Green function of a many-electron system is traditionally formulated within the self-energy picture. A different formalism was recently proposed, in which the self-energy is replaced by a dynamical exchange-correlation field, which acts on the Green function locally in both space and time. It was found that there exists a fundamental quantity, referred to as the dynamical exchange-correlation hole, which can be interpreted as effective density fluctuations induced in a many-electron system when a hole or an electron is introduced into the system, as in photoemission and inverse photoemission experiments. The dynamical exchange-correlation potential is simply the Coulomb potential of this exchange-correlation hole, which fulfils a sum rule and an exact constraint, identical to those satisfied by the static exchange-correlation hole in density-functional theory. The proposed formalism has been applied to a number of model systems such as the half-filled one-dimensional Hubbard model, the one-dimensional antiferromagnetic Heisenberg model, and the single-impurity Anderson model. The dynamical exchange-correlation hole and field of the homogeneous electron gas have also been studied with the view of constructing a density-functional approximation such as the local-density approximation. The availability of simple but accurate approximations for the exchange-correlation potential would circumvent costly computations of the traditional self-energy. The formalism may also provide new perspectives and insights into the many-body problem.

Figures

Figures reproduced from arXiv: 2505.19852 by the authors.

Figure 1
Figure 1. FIG. 1. Top: The spherical average of the xc hole of the [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The real and imaginary parts of [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The real and imaginary parts of [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (25 more)
Figure 5
Figure 5. Figure 5: FIG. 5. The real and imaginary parts of [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Definition of the radial variables [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The real part of the spherical average of the exchange [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The spherical average of the exchange hole of the [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The real part of the spherical average of the cor [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Integration over [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The real part of the exchange potential (top), the [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 15
Figure 15. Figure 15: FIG. 15. The real part of the exchange potential (blue), the [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The momentum-broadening factor [PITH_FULL_IMAGE:figures/full_fig_p031_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. The lifetime broadening factor ( [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. The energy dispersion of the homogeneous electron [PITH_FULL_IMAGE:figures/full_fig_p032_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. The total spectral function of Na: experiment [76] [PITH_FULL_IMAGE:figures/full_fig_p032_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. The spectral function [PITH_FULL_IMAGE:figures/full_fig_p033_20.png]
Figure 22
Figure 22. Figure 22: FIG. 22. The [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 21
Figure 21. Figure 21: FIG. 21. The calculated total spectral functions of the 1D [PITH_FULL_IMAGE:figures/full_fig_p033_21.png]
Figure 23
Figure 23. Figure 23: FIG. 23. The calculated band gap, [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 22
Figure 22. Figure 22: The first term gives rise to the main peak cen [PITH_FULL_IMAGE:figures/full_fig_p035_22.png]
Figure 24
Figure 24. Figure 24: FIG. 24. The calculated angle-resolved spectra with improved [PITH_FULL_IMAGE:figures/full_fig_p036_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Real part of [PITH_FULL_IMAGE:figures/full_fig_p038_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Real part of [PITH_FULL_IMAGE:figures/full_fig_p039_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Dynamic structure factor of 1D spin- [PITH_FULL_IMAGE:figures/full_fig_p040_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Dynamic structure factor of a 100-site 1D spin- [PITH_FULL_IMAGE:figures/full_fig_p040_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Illustration of the single-impurity Anderson model [PITH_FULL_IMAGE:figures/full_fig_p042_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. Top: The real part of Vxc as a function of time. [PITH_FULL_IMAGE:figures/full_fig_p042_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. The zero-temperature particle-hole symmetric SIAM [PITH_FULL_IMAGE:figures/full_fig_p043_31.png]

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Works this paper leans on

98 extracted references · 68 canonical work pages

  1. [1]

    DefiningR= r′ −r=r ′ andR ′ =r ′′ −r=r ′′ as illustrated in Fig

    Spherical average of the exchange hole For the homogeneous electron gas,rmay be chosen as the origin of coordinate and set to zero. DefiningR= r′ −r=r ′ andR ′ =r ′′ −r=r ′′ as illustrated in Fig. 7 the exchange hole in Eq. (266) becomes fort <0 ρx(R, R′, θ;t <0)×iG0(R, t <0) =− 1 Ω2 X k≤kF e−ik·Re−iεkt X k′≤kF eiq·R′ ,(275) whereq=k−k ′. The spherical av...

  2. [2]

    (287) overR ′ =r ′′ yields A1 = 4π Ω2 X k′>kF e−ik′·R X k≤kF sin (qR′) qR′ e−iεkt ×M(q, εk′ −ε k, t).(293) It can be seen from Fig

    Spherical average of the correlation hole Spherical averagingA 1 in Eq. (287) overR ′ =r ′′ yields A1 = 4π Ω2 X k′>kF e−ik′·R X k≤kF sin (qR′) qR′ e−iεkt ×M(q, εk′ −ε k, t).(293) It can be seen from Fig. 12 that for a fixedk ′, the inte- gration overkis independent of the azimuthal angle so that A1 = 1 πΩ X k′>kF e−ik′·R Z kF 0 dkk2 × Z 1 −1 dy sin (qR′) ...

  3. [3]

    Exchange potential The exchange potential is the first moment of ρx inR ′, which from Eq. (277) is given by fort <0 Vx(R, t <0) = 1 iG0(R, t) 4π Ω2 X k,k′≤kF e−ik·Re−iεkt × Z dR′ sin(qR′) q .(307) Consider the integral overR ′ with positiveα→0: lim α→0 Z ∞ 0 dR′ sin(qR′)e−αR′ = 1 q .(308) One finds Vx(R, t <0) = 1 iG0(R, t) 4π Ω2 X k,k′≤kF e−ik·Re−iεkt 1 ...

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    (300) and (301)

    Correlation potential Similarly, the correlation potential is given by the first moment inR ′ of the spherical average of the correlation hole in Eqs. (300) and (301). The integral to be evalu- ated is I(k, k′, t) = Z dR′R′Q(k, k′, R′, t) = Z dR′ Z 1 −1 dy sin (qR′) q M(q, εk′ −ε k, t) = Z 1 −1 dy 1 q2 M(q, εk′ −ε k, t),(316) whereq=|k−k ′|= p k2 +k ′2 −2...

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    (339), a possible local-density approximation [2–4] for simple metals is ΞS q (r) = [1−γ( ρ)Z(ρ)] 1 2 3π2ρ(r) 2/3 −ε q ,(349) where ρ= 1 Ω Z dr ρ(r) (350) is the average density

    Local-density approximation From the ansatz in Eq. (339), a possible local-density approximation [2–4] for simple metals is ΞS q (r) = [1−γ( ρ)Z(ρ)] 1 2 3π2ρ(r) 2/3 −ε q ,(349) where ρ= 1 Ω Z dr ρ(r) (350) is the average density. BothγandZcan be calculated as functions of the electron gas density ρwithin theGW approximation [17] or using more accurate app...

  6. [6]

    This simplest cluster reveals features which are likely to be generic

    A four-site spin chain Before considering the infinite lattice, it is instructive to consider a minimal cluster with an even number of sites for which theV xc is nonzero and can be calculated analytically. This simplest cluster reveals features which are likely to be generic. As an illustration, one of the diagonal elements is [84] V xc 11,11(t >0) =−J ( ...

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    The analytic spinon V xc in the bonding-like basis and its approximation are shown in Fig. 25. Ignoring the high-excitation factorf 3 reduces the fine-structure details inV xc. Consequently, V xc BB,BB simplifies to a constant whereasV xc BC,CB oscil- lates with a single frequency and a constant magnitude, and all other components are negligible

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    The effect ofV xc onGcan be decomposed into a static (S) and a dynamic (D) part: X q V xc(k−q, t)G(q, t) = V S(k) +V D(k, t) G(k, t)

    Extrapolation to the infinite lattice For the infinite lattice it is natural to use a Bloch basis: G(k, t) = 1 N X ij Gij(t)e−ik(i−j) ,(415) V xc(k, t) = 1 N 2 X ij V xc ii,jj (t)e−ik(i−j) .(416) The equation of motion becomes i∂tG(k, t)− X q V xc(k−q, t)G(q, t) = 2sδ(t),(417) wheres=⟨ ˆSz i ⟩, which is independent of the lattice site due to translational...

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