Engineering correlated phases through manipulation of Van Hove singularities

Chris A. Hooley; Luke C. Rhodes; Maria Ramirez; Thomas P. Sheerin

REVIEW 3 major objections 5 minor 71 references

Engineering correlated phases through manipulation of Van Hove singularities

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The type of Van Hove singularity — logarithmic, power-law, or split — selects the correlated ground state of a square-lattice Hubbard model, and tiny doping destroys the ferromagnetic state.

desk verdict Solid pRG/TUFRG comparison of Van Hove regimes, but the headline doping-fragility of ferromagnetism is likely a k-grid artifact and should not be taken as a quantitative design rule. read the letter →

arxiv 2608.07714 v1 pith:V7ASCL4B submitted 2026-08-07 cond-mat.str-el cond-mat.mtrl-sci

Thomas P. Sheerin , Maria Ramirez , Chris A. Hooley , Luke C. Rhodes This is my paper · ORCID

classification cond-mat.str-elcond-mat.mtrl-sci

keywords VanHovesingularityhigher-orderHubbardmodelfunctionalrenormalizationgroupparquetferromagnetismunconventionalsuperconductivityspin-densitywave

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 argues that the functional form of a Van Hove singularity — ordinary logarithmic, higher-order power-law, or split into two ordinary saddle points — is a practical control knob for the correlated ground state of a two-dimensional Hubbard model. Tuning the third-neighbor hopping $t_3$ through $t_{3c}=(t_1-2t_2)/4$ changes the density-of-states divergence from $\ln(W/|\xi|)$ to $|\xi|^{-1/4}$ and then back to logarithmic at two points, and the predicted weak-coupling phases rearrange accordingly: ferromagnetism, $d$-wave and $p$-wave superconductivity, and spin-density waves trade places. The paper further claims that the ferromagnetic state at Van Hove filling is extremely fragile, disappearing for Fermi-level shifts as small as $0.001\,t_1$, with electron and hole doping selecting different ordered states. The reason to care is that it turns phase engineering into a band-structure problem: instead of fine-tuning interactions, one can tune the shape and position of Van Hove singularities.

What carries the argument

The load-bearing object is the non-interacting dispersion $\xi(\mathbf{k})=-2t_1(\cos k_x+\cos k_y)-4t_2\cos k_x\cos k_y+2t_3(\cos 2k_x+\cos 2k_y)-\mu$. Increasing $t_3$ to $t_{3c}=(t_1-2t_2)/4$ turns each quadratic saddle point at the X points into a higher-order (cusp $A_3$) saddle, giving a DOS divergence $|\xi|^{-1/4}$; for $t_3>t_{3c}$ each X saddle splits into two ordinary saddle points at $\mathbf{P}=(\pi,\pm k_P)$ with $k_P=\arccos[(t_1-2t_2)/(4t_3)]$. The argument is carried by two one-loop weak-coupling RG schemes: a hot-spot parquet RG that flows a small set of interaction couplings from patches around the Van Hove points, and a truncated-unity functional RG that integrates the full Brillouin zone with a static vertex and a form-factor cutoff. The ordered phase is identified by which vertex combination diverges and by the associated susceptibility rates or gap symmetry.

What would settle it

A concrete test: compute the Van-Hove-filling phase diagram with a method that includes self-energy feedback and frequency-dependent vertices, or with a much larger form-factor cutoff, and see whether the ferromagnetic region at $t_3=0$ still disappears when the Fermi level is shifted by $0.001\,t_1$; if a quantitatively reliable calculation finds ferromagnetism stable to that doping shift, the central claim about the fragility of the ferromagnetic state is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the type of Van Hove singularity at the Fermi level determines which weak-coupling ordering tendency dominates in the $t_1$-$t_2$-$t_3$ square-lattice Hubbard model. At Van Hove filling, the single logarithmic VHS produces competition between $B_{1g}$ ($d$-wave) superconductivity and ferromagnetism, with a sizeable Fermi-liquid region; the higher-order VHS at $t_3=t_{3c}$ stabilizes ferromagnetism over most of the phase diagram; and the split-VHS regime favours a triplet $E_u$ ($p$-wave) superconducting state competing with ferromagnetism, with a small-wavevector spin-density wave appearing in the full-Brillouin-zone calculation. Away from Van Hove filling, the ferromagnetic state is destroyed by tiny doping, while the other phases persist and acquire an energy asymmetry: in the HOVHS case, electron and hole doping select different ground states because the power-law density of states is asymmetric about the singularity.

Load-bearing premise

The results depend on the assumption that the simplified renormalization-group schemes — which leave out some energy dependence and self-energy effects and, in the hot-spot version, keep only small patches of the Fermi surface — still predict the same winning ordered states as a fuller calculation at the interaction strengths used.

Editorial extensions

If this is right

In the single-VHS regime, $d$-wave superconductivity and ferromagnetism compete, and the Fermi-liquid region at intermediate $t_2/t_1$ is widened in the full-Brillouin-zone calculation compared with pRG.
In the HOVHS regime, ferromagnetism dominates most of the $(t_2/t_1, U/t_1)$ phase diagram at Van Hove filling, consistently in both RG schemes.
In the split-VHS regime, $p$-wave ($E_u$) superconductivity competes with ferromagnetism, and a small-$\mathbf{q}$ spin-density wave with $\mathbf{Q}=(\delta,\delta)$, $\delta\lesssim\pi/10$, appears only in the full-Brillouin-zone calculation.
The ferromagnetic state at Van Hove filling is destroyed by doping shifts of order $0.001\,t_1$; the phases that replace it depend on whether the VHS sits above or below the Fermi level, especially in the HOVHS case.
The phase diagram is smooth in $t_3$, so the influence of the higher-order VHS persists over a range of $t_3$ values rather than only at the critical point.

Reading between the lines

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

An implication the authors leave implicit is a design rule for materials: if a single logarithmic VHS sits at the Fermi level, sub-meV chemical-potential control is required to realize ferromagnetism, whereas the HOVHS and split-VHS regimes tolerate larger doping windows for their non-ferromagnetic phases.
The electron-hole asymmetry near a HOVHS suggests that a real material with a higher-order saddle slightly off the Fermi level could be steered into either a $p$-wave superconducting state or a small-$\mathbf{q}$ spin-density wave simply by changing the sign of the doping — a testable prediction for strained or gated quasi-2D metals.
A natural next calculation is to repeat the TUFRG flow with frequency-dependent vertices and self-energy feedback; if the ordering changes with $t_3$ survive, the design rule is not an artefact of the static-vertex truncation.

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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

Worth a look if you work on Van Hove physics. The paper does a careful pRG/TUFRG comparison for the t1-t2-t3 Hubbard model across single, higher-order, and split VHS regimes, and the authors are honest about their approximations. But the headline claim—that ferromagnetism dies for Fermi-level shifts as small as 0.001 t1—doesn't survive scrutiny of the numerics. The TUFRG loop integrals use a 40x40 k-grid. For t2/t1 = 0.375, the saddle-point mass is 0.125 t1, so the energy resolution near the saddle is (pi/40)^2/(2*0.125) ~ 0.025 t1, an order of magnitude larger than the claimed detuning. The discretized DOS cannot represent the logarithmic divergence below that scale. The FM suppression in Fig. 8(a) is therefore likely a grid artifact, not a physical effect. The authors even note that 3200x3200 k-points were needed to smoothen the split-VHS boundaries; the doping sweeps do not appear to have that resolution. Since the doping sensitivity is the paper's central design-rule claim, this is a load-bearing issue.

What is genuinely new: the continuous t3 phase diagram, the pRG-vs-TUFRG comparison across all three regimes, and the electron/hole asymmetry in the HOVHS doping phase diagram. The flow equations are derived in appendices, and the two methods agree broadly at Van Hove filling—that is real independent support for the qualitative t2 and t3 dependence. The authors are also candid about the hot-spot approximation, the one-loop truncation, and the unclear SDW origin. That honesty counts.

Other soft spots: no shipped code or data, no error bars on phase boundaries, and the high-U appendix results are presented despite being outside the stated validity range. Those are minor for a theory paper but add to the difficulty of pinning down whether the small-detuning behavior is real.

Bottom line: the paper deserves a serious referee. The comparison study is valuable and the phase diagrams at Van Hove filling are likely robust. But the doping-fragility result should be re-examined with higher k-resolution or backed by pRG detuning calculations before it becomes a design rule.

Referee Report

3 major / 5 minor

Summary. The paper studies the t1-t2-t3 square-lattice Hubbard model with the Fermi level pinned to the Van Hove singularity (VHS), using hot-spot parquet RG (pRG) and truncated-unity functional RG (TUFRG). By varying t3, the authors tune the VHS from an ordinary logarithmic singularity through a higher-order (power-law) singularity at t3 = (t1 - 2t2)/4, and then into a split-VHS regime with two logarithmic singularities per X point. They map out (t2/t1, U/t1) and (t3/t1, U/t1) phase diagrams at Van Hove filling and compare pRG with TUFRG, reporting qualitative agreement with some differences. They then study doping, parametrized by an energy shift ΔE_VHS, and find that ferromagnetic order is extremely sensitive to detuning, disappearing for shifts as small as 0.001 t1 in the single-VHS case. The central claim is that the functional form of the VHS divergence controls the correlated ground state and that the ferromagnetic state at Van Hove filling is unstable to very small Fermi-level shifts.

Significance. If established, the paper would provide a concrete design principle: tuning the VHS type via third-neighbor hopping can select between d-wave superconductivity, ferromagnetism, p-wave superconductivity, and small-Q spin-density-wave order in a single microscopic model. The paper is also valuable as a head-to-head benchmark of hot-spot pRG against full-band-structure TUFRG for the same model, including an analytic derivation of inter-patch susceptibilities in Appendix A and reproduction of known single-VHS TUFRG results. The strengths include transparent reporting of truncation choices (frequency-independent vertex, no self-energy feedback, eight-unit-cell form-factor cutoff, 80×80/40×40 grids), the use of the public divERGe package, and the explicit acknowledgment of the high-U limitations of one-loop RG. However, a headline quantitative claim about doping fragility is not supported by the numerical resolution as presented, which is a load-bearing issue for the paper's central message.

major comments (3)

[§IV B, Fig. 8(a)] The claim that ferromagnetism is suppressed for ΔE_VHS/t1 as small as 0.001 is not supported by the TUFRG numerics as presented. The loop susceptibilities are integrated on a 40×40 k-grid (Sec. III B), giving Δk ≈ π/40. With t2/t1 = 0.375 and t3 = 0, Eq. (4) gives m_X^+ = (t1 - 2t2)/2 = 0.125 t1, so the characteristic energy scale of the saddle-point neighborhood is (Δk)^2/(2m_X^+) ≈ 0.025 t1, about 25 times larger than the smallest detuning shown. A detuning of 0.001 t1 is therefore below the energy resolution with which the discretized density of states can represent the logarithmic singularity; the disappearance of the FM region may be a finite-grid artifact rather than a property of the continuum model. Since this result underlies the '1 meV' design statement and the asymmetry discussion, I ask for either (i) a convergence check with nkf = 80 or finer for the doping sweeps, (ii) a pRG doping calculation, where the DOS is treated analytically, to confirm the fragility, or (iii) a revised quantitative statement that is compatible with the grid resolution.
[Sec. V and Figs. 5, 7, 8] The paper states in Sec. V that one-loop weak-coupling RG can be trusted only up to U of roughly half the bandwidth, i.e. U ≈ 4t1 for a bandwidth of 8t1, and that at larger U results are 'likely to be significantly modified by higher-loop terms'. Nevertheless, the main-text phase diagrams in Figs. 5, 7, and 8 extend to U = 5t1 and display phase regions in the range 4t1 < U < 5t1 as if they were predictions. These high-U regions are outside the stated domain of validity of the method. The authors should either restrict the main-text diagrams to the trusted range, or clearly hatched/mark the untrusted regions and refrain from drawing conclusions from them.
[§IV A 2 and Fig. 7] The claimed 'broad qualitative agreement' between pRG and TUFRG is weakened by the complete absence in TUFRG of the large E_u (p-wave) superconducting region that pRG predicts in the split-VHS case, at the same t2/t1 used for the doping study. The authors attribute this to the pRG's neglect of the high-energy DOS step near the X-point band maxima, but no quantitative test of this explanation is provided. Since the paper uses pRG as an independent cross-check of the TUFRG phase diagrams, this qualitative discrepancy should be either resolved by a controlled comparison (e.g., including the step contribution in an extended pRG) or explicitly framed as a limitation that prevents relying on pRG for the split-VHS regime.

minor comments (5)

[§IV A 1] Typo: 'TURFG' should be 'TUFRG' in the sentence 'We find broad qualitative agreement between the pRG and TURFG phase diagrams'.
[§IV B] The sentence 'This would accord with the observation that the ferromagnetic region in Fig. 1(c) moves to more negative ΔE_VHS as U increases' appears to refer to Fig. 8(c), not Fig. 1(c), which is a non-interacting DOS plot and contains no ferromagnetic region.
[§III B] The phrase '3200×3200 k-points in the Brillouin zone' is unclear: with a 80×80 bosonic q-grid and a 40×40 loop grid, the natural count is 80×40 = 3200 k-points, not 3200×3200. Please clarify the intended meaning.
[Appendix D] The caption of Fig. D2(a) states 'U≳8 eV'; this should be 'U≳8 t1' to maintain dimensionless notation.
[Appendix B] The notation '2K1-FFLO' in the split-VHS rate table is not defined in the text; please define K1 or introduce the notation before use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all predicted phases are genuine outputs of the RG flows, not constructions from fitted target states.

full rationale

The central predictions—ordered-state identity versus t3, t2, U, and doping—are obtained by integrating one-loop RG flow equations (pRG: Eqs. (13), (18), (19); TUFRG: Eq. (21)) from the bare Hubbard interaction U, with no target phase inserted into the flows. The different Van Hove singularity types (logarithmic, power-law |ξ|^{-1/4}, and split) are computed analytically from the non-interacting dispersion (Eq. (2)) and enter as inputs, not as fitted outputs. The pRG d-functions and ordering rates are derived from the non-interacting susceptibilities in Eqs. (10) and (16) and Appendix A; the TUFRG calculation is an independent full-Brillouin-zone integration. No parameter is adjusted to reproduce any desired ordered state: the cutoff W=0.08t1, the form-factor truncation, and the k-grid sizes are stated numerical approximations, and the single-VHS TUFRG critical-scale linecut is benchmarked against existing literature (Refs. [35,36,38,43,47–49]) rather than used as a fit target. The claimed doping sensitivity of ferromagnetism is a numerical output whose resolution caveats are a matter of numerical accuracy, not circularity. Thus the derivation chain is self-contained: inputs are the model parameters and the one-loop RG scheme, and the phases emerge from solving those equations.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model and methods are standard; the central claim depends on one-loop weak-coupling RG, the hot-spot approximation for pRG, static-vertex/no-self-energy TUFRG, and hand-chosen scales (W, the subleading coefficient, and the form-factor cutoff). No free parameters are fitted to the target phases, and no new entities are introduced.

free parameters (3)

High-energy cutoff W = 0.08 t1
Set to the energetic separation between the Fermi level and the band maxima at X in the split-VHS case, and used for all regimes. Authors state it may affect Fermi-liquid region sizes but not conclusions.
Sub-leading logarithmic coefficient C of Pi_pp^0 = 0
The coefficient is unknown and of order one; the authors set it to zero after arguing that in the asymptotic RG regime C^2 is negligible compared to 4y. This affects intermediate-scale d-functions.
TUFRG form-factor cutoff = 8 unit cells (197 bonds/site)
Truncation of the real-space form-factor basis; if too small, longer-range vertex features and some ordering tendencies could be missed.

assumptions (4)

domain assumption The one-loop parquet and TUFRG flow equations capture the leading weak-coupling ordering competition when the vertex is truncated at four-fermion terms and frequency dependence is dropped.
Central results rely on these truncations; see Section III A-B, Eqs. (13), (19)-(22).
domain assumption The hot-spot approximation retains only patches around Van Hove points; all other Fermi-surface processes are neglected.
Needed for the pRG calculations; the paper itself shows this misses nesting-driven antiferromagnetism and overestimates triplet superconductivity in some regions.
ad hoc to paper The high-energy cutoff W can be set to 0.08 t1 for all regimes without changing the phase assignments.
Authors state this is a simplistic choice and argue it only affects Fermi-liquid region sizes; Section III A 1.
domain assumption A coupling exceeding 1000 t1 before the scale 10^-5 t1 signals an ordering transition, and absence of divergence indicates a Fermi liquid.
Used as the stopping criterion in both pRG and TUFRG; it is an operational definition of the phase boundary.

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Pith. "Pith review of Engineering correlated phases through manipulation of Van Hove singularities." pith.science (2026). https://pith.science/paper/V7ASCL4B

@misc{pith2026260807714,
  author       = {Pith},
  title        = {Pith review of: Engineering correlated phases through manipulation of Van Hove singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7ASCL4B}},
  note         = {Machine review of arXiv:2608.07714}
}

abstract

Controlling the ordered phases of correlated electron systems remains a central challenge in quantum materials design. Divergences in the electronic density of states, known as Van Hove singularities (VHSs), are one obvious route to such control. It is clear from recent work that the exact functional form of these divergences can profoundly affect which phases are realized; a full picture, however, remains elusive. In this work, we use both the hot-spot parquet renormalization group and the truncated-unity functional renormalization group to theoretically study the emergent correlated states of a two-dimensional square-lattice Hubbard model with VHSs at or near the Fermi level. By varying a single hopping parameter, $t_3$, we are able to change the strength of the VHS divergence in the density of states from logarithmic (for $t_3 < t_{3c}$) to power-law (for $t_3 = t_{3c}$). Further increase of $t_3$ ($t_3 > t_{3c}$) causes each original Van Hove point to split into two, both of the conventional logarithmic type. We show that which of these regimes we are in strongly influences the predicted ordered states. We also study the dependence on doping, and find that the ferromagnetic state that occurs at Van Hove filling in these models is unstable to very small shifts in the Fermi level, often giving way to distinct ordered states depending on whether the model is electron- or hole-doped. These results highlight the importance of tuning VHS properties to control ordered states in correlated materials, and offer design rules to engineer these phases in novel systems.

Figures

Figures reproduced from arXiv: 2608.07714 by the authors.

**Figure 2.** FIG. 2: The two-patch scheme used in the pRG analysis [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4: The six inequivalent four-point couplings in the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5: Phase diagrams in the ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6: Line cuts of the critical scale as a function of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: demonstrates that the dependence of the phase on t3 is mostly quite smooth, with all displayed regions having appreciable extent along the t3 direction. In particular, there are almost no discontinuous changes in the diagram at t3 = t3c, i.e. apart from the strong sup…

**Figure 8.** Figure 8: FIG. 8: Phase diagrams in the (∆ [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 53 canonical work pages

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Single-VHS case In the single-VHS case there are four non-interacting susceptibilities to be calculated, namely Π0 pp(Ω), Π0 ph(Ω), ΠQ0 pp (Ω) and Π Q0 ph (Ω). We have [22] Π0 pp(Ω)∼ν 0 ln2 W |Ω| ,Π 0 ph(Ω)∼2ν 0 ln W |Ω| , ΠQ0 pp (Ω)∼2ν 0γ1 ln W |Ω| ,Π Q0 ph (Ω)∼2ν 0γ2 ln W |Ω| , (10) whereWis a high-energy cutoff of the order of the half- bandwidth of ea...

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Further, N+ = N−√ 2 = Γ(1/4) 2 mX − 1/2 8π5/2|t3|1/4 ,(17) where Γ is the gamma function

HOVHS case For the HOVHS case there are again four independent susceptibilities, of which three are divergent [10, 30]: Π0 pp(Ω)∼ C1(N+ +N −) |Ω|1/4 ,Π 0 ph(Ω)∼ C2(N+ +N −) |Ω|1/4 , ΠQ0 pp (Ω)∼ mX − 2π ln W |Ω| ,(16) whereC 1 ≈2.17,C 2 ≈0.54. Further, N+ = N−√ 2 = Γ(1/4) 2 mX − 1/2 8π5/2|t3|1/4 ,(17) where Γ is the gamma function. We have retained all div...

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There are thus only four in- dependent susceptibilities

Split-VHS case In the split-VHS case there are,prima facie, six non- interacting susceptibilities to be calculated; however, one can show that within a patch setup Π 0 pp(Ω) = Π Q2 pp (Ω) and Π 0 ph(Ω) = Π Q2 ph (Ω). There are thus only four in- dependent susceptibilities. Furthermore, they have the same forms as those in the single-VHS case, save with th...

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+d Q1 h (g2 3 +u 2) + 2d0 h(−g1g3 −2g 2g3 +g 1e2 +g 2e2 +e 1g3), ˙e1 =−2(g 2e1 +u 2) + 2d0 h(g1e1 +e 2 2) + 2dQ1 h (g2 −e 1)e1 , ˙e2 =−2g 3e2dQ1 p + 2d0 h(g1e2 +e 1e2) + 2dQ1 h (−e2 2 +e 2g3), ˙u=−2(ue 1 +ug 2) + 2dQ1 h (2g3u−ue 2).(19) See appendix B for the rates associated to the various orders. B. TUFRG An alternative method to predict the ordered sta...

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discretely

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Single-VHS case In the single-VHS case there are four non-interacting susceptibilities to be calculated, namely Π0 pp(Ω), Π0 ph(Ω), ΠQ0 pp (Ω) and Π Q0 ph (Ω). We have [22] Π0 pp(Ω)∼ν 0 ln2 W |Ω| ,Π 0 ph(Ω)∼2ν 0 ln W |Ω| , ΠQ0 pp (Ω)∼2ν 0γ1 ln W |Ω| ,Π Q0 ph (Ω)∼2ν 0γ2 ln W |Ω| , (10) whereWis a high-energy cutoff of the order of the half- bandwidth of ea...

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[2] [2]

Further, N+ = N−√ 2 = Γ(1/4) 2 mX − 1/2 8π5/2|t3|1/4 ,(17) where Γ is the gamma function

HOVHS case For the HOVHS case there are again four independent susceptibilities, of which three are divergent [10, 30]: Π0 pp(Ω)∼ C1(N+ +N −) |Ω|1/4 ,Π 0 ph(Ω)∼ C2(N+ +N −) |Ω|1/4 , ΠQ0 pp (Ω)∼ mX − 2π ln W |Ω| ,(16) whereC 1 ≈2.17,C 2 ≈0.54. Further, N+ = N−√ 2 = Γ(1/4) 2 mX − 1/2 8π5/2|t3|1/4 ,(17) where Γ is the gamma function. We have retained all div...

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[3] [3]

There are thus only four in- dependent susceptibilities

Split-VHS case In the split-VHS case there are,prima facie, six non- interacting susceptibilities to be calculated; however, one can show that within a patch setup Π 0 pp(Ω) = Π Q2 pp (Ω) and Π 0 ph(Ω) = Π Q2 ph (Ω). There are thus only four in- dependent susceptibilities. Furthermore, they have the same forms as those in the single-VHS case, save with th...

work page

[4] [4]

+d Q1 h (g2 3 +u 2) + 2d0 h(−g1g3 −2g 2g3 +g 1e2 +g 2e2 +e 1g3), ˙e1 =−2(g 2e1 +u 2) + 2d0 h(g1e1 +e 2 2) + 2dQ1 h (g2 −e 1)e1 , ˙e2 =−2g 3e2dQ1 p + 2d0 h(g1e2 +e 1e2) + 2dQ1 h (−e2 2 +e 2g3), ˙u=−2(ue 1 +ug 2) + 2dQ1 h (2g3u−ue 2).(19) See appendix B for the rates associated to the various orders. B. TUFRG An alternative method to predict the ordered sta...

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[5] [5]

We begin with (t2/t1, U/t1) phase diagrams; those calculated using pRG are displayed in Fig

Varyingt 2/t1 In this section we discuss the phase diagrams in the case in which the Van Hove singularity is pinned at the Fermi level. We begin with (t2/t1, U/t1) phase diagrams; those calculated using pRG are displayed in Fig. 5(a), (b) and (c) for the single-VHS, HOVHS and split-VHS cases, respectively. In the single-VHS case (Fig. 5(a)), we find compe...

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[6] [6]

discretely

Varyingt 3/t1 Thus far, we have “discretely” switched between the single-VHS, HOVHS and split-VHS cases. However, one might wish to study how the ground state changes as one continuously tunest 3/t1 to move between the three cases, at fixedt 2/t1. Granted, in the previous section the value oft 3/t1 changed when we changedt 2/t1 in the HOVHS and split-VHS ...

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[7] [7]

Steppke, L

A. Steppke, L. Zhao, M. E. Barber, T. Scaffidi, F. Jerzembeck, H. Rosner, A. S. Gibbs, Y. Maeno, S. H. Simon, A. P. Mackenzie, and C. W. Hicks, Strong peak inT c of Sr 2RuO4 under uniaxial pressure, Science355, eaaf9398 (2017)

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J. G. Storey, J. L. Tallon, and G. V. M. Williams, Saddle- point van Hove singularity and the phase diagram of high-Tc cuprates, Phys. Rev. B76, 174522 (2007)

work page 2007