Stoner transitions beyond mean-field in two-dimensional electronic systems: a diagrammatic Monte Carlo study

Andrey V. Chubukov; Nikolay V. Prokof'ev; Yueh-Chen Lee

REVIEW 3 major objections 7 minor 43 references

Stoner transitions beyond mean-field in two-dimensional electronic systems: a diagrammatic Monte Carlo study

T0 review · 3 major / 7 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Stoner ferromagnetism in 2D survives beyond mean field only for interactions with a finite momentum cutoff; band anisotropy then decides between a direct spin-valley transition and a two-step cascade.

desk verdict First DiagMC treatment of Stoner transitions with a momentum-cutoff interaction; the phase diagrams are credible and useful, but the neglected Z and m* renormalizations plus the high temperature keep the quantitative boundaries provisional. read the letter →

arxiv 2607.13675 v2 pith:TKWKOJ2P submitted 2026-07-15 cond-mat.str-el

Yueh-Chen Lee , Nikolay V. Prokof'ev , Andrey V. Chubukov This is my paper · ORCID

classification cond-mat.str-el

keywords Stonertransitionferromagnetismtwo-dimensionalelectrongasdiagrammaticMonteCarlobeyondmeanfieldvalleypolarizationisospinorderinggatescreening

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 asks whether the Stoner transition to ferromagnetism, well-studied in the mean-field ladder approximation, survives once non-ladder renormalizations are included in two-dimensional electron systems with one or two valleys. Using diagrammatic Monte Carlo, the authors show that at low density the particle-particle channel logarithmically suppresses the particle-hole vertex, so a transition occurs only if the interaction has a cutoff on momentum transfer of order the Fermi momentum. For one-valley parabolic bands the transition exists for q* near kF and is mostly first-order; a flatter k6 band allows the transition at larger q*. For two-valley systems, an isotropic band orders spin and valley together, while an elliptic band first forms a valley-polarized state and then, in a window of cutoffs, a spin transition into a spin-valley polarized state. If correct, these results explain why gate-screened, finite-range interactions, as in AlAs quantum wells and graphene multilayers, show Stoner-type isospin ordering while unscreened Coulomb does not.

What carries the argument

The key object is the fully dressed two-particle-irreducible four-fermion vertex K(q), whose ladder summation is cut off by the interaction U(q) = U0 Θ(q*−q), a step-function model of a gate-screened Coulomb interaction. The spin or valley susceptibility is written as χ = χ0/(1−ψ), where ψ = χ_2PI/χ0 is the ratio of the two-particle-irreducible contribution to the bare bubble; ψ plays the role of the l = 0 Landau parameter, and the Stoner instability is ψ = 1. Diagrammatic Monte Carlo evaluates the series for ψ and for the Luttinger–Ward free energy through sixth order in U0 and then extrapolates to infinite order with a Riesz–Cesàro resummation, allowing first- and second-order transitions

What would settle it

A self-consistent susceptibility calculation (including quasiparticle residue and mass renormalization) for a single parabolic valley at q* = 1.3 kF: if the divergence ψ = 1 at finite coupling disappears once renormalizations are included, the claimed second-order window is an artifact.

Watch

Extended reading notes

Core claim

The central discovery is that beyond the ladder approximation, a Stoner instability in 2D is not generic: it requires a finite momentum-transfer cutoff q* on the interaction. Concretely, the paper finds that for a single parabolic valley, a ferromagnetic transition appears only when q* is comparable to kF (roughly 1.0–1.5 kF), with a narrow second-order window near 1.3 kF and first-order behavior elsewhere; for q* tending to infinity (contact interaction) or q* greater than 1.5 kF there is no transition. For a k6 dispersion, which is flat at small momenta, the transition persists up to q* ≈ 2 kF and can be tuned from first- to second-order. In two-valley systems with isotropic dispersion, th

Load-bearing premise

The calculation assumes that quasiparticle residue and mass renormalization can be neglected; if these are important, the critical couplings and even the existence of the transition could change.

Editorial extensions

If this is right

In a one-valley parabolic 2D electron gas, ferromagnetism is absent for unscreened or long-ranged interactions; it appears only when the interaction range is sufficiently short (q* ≲ 1.5 kF), and the transition is first order except in a narrow second-order window around q* ≈ 1.3 kF.
Flattening the band (k6 dispersion) extends the Stoner window to larger q* (up to about 2 kF) and lets one switch the transition from first to second order by decreasing the cutoff.
For two-valley systems, valley and spin orders can be tied together (isotropic dispersion) or separated into a cascade (elliptic dispersion), with a valley-polarized phase appearing first and spin polarization occurring only for a restricted set of cutoffs.
The order of the transition and the range of q* where any order exists is controlled by the ratio q*/kF, so gate distance (which sets q*) acts as a tunable knob for isospin magnetism in experiments.

Reading between the lines

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

This suggests a concrete experimental test: sweeping the gate distance in a clean 2D electron system should turn the ferromagnetic or valley-polarized state on and off as q*/kF crosses the predicted windows; a measurement of the spin susceptibility or a thermodynamic probe versus gate distance would directly check the mechanism.
The same cutoff-controlled competition likely governs other isospin orders, such as valley-coherent and charge-density-wave states in multi-valley systems; the paper's restriction to valley polarization and ferromagnetism leaves those as open predictions.
Because the results rely on the ratio q*/kF, they should hold for other dispersions with band flattening, such as moiré flat bands; the steep high-energy part of the dispersion acts as an intrinsic cutoff, which may explain why flat-band systems show robust Stoner magnetism.
A direct check of the neglected quasiparticle residue and mass renormalization could be made by a self-consistent calculation at one representative point (e.g., q* = kF); if that shifts ψ = 1 to a different coupling or erases the divergence, the window boundaries reported here would need revision.

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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

I read this paper carefully. It is a serious DiagMC study of Stoner instabilities in 2D one- and two-valley systems, with the interaction cut off at a momentum q*. The central new content is the mapping of the transition in the (λ, q*/kF) plane: in a one-valley parabolic band ferromagnetism appears only for q* between about 1.0 and 1.5 kF, with a narrow second-order window inside a mostly first-order region; k6 dispersion pushes the transition to larger cutoffs; two-valley isotropic systems order directly into a spin-valley polarized state, while elliptic systems show valley polarization first and then, in a narrower window, a spin transition. That is a genuinely new set of results, and the qualitative story is plausible. The DiagMC machinery is state of the art: they sample the irreducible vertex and Luttinger-Ward functional to sixth order and resum with Riesz-Cesàro weights, with documented extrapolation checks. The appendix is detailed enough to be reproduced by a specialist. The paper also builds honestly on prior analytic and VMC work; the heavy self-citation is to the relevant papers by the same group, not padding. So there is real value here. The soft spots are the ones the stress-test note flags, and they are not manufactured. The biggest is the neglect of quasiparticle residue Z and mass renormalization. The paper computes ψ from bare Green's functions and uses ψ=1 as the Stoner condition, but the physical Landau parameter is Z^2 (m*/m) times the vertex. The authors say they "assume these renormalizations are not critically important" and "conjecture" the Stoner physics survives. That is an honest admission, but it is untested. If the prefactor is not close to one, every critical coupling and even the order of the transition can shift. A referee should ask for this to be estimated, or at least argued. Second, the temperature T=0.25 εF is high, and the authors admit the transition location may shift. Third, the first-order boundaries come from comparing free energies of fully polarized and unpolarized states, which is only an upper bound; they use Ref. 27 to argue the partially polarized window is narrow, which is reasonable but indirect. Finally, no code or data is provided, which will limit independent checks. These are all addressable, and none of them kills the central qualitative claim, but they mean the phase boundaries as plotted should be treated as indicative rather than precise. The audience is people working on AlAs quantum wells, multilayer graphene, and beyond-mean-field Stoner physics; they will get real value from these phase diagrams. I would send this to a serious referee. It deserves to be published after the self-energy and finite-temperature concerns are dealt with, or at least clearly flagged as limitations in the paper.

Referee Report

3 major / 7 minor

Summary. The paper uses diagrammatic Monte Carlo (DiagMC) to study Stoner-type isospin transitions in two-dimensional one-valley and two-valley electron systems, modeling the gate-screened Coulomb interaction by a step-function cutoff U(q)=U0Θ(q*−q). It computes the spin/valley susceptibility from the ratio ψ=χ2PI/χ0 with bare Green's functions, extrapolating sixth-order series with a Riesz-Cesàro resummation, and compares free energies of fully polarized versus unpolarized states. The main results are: in one-valley parabolic systems, ferromagnetism occurs only for q* near kF, mostly as a first-order transition with a narrow second-order window; a k6 dispersion extends the transition to larger q*; in two-valley isotropic systems there is a direct spin-valley polarized transition, while elliptic systems show a separate valley-polarized transition followed, in a restricted q* window, by a spin transition into the spin-valley polarized state. The paper concludes that the momentum cutoff of the interaction is a key control parameter.

Significance. If the results are correct, they provide a beyond-mean-field, numerically controlled phase diagram for Stoner transitions in two dimensions and identify the interaction cutoff as the parameter that determines whether a transition exists. The study is directly relevant to recent experiments on AlAs quantum wells and rhombohedral multilayer graphene. The paper is transparent about its methods: the appendix gives a detailed account of the DiagMC updates and resummation, the central quantities are computed from first principles without fitting experimental data, and the authors explicitly acknowledge the main assumptions and their potential impact. These strengths make the work a useful contribution even where the final claims remain conditional.

major comments (3)

[Sec. II.A, Eqs. (14)–(15)] The central object ψ is computed with bare Green's functions and the Stoner condition is set at ψ=1. However, in Fermi-liquid theory the physical spin susceptibility is χs=(m*/m)χ0/(1−λs^0), where λs^0=Z^2(m*/m)Γω. The calculation omits the multiplicative factors Z^2 and m*/m, as well as self-energy insertions; the text states "We assume that these renormalizations are not critically important and just neglect them" and later calls the neglect a "conjecture." Since the phase diagrams (Figs. 1–2) are drawn in the (λ,q*) plane and λc values reach ~5–14, an omitted factor that differs from 1 in this strong-coupling regime can shift every boundary and can even remove the ψ=1 crossing. The Ward identity ZΛ=1 is also not enforced, so the computed response functions are not guaranteed to be physical. This is a load-bearing approximation, not a presentation issue. The authors should either incor
[Sec. III.A, T=0.25 εF] The numerical runs are performed at T=0.25 εF, which the authors themselves describe as "not a truly low T" and as potentially shifting the transition. The abstract and summary present the results as Stoner transitions without a temperature qualifier, but both the existence of the transition and its first- versus second-order character can change with temperature. Moreover, at finite temperature the static susceptibility does not diverge; the red arrows in Figs. 6–9 indicating a "divergence" therefore require a well-defined extrapolation procedure, which is not described. To support the zero-temperature phase diagram, the authors should show that the boundaries are stable at lower T (e.g., T=0.1εF or a T→0 extrapolation) and should clarify how λc2 is extracted from the finite-T susceptibility data.
[Appendix A.2, Fig. 17] The infinite-order extrapolation rests on an Nmax=6 series with Riesz-Cesàro weights δ=6,8,10. The only convergence check shown is for q*=kF and λ=1, which is far below the transition region. The reported critical couplings reach λc1≈14 (e.g., Fig. 8(f)), where the series is far outside its radius of convergence, and the extrapolation error cannot be inferred from a single weak-coupling example. The phase diagrams in Figs. 1–2 have no error bars on λc1 or λc2, so it is difficult to judge whether features such as the narrow second-order window at q*≈1.3kF are numerical artifacts or robust. Please provide extrapolation curves and error estimates for representative strong-coupling points, or specify a quantitative criterion for when the resummation is trusted.

minor comments (7)

[Sec. II.A] Typo: "quadiparticle" should be "quasiparticle."
[Introduction] Typo: "fist-order" should be "first-order."
[Sec. II.B] "extrapolate to infinite order in the same was as for the susceptibility" — "was" should be "way."
[Appendix A.2] In the caption of Fig. 17, "Fig. A11" should be "Eq. (A11)."
[Sec. II, Eq. (4)] The step function is defined with Θ(x)=1 for x>0 and Θ(x)=0 for x<0; the value at x=0 is irrelevant but the boundary q=q* could be specified for clarity.
[Sec. III.A and Fig. 6] The blue arrows mark where the fully polarized free energy becomes lower than the unpolarized one; this is not the same as the first-order transition coupling in the presence of a partially polarized intermediate state. The text does acknowledge this, but the phase diagrams would benefit from a clearer notation distinguishing the two quantities.
[Sec. II, definition of λ] The text says λ is defined in terms of Πph(0) but then uses the value of the polarization bubble at finite T. This should be stated unambiguously, since all numerical couplings depend on this choice.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the DiagMC susceptibilities and free energies are independently computed; the acknowledged Z/m* neglect and self-citations are limiting or interpretive, not input–output identifications.

full rationale

The central derivation is self-contained. The paper computes the two-particle irreducible susceptibility χ2PI and the Luttinger–Ward free energy by explicit diagrammatic Monte Carlo sampling to sixth order in U0, followed by a resummation (Appendix A). The Stoner criterion ψ = 1 in Eq. (15) is the divergence of the computed resummed series, not a condition imposed by the input. No experimental data or VMC results are fitted, and no normalization is tuned to reproduce a known transition; the q* scan is a parameter study of a stated step-function model U(q). The paper does rely on several results from the same group (Refs. 16, 19, 21, 27) to justify the mean-field benchmark, to motivate which orders to compare, and to interpret the first-order transition via energy crossing. These are real, published analytical results used for context and comparison, and the DiagMC data themselves independently determine the reported transitions. The main acknowledged weakness is in Sec. II.A, where the authors state: 'We assume that these renormalizations are not critically important and just neglect them' and later 'we conjecture that we capture Stoner physics even if we neglect fermionic Z-factor and mass renormalization.' This neglect of Z and m* is an explicit approximation that could shift or remove transitions, but it is not circular: the computed ψ is not redefined in terms of the output phase diagram, and the failure of the Ward-identity combination ZΛ = 1 is an accuracy risk, not a logical reduction. Likewise, the paper's use of Ref. 27 to argue that the fully polarized energy crossing is close to the actual first-order transition is a supporting assumption, not a fit to the DiagMC results. Overall, there is no step in which a prediction equals its input by construction; the score reflects only minor, non-load-bearing self-citations and the acknowledged approximation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claims rest on five unproved inputs: negligible Z/m*, finite-T equivalence, step-cutoff model, RPA-like resummation, and resummation of a finite-order series. None is independently verified; the first and third are the most fragile.

free parameters (3)

Temperature for numerical runs (T) = 0.25 ε_F
Chosen for computational convenience in Sec. III.A; the authors call it 'not a truly low T'. Since the suppression of the Stoner vertex comes from low-energy particle-particle logarithms, this value may shift or wash out the transition.
Fermi momentum k_F = 0.1 (parabolic/two-valley), 0.5 (k6)
Representative low-density momenta used in the calculations; phase boundaries are only established at these two densities.
Riesz-Cesàro exponent δ = 6, 8, 10
Arbitrary g(n,Nmax) exponent in Appendix A.2; the extrapolated result depends on it and the spread is used as the error estimate, so the final numbers carry this hand-chosen uncertainty.

assumptions (5)

domain assumption Neglect of quasiparticle residue Z and mass renormalization.
Sec. II.A: 'We assume that these renormalizations are not critically important and just neglect them' and later 'conjecture that we capture Stoner physics even if we neglect fermionic Z-factor and mass renormalization.' This connects the computed ψ to the Landau parameter and the susceptibility denominator; if Z or m* corrections are large, λc and order may change.
ad hoc to paper T = 0.25 ε_F is representative of T=0.
Sec. III.A: 'This is not a truly low T, but we expect to capture the key physics already at this T'; used for the finite-T DiagMC runs.
domain assumption Step-function interaction U(q)=U0 Θ(q*−q) represents dual-gate screened Coulomb interaction.
Sec. II Eq. (4). Real gate screening is a smooth function of q; the step cutoff changes phase boundaries and is used as a toy model while arguing it captures qualitative physics.
standard math Fermi-liquid/RPA form χ = χ0/(1−ψ) with ψ computed from 2PI diagrams.
Sec. II.A Eqs. (13)-(15); relies on Landau theory and Ward identity ZΛ=1, and assumes a K1-only vertex. This is a standard but nontrivial resummation that may fail outside the weak-coupling regime.
ad hoc to paper Sixth-order DiagMC series extrapolation is convergent and the Riesz-Cesàro resummation is adequate.
Appendix A.2; no rigorous convergence proof; different δ values give an error band, not a guarantee.

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Cite this review

Pith. "Pith review of Stoner transitions beyond mean-field in two-dimensional electronic systems: a diagrammatic Monte Carlo study." pith.science (2026). https://pith.science/paper/TKWKOJ2P

@misc{pith2026260713675,
  author       = {Pith},
  title        = {Pith review of: Stoner transitions beyond mean-field in two-dimensional electronic systems: a diagrammatic Monte Carlo study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKWKOJ2P}},
  note         = {Machine review of arXiv:2607.13675}
}

read the original abstract

Stoner instabilities for fermions with repulsive interaction have been well studied within the mean-field (ladder) approximation. In this study, we consider two-dimensional electronic systems with one or two valleys and discuss a Stoner transition beyond the ladder approximation. At weak coupling, the corrections to the ladder approximation come predominantly from the renormalization of the particle-hole vertex in the particle-particle channel, and the lowest-order corrections are logarithmically singular in the low-density limit. To investigate the problem beyond the lowest order, we apply the diagrammatic Monte Carlo algorithm and treat ladder and non-ladder renormalizations on equal footing. We find that in a one-valley system, a Stoner transition to a ferromagnetism occurs at low density only if there is a cutoff on the momentum transfer carried by the interaction. In a two-valley system, the restriction is less severe. Here we find either a direct Stoner transition into a spin- and valley-polarized state, or a set of two Stoner transitions via an intermediate valley-polarized state. The pathway is controlled by the anisotropy of the fermionic dispersion.

Figures

Figures reproduced from arXiv: 2607.13675 by the authors.

**Figure 2.** FIG. 2. Phase diagram of a two-valley system with an isotropic dispersion relation (panel (a) ), and elliptic dispersion [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Summation of the ladder diagrams. The second term is what we define as [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: FIG. 4. The two-particle irreducible four point function [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. Ladder diagrams for the spin susceptibility in a Fermi-liquid theory. The black-shaded area is the side vertex [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. (a)-(e) Full susceptibility in units of its bare value for a single Fermi surface with isotropic parabolic dispersion [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7. (a)-(e) Full susceptibility in units of its bare value for a single circular Fermi surface with [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: FIG. 8. (a) - (e) Full susceptibility for a two valley system with isotropic dispersion with [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: FIG. 9. (a) - (e) Full susceptibility for a two valley system with elliptic dispersion with [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: FIG. 10. Full spin susceptibility for a transition from VP to SVP state with [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: FIG. 11. Ferromagnetic (blue) and valley (green) susceptibilities for [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: FIG. 12. Illustration of the Add Hartree/ [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]

**Figure 13.** Figure 13: FIG. 13. Demonstration of adding a Hartree bubble on a Green’s function which is a part of a Hartree bubble eliminates [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]

**Figure 14.** Figure 14: FIG. 14. Illustration of updates that swap the interaction lines (top panel) and the Green’s functions (bottom panel). [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]

**Figure 15.** Figure 15: FIG. 15. The Swap G update that results in intervalley exchange interaction. Here two distinct valleys are labeled [PITH_FULL_IMAGE:figures/full_fig_p030_15.png]

**Figure 16.** Figure 16: FIG. 16. An example of obtaining a term in Luttinger-Ward functional in a two-valley system with elliptic dispersion [PITH_FULL_IMAGE:figures/full_fig_p031_16.png]

**Figure 17.** Figure 17: FIG. 17. The ratio [PITH_FULL_IMAGE:figures/full_fig_p032_17.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 3 canonical work pages

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𝜆!# 𝜆!#𝜆!

The extended ranges of q∗ where either first- or second-order transitions occur for two complementary reasons: the DOS gets larger at small k, where the dispersion is flat, and the steep dispersion for k> 1 18 (a) (d) (e) (b) (c) (f) Notransition𝜆!" 𝜆!# 𝜆!#𝜆!" 𝜆!" 𝜆!# 𝜆!" FIG. 9. (a) - (e) Full susceptibility for a two valley system with elliptic dispersi...
[2]

𝜆!# 𝜆!" 𝜆!

The density of states (DOS) per spin projection is N(ϵ)= 1 4παc1/α 1 ϵ(α−1)/α (1) The Fermi energy isϵ F =ck 2α F and N(k F)=N F = 1 4παc 1 k2(α−1) F (2) At T =0, the dimensionless coupling relevant to Stoner transition is λ = U0NF = U0Πph(0), whereΠ ph(0) is 3 a static polarization bubble for free fermions in the limit of zero momentum. For our calculati...
[3]

The easiest way to see why the valley polarization transition in systems with elliptic dispersion is present at a smaller q∗ than in systems with isotropic dispersion is to calculate the kinetic energy that the interaction has to overcome to obtain the valley polarization. For a system with two spins and two valleys, the density of a component with a give...
[4]

Updates and measurement Suppose we want to calculate an observableO perturbatively using the diagrammatic expansion. We can write down the general expression as O= ∞X n=0 X Tn Z DTn(x1,...,x n)dx1dx2...dx n,(A1) where the sum is the order of the expansion, n and over diagrams of different topologies at a given n, Tn, and the integral is over internal vari...
[5]

Since exponential factors in Eq

Changeτ: This update is implemented only for the lowest-order contribution to susceptibility whenD is the product of G(k,τ ) and G(k,−τ ). Since exponential factors in Eq. 11 cancel out, this update is always accepted. The lowest-order LW diagrams are only Hartree and Fock diagrams, both of which only involve Green’s function atτ=0
[6]

Change k: This update is also implemented only at the lowest order. For susceptibility, we change the momenta of both Green’s functions fromktok ′, and the acceptance probability is P= nF(ϵk′,σ,γ)(1−n F(ϵk′,σ,γ)) nF(ϵk,σ,γ)(1−n F(ϵk,σ,γ)) .(A3) For the LW functional, we only change momentum for one G. For the Hartree diagram, the acceptance probability is...
[7]

This is similar to changingkexcept here we change the valley index from γ to−γ and substitute the samek into the dispersion of the other valley

Swap valley: This update is necessary for two-valley systems with elliptic dispersion and is allowed only in the bare susceptibility and lowest order Hartree diagram of Luttinger-Ward functional. This is similar to changingkexcept here we change the valley index from γ to−γ and substitute the samek into the dispersion of the other valley. For the bare sus...
[8]

Add Hartree: This update allows us to change the diagram order, although diagrams with any Hartree bubbles are not included in the measurements since the renormalization of G is not considered. We randomly choose a Green’s function in the present configuration, an imaginary timeτ∈ (0,β =1/T), and a momentum k from a circle of radius 1, which is the high-m...

Show all 43 references

[9]

Add Hartree

Remove Hartree: This update removes a Hartree bubble selected at random from the diagram. This update decreases the diagram by one and is complementary to the "Add Hartree" update. See Fig. 12 for illustration of both updates. In the same notation, the acceptance probability i...
[10]

14 the acceptance probability is given by P= Gσ,γ(k,τ k−τ i)Gσ,γ(k+p−q,τ j−τ k)Gσ,γ(q,τ l−τ j) Gσ,γ(k,τ j−τ i)Gσ,γ(p,τ k−τ j)Gσ,γ(q,τ l−τ k) .(A9)

SwapU: The proposal is to swap end points of two interaction lines connected by a Green’s function (this description is topological; interaction lines do not change their variables in the update, only Green’s functions are modified) With the imaginary time and momenta specifie...
[11]

We randomly choose an interaction line and propose to swap the destination points for two outgoing Green’s functions

Swap G: This update changes the number of fermion loops. We randomly choose an interaction line and propose to swap the destination points for two outgoing Green’s functions. Since the values of the Green’s functions remain the same, the acceptance probability depends only on ...
[12]

The series will oscillate with ever-increasing amplitude when they happen to be outside of the radius of convergence, in which case the 28 Add FIG

Resummation scheme In practice, one can only compute expansion coefficients up to a finite order,Nmax. The series will oscillate with ever-increasing amplitude when they happen to be outside of the radius of convergence, in which case the 28 Add FIG. 13. Demonstration of addin...
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For larger q∗, the region, over which p is integrated, covers the entire Fermi surface, and ˜Πph(0) is the same asΠ ph(0)

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

𝜆!# 𝜆!#𝜆!

The extended ranges of q∗ where either first- or second-order transitions occur for two complementary reasons: the DOS gets larger at small k, where the dispersion is flat, and the steep dispersion for k> 1 18 (a) (d) (e) (b) (c) (f) Notransition𝜆!" 𝜆!# 𝜆!#𝜆!" 𝜆!" 𝜆!# 𝜆!" FIG. 9. (a) - (e) Full susceptibility for a two valley system with elliptic dispersi...

[2] [2]

𝜆!# 𝜆!" 𝜆!

The density of states (DOS) per spin projection is N(ϵ)= 1 4παc1/α 1 ϵ(α−1)/α (1) The Fermi energy isϵ F =ck 2α F and N(k F)=N F = 1 4παc 1 k2(α−1) F (2) At T =0, the dimensionless coupling relevant to Stoner transition is λ = U0NF = U0Πph(0), whereΠ ph(0) is 3 a static polarization bubble for free fermions in the limit of zero momentum. For our calculati...

[3] [3]

The easiest way to see why the valley polarization transition in systems with elliptic dispersion is present at a smaller q∗ than in systems with isotropic dispersion is to calculate the kinetic energy that the interaction has to overcome to obtain the valley polarization. For a system with two spins and two valleys, the density of a component with a give...

[4] [4]

Updates and measurement Suppose we want to calculate an observableO perturbatively using the diagrammatic expansion. We can write down the general expression as O= ∞X n=0 X Tn Z DTn(x1,...,x n)dx1dx2...dx n,(A1) where the sum is the order of the expansion, n and over diagrams of different topologies at a given n, Tn, and the integral is over internal vari...

[5] [5]

Since exponential factors in Eq

Changeτ: This update is implemented only for the lowest-order contribution to susceptibility whenD is the product of G(k,τ ) and G(k,−τ ). Since exponential factors in Eq. 11 cancel out, this update is always accepted. The lowest-order LW diagrams are only Hartree and Fock diagrams, both of which only involve Green’s function atτ=0

[6] [6]

Change k: This update is also implemented only at the lowest order. For susceptibility, we change the momenta of both Green’s functions fromktok ′, and the acceptance probability is P= nF(ϵk′,σ,γ)(1−n F(ϵk′,σ,γ)) nF(ϵk,σ,γ)(1−n F(ϵk,σ,γ)) .(A3) For the LW functional, we only change momentum for one G. For the Hartree diagram, the acceptance probability is...

[7] [7]

This is similar to changingkexcept here we change the valley index from γ to−γ and substitute the samek into the dispersion of the other valley

Swap valley: This update is necessary for two-valley systems with elliptic dispersion and is allowed only in the bare susceptibility and lowest order Hartree diagram of Luttinger-Ward functional. This is similar to changingkexcept here we change the valley index from γ to−γ and substitute the samek into the dispersion of the other valley. For the bare sus...

[8] [8]

Add Hartree: This update allows us to change the diagram order, although diagrams with any Hartree bubbles are not included in the measurements since the renormalization of G is not considered. We randomly choose a Green’s function in the present configuration, an imaginary timeτ∈ (0,β =1/T), and a momentum k from a circle of radius 1, which is the high-m...