# Ferromagnetic instability in itinerant fcc lattice electron systems
with higher order van Hove singularities:
Functional renormalization group study

P. A. Igoshev  Institute of Metal Physics,
620108, Kovalevskaya str. 18, Ekaterinburg, Russia  A. A. Katanin  Center for
Photonics and 2D Materials, Moscow Institute of Physics and Technology,
Institutsky lane 9, Dolgoprudny, 141700 Moscow region, Russia  Institute of
Metal Physics,
620108, Kovalevskaya str. 18, Ekaterinburg, Russia

###### Abstract

We investigate the possibility of ferromagnetic ordering in the non-degenerate
Hubbard model on the face-centered cubic lattice within the functional
renormalization group technique using temperature as a scale parameter. We
assume the relations between nearest, next-nearest, and next-next-nearest
hopping parameters providing higher order (giant) van Hove singularity of the
density of states. The ferromagnetic instability formation with lowering
temperature is described consistently in the one-loop approximation for a one-
particle irreducible vertex of two-particle electron interaction. The chemical
potential versus temperature phase diagrams are calculated. We find
ferromagnetic order only for sufficiently strong divergence of the density of
states and fillings in the vicinity of van Hove singularity. The obtained
Curie temperature is more than an order of magnitude smaller than the results
of the random-phase approximation. The main origin of the suppression of
ferromagnetism is the screening of interaction in the particle-particle
channel. We also do not find the pronounced tendency towards incommensurate
order when the Fermi level is moved away from a van Hove singularity, such
that the first order quantum phase transitions from the ferro- to paramagnetic
phase are obtained.

##  I Introduction

Weak itinerant magnets, such as ZrZn2 1958:ZrZn2_discovery ; 2020:Skornyakov ,
Ni3Al 1984:Ni3Al:Sasakura ; 2005:Niklowitz , and some other compounds,
represent substances having low magnetic transition temperatures and small
saturation magnetic moment. The problem of description of itinerant
ferromagnetism is a long-standing problem, which started from the Stoner
theory Stoner ; 1966:Penn . Although this theory describes main features of
magnetic transitions, it strongly overestimates magnetic transition
temperatures, does not explain the Curie-Weiss behavior of magnetic
susceptibilities, and yields an incorrect critical behavior. These effects
occur due to electronic and magnetic correlations, which are not accounted by
the Stoner theory.

Some drawbacks of the Stoner theory were improved by spin-fluctuation
approaches, such as Murata-Doniach 1972:Murata , Moriya Moriya_book  theory,
etc. These approaches account for the effect of correlations by considering
the contribution of the paramagnon interaction to the magnetic susceptibility.
While Murata-Doniach theory reproduces the Curie-Wess law, it accounts only
for the static spin correlations. The dynamic spin correlations were accounted
by Moriya theory, which yields the temperature $T$ dependence of magnetic
susceptibility $\propto 1/(T^{4/3}-T_{\rm C}^{4/3})$ ($T_{\rm C}$ being the
Curie temperature), which is close to the Curie-Weiss law. The Moriya theory,
based on the bosonic mean-field approximation, was justified within the Hertz-
Millis renormalization group approach 1976:Hertz ; 1993:Millis  and it was
shown that Moriya theory yields qualitatively correct results since the
effective dimension $d+z=6$ for the cubic itinerant ferromagnets ($z=3$ is the
dynamic critical exponent) is larger than the upper critical dimension
$d_{c}=4$ of the $\phi^{4}$ theory.

Yet, the spin-fluctuation approaches consider an effect of only a single
(particle-hole) channel of electron interaction, which represents the
paramagnon contribution. At the same time, it was argued that for two-
dimensional systems, the singular behavior of the density of states near the
Fermi level yields an interplay of different channels of electron interactions
2001:Katanin ; 2001:Salmhofer ; Kampf . In three dimensions, peaks of the
density of states that originate from the so called “giant” or “higher-order”
van Hove singularities (vHs), play the important role of the ferromagnetism
formation 1993:Katsnelson ; 2019:Igoshev_PMM ; 2019:Igoshev_JETP ;
2020:Classen ; 2022:Igoshev ; 2022:Betouras . These singularities originate
from vanishing of some inverse masses of electronic dispersion at van Hove
points, which makes the dispersion effectively two- or one-dimensional. A
number of band structure investigations point out appearing van Hove
singularities of electron density of states, which play the important role for
the physics of itinerant magnets. For ZrZn2, the peak of density of states is
formed by the vicinity of L point of the Brillouin zone 2007:Yamaji ;
2004:Major ; 2003:Yates ; 2020:Skornyakov , while for Ni3Al, the van Hove
point R produces the peak in the density of states 2011:Hamid . In nickel,
which is a stronger metallic magnet, there is a flat band in the vicinity of L
point of the Brillouin zone 2017:Hausoel . For bcc (body centered cubic) iron
$e_{\rm g}$ derived band appears to be flat in several direction which,
together with Hund coupling $J_{\rm H}$, results in the local moment formation
OurFe ; 2015:Efremov .

A large enough value of the density of states (DOS) at the Fermi level,
occurring due to vHs, leads to the ferromagnetic ground state according to the
Stoner theory Stoner ; 1966:Penn . Therefore, on one hand, the fundamental
origin of the itinerant ferromagnetism is the presence of van Hove
singularities of itinerant electron electronic dispersion. On the other, the
van Hove singularities induce the interplay of different channels of electron
scattering and, therefore, require going beyond the spin fluctuation
approaches. In particular, the Kanamori screening Kanamori  is expected to be
particularly important near van Hove singularities, see also Refs. Tmatrix1, ;
3body1, ; 3body2, .

In this respect, dynamical mean-field theory provides a possibility for
studying the effect of local electronic correlations on ferromagnetism in the
single-band Ulmke ; Pruschke  and multiband Arita ; Licht ; OurFe ;
2020:Skornyakov ; 2017:Hausoel  models. In particular, the problem of
ferromagnetism formation in the single-band Hubbard model on fcc (face
centered cubic) lattices was considered within DMFT [31] in cases (i)
$d=\infty$ at the ratio of next-nearest and nearest neighbor hopping
parameters $t^{\prime}/t=1/2$ (which provides a giant one-dimensional-like van
Hove singularity at the bottom of the band) and (ii) $d=3$ at $t^{\prime}=t/4$
which yields a wide and moderate-valued DOS plateau in the vicinity of band
bottom (see also Ref. 2022:Igoshev, ). While in case (i) wide ferromagnetic
region was found for Coulomb interaction parameter $U\sim 7t$, in case (ii),
ferromagnetism was not found at $U\sim 14t$, but found at $U\sim 21t$. These
DMFT results imply that ferromagnetic ordering for the non-degenerate model in
the absence of strong peak of density of states can be stable only at very
large interaction parameter $U$ values. However, the treatment of the non-
local effects is beyond the scope of the dynamical mean-field theory.
Moreover, a generalization of the Stoner theory to treat the spin spiral
instability and Néel antiferromagnetism reveals also that in a major part of
the phase diagram, the spiral phase is even more preferable than the
ferromagnetic one Schulz ; 2010:Igoshev ; 2015:Igoshev . Therefore the theory
of itinerant ferromagnetism should also treat its competition with the other
type of magnetic orderings, including incommensurate ones.

A tool allowing to treat both, the interplay of different channels of electron
interaction and the competition of ferromagnetic and incommensurate
correlations is the functional renormalization group (fRG) technique, which is
a powerful method for treatment of correlation effects fRG ; Zanchi ; Halboth
; 2001:Salmhofer ; Kampf ; Honerkamp ; Rohe ; Lauscher ; 2009:Husemann ;
Katanin ; 2013:Eberlein ; 2017:TUfRG ; 2022:BFfRG ; 2011:Igoshev ; Yuki ;
Honerkamp01 . The fRG directly takes into account non-local electron
correlations which are of a great importance in the case of weak itinerant
magnetism. This approach manifested its success for consideration of the
problem of ferromagnetism of two-dimensional Hubbard model due to the
closeness of the Fermi level to logarithmic van Hove singularity 2001:Katanin
; Honerkamp ; Yuki ; Kampf ; 2011:Igoshev . It was found that necessary
condition for ferromagnetism formation is its “parameteric separation” from
other types of instabilities like superconducting or antiferromagnetic one,
which can be achieved by a strong curvature of the Fermi surface caused by
large $t^{\prime}/t$, where $t(t^{\prime})$ is the nearest- (next-nearest-)
neighbor hopping parameter (see also Refs. 2007:Igoshev, ; 2009:Igoshev, ;
2010:Igoshev, ). The ferromagnetic instability is enhanced in the flat band
case, when one of the inverse masses of the electronic dispersion vanishes.
This case in three dimensions, corresponding to higher-order van Hove
singularity, represents therefore a certain interest for the investigation.

While the early fRG approaches for fermionic systems were based on the patch
parametrization of electron interaction vertices, it was suggested by Husemann
and Salmhofer 2009:Husemann  to parametrize the flow of each interaction
channel by the bosonic momentum (and possibly frequency) transfer, while the
remaining dependence on two fermionic momenta and frequencies can be accounted
less accurately via their projections onto some restricted set of basis
functions. This idea was further developed within the truncated unity
functional renormalization group approach 2017:TUfRG , which suggests
projection of the “transverse” channels onto the same set of basis functions,
which greatly simplifies calculations. Finally, recently the formulation of
the approach of Husemann and Salmhofer in terms of the triangular boson-
fermion vertices and the reminder was proposed 2022:BFfRG .

These recent developments give a possibility to study itinerant ferromagnetism
of the three-dimensional fermionic systems with sufficiently small number of
flowing vertices. In the present paper, we investigate the formation of
ferromagnetism on fcc lattice in the presence of extended van Hove
singularities. In particular, we study (i) how the strength of the van Hove
singularity affects stability of the ferromagnetism; (ii) what is the main
destructive factor of ferromagnetism in three dimensions: a competition with
incommensurate magnetic order or the competition with the paramagnetic phase,
and (iii) how important is the effect of screening of Coulomb interaction.

To study the above mentioned problems, we present a study of the evolution of
magnetic and electronic properties with decreasing temperature within 1PI fRG
for the non-degenerate Hubbard model on the fcc lattice. The plan of the paper
is the following. In Sec. II, the model and fRG equations are presented. In
Sec. III, we present and discuss the numerical results. In Sec. IV, we
conclude the paper.

##  II The model and RG equations

###  II.1 The model

We consider the Hubbard model with the Hamiltonian

$\mathcal{H}=\sum_{{\bf k}\sigma}\epsilon_{\mathbf{k}}c_{{\bf
k}\sigma}^{+}c_{{\bf k}\sigma}+\frac{U}{2N}\sum_{{\bf k}_{1}{\bf k}_{2}{\bf
k}_{3}{\bf
k}_{4},\sigma\sigma^{\prime}}c_{\mathbf{k}_{1}\sigma}^{+}c_{\mathbf{k}_{2}\sigma^{\prime}}^{+}c_{\mathbf{k}_{3}\sigma^{\prime}}c_{\mathbf{k}_{4}\sigma}\delta_{{\bf
k}_{1}+\mathbf{k}_{2},\mathbf{k}_{3}+\mathbf{k}_{4}},$ (1)

where $\epsilon_{\mathbf{k}}$ is an electronic dispersion, $U$ is the Hubbard
on–site interaction, $N$ is lattice site number, $\delta$ is the Kronecker
$\delta$-symbol, $c_{{\bf k}\sigma}$ and $c^{+}_{{\bf k}\sigma}$ are the
electron destruction and creation operators, respectively. We consider
electronic dispersion on the face-centered cubic lattice with nearest-
(unity), next-nearest $\tau$, and next-next-nearest $\tau^{\prime}$ hoppings:

$\epsilon_{{\mathbf{k}}}(\tau,\tau^{\prime})=-4\left(\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}+\cos\frac{k_{x}}{2}\cos\frac{k_{z}}{2}+\cos\frac{k_{y}}{2}\cos\frac{k_{z}}{2}\right)\\\
+\vphantom{\frac{k_{x}}{2}}2\tau\left(\cos k_{x}+\cos k_{y}+\cos
k_{z}\right)+4\tau^{\prime}\left(\cos k_{x}\cos k_{y}+\cos k_{x}\cos
k_{z}+\cos k_{y}\cos k_{z}\right),$ (2)

where the lattice constant is put to unity.

###  II.2 van Hove singularities

We are interested in studying itinerant magnetism induced by higher order van
Hove singularities.

Figure 1: $\tau^{\prime}$ dependence of van Hove energies at $\tau=-0.13$.

For itinerant ferromagnets on fcc lattice peculiarities of the dispersion in
the vicinity of L$(\pi,\pi,\pi)$ point represent certain interest. To
investigate the contribution of the vicinity of L$(\pi,\pi,\pi)$ point to DOS,
we expand the Eq. (2) as
$\epsilon_{{\mathbf{k}}}(\tau,\tau^{\prime})=\epsilon_{{\mathbf{k}}_{\rm
L}}(\tau,\tau^{\prime})+(\tau-4\tau^{\prime})q^{2}-q_{y}q_{z}-q_{z}q_{x}-q_{x}q_{y}$,
where ${\mathbf{k}}={\mathbf{k}}_{\rm L}+{\mathbf{q}}$ and
$\epsilon_{{\mathbf{k}}_{\rm L}}(\tau,\tau^{\prime})=6(-\tau+2\tau^{\prime})$.
Since the eigenvalues of this ${\mathbf{q}}$-quadratic form (inverse masses)
are $m^{-1}_{\rm L,1}=1+2\tau-8\tau^{\prime}$ (doubly degenerate) and
$m^{-1}_{\rm L,2}=2(-1+\tau-4\tau^{\prime})$, we obtain the condition on giant
(quasi-one-dimensional-like) van Hove singularity at

$\tau^{\prime}_{\rm c}(\tau)=\frac{1}{8}+\frac{\tau}{4}.$ (3)

Figure 2: (a) Density of states $\rho(\epsilon,\tau,\tau^{\prime})$ plot as a
function of $\epsilon$ for $\tau=-0.13$, $\tau^{\prime}=0.0925$ and
$\tau^{\prime}=0.13$. In the inset, $\rho(\epsilon,\tau,\tau^{\prime})$ at
$\tau^{\prime}=0.13$ in the energy in the window near the van Hove structure,
corresponding to the energies $E_{\rm L_{1}}$ and $E_{\rm L_{2}}$ (marked by
dashed vertical lines) is shown. (b) The density of states in the vicinity of
the bottom of the band for $\tau^{\prime}=0$, $\tau=1$ (red solid line) and
$\tau=1.05$ (black solid line). For the case $\tau=1.05$, the positions of van
Hove structure energies 2022:Igoshev  $E_{\varSigma^{\ast}}=-5.736$,
$E_{\varDelta^{\ast}}=-5.71$ and $E_{\Gamma}=-5.7$ are shown by vertical
dashed lines.

The investigation of the electronic dispersion in the vicinity of L point at
$\tau^{\prime}\neq\tau^{\prime}_{\rm c}(\tau)$ (see Appendix A) yields the
following topological transition in the profile of isoenergy surfaces: While
at $\tau^{\prime}<\tau^{\prime}_{\rm c}(\tau)$ we have single van Hove point L
with finite masses, at $\tau^{\prime}>\tau^{\prime}_{\rm c}(\tau)$ the van
Hove point L splits into two van Hove points L${}_{1}=(\pi-\delta
k_{1},\pi+\delta k_{1},\pi)$ (on the face of the Brillouin zone) and
L${}_{2}=(\pi-\delta k_{2},\pi-\delta k_{2}^{\prime},\pi-\delta
k_{2}^{\prime})$, where $\delta k_{1}\simeq
2\sqrt{\delta\tau^{\prime}/\tau^{\prime}}$, $\delta
k_{2}(\tau,\tau^{\prime})\approx-4\sqrt{{\delta\tau^{\prime}}/({3\tau^{\prime}})},$
$\delta k_{2}^{\prime}(\tau,\tau^{\prime})\approx
2\sqrt{{\delta\tau^{\prime}}/({3\tau^{\prime}})}$ at small
$\delta\tau^{\prime}=\tau^{\prime}-\tau^{\prime}_{\rm c}$. The energies of
these van Hove points as functions of $\tau^{\prime}$ at $\tau=-0.13$ are
shown in Fig. 1. One can see that despite the fact that $L_{1,2}$ points are
well separated in reciprocal space, their energies $E_{L_{1,2}}$ are very
close and their difference increases only slowly with increasing of
$\tau^{\prime}$. This scenario of high-order van Hove point splitting was
earlier found for cubic lattices with the dispersion, containing only nearest-
and next-nearest neighbor hoppings 2019:Igoshev_PMM ; 2019:Igoshev_JETP ;
2022:Igoshev . Using the tetrahedron method 1994:Andersen , we calculate
numerically the density of states
$\rho(\epsilon,\tau,\tau^{\prime})=\sum_{{\mathbf{k}}}\delta(\epsilon-\epsilon_{{\mathbf{k}}}(\tau,\tau^{\prime}))$,
see Fig. 2(a). Here and below we include the factor $1/N$ into the definition
of the sum over Brillouin zone. One can see that the plateau with finite but
rather high value of density of states occurs in the energy interval
$\left[E_{\rm L_{1}},E_{\rm L_{2}}\right]$. At
$\tau^{\prime}=\tau^{\prime}_{\rm c}(\tau)$, where $\tau^{\prime}_{\rm
c}(\tau)$ is determined by the Eq. (3), the DOS has a logarithmic van Hove
singularity: $\rho(\epsilon,\tau,\tau^{\prime}_{\rm
c}(\tau))\sim{8}/({\pi^{2}{{\sqrt{3(1+2\tau)}}}})\ln\left({4}/{|\epsilon-\epsilon_{\mathbf{k}_{\rm
L}}(\tau,\tau^{\prime}_{\rm c})|}\right)$ (see Appendix A).

We also compare the above discussed van Hove singularities to the case of
high-order van Hove singularity for the dispersion on the fcc lattice with
only nearest and next-nearest hopping ($\tau^{\prime}=0$) 2022:Igoshev . In
this case, the high-order van Hove point is the $\Gamma$ point of the
Brillouin zone; the van Hove singularity occurs when $\tau=1$ and it is
located at the bottom of the band,
$\rho(\epsilon,1,0)\propto(\epsilon+6)^{-1/4}$, see Fig. 2(b). While $\tau$
deviates from 1, the van Hove point splits onto the van Hove structure
corresponding to the points ${\varSigma^{\ast}}$ and ${\varDelta^{\ast}}$,
which lie on high-symmetry directions $\varSigma(p,p,0)$ and
$\varDelta(p,0,0)$ in the vicinity of the $\Gamma$ point and have close
energies provided that $\tau-1\ll 1$. While the energy interval
$[E_{\varSigma^{*}},E_{\varDelta^{*}}]$ does not formally correspond to
divergence of DOS, it can provide rather high value of DOS, which is of a
great importance for itinerant ferromagnetism stability. See an example of DOS
in Fig. 2(b) at $\tau=1.05$ and more details in Ref. 2022:Igoshev, .

###  II.3 fRG equations

We use the 1PI fRG equations fRG  for 1PI 2-particle vertex, which in the
absence of magnetic field and magnetic order (i.e., for the SU(2) symmetry)
has the form

$\Gamma_{\sigma_{1}\sigma_{2};\sigma_{3}\sigma_{4}}(p_{1},p_{2},p_{3})=V(p_{1},p_{2},p_{3})\delta_{\sigma_{1}\sigma_{3}}\delta_{\sigma_{2}\sigma_{4}}-V(p_{1},p_{2},p_{4})\delta_{\sigma_{1}\sigma_{4}}\delta_{\sigma_{2}\sigma_{3}}.$
(4)

The momentum dependence of the vertex $V$ is of particular interest. To
parametrize this, we introduce following C. Husemann and M. Salmhofer
2009:Husemann  bosonic ($l$) and fermionic ($k_{1,2}$) momenta in various
channels:

$\displaystyle l^{\rm P}$ $\displaystyle=$ $\displaystyle p_{1}+p_{2},k^{\rm
P}_{1}=(p_{1}-p_{2})/2,k^{\rm P}_{2}=(p_{4}-p_{3})/2,$ (5) $\displaystyle
l^{\rm M}$ $\displaystyle=$ $\displaystyle p_{1}-p_{3},k^{\rm
M}_{1}=(p_{1}+p_{3})/2,k^{\rm M}_{2}=(p_{2}+p_{4})/2,$ (6) $\displaystyle
l^{\rm K}$ $\displaystyle=$ $\displaystyle p_{3}-p_{2},k^{\rm
K}_{1}=(p_{1}+p_{4})/2,k^{\rm K}_{2}=(p_{2}+p_{3})/2.$ (7)

Within the channel decoupling representation

$\displaystyle V(p_{1},p_{2},p_{3})$ $\displaystyle=U-\Phi_{\rm SC}(l^{\rm
P},k_{1}^{\rm P},k_{2}^{\rm P})+\Phi_{\rm M}(l^{\rm M},k_{1}^{\rm
M},k_{2}^{\rm M})$ $\displaystyle+\dfrac{1}{2}\Phi_{\rm M}(l^{\rm
K},k_{1}^{\rm K},k_{2}^{\rm K})-\dfrac{1}{2}\Phi_{\rm K}(l^{\rm K},k_{1}^{\rm
K},k_{2}^{\rm K}),$ (8)

we have the following equations for constituents of the vertex $V$

$\displaystyle\dot{\Phi}_{\rm SC}(l^{\rm P},k_{1}^{\rm P},k_{2}^{\rm P})$
$\displaystyle=$ $\displaystyle-\tau_{\rm pp}(p_{1},p_{2},p_{3}),$ (9)
$\displaystyle\dot{\Phi}_{\rm M}(l^{\rm M},k_{1}^{\rm M},k_{2}^{\rm M})$
$\displaystyle=$ $\displaystyle\tau_{\rm ph}^{\rm cr}(p_{1},p_{2},p_{3}),$
(10) $\displaystyle\dot{\Phi}_{\rm K}(l^{\rm K},k_{1}^{\rm K},k_{2}^{\rm K})$
$\displaystyle=$ $\displaystyle-2\tau_{\rm ph}^{\rm
d}(p_{1},p_{2},p_{3})+\tau_{\rm ph}^{\rm cr}(p_{1},p_{2},p_{4}),$ (11)

where the derivative is taken with respect to the flow parameter $s=\ln(t/T)$
and explicit expressions for $\tau_{\rm pp}(p_{1},p_{2},p_{3})$, $\tau_{\rm
ph}^{\rm d}(p_{1},p_{2},p_{3})$, and $\tau_{\rm ph}^{\rm
cr}(p_{1},p_{2},p_{3})$ are given in Appendix B.

In the following, we neglect the frequency dependencies of the vertices, which
is justified in the weak-coupling regime. To treat the momentum dependencies,
we apply truncated unity fRG approach 2017:TUfRG . To this end, we define the
projection operators acting on functions of the momenta $F(p_{1},p_{2},p_{3})$
as

$\displaystyle F^{\rm P}_{n,m}(\mathbf{l})$ $\displaystyle=$
$\displaystyle\sum_{{\mathbf{k}}{\mathbf{k}}^{\prime}}f_{n}(\textbf{k})f_{m}(\textbf{k}^{\prime})F\left(\dfrac{\mathbf{l}}{2}+\textbf{k},\dfrac{\mathbf{l}}{2}-\textbf{k},\dfrac{\mathbf{l}}{2}-\textbf{k}^{\prime}\right),$
(12) $\displaystyle F^{\rm M}_{n,m}(\mathbf{l})$ $\displaystyle=$
$\displaystyle\sum_{{\mathbf{k}}{\mathbf{k}}^{\prime}}f_{n}(\textbf{k})f_{m}(\textbf{k}^{\prime})F\left(\dfrac{\mathbf{l}}{2}+\textbf{k},{\mathbf{k}}^{\prime}-\dfrac{\mathbf{l}}{2},{\mathbf{k}}-\dfrac{\mathbf{l}}{2}\right),$
(13) $\displaystyle F^{\rm K}_{n,m}(\mathbf{l})$ $\displaystyle=$
$\displaystyle\sum_{{\mathbf{k}}{\mathbf{k}}^{\prime}}f_{n}(\textbf{k})f_{m}(\textbf{k}^{\prime})F\left(\dfrac{\mathbf{l}}{2}+\textbf{k},\textbf{k}^{\prime}-\dfrac{\mathbf{l}}{2},\dfrac{\mathbf{l}}{2}+\textbf{k}^{\prime}\right),$
(14)

where a set of functions $f_{n}({\mathbf{k}})$ constitute a basis in the
corresponding Hilbert space. We apply these operators to Eqs. (9)–(11). Using
the projection rules (12) and (13), and inserting representation of unity in
the right-hand sides (r.h.s.), we obtain the projected components of the
r.h.s. (see Appendix B)

$\displaystyle(\tau_{\rm pp})^{\rm P}_{n_{1},n_{2}}(\mathbf{l})$
$\displaystyle=\sum_{m_{1},m_{2}}V^{\rm P}_{n_{1},m_{1}}(\mathbf{l})\chi^{\rm
pp}_{m_{1},m_{2}}(\mathbf{l})V^{\rm P}_{m_{2},n_{2}}(\mathbf{l}),$ (15)
$\displaystyle(\tau^{\rm cr}_{\rm ph})^{\rm M}_{n_{1},n_{2}}(\mathbf{l})$
$\displaystyle=\sum_{m_{1},m_{2}}V^{\rm M}_{n_{1},m_{1}}(\mathbf{l})\chi^{\rm
ph}_{m_{1},m_{2}}(\mathbf{l})V^{\rm M}_{m_{2},n_{2}}(\mathbf{l}),$ (16)

where the explicit form of $V^{\rm P,M}(\mathbf{l})$ is given by the Eqs. (12)
and (13) with $F=V$,

$\displaystyle\chi^{\rm pp,ph}_{n,m}(\mathbf{l})$
$\displaystyle=\sum_{\mathbf{p}}f_{n}(\mathbf{p})f_{m}(\mathbf{p})\sum_{\nu}L_{\rm
pp,ph}(\mathbf{l},p),$ (17) $\displaystyle L_{\rm pp,ph}(\mathbf{l},p)$
$\displaystyle=-\partial_{s}\left[G(\mathbf{l}/2+{\bf
p},\nu)G(\pm\mathbf{l}/2\mp{\bf p},\mp\nu)\right],$ (18)

$G({\bf p},\nu)$ is the electron Green’s function. Finally,

$(\tau^{\rm d}_{\rm ph})^{\rm
K}_{n_{1},n_{2}}(\mathbf{l})=-2\sum_{m_{1},m_{2}}V^{\rm
K}_{n_{1},m_{1}}(\mathbf{l})\chi^{\rm ph}_{m_{1},m_{2}}(\mathbf{l})V^{\rm
K}_{m_{2},n_{2}}(\mathbf{l})\\\ +\sum_{m_{1},m_{2}}[V_{\rm ex}]^{\rm
K}_{n_{1},m_{1}}(\mathbf{l})\chi^{\rm ph}_{m_{1},m_{2}}(\mathbf{l})V^{\rm
K}_{m_{2},n_{2}}(\mathbf{l})+\sum_{m_{1},m_{2}}V^{\rm
K}_{n_{1},m_{1}}(\mathbf{l})\chi^{\rm ph}_{m_{1},m_{2}}(\mathbf{l})[V_{\rm
ex}]^{\rm K}_{m_{2},n_{2}}(\mathbf{l}),$ (19)

where the index “ex” corresponds to an interchange of third and forth 4-vertex
argument: $V_{\rm
ex}(\mathbf{l}/2+\mathbf{k}_{1},\mathbf{k}_{2}-\mathbf{l}/2,\mathbf{l}/2+\mathbf{k}_{2})=V(\mathbf{l}/2+\mathbf{k}_{1},\mathbf{k}_{2}-\mathbf{l}/2,\mathbf{k}_{1}-\mathbf{l}/2)$,
and

$(\tau^{\rm cr}_{\rm ph,ex})^{\rm
K}_{n_{1},n_{2}}(\mathbf{l})=\sum_{m_{1},m_{2}}V^{\rm
M}_{n_{1},m_{1}}(\mathbf{l})\chi^{\rm ph}_{m_{1},m_{2}}(\mathbf{l})V^{\rm
M}_{m_{2},n_{2}}(\mathbf{l}),$ (20)

the explicit form of $V^{\rm K}(\mathbf{l})$ is given by the Eq. (14) with
$F=V$ and we have used ${F}^{\rm K}_{\rm ex}={F}^{\rm M}$.

Denoting matrices with respect to the projection indexes by bold capital
letters, we rewrite Eqs. (9)–(11) in the matrix form in terms of projections:

$\displaystyle\dot{\bm{\Phi}}^{\rm SC}(\mathbf{l})$ $\displaystyle=$
$\displaystyle-\mathbf{V}^{\rm P}(\mathbf{l})\cdot{\bm{\chi}}^{\rm
pp}(\mathbf{l})\cdot\mathbf{V}^{\rm P}(\mathbf{l}),$ (21)
$\displaystyle\dot{\bm{\Phi}}^{\rm M}(\mathbf{l})$ $\displaystyle=$
$\displaystyle\mathbf{V}^{\rm M}(\mathbf{l})\cdot{\bm{\chi}}^{\rm
ph}(\mathbf{l})\cdot\mathbf{V}^{\rm M}(\mathbf{l}),$ (22)
$\displaystyle\dot{\bm{\Phi}}^{\rm K}(\mathbf{l})$ $\displaystyle=$
$\displaystyle(2\mathbf{V}^{\rm K}(\mathbf{l})-\mathbf{V}^{\rm
M}(\mathbf{l}))\cdot{\bm{\chi}}^{\rm ph}(\mathbf{l})\cdot(2\mathbf{V}^{\rm
K}(\mathbf{l})-\mathbf{V}^{\rm M}(\mathbf{l})),$ (23)

where $\cdot$ means matrix multiplication. We consider the temperature
flow2001:Salmhofer  with $s=-\ln T$, $G({\bf p},\nu)={T^{1/2}}/({{\rm
i}\nu-\xi_{\mathbf{p}}})$, and

$\sum_{\nu}L_{\rm pp,ph}({\bf
l},p)=\pm\partial_{s}\frac{f[\xi_{\mathbf{p}+\mathbf{l}/2}]-f[\mp\xi_{\mathbf{p}-\mathbf{l}/2}]}{\xi_{\mathbf{p}+\mathbf{l}/2}\pm\xi_{\mathbf{p}-\mathbf{l}/2}}.$
(24)

Here $\xi_{\mathbf{p}}=\varepsilon_{{\mathbf{p}}}-\mu$ ($\mu$ being the
chemical potential), $f[\xi]=1/(\exp(\xi/T)+1)$ is the Fermi function. Since
$\partial_{s}{f}[\xi]=\xi f^{\prime}[\xi]$, we get

$\chi^{\rm
pp,ph}_{n,m}(\mathbf{l})=\pm\sum_{\mathbf{p}}f_{n}(\mathbf{p})f_{m}(\mathbf{p})\frac{\xi_{\mathbf{p}+\mathbf{l}/2}f^{\prime}[\xi_{\mathbf{p}+\mathbf{l}/2}]\pm\xi_{\mathbf{p}-\mathbf{l}/2}f^{\prime}[\mp\xi_{\mathbf{p}-\mathbf{l}/2}]}{\xi_{\mathbf{p}+\mathbf{l}/2}\pm\xi_{\mathbf{p}-\mathbf{l}/2}},$
(25)

We restrict consideration to the trivial set of the ($s$-wave) basis functions
$f_{m}={\rm const}$, which corresponds to momentum averaging of the vertex
taken in the “transverse” channels. The fRG equations can be written in the
following explicit form within the channel decoupling scheme

$\displaystyle\dot{\Phi}_{\rm SC}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle-V_{\rm SC}^{2}({\mathbf{q}})L_{+}({\mathbf{q}}),$ (26)
$\displaystyle\dot{\Phi}_{\rm M}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle+V_{\rm M}^{2}({\mathbf{q}})L_{-}({\mathbf{q}}),$ (27)
$\displaystyle\dot{\Phi}_{\rm K}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle+V_{\rm K}^{2}({\mathbf{q}})L_{-}({\mathbf{q}}),$ (28)

where

$\displaystyle V_{\rm SC}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle
U-\Phi_{\rm SC}({\mathbf{q}})+\frac{3}{2}\bar{\Phi}_{\rm
M}-\frac{1}{2}\bar{\Phi}_{\rm K},$ (29) $\displaystyle V_{\rm
M}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle U+\Phi_{\rm
M}({\mathbf{q}})-\bar{\Phi}_{\rm SC}+\frac{1}{2}\bar{\Phi}_{\rm
M}-\frac{1}{2}\bar{\Phi}_{\rm K},$ (30) $\displaystyle V_{\rm
K}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle U-\Phi_{\rm
K}({\mathbf{q}})-\bar{\Phi}_{\rm SC}+\frac{3}{2}\bar{\Phi}_{\rm
M}+\frac{1}{2}\bar{\Phi}_{\rm K},$ (31)
$L_{\pm}({\mathbf{q}})=\pm\sum_{{\mathbf{p}}}\frac{\xi_{{\mathbf{p}}+{\mathbf{q}}/2}f^{\prime}_{{\mathbf{p}}+{\mathbf{q}}/2}\pm\xi_{{\mathbf{p}}-{\mathbf{q}}/2}f^{\prime}_{{\mathbf{p}}-{\mathbf{q}}/2}}{\xi_{{\mathbf{p}}+{\mathbf{q}}/2}\pm\xi_{{\mathbf{p}}-{\mathbf{q}}/2}},$
(32)

and

$\bar{\Phi}_{c}\equiv\sum_{{\mathbf{q}}}\Phi_{c}({\mathbf{q}}),\;c=\text{SC,
M, K}.$ (33)

The fRG equations (26)–(28) form the system of ordinary functional
differential equations. These equations also allow single-boson exchange
decomposition 2022:BFfRG , see Appendix C. Below we present the procedure of
numerical solution of the equations (26)–(28) in details.

###  II.4 Numerical implementation

Since the polarization functions derivatives $L_{\pm}({\bf q})$ are
independent of $\Phi_{\rm SC,M,K}({\mathbf{q}})$, we calculate the scale $s$
dependence of $L_{\pm}({\bf q})$ on a grid of $s$ and ${\mathbf{q}}$. The $s$
dependence of $L_{\pm}({\bf q})$ is further interpolated by splines. We use
the ${\mathbf{q}}$ grid in the Brillouin zone with the number of reciprocal
vector division $n_{{\mathbf{q}}}=8$, which corresponds to 29 different points
in the part of the Brillouin zone, irreducible by the point group symmetry. We
have verified that the results do not change qualitatively for
$n_{{\mathbf{q}}}=16$. To perform integration over ${\mathbf{k}}$ in Eq. (32)
we consider adaptive division of each of the $6n^{3}_{{\mathbf{q}}}$
tetrahedrons in the standard uniform grid of the Brillouin zone using TOMS
algorithm 1993:Berntsen . Here we take into in account the fact that
$\Phi_{\rm SC,M,K}(\mathbf{q})$ is a more smooth function of $\mathbf{q}$,
than the electronic dispersion $\xi_{\mathbf{k}}$, which details are of a
great importance and require using very dense (adaptive) $\mathbf{k}$ grid for
integration (see Eq. (32)). The obtained functions $L_{\pm}({\bf q})$ are used
for the solution of ordinary differential equations by Runge-Kutta procedure
for different values of $U$ without recalculation of $L_{\pm}$. The divergence
of $\Phi_{\rm M}(\mathbf{Q})$ at some scale $s=s_{0}$ corresponds to the phase
transition into the ordered phase with the wave vector $\mathbf{Q}$ and the
corresponding critical temperature (Curie temperature $T_{\rm C}=\exp(-s_{0})$
at $\mathbf{Q}=0$).

Figure 3: The $s$ dependence of $L_{-}(\mathbf{q}=0)$, see definition (32), at
$\tau=-\tau^{\prime}=-0.13$, and different values of $\mu$. Numbers in the
legend denote $\mu$.

The $s$ dependence of quantities $L_{\pm}(\mathbf{q})$ determines the fRG flow
for specified $U$. The most important is $L_{-}(\mathbf{q}=0)$ dependence, see
the examples of its scale dependence in Fig. 3. Non-monotonic $s$-dependence
of $L_{-}(0)$ originates from the presence of several energy scales of DOS.
The first pair of extema ($0.5\lesssim s\lesssim 1.5$) corresponds to large
scale feature of DOS at $2.5\lesssim\epsilon\lesssim 3.5$ and its is not
sensitive to the variation of the Fermi level position. The second pair of
extema (at $3\lesssim s\lesssim 5$), which is present for $\mu=2.123$, is
sensitive to the Fermi level position and originates from the small scale
feature (van Hove plateau) of density of states at
$2.167\lesssim\epsilon\lesssim 2.172$. One can see that for $\mu$ far away
from van Hove plateau, $L_{-}(\mathbf{q}=0)$ becomes negative below some
$\mu$-dependent temperature. Therefore below this temperature the formation of
ferromagnetic instability becomes impossible, see Eq. (27). When for fixed $U$
and the temperature $T_{\rm C}(\mu,U)$ the condition $L_{-}(\mathbf{q}=0)=0$
is fulfilled, $T_{\rm C}$ jumps to zero which indicates the first order
quantum phase transition.

Since the numerical complexity of calculation of $L_{\pm}(\mathbf{Q})$ grows
with decreasing temperature due to narrowing of actual integration region
domain, only $s\leq s_{\rm max}\simeq 6$ region is accessible. For calculation
of the phase diagrams we calculate fRG flows at several values of the chemical
potential $\mu$ and the dense grid of $U$, determining Curie temperature
$T_{\rm C}(\mu,U)$, if it exists. Generally, the flow is stopped when the
vertices become large, which corresponds to magnetic instability, or maximal
available scale $s_{\rm max}$ is achieved. If the ferromagnetic component of
the vertex $\Phi_{\rm M}(\mathbf{q}=0)$ is strongly enhanced at the end of the
flow, the extrapolation procedure is employed for $\Phi_{\rm M}(\mathbf{q}=0)$
to determine $T_{\rm C}$ beyond the available temperature region.

##  III Results

In this section we present and discuss numerical results of the present fRG
approach for different position of the Fermi level and Coulomb interaction
$U$.

Figure 4: Scale (temperature) dependence for components of the vertex
$\Phi_{\rm SC}(\mathbf{0})$ and $\Phi_{\rm M}(\mathbf{q})$ in units of $U$ for
various symmetric ${\mathbf{q}}$ vectors for $\tau=-\tau^{\prime}=-0.13$,
$\mu=E_{\rm L_{2}}=2.17335$, $U=4t$. Double-dot-dashed line shows the RPA
result. The Fermi level corresponds to right edge of van Hove plateau, see
Fig. 2(b). The divergence of $\Phi_{M}(\mathbf{Q}=0)$ corresponds to $s=s_{\rm
C}=5.0$.

We consider first the flow for $\tau=-\tau^{\prime}=-0.13$. For the Fermi
level $\mu=E_{L_{2}}=2.17335t$, corresponding to the right edge of van Hove
plateau (see Fig. 2), the fRG flow of the vertices $\Phi_{\rm SC}(0)$ and
$\Phi_{\rm M}({\bf Q})$ for $U=4t$ and symmetric wave vectors
$\mathbf{Q}=0,\mathbf{Q}_{\rm X}=(2\pi,0,0),\mathbf{Q}_{\rm
L}=(\pi,\pi,\pi),\mathbf{Q}_{\rm W}=(2\pi,\pi,0),\mathbf{Q}_{{\rm
X}_{1}}=\mathbf{Q}_{\rm X}/2$ is shown in Fig. 4. The critical temperature of
ferromagnetic instability $T_{\rm C}$ manifests itself by the diverging
$\Phi_{\rm M}(\mathbf{Q}=0)$ component of the vertex at $s=s_{\rm C}$. The
substantial screening particle-particle component of the vertex, $\Phi_{\rm
SC}(\mathbf{Q}=0)\approx(1.5-2)U$ is found in the end of the flow. We actually
do not find any noticeable incommensurate or antiferromagnetic fluctuation to
be enhanced and competing with ferromagnetism formation. In particular, from
Fig. 4 one can see that $\Phi_{\rm M}(\mathbf{q})$ is not enhanced remarkable
when $\mathbf{q}\neq 0$ for the chosen $\mathbf{q}$ points; we have also
verified that this holds in the entire Brillouin zone. Therefore the
ferromagnetic ordering is suppressed due $\bar{\Phi}$ contributions into Eq.
(22), but not due to competition with incommensurate $\Phi(\mathbf{q}\neq 0)$,
in contrast to the mean-field studies 2010:Igoshev ; 2015:Igoshev . One can
also see that the obtained Curie temperature $T_{\rm C}$ is much lower than
the corresponding temperature in the random phase approximation (RPA) due to
screening of the interaction in the particle-particle channel.

Figure 5: $\tau=-0.13$, $\tau^{\prime}=+0.0925$, $\mu=1.89$. Scale
(temperature) dependence for components of 4-vertex $\Phi_{\rm
SC}(\mathbf{Q}=0)$, $\Phi_{\rm M}(\mathbf{q})$ in units of $U$ for symmetric
vectors ${\bf q}$, $U=5t$. Double-dot-dashed line shows the RPA result. The
Fermi level corresponds to van Hove singularity at $\mu=1.89t$, see Fig. 2(a).

To show the necessity of having a pair of van Hove points, we contrast the
above discussed case $\tau^{\prime}=0.13$ (a pair of giant van Hove
singularities yielding plateau of of DOS) to the case
$\tau^{\prime}=\tau^{\prime}_{\rm c}=0.0925$ (weak van Hove singularity
$\rho(\epsilon,\tau,\tau^{\prime}_{\rm
c}(\tau))\sim{8}/({\pi^{2}{{\sqrt{3(1+2\tau)}}}})\ln\left({4}/{|\epsilon-\epsilon_{\mathbf{k}_{\rm
L}}(\tau,\tau^{\prime}_{\rm c})|}\right)$, see Sect. II.2). One can see from
Fig. 5 that at $U=5t$ no divergence of $\Phi_{\rm M}(\mathbf{Q}=0)$ is
achieved. In this case, the divergence is suppressed by faster growth of the
screening component $\Phi_{\rm SC}(\mathbf{Q}=0)$, which also reaches
considerable value $\sim(1.5-2)U$. The role of incommensurate or
antiferromagnetic correlations is the same as in the case
$\tau=-\tau^{\prime}=-0.13$, none of them compete with the ferromagnetic
instability. Therefore the plateau of the density of states, or, at least,
stronger than logarithmic divergence of the density of states, seems to be the
necessary condition for ferromagnetism formation in three dimensions in the
single-band Hubbard model. This conclusion agrees with the results of Ref.
Ulmke, .

We show the obtained phase diagram $T_{\rm C}(\mu,U)$ for the case
$\tau=-\tau^{\prime}=-0.13$ in Fig. 6. The chosen interval of the chemical
potentials $\mu$ is taken around vHs plateau, corresponding to the fillings
$n\in(0.65,0.68)$ per one spin projection. We also show the line
$L_{-}(\mathbf{q}=0)=0$ as a function of temperature and $\mu$.

Figure 6: Phase diagram $\mu$-$T_{\rm C}$ for $\tau=-\tau^{\prime}=-0.13$ at
different $U$. $T^{\rm RPA}_{\rm C}(\mu,U_{\rm eff})$ is shown by dashed lines
(see text). The arrows mark position of the plateau of the density of states.

One can see that as expected, the phase transition temperatures increase with
$U$; $T_{\rm C}(\mu,U)$ is also largest for the Fermi level at the narrow van
Hove plateau and decrease when moving away from the plateau. The obtained
quantum phase transitions are of the first order (corresponding to the sudden
disappearance of $T_{\rm C}$ with changing $\mu$) due to change of the sign of
$L_{-}({\bf q}=0)$. It is interesting that in the considered case the lines of
the first-order transitions form themselves the “quantum critical fan” which
begins at $T=0$ at the position of a narrow van Hove plateau. At strong $U$
substantial values of $T_{\rm C}(\mu,U)$ are obtained therefore far beyond the
position of the plateau.

Figure 7: $U$ dependence of $U_{\rm eff}$ for the cases
$\tau=\tau^{\prime}=-0.13$ and $\tau=1.05$, $\tau^{\prime}=0$.  Figure 8: The
same as Fig. 6 at $\tau=1.05$ and $\tau^{\prime}=0$. The value $U_{\rm eff}$
for RPA is adjusted to reproduce the same $T_{\rm C}$ at $\mu=-5.72t$. The
arrows mark position of the plateau of the density of states.

The obtained values of Curie temperature are almost an order of magnitude
smaller than the Curie temperature in RPA. To compare the obtained results
with RPA we therefore introduce an effective Coulomb interaction $U_{\rm eff}$
(cf. Ref. Yuki, ). The interaction $U_{\rm eff}$ is determined to match
$T^{\rm RPA}_{\rm C}(\mu_{*},U_{\rm eff})=T_{\rm C}(\mu_{*},U)$ for some fixed
$\mu=\mu_{*}$ (for $\tau=-\tau^{\prime}=-0.13$ we choose $\mu_{*}=E_{L_{2}}$).
From Fig. 6 one can see that $T^{\rm RPA}_{\rm C}(\mu,U_{\rm eff})$ only
weakly deviates from the results of the fRG approach. The $U$ dependence of
the resulting $U_{\rm eff}$ is shown in Fig. 7. We have verified that simple
estimate of $U_{\rm eff}=U/(1+U\Pi_{\rm pp}(\mathbf{q}=0))$ with $\Pi_{\rm
pp}(\mathbf{q})=\sum_{\mathbf{p}}\\{{f[-\xi_{\mathbf{p}-\mathbf{q}/2}]-f[\xi_{\mathbf{p}+\mathbf{q}/2}]}\\}/({\xi_{\mathbf{p}+\mathbf{q}/2}+\xi_{\mathbf{p}-\mathbf{q}/2}})$,
corresponding to the summation of ladder particle-particle diagrams, yields
much smaller (up to 4 times) value $U_{\rm eff}$, than obtained within the fRG
approach. Despite the fact that this approach was often used for screening
strength estimation, our results imply impossibility of separate
renormalization of Coulomb interaction in different channels, which also shows
important effect of the interference of different electron scattering
channels.

In Fig. 8 we show that the similar dome-shaped form of the phase diagram in
terms of $\mu$ and $T_{\rm C}$ is obtained for $\tau=1.05$ and
$\tau^{\prime}=0$. The chosen interval of chemical potentials $\mu$ is taken
around vHs plateau, corresponding to fillings $n\in(0.031,0.056)$ per one spin
projection. In this case despite, the relatively broad van Hove plateau, the
deviation of $\mu$ from van Hove structure (see Fig. 2(b)) results in a rapid
decay of $T_{\rm C}$. The role of incommensurate correlations here is the same
as in above considered cases for $\tau=-0.13$, they are not enhanced
considerably. Choosing $\mu_{*}=-5.72t$ for the determination of $U_{\rm
eff}$, we see from Fig. 8 that the obtained $T^{\rm RPA}_{\rm C}(\mu,U_{\rm
eff})$ for the case $\tau=1.05$, $\tau^{\prime}=0$ almost perfectly reproduces
the results of fRG approach in the region $T_{\rm C}>0$. For the case
$\tau=1.05$, $\tau^{\prime}=0$ the renormalization of the interaction $U$ is
weaker than for the case $\tau=-\tau^{\prime}=-0.13$ (see Fig. 7), which seems
to be related to smaller density of states in the former case.

##  IV Conclusions

In this study, we present the fRG treatment of an instability towards
ferromagnetic order in the Hubbard model, controlled by the Fermi level being
in the vicinity of van Hove singularity and the on–site Coulomb interaction.

For the densities of states with the plateau, formed by the higher-order van
Hove singularities, we find ferromagnetic instability for the Fermi level
being in the vicinity of the plateau, with the Curie temperature decreasing
with moving the Fermi level away from the plateau. We find the first-order
quantum phase transition to paramagnetic state with further moving the Fermi
level away from the plateau.

In the considered case of van Hove singularities in fcc lattice,
incommensurate correlations are not remarkably enhanced. This considerably
distinguishes from the case of ferromagnetism formation in two-dimensional
case with large $t^{\prime}/t\sim 0.4-0.5$ where strong competition with long-
wave incommensurate magnetic fluctuations was found 2007:Igoshev ;
2009:Igoshev ; 2011:Igoshev . We therefore find that in the three-dimensional
case the main cause of ferromagnetism suppression is the particle-particle
screening.

The analysis of fRG flows in the the case of logarithmic singularity of
density of states ($\tau=-0.13$ and $\tau^{\prime}=0.0925$) with the Fermi
level being at van Hove singularity yields the absence of ferromagnetic
instability well below available temperature scales ($s<s_{\rm max}=5.5-6.0$).
One can state that wide plateau with large value of density of states caused
by van Hove structure (or, at least, stronger divergence of the density of
states) is necessary to stabilize the ferromagnetic instability in the three-
dimensional case, and logarithmic van Hove singularity is not sufficient due
to strong particle-particle screening.

In the present paper we have considered $s$-wave basis function only, which
implies averaging of the dependence of vertices on fermionic momenta. In
future studies other basis functions, e.g. peaked at the van Hove singularity
points, can be included. This may allow to describe the effect of stronger
fermionic momentum dependence of vertices near van Hove points.

In realistic materials, the Hund exchange interaction plays important role. On
one hand, it provides (partial) formation of local magnetic moments, such that
the local dynamic two-particle correlations, not accounted in the present
study become important (see, e.g., Refs. 2017:Hausoel, ; OurFe, ). On the
other hand, Hund exchange enhances the tendency towards ferromagnetic order,
see, in particular, Ref. Arita, . Studying the effect of Hund exchange in
realistic materials within the non-local extensions of dynamical mean-field
theory OurRev , in particular, the DMF2RG approach DMF2RG , is therefore of
certain interest.

##  V Acknowledgments

The research was carried out within the state assignment of the Ministry of
Science and Higher Education of the Russian Federation (theme “Quant” No.
122021000038-7) and partly supported by RFBR grant 20-02-00252a. A. K. also
acknowledges the support from the Ministry of Science and Higher Education of
the Russian Federation (Agreement No. 075-15-2021-606). The calculations were
performed on the Uran supercomputer at the IMM UB RAS.

##  Appendix A Electron dispersion and giant van Hove singularities of DOS

At $\tau^{\prime}<\tau^{\prime}_{\rm c}$, only van Hove L point is present.
The investigation of the dispersion (2) in the vicinity of L point at
$\tau^{\prime}_{\rm c}<\tau^{\prime}$ yields the following van Hove points:
(i) L${}_{1}(\pi-\delta k_{1},\pi+\delta k_{1},\pi)$ (on the face of Brillouin
zone), where

$\delta k_{1}(\tau,\tau^{\prime})=2\arcsin\sqrt{1-\frac{\tau^{\prime}_{\rm
c}(\tau)}{\tau^{\prime}}}.$ (34)

(ii) L${}_{2}(\pi-\delta k_{2}(\tau,\tau^{\prime}),\pi-\delta
k_{2}^{\prime}(\tau,\tau^{\prime}),\pi-\delta
k_{2}^{\prime}(\tau,\tau^{\prime}))$, where

$\displaystyle\delta k_{2}(\tau,\tau^{\prime})$ $\displaystyle=$
$\displaystyle-2\arcsin\frac{2\mathfrak{s}(\tau,\tau^{\prime})}{1+8\delta\tau^{\prime}(\tau)-16\tau^{\prime}\mathfrak{s}^{2}(\tau,\tau^{\prime})},$
(35) $\displaystyle\delta k_{2}^{\prime}(\tau,\tau^{\prime})$ $\displaystyle=$
$\displaystyle 2\arcsin\mathfrak{s}(\tau,\tau^{\prime}),$ (36)

where $\delta\tau^{\prime}(\tau)=\tau^{\prime}-\tau^{\prime}_{\rm c}(\tau)$
and $\mathfrak{s}=\mathfrak{s}(\tau,\tau^{\prime})$ is a root of the equation

$\delta\tau^{\prime}+\tau^{\prime}\mathfrak{s}^{2}+4\left(1+4\delta\tau^{\prime}(\tau)-4\tau^{\prime}\mathfrak{s}^{2}\right)(-\delta\tau^{\prime}(\tau)+2\tau^{\prime}\mathfrak{s}^{2})(1+4\delta\tau^{\prime}(\tau)-8\tau^{\prime}\mathfrak{s}^{2})=0.$
(37)

At small $\delta\tau^{\prime}$, we can simplify this equation by neglecting
small terms in brackets compared to 1, to obtain

$-3\delta\tau^{\prime}(\tau)+9\tau^{\prime}\mathfrak{s}^{2}=0,$ (38)
$\displaystyle\delta k_{2}(\tau,\tau^{\prime})$ $\displaystyle\approx$
$\displaystyle-2\arcsin 2\sqrt{\frac{\delta\tau^{\prime}}{3\tau^{\prime}}},$
(39) $\displaystyle\delta k_{2}^{\prime}(\tau,\tau^{\prime})$
$\displaystyle\approx$ $\displaystyle
2\arcsin\sqrt{\frac{\delta\tau^{\prime}}{3\tau^{\prime}}}.$ (40)

Note that at small $\delta\tau^{\prime}$ we have $\delta
k_{2}(\tau,\tau^{\prime})\approx 2\delta k_{2}^{\prime}(\tau,\tau^{\prime})$
which implies that L2 point stays closely to the face of Brillouin zone at
small $\delta\tau^{\prime}$. The energies of these van Hove points as a
function of $\tau^{\prime}$ are shown in Fig. 1 in the main text. The
deviations $\delta k_{1}(\tau,\tau^{\prime})$, $\delta
k_{2}(\tau,\tau^{\prime})$, $\delta k_{2}^{\prime}(\tau,\tau^{\prime})$ as
functions of $\tau$ are shown in Fig. 9.

Figure 9: $k$ deviations of van Hove position, see Eqs. (34)–(36).

One can see that the energy levels of ${\mathbf{k}}_{\rm L_{1}}$,
${\mathbf{k}}_{\rm L_{2}}$ points are very close and the distance between
these points increases slowly with increasing of $\tau^{\prime}$. This
scenario of high-order van Hove point destruction was earlier found for cubic
lattices for the dispersion in the nearest and next-nearest approximation
2019:Igoshev_PMM ; 2019:Igoshev_JETP ; 2022:Igoshev .

In the case $\tau^{\prime}=\tau^{\prime}_{\rm c}$, see Eq. (3), we expand the
dispersion in the vicinity of L point up to fourth-order terms with respect to
both $k_{\Lambda}$ and $q_{\perp}$

$\epsilon_{\mathbf{k}_{\rm
L}+\Delta\mathbf{k}}=\frac{3}{2}(1-2\tau)-\frac{3}{2}k^{2}_{\varLambda}+\frac{1}{96}\left(12k^{2}_{\varLambda}q^{2}_{\perp}+8(1+\tau)k^{4}_{\varLambda}+3(1+2\tau)q^{4}_{\perp}+2\sqrt{2}(1+4\tau)\sin
3\varphi k_{\varLambda}q^{3}_{\perp}\right),$ (41)

here we parameterize the deviation from L point as
$\Delta\mathbf{k}=q_{\perp}\cos\varphi\mathbf{e}_{1}+q_{\perp}\sin\varphi\mathbf{e}_{2}+k_{\varLambda}\mathbf{e}_{3}$,
where the orthonormal basis $\mathbf{e}_{1}(+1/\sqrt{2},0,-1/\sqrt{2})$,
$\mathbf{e}_{2}(-1/\sqrt{6},+2/\sqrt{6},-1/\sqrt{6})$,
$\mathbf{e}_{3}(+1/\sqrt{3},+1/\sqrt{3},+1/\sqrt{3})$ was used. For
convenience, we retain only leading contributions into
$\epsilon_{\mathbf{k}_{\rm L}+\Delta\mathbf{k}}$ with respect to $k_{\Lambda}$
and $q_{\perp}$ dependence:

$\delta\epsilon_{\mathbf{k}_{\rm
L}+\Delta\mathbf{k}}\approx\epsilon_{\mathbf{k}_{\rm
L}+\Delta\mathbf{k}}-\frac{3}{2}(1-2\tau)=-\frac{3}{2}k^{2}_{\varLambda}+\frac{1+2\tau}{32}q^{4}_{\perp}.$
(42)

It can be verified that neglected terms in Eq. (42) do not change
substantially the $\delta\epsilon_{\mathbf{k}_{\rm L}+\Delta\mathbf{k}}$ in a
rather extended vicinity of the L point. Since here $q_{\perp}^{2}$ plays a
similar role as $k_{\Lambda}$ we introduce the same (up to numerical factors)
cutoff $\kappa$ for them:

$(3/2)k_{\Lambda}^{2},\frac{1+2\tau}{32}q^{4}_{\perp}<\kappa^{2}\lesssim 1,$
(43)

From the DOS definition we get the contribution of 8 regions near the L point
and corresponding points shifted by reciprocal lattice constants. We obtain:

$\Delta\rho(\delta\epsilon)\approx\frac{8\pi}{V_{\rm
BZ}}\int\limits_{-\sqrt{2/3}\kappa}^{\sqrt{2/3}\kappa}dk_{\Lambda}\int\limits_{0}^{\sqrt{32\kappa/(1+2\tau)}}dq_{\perp}q_{\perp}\delta\left(\delta\epsilon+\frac{3}{2}k^{2}_{\varLambda}-\frac{1+2\tau}{32}q^{4}_{\perp}\right),$
(44)

where $V_{\rm BZ}=2\pi^{3}$ being Brillouin zone volume. So we obtain that in
this case the logarithmic giant van Hove singularity

$\Delta\rho(\delta\epsilon)\approx\frac{8}{\pi^{2}\sqrt{3}\sqrt{1+2\tau}}\ln\frac{4\kappa^{2}}{|\delta\epsilon|}.$
(45)

##  Appendix B RG equations

We consider the one-loop truncation of fRG equations for
$V(p_{1},p_{2},p_{3})$ fRG .

$\partial_{s}V(p_{1},p_{2},p_{3})=\tau_{\rm pp}(p_{1},p_{2},p_{3})+\tau^{\rm
d}_{\rm ph}(p_{1},p_{2},p_{3})+\tau^{\rm cr}_{\rm ph}(p_{1},p_{2},p_{3}),$
(46)

where

$\displaystyle\tau_{\rm pp}(p_{1},p_{2},p_{3})$
$\displaystyle=-\sum_{p}\partial_{s}\left[G(p)G(p_{1}+p_{2}-p)\right]V\left(p_{1},p_{2},p\right)V\left(p_{1}+p_{2}-p,p,p_{3}\right),$
(47) $\displaystyle\tau_{\rm ph}^{\rm cr}(p_{1},p_{2},p_{3})$
$\displaystyle=-\sum_{p}\partial_{s}\left[G(p)G(p+p_{3}-p_{1})\right]V\left(p_{1},p+p_{3}-p_{1},p_{3}\right)V\left(p,p_{2},p+p_{3}-p_{1}\right)$
(48) $\displaystyle\tau_{\rm ph}^{\rm d}(p_{1},p_{2},p_{3})$
$\displaystyle=\sum_{p}\partial_{s}\left[G(p)G(p+p_{2}-p_{3})\right][2V\left(p_{1},p+p_{2}-p_{3},p\right)V\left(p,p_{2},p_{3}\right)$
(49)
$\displaystyle-V\left(p_{1},p+p_{2}-p_{3},p_{1}+p_{2}-p_{3}\right)V\left(p,p_{2},p_{3}\right)$
$\displaystyle-V\left(p_{1},p+p_{2}-p_{3},p\right)V\left(p,p_{2},p+p_{2}-p_{3}\right)].$

Here $s$ is fRG scale parameter, $p=({\rm i}\nu_{n},\mathbf{p})$ is a
3-momentum, $\nu_{n}=\pi(2n+1)T$ being the fermionic Matsubara frequency with
integer $n_{p}$ and $\mathbf{p}$ being a vector in the Brillouin zone.
Analogous notations hold for $p_{i}$. We use the shorthand notation

$\sum_{p}={T}\sum_{\nu_{n},\mathbf{p}}.$ (50)

Three contribution in Eq. (46) corresponds to particle-particle $\tau_{\rm
pp}$ (see Eq. (47)), direct particle-hole $\tau_{\rm ph}^{\rm d}$ (see Eq.
(49)) and crossed particle-hole direct particle-hole $\tau_{\rm ph}^{\rm cr}$
(see Eq. (48)).

We rewrite the Eq. (47) through momenta $l,k_{1},k_{2}$, introduced for
particle-particle (P) channel by Eq. (5), performing the variable change
$p\rightarrow l/2-p$

$\displaystyle\tau_{\rm pp}(l/2+k_{1},l/2-k_{1},l/2-k_{2})$
$\displaystyle=\sum_{p}L_{\rm pp}(l,p)V\left(l/2+k_{1},l/2-k_{1},l/2-p\right)$
$\displaystyle\times V\left(l/2+p,l/2-p,l/2-k_{2}\right).$ (51)

Here arguments of both $V$ factors are the same as the arguments on the left
hand side. Following Ref. 2017:TUfRG, , we use the identity

$\delta_{\mathbf{k}\mathbf{k}^{\prime}}=\sum_{n}f_{n}(\mathbf{k})f_{n}(\mathbf{k}^{\prime})$
(52)

to obtain

$\displaystyle\tau_{\rm pp}(l/2+k_{1},l/2$ $\displaystyle-
k_{1},l/2-k_{2})=\sum_{m_{1},m_{2},n_{1},n_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\sum_{p}f_{m_{1}}(\mathbf{p})L_{\rm
pp}(l,p)f_{m_{2}}(\mathbf{p})$
$\displaystyle\times\sum_{\mathbf{k}_{1}^{\prime}\mathbf{p}_{1}^{\prime}}f_{n_{1}}(\mathbf{k}_{1}^{\prime})V\left(l/2+\mathbf{k}_{1}^{\prime},l/2-\mathbf{k}_{1}^{\prime},l/2-\mathbf{p}_{1}^{\prime}\right)f_{m_{1}}(\mathbf{p}_{1}^{\prime})$
$\displaystyle\times\sum_{\mathbf{k}_{2}^{\prime}\mathbf{p}_{2}^{\prime}}f_{n_{2}}(\mathbf{k}_{2}^{\prime})V\left(l/2+\mathbf{p}_{2}^{\prime},l/2-\mathbf{p}_{2}^{\prime},l/2-\mathbf{k}_{2}^{\prime}\right)f_{m_{2}}(\mathbf{p}_{2}^{\prime}).$
(53)

Following the same algorithm for the crossed particle-hole (M) channel, for
the contribution (48), we get

$\displaystyle\tau_{\rm ph}^{\rm cr}(l/2+k_{1},k_{2}$
$\displaystyle-l/2,k_{1}-l/2)=\sum_{m_{1},m_{2},n_{1},n_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\sum_{p}f_{m_{1}}(\mathbf{p})L_{\rm
ph}(l,p)f_{m_{2}}(\mathbf{p})$
$\displaystyle\times\sum_{\mathbf{k}_{1}^{\prime}\mathbf{p}_{1}^{\prime}}f_{n_{1}}(\mathbf{k}_{1}^{\prime})V\left(l/2+\mathbf{k}_{1}^{\prime},\mathbf{p}_{1}^{\prime}-l/2,\mathbf{k}_{1}^{\prime}-l/2\right)f_{m_{1}}(\mathbf{p}_{1}^{\prime})$
$\displaystyle\times\sum_{\mathbf{k}_{2}^{\prime}\mathbf{p}_{2}^{\prime}}f_{n_{2}}(\mathbf{k}_{2}^{\prime})V\left(l/2+\mathbf{p}_{2}^{\prime},\mathbf{k}_{2}^{\prime}-l/2,\mathbf{p}_{2}^{\prime}-l/2\right)f_{m_{2}}(\mathbf{p}_{2}^{\prime}).$
(54)

Now we consider the projection of the expression (49) onto the channel K (see
Eq. (7)). We have

$\displaystyle\tau_{\rm ph}^{\rm d}(l/2+k_{1},k_{2}$
$\displaystyle-l/2,l/2+k_{2})=-\sum_{m_{1},m_{2},n_{1},n_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\sum_{p}f_{m_{1}}(\mathbf{p})L_{\rm
ph}(l,p)f_{m_{2}}(\mathbf{p})$
$\displaystyle\times\sum_{\mathbf{k}_{1}^{\prime}\mathbf{p}_{1}^{\prime}\mathbf{k}_{2}^{\prime}\mathbf{p}_{2}^{\prime}}\left[2f_{n_{1}}(\mathbf{k}_{1}^{\prime})V\left(l/2+\mathbf{k}_{1}^{\prime},\mathbf{p}_{1}^{\prime}-l/2,\mathbf{p}_{1}^{\prime}+l/2\right)f_{m_{1}}(\mathbf{p}_{1}^{\prime})\right.$
$\displaystyle\left.\times
f_{n_{2}}(\mathbf{k}_{2}^{\prime})V\left(\mathbf{p}_{2}^{\prime}+l/2,\mathbf{k}_{2}^{\prime}-l/2,l/2+\mathbf{k}_{2}^{\prime}\right)f_{m_{2}}(\mathbf{p}_{2}^{\prime})\right.$
$\displaystyle\left.-f_{n_{1}}(\mathbf{k}_{1}^{\prime})V\left(l/2+\mathbf{k}_{1}^{\prime},\mathbf{p}_{1}^{\prime}-l/2,\mathbf{k}_{1}^{\prime}-l/2\right)f_{m_{1}}(\mathbf{p}_{1}^{\prime})\right.$
$\displaystyle\left.\times
f_{n_{2}}(\mathbf{k}_{2}^{\prime})V\left(\mathbf{p}_{2}^{\prime}+l/2,\mathbf{k}_{2}^{\prime}-l/2,l/2+\mathbf{k}_{2}^{\prime}\right)f_{m_{2}}(\mathbf{p}_{2}^{\prime})\right.$
$\displaystyle\left.-f_{n_{1}}(\mathbf{k}_{1}^{\prime})V\left(l/2+\mathbf{k}_{1}^{\prime},\mathbf{p}_{1}^{\prime}-l/2,\mathbf{p}_{1}^{\prime}+l/2\right)f_{m_{1}}(\mathbf{p}_{1}^{\prime})\right.$
$\displaystyle\left.\times
f_{n_{2}}(\mathbf{k}_{2}^{\prime})V\left(\mathbf{p}_{2}^{\prime}+l/2,\mathbf{k}_{2}^{\prime}-l/2,\mathbf{p}_{2}^{\prime}-l/2\right)f_{m_{2}}(\mathbf{p}_{2}^{\prime})\right].$
(55)

The second term in Eq. (11) contains momentum arguments suitable for
projection in K channel

$\tau^{\rm cr}_{\rm ph}(k_{1}+l/2,k_{2}-l/2,k_{1}-l/2)=\sum_{p}L_{\rm
ph}(l,p)V\left(k_{1}+l/2,p-l/2,k_{1}-l/2\right)\\\ \times
V\left(p+l/2,k_{2}-l/2,p-l/2\right).$ (56)

The third momentum argument of both $V$ factors does not have the form of
K-channel, see Eq. (14), therefore we, as above, interchange third and forth
momentum, $V\left(k_{1}+l/2,p-l/2,k_{1}-l/2\right)=V_{\rm
ex}\left(k_{1}+l/2,p-l/2,p+l/2\right)$,
$V\left(l/2+p,k_{2}-l/2,p-l/2\right)=V_{\rm
ex}\left(l/2+p,k_{2}-l/2,k_{2}+l/2\right)$. Again using the identity (52), we
get

$\displaystyle\tau^{\rm cr}_{\rm ph}(l/2+k_{1},k_{2}$
$\displaystyle-l/2,k_{1}-l/2)=\sum_{n_{1},n_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\sum_{p}f_{m_{1}}(\mathbf{p})f_{m_{2}}(\mathbf{p})L_{\rm
ph}(l,p)$
$\displaystyle\times\sum_{\mathbf{k}_{1}^{\prime}\mathbf{p}_{1}^{\prime}}f_{n_{1}}(\mathbf{k}_{1}^{\prime})f_{m_{1}}(\mathbf{p}_{1}^{\prime})V_{\rm
ex}\left(l/2+\mathbf{k}_{1}^{\prime},\mathbf{p}_{1}^{\prime}-l/2,\mathbf{p}_{1}^{\prime}+l/2\right)$
$\displaystyle\times\sum_{\mathbf{k}_{2}^{\prime}\mathbf{p}_{2}^{\prime}}f_{n_{2}}(\mathbf{p}_{2}^{\prime})f_{m_{2}}(\mathbf{k}_{2}^{\prime})V_{\rm
ex}\left(\mathbf{p}_{2}^{\prime}+l/2,\mathbf{k}_{2}^{\prime}-l/2,\mathbf{k}_{2}^{\prime}+l/2\right).$
(57)

We consider the projection of the interaction in Eq. (II.3). The dependence of
vertex $V$ on fermionic frequencies of $k_{1}$ and $k_{2}$ in all channels is
out of the scope within this projection procedure and is not considered in our
study. For convenience we introduce

$\Phi_{\rm M-K}=\Phi_{\rm M}-\Phi_{\rm K},$ (58)

using the fact that the contributions of channels M and K enters the last two
terms of Eq. (II.3) in the same way. For P-channnel projection we use the
representation

$V_{\rm P}(\mathbf{l};\mathbf{k}_{1},\mathbf{k}_{2})=U-\Phi_{\rm
SC}\left(\mathbf{l},\mathbf{k}_{1},\mathbf{k}_{2}\right)+\Phi_{\rm
M}\left(\mathbf{k}_{1}+\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2},\dfrac{\mathbf{l}-\mathbf{k}_{1}+\mathbf{k}_{2}}{2}\right)\\\
+\dfrac{1}{2}\Phi_{\rm
M-K}\left(\mathbf{k}_{1}-\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2},\dfrac{\mathbf{l}-\mathbf{k}_{1}-\mathbf{k}_{2}}{2}\right),$
(59)

therefore we derive Eq. (21) with

$V^{\rm
P}_{n_{1}n_{2}}(\mathbf{l})=U\delta_{n_{1},0}\delta_{n_{2},0}-\Phi^{\rm
SC}_{n_{1}n_{2}}(\mathbf{l})+\sum_{{\mathbf{k}}_{1}{\mathbf{k}}_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\Phi_{\rm
M}\left(\mathbf{k}_{1}+\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2},\dfrac{\mathbf{l}-\mathbf{k}_{1}+\mathbf{k}_{2}}{2}\right)\\\
+\dfrac{1}{2}\sum_{{\mathbf{k}}_{1}{\mathbf{k}}_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\Phi_{\rm
M-K}\left(\mathbf{k}_{1}-\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2},\dfrac{\mathbf{l}-\mathbf{k}_{1}-\mathbf{k}_{2}}{2}\right).$
(60)

Analogously for M-channel we derive Eq. (22) with

$V_{\rm M}(\mathbf{l};\mathbf{k}_{1},\mathbf{k}_{2})=U-\Phi_{\rm
SC}\left(\mathbf{k}_{1}+\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2},\dfrac{\mathbf{l}+\mathbf{k}_{2}-\mathbf{k}_{1}}{2}\right)+\Phi^{\rm
M}\left(\mathbf{l},\mathbf{k}_{1},\mathbf{k}_{2}\right)\\\
+\dfrac{1}{2}\Phi_{\rm
M-K}\left(\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2},\dfrac{-\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2}\right),$
(61)

and

$V^{\rm
M}_{n_{1}n_{2}}(\mathbf{l})=U\delta_{n_{1},0}\delta_{n_{2},0}-\sum_{{\mathbf{k}}_{1}{\mathbf{k}}_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\Phi_{\rm
SC}\left(\mathbf{k}_{1}+\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2},\dfrac{\mathbf{l}+\mathbf{k}_{2}-\mathbf{k}_{1}}{2}\right)+\Phi^{\rm
M}_{n_{1}n_{2}}(\mathbf{l})\\\
+\dfrac{1}{2}\sum_{{\mathbf{k}}_{1}{\mathbf{k}}_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\Phi_{\rm
M-K}\left(\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2},\dfrac{-\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2}\right).$
(62)

Analogously for K-channel we derive Eq. (23) with

$V_{\rm K}(\mathbf{l};\mathbf{k}_{1},\mathbf{k}_{2})=U-\Phi_{\rm
SC}\left(\mathbf{k}_{1}+\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2},\dfrac{-\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2}\right)\\\
+\Phi_{\rm
M}\left(\mathbf{k}_{1}-\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2},\dfrac{-\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2}\right)+\dfrac{1}{2}\Phi_{\rm
M-K}\left(\mathbf{l},\mathbf{k}_{1},\mathbf{k}_{2}\right)$ (63)

and

$V^{\rm
K}_{n_{1}n_{2}}(\mathbf{l})=U\delta_{n_{1},0}\delta_{n_{2},0}-\sum_{{\mathbf{k}}_{1}{\mathbf{k}}_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{\mathbf{k}}_{2})\Phi_{\rm
SC}\left(\mathbf{k}_{1}+\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2},\dfrac{-\mathbf{l}+\mathbf{k}_{1}-\mathbf{k}_{2}}{2}\right)\\\
+\sum_{{\mathbf{k}}_{1}{\mathbf{k}}_{2}}f_{n_{1}}(\mathbf{k}_{1})f_{n_{2}}(\mathbf{k}_{2})\Phi_{\rm
M}\left(\mathbf{k}_{1}-\mathbf{k}_{2},\dfrac{\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2},\dfrac{-\mathbf{l}+\mathbf{k}_{1}+\mathbf{k}_{2}}{2}\right)+\dfrac{1}{2}\Phi^{\rm
M-K}_{n_{1}n_{2}}(\mathbf{l}).$ (64)

##  Appendix C Single-boson exchange decomposition

Eqs. (26)-(28) allow single-boson exchange decomposition of the vertex
function through triangular (Hedin, “Yukawa”) vertex functions $\gamma_{\rm
SC,M,K}({\mathbf{q}})$ and bosonic propagators $D_{\rm SC,M,K}({\mathbf{q}})$
2022:BFfRG

$\displaystyle U-\Phi_{\rm SC}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle\gamma^{2}_{\rm SC}({\mathbf{q}})D_{\rm SC}({\mathbf{q}})+R_{\rm
SC}({\mathbf{q}}),$ (65) $\displaystyle U+\Phi_{\rm M}({\mathbf{q}})$
$\displaystyle=$ $\displaystyle\gamma^{2}_{\rm M}({\mathbf{q}})D_{\rm
M}({\mathbf{q}})+R_{\rm M}({\mathbf{q}}),$ (66) $\displaystyle U-\Phi_{\rm
K}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle\gamma^{2}_{\rm
K}({\mathbf{q}})D_{\rm K}({\mathbf{q}})+R_{\rm K}({\mathbf{q}}).$ (67)

where the triangular vertex functions assumed to be independent on fermionic
momenta. Retaining of residue contributions $R_{\rm SC,M,K}({\mathbf{q}})$
correspond to taking into account of so-called “box” diagrams for 4-vertex
(their dependence on fermionic momenta is also neglected).

Instead of one equation in each channel we get 3 ones (for $D$, $\gamma$ and
$R$). The equations for $D$ read

$\displaystyle\dot{D}_{\rm SC}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle\gamma^{2}_{\rm SC}({\mathbf{q}})D_{\rm
SC}^{2}({\mathbf{q}})L_{+}({\mathbf{q}}),$ (68) $\displaystyle\dot{D}_{\rm
M}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle\gamma^{2}_{\rm
M}({\mathbf{q}})D_{\rm M}^{2}({\mathbf{q}})L_{-}({\mathbf{q}}),$ (69)
$\displaystyle\dot{D}_{\rm K}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle-\gamma^{2}_{\rm K}({\mathbf{q}})D_{\rm
K}^{2}({\mathbf{q}})L_{-}({\mathbf{q}}).$ (70)

The equation for $\gamma$ read

$\displaystyle\dot{\gamma}_{\rm SC}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle\gamma_{\rm SC}({\mathbf{q}})\left(R_{\rm
SC}({\mathbf{q}})+\frac{3}{2}\bar{\Phi}_{\rm M}-\frac{1}{2}\bar{\Phi}_{\rm
K}\right)L_{+}({\mathbf{q}}),$ (71) $\displaystyle\dot{\gamma}_{\rm
M}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle\gamma_{\rm
M}({\mathbf{q}})\left(R_{\rm M}({\mathbf{q}})-\bar{\Phi}_{\rm
SC}+\frac{1}{2}\bar{\Phi}_{\rm M}-\frac{1}{2}\bar{\Phi}_{\rm
K}\right)L_{-}({\mathbf{q}}),$ (72) $\displaystyle\dot{\gamma}_{\rm
K}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle-\gamma_{\rm
K}({\mathbf{q}})\left(R_{\rm K}({\mathbf{q}})-\bar{\Phi}_{\rm
SC}+\frac{3}{2}\bar{\Phi}_{\rm M}+\frac{1}{2}\bar{\Phi}_{\rm
K}\right)L_{-}({\mathbf{q}}).$ (73)

The equation for $R$ read

$\displaystyle\dot{R}_{\rm SC}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle\left(R_{\rm SC}({\mathbf{q}})+\frac{3}{2}\bar{\Phi}_{\rm
M}-\frac{1}{2}\bar{\Phi}_{\rm K}\right)^{2}L_{+}({\mathbf{q}}),$ (74)
$\displaystyle\dot{R}_{\rm M}({\mathbf{q}})$ $\displaystyle=$
$\displaystyle\left(R_{\rm M}({\mathbf{q}})-\bar{\Phi}_{\rm
SC}+\frac{1}{2}\bar{\Phi}_{\rm M}-\frac{1}{2}\bar{\Phi}_{\rm
K}\right)^{2}L_{-}({\mathbf{q}}),$ (75) $\displaystyle\dot{R}_{\rm
K}({\mathbf{q}})$ $\displaystyle=$ $\displaystyle-\left(R_{\rm
K}({\mathbf{q}})-\bar{\Phi}_{\rm SC}+\frac{3}{2}\bar{\Phi}_{\rm
M}+\frac{1}{2}\bar{\Phi}_{\rm K}\right)^{2}L_{-}({\mathbf{q}}).$ (76)

## References

  * (1) B. T. Matthias and R. M. Bozorth, Phys. Rev. 109, 604 (1958).
  * (2) S. L. Skornyakov, V. S. Protsenko, V. I. Anisimov, and A. A. Katanin, Phys. Rev. B 102, 085101 (2020).
  * (3) H. Sasakura, K. Suzuki, Y. Masuda, Journal of the Physical Society of Japan 53, 352 (1984).
  * (4) P. G. Niklowitz, F. Beckers, G. G. Lonzarich, G. Knebel, B. Salce, J. Thomasson, N. Bernhoeft, D. Braithwaite, and J. Flouquet, Phys. Rev. B 72, 024424 (2005).
  * (5) E. C. Stoner Proc. Roy. Soc. A 165 (922), 372 (1938); 169 (938), 339 (1939).
  * (6) D. R. Penn, Phys. Rev. 142, 350 (1966).
  * (7) K. K. Murata and S. Doniach, Phys. Rev. Lett. 29, 285 (1972).
  * (8) T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism, Springer: Berlin, 1985.
  * (9) J. A. Hertz, Phys. Rev. B 14, 1165 (1976).
  * (10) A. J. Millis, Phys. Rev. B 48, 7183 (1993).
  * (11) V. Yu. Irkhin, A. A. Katanin, and M. I. Katsnelson, Phys. Rev. B 64, 165107 (2001).
  * (12) C. Honerkamp and M. Salmhofer, Phys. Rev. Lett. 87, 187004 (2001); Phys. Rev. B 64, 184516 (2001).
  * (13) A. A. Katanin and A. P. Kampf, Phys. Rev. B 68, 195101 (2003).
  * (14) S. V. Vonsovskii, M. I. Katsnelson, and A. V. Trefilov, Fiz. Met. Metalloved. 76 (3) 3 (1993); 76 (4), 3 (1993).
  * (15) P. A. Igoshev and V. Yu. Irkhin, Phys. Met. Metallogr. 120, 1282 (2019).
  * (16) P. A. Igoshev and V. Yu. Irkhin, JETP Lett. 110, 727 (2019).
  * (17) L. Classen, A. V. Chubukov, C. Honerkamp, M. M. Scherer, Phys. Rev. B 102, 125141 (2020).
  * (18) P.A. Igoshev, V.Yu. Irkhin, Physics Letters A 438, 128107 (2022).
  * (19) A. Chandrasekaran, J. J. Betouras, ArXiv: 2207.06099.
  * (20) Y. Yamaji, T. Misawa, and M. Imada, Journal of the Physical Society of Japan 76, 063702 (2007).
  * (21) Zs. Major, S. B. Dugdale, R. J. Watts, G. Santi, M. A. Alam, S. M. Hayden, J. A. Duffy, J. W. Taylor, T. Jarlborg, E. Bruno, D. Benea, and H. Ebert, Phys. Rev. Lett. 92, 107003 (2004).
  * (22) S. J. C. Yates, G. Santi, S. M. Hayden, P. J. Meeson, and S. B. Dugdale, Phys. Rev. Lett. 90, 057003 (2003).
  * (23) A. S. Hamid, A. Uedono, Zs. Major, T. D. Haynes, J. Laverock, M. A. Alam, S. B. Dugdale, and D. Fort, Phys. Rev. B 84, 235107 (2011).
  * (24) A. Hausoel, M. Karolak, E. Sasioglu, A. Lichtenstein, K. Held, A. Katanin, A. Toschi, and G. Sangiovanni, Nat. Commun. 8, 16062 (2017).
  * (25) A. A. Katanin, A. I. Poteryaev, A. V. Efremov, A. O. Shorikov, S. L. Skornyakov, M. A. Korotin, and V. I. Anisimov, Phys. Rev. B 81, 045117 (2010).
  * (26) P. A. Igoshev, A. V. Efremov, A. A. Katanin, Phys. Rev. B 91, 195123 (2015).
  * (27) J. Kanamori, Progr. Theor. Phys. 30, 275 (1963).
  * (28) R. Hlubina, Phys. Rev. B 59, 9600 (1999).
  * (29) C. Calandra and F. Manghi, Phys. Rev. B 50, 2061 (1994); F. Manghi, V. Bellini and C. Arcangeli, Phys. Rev. B 56, 7149 (1997).
  * (30) J. Igarashi, J. Phys. Soc. Jpn. 52, 2827 (1983); J. Igarashi, M. Takahashi, and T. Nagano, J. Phys. Soc. Jpn. 68, 3682 (1999).
  * (31) M. Ulmke, Eur. Phys. J. B 1, 301 (1998); D. Vollhardt, N. Blümer, K. Held, M. Kollar, J. Schlipf, M. Ulmke, J. Wahle, Advances in Solid State Physics 38 (Vieweg, Wiesbaden, 1999), p. 383.
  * (32) R. Zitzler, Th. Pruschke, and R. Bulla, Eur. Phys. J. B 27, 473-481 (2002).
  * (33) A. I. Lichtenstein, M. I. Katsnelson, and G. Kotliar, Phys. Rev. Lett. 87, 067205 (2001).
  * (34) S. Sakai, R. Arita, and H. Aoki, Phys. Rev. Lett. 99, 216402 (2007).
  * (35) H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990).
  * (36) P. A. Igoshev, M. A. Timirgazin, A. A. Katanin, A. K. Arzhnikov, and V. Yu. Irkhin, Phys. Rev. B 81, 094407 (2010)
  * (37) P.A. Igoshev, M.A. Timirgazin, V.F. Gilmutdinov, A.K. Arzhnikov and V. Yu. Irkhin, J. Phys.: Cond. Matt. 27, 446002 (2015).
  * (38) M. Salmhofer and C. Honerkamp, Prog. Theor. Phys. 105, 1 (2001).
  * (39) D. Zanchi and H. J. Schulz, Phys. Rev. B 61, 13609 (2000).
  * (40) C. J. Halboth and W. Metzner, Phys. Rev. B 61, 7364 (2000).
  * (41) C. Honerkamp, M. Salmhofer, N. Furukawa, and T. M. Rice, Phys. Rev. B 63, 035109 (2001).
  * (42) C. Honerkamp, D. Rohe, S. Andergassen, and T. Enss, Phys. Rev. B 70, 235115 (2004).
  * (43) W. Metzner, J. Reiss, and D. Rohe, Phys. Stat. Sol.(b) 243, 46 (2005).
  * (44) M. Salmhofer, C. Honerkamp, W. Metzner, and O. Lauscher, Prog. Theor. Phys. 112, 943 (2004); R. Gersch, C. Honerkamp, D. Rohe, W. Metzner, Eur. Phys. J. B. 48, 349 (2005); R. Gersch, J. Reiss, C. Honerkamp, New J. Phys. 8, 320 (2006); R. Gersch, C. Honerkamp, and W. Metzner, New J. Phys. 10, 045003 (2008).
  * (45) C. Husemann and M. Salmhofer, Phys. Rev. B 79, 195125 (2009).
  * (46) A. Katanin, Phys. Rev. B 81, 165118 (2010).
  * (47) P. A. Igoshev, V. Yu. Irkhin, and A. A. Katanin, Phys. Rev. B 83, 245118 (2011).
  * (48) A. A. Katanin, H. Yamase, V. Yu. Irkhin, J. Phys. Soc. Jpn. 80, 063702 (2011).
  * (49) A. Eberlein and W. Metzner, Phys. Rev. B 87, 174523 (2013).
  * (50) J. Lichtenstein, D. Sánchez de la Pena, D. Rohe, E. Di Napoli, C. Honerkamp, S. A. Maier, Computer Physics Communications 213, 100 (2017)
  * (51) P. M. Bonetti, A. Toschi, C. Hille, S. Andergassen, and D. Vilardi, Phys. Rev. Research 4, 013034 (2022).
  * (52) P. A. Igoshev, A. A. Katanin, and V. Yu. Irkhin, Journ. Exp. Theor. Phys. 105, 1043 (2007).
  * (53) P. A. Igoshev, A. A. Katanin, H. Yamase, V. Yu. Irkhin, J. Magn. Magn. Mat. 321, 899 (2009).
  * (54) P. E. Blöchl, O. Jepsen, O. K. Andersen, Phys. Rev. B 49, 16223 (1994).
  * (55) J. Berntsen, R. Cools, and T. O. Espelid, ACM Transactions on Mathematical Software 19, 320 (1993).
  * (56) G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, K. Held, Rev. Mod. Phys. 90, 025003 (2018).
  * (57) C. Taranto, S. Andergassen, J. Bauer, K. Held, A. Katanin, W. Metzner, G. Rohringer, and A. Toschi, Phys. Rev. Lett. 112, 196402 (2014); N. Wentzell, C. Taranto, A. A. Katanin, A. Toschi, and S. Andergassen, Phys. Rev. B 91, 045120 (2015).