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A key role of correlation effects in the Lifshitz transition in Sr$_2$RuO$_4$

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Electronic correlations, and specifically the orbital anisotropy of the Coulomb interaction, determine the uniaxial strain at which Sr2RuO4 undergoes a Lifshitz transition; new measurements put that strain at $(-0.44 \pm 0.06)\times…

desk verdict A careful experimental refinement of the Lifshitz strain in Sr2RuO4, paired with a plausible but not yet fully independent DMFT mechanism for why correlations reduce it. read the letter →

arxiv 1909.02743 v1 pith:TIQVJM6N submitted 2019-09-06 cond-mat.str-el

Mark E. Barber , Frank Lechermann , Sergey V. Streltsov , Sergey L. Skornyakov , Sayak Ghosh , B. J. Ramshaw , Naoki Kikugawa , Dmitry A. Sokolov
show 3 more authors
Andrew P. Mackenzie Clifford W. Hicks Igor I. Mazin
This is my paper · ORCID
classification cond-mat.str-el
keywords LifshitztransitionSr2RuO4uniaxialstrainVanHovesingularitydynamicalmean-fieldtheoryorbitalanisotropyCoulombcorrelationsFermisurface
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 pins down the uniaxial strain at which the largest Fermi surface of the superconductor Sr2RuO4 changes topology — a Lifshitz transition — and shows that electronic correlations determine that strain. New measurements place the critical longitudinal strain at $\varepsilon_{xx} = (-0.44 \pm 0.06) \times 10^{-2}$, corresponding to a B$_{1g}$ strain of $(-0.66 \pm 0.09) \times 10^{-2}$. Density functional theory alone predicts a strain more than twice as large. The authors show, with dynamical mean-field theory, that Coulomb repulsion shrinks the predicted strain, and that only an orbital-dependent Coulomb interaction on the ruthenium site brings theory into agreement with experiment. The result makes high-precision strain measurements a sharp test of correlated-electron calculations.

What carries the argument

The central object is the energy of the $xy$ band at the Y point of the Brillouin zone, $E_{xy} - E_F$, whose zero crossing as a function of compressive $\varepsilon_{xx}$ defines the Van Hove strain $\varepsilon_{\rm VHS}$ and the Lifshitz transition. The argument is carried by the orbital anisotropy parameter $u = U_{xy,xy} - U_{\{xz,yz\},\{xz,yz\}}$, which raises the Coulomb repulsion felt by electrons in the $xy$ orbital (the $\gamma$ sheet) relative to the $xz,yz$ orbitals. Correlations grow the $\gamma$ sheet and, for isotropic $U$, push $\varepsilon_{\rm VHS}$ essentially to zero; a nonzero $u$ partially counteracts that growth, moving the crossing back to the experimentally observed strain. This parameterization lets the calculation connect the measured critical strain to a microscopic Coulomb quantity.

What would settle it

Measure the Fermi-surface topology directly — for example with quantum oscillations or ARPES on a sample under the same uniaxial strain — at strains between the resistivity peak and the Tc peak. If no topology change occurs until the Tc peak strain, the paper's assignment of $\varepsilon_{\rm VHS}$ and the inferred value of $u$ would need revision.

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Extended reading notes

Core claim

The paper establishes that the strain-induced Lifshitz transition in Sr2RuO4 — where the $\gamma$ Fermi-surface sheet becomes open and a Van Hove singularity crosses the Fermi level — occurs at a longitudinal strain $\varepsilon_{\rm VHS} = (-0.44 \pm 0.06)\times 10^{-2}$, corresponding to a B$_{1g}$ strain of $(-0.66 \pm 0.09)\times 10^{-2}$. This value is considerably smaller than density functional theory predicts, even with spin-orbit coupling. Using DFT+DMFT with a rotationally invariant slave-boson solver, the authors show that Coulomb correlations reduce the critical strain, but that an isotropic Coulomb interaction strong enough to match measured mass renormalizations overshoots, pushing the transition essentially to zero strain. Adding an orbital anisotropy $u = U_{xy,xy} - U_{\{xz,yz\},\{xz,yz\}}$ in the range 0.10–0.15 eV restores agreement with experiment. The paper concludes that the orbital anisotropy of the local Coulomb interaction on the Ru site is essential, and that the measured critical strain provides a new quantitative constraint on electronic structure theory.

Load-bearing premise

The paper identifies the peak in low-temperature resistivity with the Lifshitz transition; if the true transition occurs at the slightly larger strain where Tc peaks, the experimental benchmark and the fitted Coulomb anisotropy would shift.

Editorial extensions

If this is right

  • The measured $\varepsilon_{\rm VHS}$ becomes a benchmark for future electronic-structure methods applied to Sr2RuO4: a method must reproduce both the mass renormalization and the critical strain.
  • Strain tuning toward a Van Hove singularity in other multi-orbital correlated metals will require correlation-aware methods, not just DFT, to predict the required strain.
  • The orbital anisotropy of the Coulomb interaction is an essential ingredient of the Hubbard model for Sr2RuO4; an isotropic $U$ misplaces the Lifshitz transition even when it reproduces band masses.
  • Because the Tc peak and the resistivity peak occur at slightly different strains, identifying $\varepsilon_{\rm VHS}$ with the resistivity peak sharpens the separation between the topology change and the pairing-enhancement mechanism.

Reading between the lines

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

  • If the same anisotropy mechanism operates in other layered ruthenates, strain experiments could serve as a direct probe of orbital-resolved Coulomb repulsion, complementing spectroscopy.
  • The 5–12% offset between the Tc peak and the resistivity peak leaves open the possibility that the true topology change sits at the Tc peak; a direct Fermi-surface probe in the same stress cell could settle which feature marks the transition.
  • Because the fitted $u$ lies in the range predicted by constrained-RPA calculations, the experiment effectively measures a microscopic interaction parameter; similar strain metrology could test Hubbard parameters in other correlated oxides.
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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

3 major / 4 minor

Summary. The paper reports a refined experimental determination of the strain at which the gamma Fermi surface sheet of Sr2RuO4 undergoes a Lifshitz transition under uniaxial pressure, yielding a longitudinal strain of (-0.44 +/- 0.06) x 10^-2 and a B1g strain of (-0.66 +/- 0.09) x 10^-2. The measurement uses a force-sensor-based stress cell and new low-temperature elastic moduli. The authors then compare this experimental benchmark with density functional theory (DFT) and DFT+DMFT calculations. DFT with and without spin-orbit coupling overestimates the critical strain in magnitude. DFT+DMFT with isotropic Coulomb interactions (U = 3.1 eV, J_H = 0.7 eV) overcorrects, giving a critical strain close to zero. Introducing an orbital anisotropy u = U_xy,xy - U_{xz,yz},{xz,yz} of 0.10-0.15 eV (with inter-orbital anisotropy u' = u/3) brings the calculated critical strain into agreement with experiment. The authors conclude that electronic correlations reduce the critical strain and that orbital anisotropy of the local Coulomb interaction is required to reproduce the experimental value.

Significance. The experimental part is careful and provides a reliable benchmark: two samples, force-sensor calibration, and new low-temperature elastic moduli give a credible value of the Lifshitz strain. The qualitative finding that correlations reduce the critical strain relative to DFT is robust and important, as it reconciles the observed strain with theoretical expectations. The demonstration that the orbital anisotropy of the Hubbard interaction can shift the Lifshitz strain by several tenths of a percent is a valuable constraint on electronic-structure methods. However, the quantitative agreement is obtained by fitting the anisotropy parameter u, not by an ab initio calculation, so the paper's central quantitative claim is a model-dependent inference rather than a prediction. The paper is honest about this limitation, and it provides a clear motivation for future constrained-RPA or other calculations of the orbital-dependent Coulomb interaction.

major comments (3)
  1. [Calculation results, Eq. (1)] The orbital-anisotropy parameter u is introduced as an adjustable parameter, and the agreement with the experimental epsilon_VHS is used to set u ~ 0.10-0.15 eV. Reference 18 is cited only for the qualitative statement that U_xy exceeds U_{xz,yz}; no independent ab initio value for this anisotropy in Sr2RuO4 is provided. If a constrained-RPA calculation yielded u outside this range, the agreement would be a fitting artifact rather than evidence for the anisotropy mechanism. The authors should either report such a calculation (or a strong estimate from other sources) or explicitly reframe the conclusion as a demonstration of sensitivity rather than a quantitative prediction.
  2. [Conclusions and isotropic parameter comparison] The claim that agreement with experiment 'can be achieved only by accounting for orbital anisotropy' is supported by only two isotropic parameter sets (U = 3.1 eV, J_H = 0.7 eV and U = 2.3 eV, J_H = 0.4 eV). It is not demonstrated that no isotropic (U, J_H) combination consistent with the observed mass renormalization (m*/m ~ 2.5-4.4) can reproduce the experimental critical strain. A more systematic scan over the isotropic parameter space, or a mapping of the region consistent with the experimental mass renormalization, is needed to establish that anisotropy is truly necessary rather than merely a convenient knob.
  3. [Appendix (RISB solver)] The RISB impurity solver has a restricted self-energy form (static plus linear in frequency), which could bias the orbital-dependent renormalizations that are central to the proposed mechanism. The paper itself recommends verification with other impurity solvers for the carrier redistribution, but no cross-check is provided for the key quantity epsilon_VHS. A test with a more exact solver (e.g., continuous-time QMC) for at least one strain and one u value would materially strengthen the central quantitative claim.
minor comments (4)
  1. [Fig. 4 discussion] In the text near Fig. 4, 'For both u = 0.10 and 0.15 meV' should read 'eV', since the parameter u is introduced in electron-volt units.
  2. [Strain conversion paragraph] The sentence 'for comparison with experiment they should be scaled by 1.39/1.51' is clear, but it would help to state explicitly whether the quoted DMFT values (-0.38 x 10^-2 and -0.70 x 10^-2) have already been scaled or should be scaled by the reader; the current wording leaves some ambiguity.
  3. [Experimental error bars] The statement 'all experimental error bars are 2 sigma' appears in the experimental section; it would be useful to remind the reader of this convention when the final epsilon_VHS uncertainty is quoted in the abstract and conclusions.
  4. [Abstract] The phrase 'new uniaxial stress apparatus' could be clarified as 'recently developed' because the apparatus was previously described in Ref. 7.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the u-scan is an openly labeled parameter study, not a disguised prediction.

full rationale

The paper's central chain is not circular. The claim that electronic correlations reduce the Lifshitz strain is made by comparing DFT (εVHS at least −1.1×10⁻²) with isotropic DMFT (εVHS shifted essentially to zero), a comparison that does not involve the adjustable anisotropy u. The anisotropic term is then introduced explicitly as a parameter: 'We treat the intra-orbital Coulumb anisotropy u = U_xy,xy − U_{xz,yz},{xz,yz} as a parameter.' The quantitative agreement is a postdiction: after the experimental εVHS is known, the paper shows that u ≈ 0.10–0.15 eV reproduces it ('the experimental εVHS is reproduced for u ≡ U_xy,xy − U_{xz,yz},{xz,yz} ∼ 0.10–0.15 eV'). The paper does not rename this fit as an ab initio prediction; it labels u as a parameter and cites independent cRPA work (Refs. 17 and 18, by other author groups) for the qualitative ordering U_xy > U_{xz,yz}. The computed epsilon_VHS as a function of u is a genuine model calculation rather than a definitional identity, so the matching step is not equivalent to its inputs by construction. The experimental assumption that the resistivity peak marks the Lifshitz transition affects the benchmark value but is not fed into the definition of the theoretical Van Hove strain; it is an experimental interpretation, not a circular input. The main caveat is scientific rather than logical: the strength of the anisotropy conclusion would be increased by an independent cRPA or QMC value of u, but its absence is a validation gap, not circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central quantitative result depends on one fitted parameter, the orbital anisotropy u, plus several assumptions about the impurity solver, interaction parameters, relaxed structures, and the identification of the transition with the resistivity peak. None of these are invented entities; they are standard modeling choices. The fitted u is the main free parameter.

free parameters (1)
  • u (orbital anisotropy of intra-orbital Coulomb interaction) = 0.10 to 0.15 eV (range chosen to match experiment)
    The difference Uxy,xy minus U{xz,yz},{xz,yz} is treated as a parameter. It is varied and the critical strain is interpolated; values 0.10 and 0.15 eV bracket the experimental value. It is motivated by cRPA and Fermi surface fits but not independently fixed.
assumptions (5)
  • domain assumption The peak in low-temperature resistivity marks the Lifshitz transition.
    This is explicitly assumed when assigning the experimental critical strain. It is justified by transport theory references but not proven, and Tc peaks at a strain 5 to 12% larger.
  • domain assumption The RISB approximation to DMFT is sufficiently accurate for the correlated subspace.
    The paper uses the rotationally invariant slave-boson solver at saddle point, which has a simplified self-energy. Results may depend on the solver; the authors suggest verification with other solvers.
  • domain assumption The Slater-Kanamori parameters U=3.1 eV and JH=0.7 eV are appropriate for Ru t2g.
    These are taken from earlier works and match the lower end of the experimental mass renormalization; the paper does not derive them.
  • domain assumption Crystal structures optimized in Ref. 2 as a function of strain are valid inputs.
    All calculations use these structures; errors in the relaxed structures would propagate to the critical strain.
  • domain assumption Exy minus EF varies linearly between the three computed strains.
    The critical strain is obtained by linear interpolation of Exy-EF at strains 0, -0.4%, -0.9%; this may introduce error if nonlinear.

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Pith. "Pith review of A key role of correlation effects in the Lifshitz transition in Sr$_2$RuO$_4$." pith.science (2026). https://pith.science/paper/TIQVJM6N

@misc{pith2026190902743,
  author       = {Pith},
  title        = {Pith review of: A key role of correlation effects in the Lifshitz transition in Sr$_2$RuO$_4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TIQVJM6N}},
  note         = {Machine review of arXiv:1909.02743}
}
abstract

Uniaxial pressure applied along an Ru-Ru bond direction induces an elliptical distortion of the largest Fermi surface of Sr$_2$RuO$_4$, eventually causing a Fermi surface topological transition, also known as a Lifshitz transition, into an open Fermi surface. There are various anomalies in low-temperature properties associated with this transition, including maxima in the superconducting critical temperature and in resistivity. In the present paper, we report new measurements, employing new uniaxial stress apparatus and new measurements of the low-temperature elastic moduli, of the strain at which this Lifshitz transition occurs: a longitudinal strain $\varepsilon_{xx}$ of $(-0.44\pm0.06)\cdot10^{-2}$, which corresponds to a B$_{1g}$ strain $\varepsilon_{xx} - \varepsilon_{yy}$ of $(-0.66\pm0.09)\cdot10^{-2}$. This is considerably smaller than the strain corresponding to a Lifshitz transition in density functional theory calculations, even if the spin-orbit coupling is taken into account. Using dynamical mean-field theory we show that electronic correlations reduce the critical strain. It turns out that the orbital anisotropy of the local Coulomb interaction on the Ru site is furthermore important to bring this critical strain close to the experimental number, and thus well into the experimentally accessible range of strains.

Figures

Figures reproduced from arXiv: 1909.02743 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic of the uniaxial stress cell incorporat [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Band structure calculated within GGA [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Energy of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of the correlated Fermi surface with [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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