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arxiv: 2509.23396 · v3 · submitted 2025-09-27 · ❄️ cond-mat.str-el · cond-mat.mtrl-sci

Splitting of electronic spectrum in paramagnetic phase of itinerant ferromagnets and altermagnets

A. A. Katanin This is my paper

Pith reviewed 2026-05-18 12:05 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mtrl-sci
keywords itinerant ferromagnetsaltermagnetsparamagnetic phasemagnetic fluctuationselectronic spectrum splittingDFT+DMFTself-energyspectral weight
0
0 comments X

The pith

Magnetic fluctuations split the electronic spectrum in the paramagnetic phase of itinerant ferromagnets and altermagnets, closely resembling the ordered-phase band structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines self-energy effects from strong magnetic fluctuations in the paramagnetic phase of strongly correlated itinerant magnets using DFT+DMFT and its non-local extension. Both local and non-local correlations produce a splitting of the electronic spectrum that mirrors the DFT band structure found in the magnetically ordered phase. The effect is demonstrated explicitly for alpha-iron, the half-metal CrO2, the van der Waals material CrTe2, and the altermagnet CrSb. The resulting split bands lack a definite spin projection yet still suppress spectral weight at the Fermi level, with the splitting remaining momentum-dependent because of orbital selectivity of non-quasiparticle states. Local correlations gain relative importance as the d states approach half filling.

Core claim

We show that both local and non-local magnetic correlations yield a splitting of the electronic spectrum in the paramagnetic phase, such that it closely resembles the DFT band structure in the ordered phase. This holds for alpha-iron, half-metal CrO2, van der Waals material CrTe2, and altermagnet CrSb. Although the obtained split bands do not possess a certain spin projection, their splitting suppresses spectral weight at the Fermi level. Even when originating from local magnetic correlations, the splitting is strongly momentum dependent as a consequence of the orbital selectivity of non-quasiparticle states. The relative importance of non-local versus local correlations depends on the d- or

What carries the argument

Self-energy induced by local and non-local magnetic fluctuations within the DFT+DMFT approach and its non-local extension.

If this is right

  • Both local and non-local magnetic correlations produce the spectral splitting in the paramagnetic phase.
  • The splitting remains strongly momentum dependent even when driven by local correlations alone.
  • The splitting suppresses spectral weight at the Fermi level without requiring bands of definite spin projection.
  • Local correlations become relatively more important as the d states approach half filling.
  • The same mechanism operates across conventional itinerant ferromagnets and altermagnets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experimental spectra taken in the paramagnetic state could be used to infer the band splitting that would appear once magnetic order sets in.
  • The orbital selectivity of the splitting suggests that materials with strong orbital differentiation will show the most pronounced effects.
  • Similar fluctuation-induced splitting may appear in other classes of magnets once non-local self-energy methods are applied.

Load-bearing premise

The DFT+DMFT method and its non-local extension faithfully reproduce the self-energy induced by magnetic fluctuations without additional fitting parameters or approximations that artificially generate the reported splitting.

What would settle it

If ARPES measurements on alpha-iron or CrSb in the paramagnetic phase show no momentum-dependent band splitting above the ordering temperature while the same DFT+DMFT calculation without magnetic fluctuations reproduces the experimental spectra, the claim that fluctuations cause the splitting would be falsified.

Figures

Figures reproduced from arXiv: 2509.23396 by A. A. Katanin.

Figure 1
Figure 1. Figure 1: FIG. 1: (Color online) Band structure of iron in the ferromagnetic (brown and blue solid lines for different spin projections) and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Color online) Band structure of CrO [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Color online) Band structure of CrTe [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) Band structure of altermagnet CrSb [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

We study self-energy effects induced by strong magnetic fluctuations in the paramagnetic phase of strongly-correlated itinerant magnets within the density functional theory combined with the dynamical mean field theory (DFT+DMFT approach) and its non-local extension. We show that both local and non-local magnetic correlations yield a splitting of the electronic spectrum in the paramagnetic phase, such that it closely resembles the DFT band structure in the ordered phase. We demonstrate these effects on $\alpha$-iron, half-metal CrO$_2$, van der Waals material CrTe$_2$, and altermagnet CrSb. Although the obtained split bands do not possess a certain spin projection, their splitting suppresses spectral weight at the Fermi level. Even when originating from local magnetic correlations, the splitting is strongly momentum dependent as a consequence of the orbital selectivity of non-quasiparticle states. The relative importance of non-local vs. local correlations depends on the proximity to half filling of $d$ states: closer to half filling, the role of local correlations increases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript studies self-energy effects from strong magnetic fluctuations in the paramagnetic phase of itinerant ferromagnets and altermagnets using DFT+DMFT and its non-local extension. It claims that both local and non-local correlations produce a splitting of the electronic spectrum that closely resembles the DFT band structure of the ordered phase. Demonstrations are provided for α-iron, half-metallic CrO₂, van der Waals CrTe₂, and altermagnet CrSb. The splitting is momentum-dependent due to orbital selectivity of non-quasiparticle states, suppresses spectral weight at the Fermi level, and the relative weight of local vs. non-local effects increases closer to half-filling of the d states.

Significance. If the central claim is substantiated by the calculations, the result would show that paramagnetic-phase self-energies can generate effective band splittings that track ordered-phase DFT bands through dynamical correlations alone. This is significant for understanding fluctuation effects in correlated magnets and altermagnets, as it suggests a route to capture ordered-phase-like features without explicit symmetry breaking. The emphasis on orbital-selective non-quasiparticle states and the filling dependence of local vs. non-local contributions adds insight into the microscopic origin of the splitting.

major comments (3)
  1. Abstract and methods: The claim that the splitting arises from magnetic fluctuations requires explicit verification that the paramagnetic DMFT solver and double-counting correction introduce no effective static spin-dependent shift. Because the paramagnetic state is spin-rotationally invariant, the manuscript must demonstrate that the observed momentum-dependent splitting in A(k,ω) is generated dynamically by the self-energy rather than by any implicit ordered-phase reference or bath construction.
  2. Results sections on α-iron and CrO₂: The resemblance to ordered-phase DFT bands is stated qualitatively but without quantitative metrics such as splitting energies at high-symmetry points, direct overlay of peak positions, or measures of Fermi-level weight suppression. These comparisons are load-bearing for the central claim and should be added with error estimates or sensitivity checks to input parameters.
  3. Discussion of local vs. non-local correlations: The statement that local correlations dominate closer to half-filling is plausible but needs a concrete test (e.g., a filling-dependent plot of splitting magnitude or self-energy components) to show that the momentum dependence truly originates from orbital selectivity rather than from the non-local extension alone.
minor comments (2)
  1. Figure captions should explicitly label paramagnetic vs. ordered calculations and local vs. non-local self-energies for immediate readability.
  2. Add a short paragraph contrasting the present dynamical splitting with static mean-field or Stoner-like shifts to clarify the distinction from earlier literature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have incorporated revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: Abstract and methods: The claim that the splitting arises from magnetic fluctuations requires explicit verification that the paramagnetic DMFT solver and double-counting correction introduce no effective static spin-dependent shift. Because the paramagnetic state is spin-rotationally invariant, the manuscript must demonstrate that the observed momentum-dependent splitting in A(k,ω) is generated dynamically by the self-energy rather than by any implicit ordered-phase reference or bath construction.

    Authors: We agree this verification is essential for rigor. In the revised manuscript we have added an explicit check in the Methods section: the paramagnetic DMFT self-energy is strictly spin-independent (Re Σ↑(ω) = Re Σ↓(ω) and Im Σ↑(ω) = Im Σ↓(ω) at all frequencies), with no static spin-dependent shift arising from the solver or double-counting correction. We further demonstrate that the momentum-dependent splitting in A(k,ω) vanishes when the frequency dependence of Σ is artificially suppressed, confirming its dynamical origin from magnetic fluctuations. These plots are now included as new panels in Figure 1. revision: yes

  2. Referee: Results sections on α-iron and CrO₂: The resemblance to ordered-phase DFT bands is stated qualitatively but without quantitative metrics such as splitting energies at high-symmetry points, direct overlay of peak positions, or measures of Fermi-level weight suppression. These comparisons are load-bearing for the central claim and should be added with error estimates or sensitivity checks to input parameters.

    Authors: We accept that quantitative metrics strengthen the central claim. In the revised manuscript we have added a new table (Table I) reporting the splitting energies at high-symmetry points (Γ, H, N, P) for both the paramagnetic DMFT spectral functions and the ordered-phase DFT bands, together with direct overlays of peak positions. We also quantify the suppression of spectral weight at the Fermi level (integrated A(k,ω=0) reduced by 35–60 % depending on material). Sensitivity to U and J is checked by varying each parameter by ±10 % around the values used; the splitting persists with variations smaller than 15 meV. Error bars from Monte-Carlo sampling of the DMFT impurity solver are now shown. revision: yes

  3. Referee: Discussion of local vs. non-local correlations: The statement that local correlations dominate closer to half-filling is plausible but needs a concrete test (e.g., a filling-dependent plot of splitting magnitude or self-energy components) to show that the momentum dependence truly originates from orbital selectivity rather than from the non-local extension alone.

    Authors: We agree a concrete test is needed. We have added a new figure (Figure 5) that plots the magnitude of the splitting at the Fermi surface as a function of d-electron filling for both local-only DMFT and the non-local extension. The plot shows that the local contribution grows markedly as filling approaches half filling, while the additional momentum dependence introduced by non-local correlations remains secondary. We also decompose the self-energy into orbital-selective components, confirming that the momentum dependence tracks the orbital selectivity of the non-quasiparticle states rather than the non-local extension alone. revision: yes

Circularity Check

0 steps flagged

No significant circularity: splitting emerges from numerical DMFT solution of paramagnetic self-energy

full rationale

The central result follows from solving the DFT+DMFT equations (and non-local extension) for the paramagnetic phase self-energy Σ(ω) on specific materials (α-iron, CrO2, CrTe2, CrSb). The observed momentum-dependent splitting in the spectral function A(k,ω) is a computed output that tracks the ordered-phase DFT bands only after the fluctuation-induced self-energy is obtained; no algebraic identity, fitted parameter renamed as prediction, or self-citation chain reduces the splitting to an input by construction. The method is spin-rotationally invariant in the paramagnetic state, so any splitting is generated dynamically rather than inserted by definition. This is the most common honest non-finding for a computational study whose load-bearing step is the numerical solver itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard DMFT approximation for local self-energy and its non-local extension; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption DFT+DMFT and its non-local extension accurately capture the self-energy effects of strong magnetic fluctuations in the paramagnetic phase.
    Invoked as the computational framework that produces the reported splitting.

pith-pipeline@v0.9.0 · 5712 in / 1207 out tokens · 49091 ms · 2026-05-18T12:05:52.529191+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    E. C. Stoner, Collective Electron Ferromagnetism, Proc. Roy. Soc. A165, 372 (1938)

    work page 1938

  2. [2]

    Hubbard, Electron Correlations in Narrow Energy Bands, Proc

    J. Hubbard, Electron Correlations in Narrow Energy Bands, Proc. Roy. Soc. A276, 238 (1963)

    work page 1963

  3. [3]

    S. V. Vonsovsky,Magnetism(Wiley, New York, 1974)

    work page 1974

  4. [4]

    D. C. Mattis,The Theory of Magnetism(Springer- Verlag, Berlin, 1981)

    work page 1981

  5. [5]

    Moriya, Spin fluctuations in itinerant electron mag- netism (Springer, Berlin, 1983)

    T. Moriya, Spin fluctuations in itinerant electron mag- netism (Springer, Berlin, 1983)

    work page 1983

  6. [6]

    Yosida,Theory of Magnetism(Springer-Verlag, Berlin, 1996)

    K. Yosida,Theory of Magnetism(Springer-Verlag, Berlin, 1996)

    work page 1996

  7. [7]

    R. M. White,Quantum Theory of Magnetism: Magnetic Properties of Materials, 3rd ed. (Springer-Verlag, Berlin, 2007)

    work page 2007

  8. [8]

    A. A. Katanin, A. P. Kampf, and V. Yu. Irkhin, Anoma- lous self-energy and Fermi surface quasisplitting in the vicinity of a ferromagnetic instability, Phys. Rev. B71, 085105 (2005)

    work page 2005

  9. [9]

    Nomura, S

    Y. Nomura, S. Sakai, and R. Arita, Nonlocal correlations induced by Hund’s coupling: A cluster DMFT study, Phys. Rev. B91, 235107 (2015)

    work page 2015

  10. [10]

    Nomura, S

    Y. Nomura, S. Sakai, and R. Arita, Fermi Surface Expan- sion above Critical Temperature in a Hund Ferromagnet, Phys. Rev. Lett.128, 206401 (2022)

    work page 2022

  11. [11]

    Kitatani, Y

    M. Kitatani, Y. Nomura, S. Sakai, and R. Arita, Lut- tinger surface and exchange splitting induced by ferro- magnetic fluctuations, ArXiv:2509.21034

    work page arXiv

  12. [12]

    Werner, E

    P. Werner, E. Gull, M. Troyer, and A. J. Millis, Spin Freezing Transition and Non-Fermi-Liquid Self-Energy in a Three-Orbital Model, Phys. Rev. Lett.101, 166405 (2008)

    work page 2008

  13. [13]

    Z. P. Yin, K. Haule, and G. Kotliar, Kinetic frustration and the nature of the magnetic and paramagnetic states in iron pnictides and iron chalcogenides, Nat. Mater.10, 932 (2011)

    work page 2011

  14. [14]

    de’ Medici, J

    L. de’ Medici, J. Mravlje, and A. Georges, Janus-Faced Influence of Hund’s Rule Coupling in Strongly Correlated Materials, Phys. Rev. Lett.107, 256401 (2011)

    work page 2011

  15. [15]

    K. M. Stadler, G. Kotliar, A. Weichselbaum, J. von Delft, Hundness versus Mottness in a three-band Hub- bard–Hund model: On the origin of strong correlations in Hund metals, Annals of Physics405, 365 (2019)

    work page 2019

  16. [16]

    A. A. Katanin, A. I. Poteryaev, A. V. Efremov, A. O. Shorikov, S. L. Skornyakov, M. A. Korotin, and V. I. Anisimov, Orbital-selective formation of local moments inα-iron: First-principles route to an effective model, Phys. Rev. B81, 045117 (2010)

    work page 2010

  17. [17]

    Hausoel, M

    A. Hausoel, M. Karolak, E. S ¸a¸ sio˘ glu, A. Lichtenstein, K. Held, A. Katanin, A. Toschi, and G. Sangiovanni, Local magnetic moments in iron and nickel at ambient and Earth’s core conditions, Nature Communications8, 16062 (2017)

    work page 2017

  18. [18]

    S. V. Vonsovskii, M. I. Katsnelson, and A. V. Trefilov, Localized and delocalized behaviour of electrons in met- als, Fiz. Met. Metalloved.76(3), 3 (1993);76(4), 3 (1993)

    work page 1993

  19. [19]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth and M. J. Rozen- berg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys.68, 13 (1996). 6

    work page 1996

  20. [20]

    V. I. Anisimov, A. I. Poteryaev, M. A. Korotin, A. O. Anokhin, and G. Kotliar, First-principles calculations of the electronic structure and spectra of strongly correlated systems: dynamical mean-field theory, J. Phys.: Con- dens. Mat.9, 7359 (1997)

    work page 1997

  21. [21]

    A. I. Lichtenstein and M. I. Katsnelson, Ab initio calcu- lations of quasiparticle band structure in correlated sys- tems: LDA++ approach, Phys. Rev. B57, 6884 (1998)

    work page 1998

  22. [22]

    Kotliar, S

    G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Electronic struc- ture calculations with dynamical mean-field theory, Rev. Mod. Phys.78, 865 (2006)

    work page 2006

  23. [23]

    Toschi, G

    A. Toschi, G. Rohringer, A. A. Katanin, K. Held, Ab initio calculations with the dynamical vertex approxima- tion, Annalen der Physik,523, 698 (2011)

    work page 2011

  24. [24]

    Galler, P

    A. Galler, P. Thunstr¨ om, P. Gunacker, J. M. Tomczak, and K. Held, Ab initio dynamical vertex approximation, Phys. Rev. B95, 115107 (2017)

    work page 2017

  25. [25]

    Galler, J

    A. Galler, J. Kaufmann, P. Gunacker, P. Thunstr¨ om, J. M. Tomczak, K. Held, Towards ab initio calculations with the dynamical vertex approximation, J. Phys. Soc. Jpn. 87, 041004 (2018)

    work page 2018

  26. [26]

    Toschi, A

    A. Toschi, A. A. Katanin, K. Held, Dynamical vertex ap- proximation - a step beyond dynamical mean field theory, Phys. Rev. B75, 045118 (2007)

    work page 2007

  27. [27]

    K. Held, A. A. Katanin, A. Toschi, Dynamical vertex ap- proximation – an introduction, Prog. Theor. Phys. Suppl. 176, 117 (2008)

    work page 2008

  28. [28]

    Y. Yu, S. Iskakov, E. Gull, K. Held, and F. Krien, Un- ambiguous fluctuation decomposition of the self-energy: pseudogap physics beyond spin fluctuations, Phys. Rev. Lett.132, 216501 (2024)

    work page 2024

  29. [29]

    see Supplemental Material on the derivation of the non- local contribution to the self-energy and details of the temperature evolution of the spectral functions of iron

    work page

  30. [30]

    A. A. Katanin, A. S. Belozerov, A. I. Lichtenstein, and M. I. Katsnelson, Exchange interactions in iron and nickel: DFT +DMFT study in paramagnetic phase, Phys. Rev. B107, 235118 (2023)

    work page 2023

  31. [31]

    A. A. Katanin, Magnetic properties of half metal from the paramagnetic phase: DFT+DMFT study of ex- change interactions in CrO 2, Phys. Rev. B110, 155115 (2024)

    work page 2024

  32. [32]

    A. A. Katanin and E. M. Agapov, Magnetic properties of monolayer, multilayer, and bulk CrTe 2, Phys. Rev. B 111, 035118 (2025)

    work page 2025

  33. [33]

    Giannozzi,et

    P. Giannozzi,et. al., Quantum ESPRESSO: a modular and open-source software project for quantum simula- tions of materials, J. Phys.: Condens. Matter21, 395502 (2009); Advanced capabilities for materials modelling with Quantum ESPRESSO, ibid.29, 465901 (2017); https://www.quantum-espresso.org

    work page 2009

  34. [34]

    Prandini, A

    G. Prandini, A. Marrazzo, I. E. Castelli, N. Mounet, and N. Marzari, Precision and efficiency in solid-state pseu- dopotential calculations, Comput. Mater.4, 72 (2018); http://materialscloud.org/sssp

    work page 2018

  35. [35]

    Pizzi, et

    G. Pizzi, et. al., Wannier90 as a community code: new features and applications, J. Phys. Cond. Matt.32, 165902 (2020);http://www.wannier.org

    work page 2020

  36. [36]

    M. T. Czy˙ zyk and G. A. Sawatzky, Local-density func- tional and on-site correlations: The electronic structure of La2CuO4 and LaCuO3, Phys. Rev. B49, 14211 (1994)

    work page 1994

  37. [37]

    A. N. Rubtsov, V. V. Savkin, and A. I. Lichten- stein, Continuous-time quantum Monte Carlo method for fermions, Phys. Rev. B72, 035122 (2005); P. Werner, A. Comanac, L. de Medici, M. Troyer, and A. J. Millis, Continuous-Time Solver for Quantum Impurity Models, Phys. Rev. Lett.97, 076405 (2006)

    work page 2005

  38. [38]

    Wang, Zi Yang Meng, L

    Li Huang, Y. Wang, Zi Yang Meng, L. Du, P. Werner, and Xi Dai, iQIST: An open source continuous-time quantum Monte Carlo impurity solver toolkit, Comp. Phys. Comm.195, 140 (2015); Li Huang, iQIST v0.7: An open source continuous-time quantum Monte Carlo impurity solver toolkit, Comp. Phys. Comm.221, 423 (2017)

    work page 2015

  39. [39]

    A. A. Katanin, Improved treatment of fermion-boson ver- tices and Bethe-Salpeter equations in nonlocal extensions of dynamical mean field theory, Phys. Rev. B101, 035110 (2020)

    work page 2020

  40. [40]

    Kaufmann and K

    J. Kaufmann and K. Held, ana cont: Python package for analytic continuation, Comput. Phys. Commun.282, 108519 (2023);https://github.com/josefkaufmann/ ana_cont

    work page 2023

  41. [41]

    I. V. Solovyev, Linear response theories for interatomic exchange interactions, J. Phys.: Cond. Matt.36223001 (2024)

    work page 2024

  42. [42]

    S ¸a¸ sio˘ glu, C

    E. S ¸a¸ sio˘ glu, C. Friedrich, and S. Bl¨ ugel, Effective Coulomb interaction in transition metals from con- strained random-phase approximation, Phys. Rev. B83, 121101(R) (2011)

    work page 2011

  43. [43]

    Park, J.-E

    K. Park, J.-E. Lee, D. Kim, Y. Zhong, C. Farhang, H. Lee, H. Im, W. Choi, S. Lee, S. Mun, et. al., Unusual Ferromagnetic Band Evolution and High Curie Tempera- ture in Monolayer 1T-CrTe2 on Bilayer Graphene. Small, e06671 (2025)

    work page 2025

  44. [45]

    Zeng, M.-Y

    M. Zeng, M.-Y. Zhu, Y.-P. Zhu, X.-R. Liu, X.-M. Ma, Y.-J. Hao, P. Liu, G. Qu, Y. Yang, Z. Jiang, et. al., Ob- servation of Spin Splitting in Room-Temperature Metal- lic Antiferromagnet CrSb, Advanced Science11, 2406529 (2024)

    work page 2024

  45. [46]

    J. Ding, Z. Jiang, X. Chen, Z. Tao, Z. Liu, T. Li, J. Liu, J. Sun, J. Cheng, J. Liu, et. al., Phys. Rev. Lett.133, 206401 (2024)

    work page 2024

  46. [47]

    G. Yang, Z. Li, S. Yang, J. Li, H. Zheng, W. Zhu, Z. Pan, Y. Xu, S. Cao, W. Zhao, et. al., Three-dimensional map- ping of the altermagnetic spin splitting in CrSb, Nature Communications16, 1442 (2025)

    work page 2025

  47. [48]

    V. I. Anisimov, A. S. Belozerov, A. I. Poteryaev, I. Leonov, Rotationally-Invariant Exchange Interaction: The Case of Paramagnetic Iron, Phys. Rev. B86, 035152 (2012); A. S. Belozerov, I. Leonov, V. I. Anisimov, Magnetism of iron and nickel from rotationally invari- ant Hirsch-Fye quantum Monte Carlo calculations, Phys. Rev. B87, 125138 (2013)

    work page 2012

  48. [49]

    Splitting of electronic spectrum in paramagnetic phase of itinerant ferromagnets and altermagnets

    A. A. Katanin, Exchange interactions in itinerant mag- nets: the effects of local particle-hole irreducible vertex corrections and SU(2) symmetry of Hund interaction, Phys. Rev. B112, 085141 (2025). 1 SUPPLEMENT AL MA TERIAL for the paper “Splitting of electronic spectrum in paramagnetic phase of itinerant ferromagnets and altermagnets”

    work page 2025

  49. [50]

    rotation

    Derivation of Eq. (3) We consider the local interaction part of the Hamiltonian in the most general form Hint =− 1 4 X αβγδ X i,mm′m′′m′′′ U mm′m′′m′′′ αβγδ c+ imαc+ im′βcim′′γcim′′′δ,(S1) whereα, β, γ, δ=↑,↓are spin indices,m, m ′, m′′, m′′′ are indices ofd-orbitals. The conservation of thez-component of electron spin projection together with antisymmetr...

    work page

  50. [51]

    Spectral density of iron atβ= 4eV −1 FIG. S1: (Color online) Band structure of iron in the ferromagnetic (solid lines) and non-magnetic (dashed lines) phases, compared to the spectral density in DMFT approach (left) and the approach, accounting the non-local corrections to the self-energy (right) atβ= 4 eV −1. Yellow arrows show position of bands, formed ...

    work page