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Brueckner-Goldstone perturbation theory for the half-filled Hubbard model in infinite dimensions

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Daniel Ruhl, Florian Gebhard

Strongly Correlated Electrons · Submitted 28 Feb 2006 · 21 references

Pith's verdict: A careful fourth-order perturbative calculation that corrects a known energy coefficient; the main caveat is an asserted locality classification that deserves a more explicit proof.

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CitationDaniel Ruhl, Florian Gebhard. "Brueckner-Goldstone perturbation theory for the half-filled Hubbard model in infinite dimensions." arXiv:cond-mat/0602654 (2006). https://pith.science/paper/Q7LYAQZF
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The paper

The paper claims

Brueckner–Goldstone perturbation theory fixes the fourth-order ground-state energy of the half-filled Hubbard model in infinite dimensions to $7.2475\times10^{-6}$ per site (in units of $t=1$), correcting an earlier Feynman–Dyson value…

The reading

The paper calculates the ground-state energy, momentum distribution, and quasiparticle weight of the half-filled Hubbard model on an infinite-dimensional Bethe lattice through fourth order in the interaction $U$, using Brueckner–Goldstone perturbation theory. Its central result is the fourth-order coefficient of the ground-state energy: $E_0^{(4)}/L = 7.2475\times10^{-6}$ (in units of the hopping amplitude $t=1$), a tiny positive number that corrects an earlier Feynman–Dyson calculation and agrees with quantum Monte Carlo data. Because the coefficient is so small, earlier attempts to extract it from moments of the single-particle spectral function produced the wrong value; the linked-cluster expansion used here gives it directly as a Taylor coefficient. If correct, the paper provides a controlled weak-coupling benchmark for approximate methods aimed at the Mott transition.

What carries the argument

The central object is the Brueckner–Goldstone linked-cluster expansion of the ground-state energy, expressed as a sum over connected Goldstone diagrams. Its power here comes from two facts valid in infinite dimensions: the joint density of states factorizes, $D_q(\epsilon_1,\epsilon_2) = D_0(\epsilon_1)D_0(\epsilon_2)$ for almost all momentum transfers $q$, and the density of states is semi-circular on the Bethe lattice. This reduces the momentum sums to products of particle and hole propagators $F_p(\lambda)$, $F_h(\lambda)$, with the energy denominators handled by the Feynman trick $1/x = \int_0^\infty d\lambda\, e^{-\lambda x}$. The fourth-order calculation enumerates all 81 diagram pairs, discards disconnected and vanishing ones, and identifies the three classes that remain non-local even in infinite dimensions—the self-energy insertion $(a)_\sigma\times(i)_{-\sigma}$ and two classes of Pauli-forbidden diagrams $(d)_\sigma\times(h)_{-\sigma}$ and $(e)_\sigma\times(i)_{-\sigma}$—whose contributions must replace the corresponding local-diagram terms (Eqs. 43–52).

Extended reading notes

Core claim

The paper's discovery is that Brueckner–Goldstone perturbation theory, applied to the half-filled Hubbard model in infinite dimensions, yields the fourth-order ground-state energy coefficient $E_0^{(4)}/L = 7.2475 \times 10^{-6}$ for a semi-circular density of states (with $t=1$, bandwidth $W=4$), i.e., $0.00046384\,U^4/W^3$ in bandwidth units. This replaces the fourth-order coefficient derived in Ref. [7] from Feynman–Dyson self-energy data, which had the wrong value (indeed, the wrong sign) because the self-energy was not accurate enough to determine the tiny coefficient from spectral moments. The corrected coefficient is consistent with quantum Monte Carlo estimates $E_0^{(4),\mathrm{QMC}}/L = (5\pm2)\times10^{-4}$. In addition, the paper derives the momentum distribution up to fourth order as a functional derivative of the ground-state energy with respect to the bare dispersion, $n_\sigma(\epsilon(k)) = \frac12\, \delta E_0/\delta \epsilon(k)$, and obtains the quasiparticle weight $Z(U) = 1 - 0.0817484\,U^2/t^2 + 0.00380158\,U^4/t^4$, in agreement with previous Feynman–Dyson results.

Load-bearing premise

The calculation assumes that in infinite dimensions the only fourth-order Goldstone diagrams that keep a momentum-conservation constraint are the self-energy insertion and two specific Pauli-forbidden diagram classes; if any other diagram fails to factorize into products of single-particle densities of states, the fourth-order coefficient $7.2475\times10^{-6}$ changes.

Editorial extensions

If this is right

  • The corrected coefficient $E_0^{(4)}/L = 7.2475\times10^{-6}$ (with $t=1$) gives a weak-coupling benchmark for dynamical mean-field theory solvers and for quantum Monte Carlo at least up to $U \approx W/2$.
  • The momentum distribution to fourth order, obtained via the functional derivative $n_\sigma(\epsilon)=\frac12\,\delta E_0/\delta\epsilon$, is now available as a Taylor series and agrees with Rayleigh–Schrödinger perturbation theory.
  • The quasiparticle weight series $Z(U)=1-0.0817484\,U^2/t^2+0.00380158\,U^4/t^4$ confirms the Feynman–Dyson result and delimits the reliable weak-coupling range to $U \lesssim W/2$.
  • Calculating the sixth-order term is impractical: the number of local diagrams alone is of order $200^2$ and the required integrals become five-fold in $\lambda$.

Reading between the lines

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

  • If the factorization of the joint density of states applies to all higher orders beyond the three non-local classes identified here, the $\lambda$-integral representation should extend to sixth order, making an automated enumeration of Goldstone diagrams a viable route to the next coefficient (the authors estimate the diagram count but do not pursue it).
  • The unusually small fourth-order coefficient suggests the ground-state energy series is remarkably flat in $U$ at weak coupling; this may explain why the Mott transition, which occurs at strong coupling, is invisible to low-order perturbation theory and why energy-based benchmarks are insensitive to the quality of an approximate solution.
  • One could test the functional-derivative relation for the double occupancy $D(U)=\partial E_0/\partial U$ in the same linked-cluster framework, yielding a direct Taylor series for a local correlation function that complements the momentum-distribution result.
Original abstract

We use Brueckner-Goldstone perturbation theory to calculate the ground-state energy of the half-filled Hubbard model in infinite dimensions up to fourth order in the Hubbard interaction. We obtain the momentum distribution as a functional derivative of the ground-state energy with respect to the bare dispersion relation. The resulting expressions agree with those from Rayleigh-Schroedinger perturbation theory. Our results for the momentum distribution and the quasi-particle weight agree very well with those obtained earlier from Feynman-Dyson perturbation theory for the single-particle self-energy. We give the correct fourth-order coefficient in the ground-state energy which was not calculated accurately enough from Feynman-Dyson theory due to the insufficient accuracy of the data for the self-energy, and find a good agreement with recent estimates from Quantum Monte-Carlo calculations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,331 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.25 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We use Brueckner-Goldstone perturbation theory to calculate the ground-state energy of the half-filled Hubbard model in infinite dimensions up to fourth order in the Hubbard interaction. We obtain the momentum distribution as a functional derivative of the ground-state energy with respect to the bare dispersion relation. The resulting expressions agree with those from Rayleigh-Schroedinger perturbation theory. Our results for the momentum distribution and the quasi-particle weight agree very well with those obtained earlier from Feynman-Dyson perturbation theory for the single-particle self-energy. We give the correct fourth-order coefficient in the ground-state energy which was not calculated accurately enough from Feynman-Dyson theory due to the insufficient accuracy of the data for the self-energy, and find a good agreement with recent estimates from Quantum Monte-Carlo calculations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We use Brueckner-Goldstone perturbation theory to calculate the ground-state energy of the half-filled Hubbard model in infinite dimensions up to fourth order in the Hubbard interaction. contribution 0.69 0.97 0.15 asserted 0.99
s1 We obtain the momentum distribution as a functional derivative of the ground-state energy with respect to the bare dispersion relation. method 0.97 0.95 0.06 asserted 1.00
s2 The resulting expressions agree with those from Rayleigh-Schroedinger perturbation theory. result 0.94 0.71 0.12 asserted 1.00
s3 Our results for the momentum distribution and the quasi-particle weight agree very well with those obtained earlier from Feynman-Dyson perturbation theory for the single-particle self-energy. result 0.96 0.73 0.19 asserted 1.00
s4 We give the correct fourth-order coefficient in the ground-state energy which was not calculated accurately enough from Feynman-Dyson theory due to the insufficient accuracy of the data for the self-energy, and find a good agreement with recent estimates from Quantum Monte-Carlo calculations. result 0.78 0.95 0.46 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.93.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Second-order Goldstone diagram. 3.1. Second order The ground-state energy to second order is given by E (2) 0 = hFS|Dˆ (E (0) 0 − Tˆ) −1Dˆ |FSic , (20) to which only a single diagram contributes, see
Figure 2. Diagram parts for one spin species for the fourth-order Goldstone diagrams. In order to generate all possible diagrams, we draw all 81 possible pairs of diagrams from (a)↑ × (a)↓, (a)↑ × (b)↓, . . . , to (i)↑ × (i)↓, and connect all vertices with horizontal interaction lines. As an example, the diagram (a)↑ × (i)↓ is shown in
Figure 3. Fourth-order Goldstone diagram (a)σ × (i)−σ with ‘self-energy insertion’. therefore vanish due to the factor n (0) σ (ǫ)(1 − n (0) σ (ǫ)). Nevertheless, it is quite tedious to work out the remaining 66 diagrams even when the spin-flip symmetry is employed. We proceed differently. If all 66 diagrams were local diagrams, we could ignore the momentum conservation at the vertices as in the case of the second-order diagram in Sect. 3.1. The contribution of all non-vanishing connected diagrams would be Ee(4) 0 L = (−1)3 Z ∞ 0 dλ1 Z ∞ 0 dλ2 Z ∞ 0 dλ3 [a + b + c + d + e + f + g + h + i] 2 −
Figure 4. ‘Pauli-forbidden’ fourth-order Goldstone diagram (d)σ × (h)−σ. The two hole-lines in the right part of the figure carry the same momentum. Second, different contributions result from the ‘Pauli-forbidden diagrams’. The first class of Pauli-forbidden diagrams is generated by (d)σ ×(h)−σ, as shown in
Figure 5. ‘Pauli-forbidden’ fourth-order Goldstone diagram (e)σ × (i)−σ. The two hole lines in the left part of the figure carry the same momentum. The second class of Pauli-forbidden diagrams results from (e)σ × (i)−σ, as shown in
Figure 6. Second-order contribution to the momentum distribution for ǫ > 0
Figure 7. Second-order contribution to the momentum distribution for ǫ > 0. In
Figure 8. Quasi-particle weight as a function of the interaction strength from (78). We show Z(U) as a function of U/W in

Figures reproduced from arXiv:cond-mat/0602654 by the authors.

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Discussion

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