# Morphological classification of astronomical images with limited labelling

Andrey Soroka,1 Alex Meshcheryakov,2 and Sergey Gerasimov1

###### Abstract

The task of morphological classification is complex for simple
parameterization, but important for research in the galaxy evolution field.
Future galaxy surveys (e.g. EUCLID) will collect data about more than a
$10^{9}$ galaxies. To obtain morphological information one needs to involve
people to mark up galaxy images, which requires either a considerable amount
of money or a huge number of volunteers. We propose an effective semi-
supervised approach for galaxy morphology classification task, based on active
learning of adversarial autoencoder (AAE) model. For a binary classification
problem (top level question of Galaxy Zoo 2 decision tree) we achieved
accuracy 93.1% on the test part with only 0.86 millions markup actions, this
model can easily scale up on any number of images. Our best model with
additional markup achieves accuracy of 95.5%. To the best of our knowledge it
is a first time AAE semi-supervised learning model used in astronomy.

1Moscow State University CMC, Moscow, Russia<EMAIL_ADDRESS>
2Space Research Institute of RAS, Moscow, Russia<EMAIL_ADDRESS>
##  1 Introduction

Galaxies exhibit a wide variety of shapes, colors and sizes. These properties
are indicative of their age, formation conditions, and interactions with other
galaxies over the course of many Gyr. Studies of galaxy formation and
evolution use morphology to probe the physical processes that give rise to
them. Traditionally, morphological classification has been carried out by
professional astronomers using visual inspections of images obtained from
telescopes. The number of detected galaxies in modern sky surveys and computer
simulations has been found to be growing rapidly. The number of images in
surveys (SDSS, DESI LIS, PanSTARRS) is billions of galaxies, future surveys
such as Euclid (Refregier et al. 2010) will provide images of more than
billion of galaxies. Manually inspecting all these images to annotate them
with morphological information is impractical for either individual
astronomers or small teams.

An attempt to solve this problem was the Galaxy Zoo project by Willett et al.
(2013): volunteers classified images of galaxies through a web interface.
Participants were asked various questions such as "How rounded is the galaxy?"
or "Is there a convex in the center?" Depending on the answer, it was
determined which question would be asked next. The questions formed a decision
tree that covered a wide range of morphological categories. The crowdsourcing
project also faces the problem of a large amount of data: if we try to study
the morphology of only 1% of the volume of images from Euclid, we will get
millions of objects for morphological classification. A project of this
magnitude would take over sixty years to process all the data at the pace of
the Galaxy Zoo 2.

There are approaches to galaxy classification based on the ideas of active
learning. They allow to increase the classification speed by 8 times at the
expense of a slight loss of quality, but even at this marking rate, it will
take 7.5 years for the SWAP by Beck et al. (2018) (SoTA model) to handle 1% of
all Euclid data.

The main objectives of our research: 1) create a model for the morphological
classification of galaxies, 2) the model should use small as possible amount
of labeled data, 3) predictive ability should be better than current SoTA
solutions, 4) use of active learning technique. The second task is the main
one, since reducing the number of required markup images will reduce the
number of primary classification actions carried out by volunteers / research
groups. This, in turn, will significantly speed up the marking of
morphological annotations of modern surveys.

##  2 Data

We chose Kaggle Galaxy Zoo dataset for our experiments. With 138000 images of
galaxies in jpg format. As mentioned above, Galaxy Zoo is a crowdsourcing
project, each image in it was classified based on the votes of volunteers, an
average of 42 people per galaxy. Thus, we have a dependence between amount of
labeled data and markup actions: 1 labeled image $\approx 42$ markup actions.
Annotation of data was provided by Kyle Willett, KGZ organizer. Authors are
grateful for his help and kind words.

##  3 Methods

In this work, we propose semi-supervised algorithm, modernization of the
adversarial autoencoder by Makhzani et al. (2016). We did classification over
a continuous latent representation using additional MLP layer. We trained
encoder in semi-supervised mode with both labeled and unlabeled data, the
second was an order of magnitude more. In the process of training adversarial
autoencoder, in addition to the classical "uniform" selection of training
samples, we also used active learning approach. The training examples were
selected sequentially, based on the prediction of the model trained in the
previous step with fewer examples. At the first step, 4% of the total number
of images were randomly marked as a training sample, model was trained on
them, other images were classified with that model, and the next 1% of the
total number of images was marked, for which algorithm predicted the
probability closest to 0.5 (random prediction). The model was retrained,
process continued until the total amount of labeled data was equal to 10% of
all training data. We approximated the results to the dataset size used by
Beck et al. (2018). To do this, we took the accuracy obtained on the unlabeled
part and transferred it to an enlarged one in accordance with the size of the
SWAP dataset minus 20k labeled images and calculated the final results as a
composition of these values.

Figure 1.: Architecture of semi-supervised AAE model.

##  4 Results

The main learning outcomes: 1) Semi-supervised model (AAE) achieves an
accuracy of 91% on 10% labeled data ($\sim$850,000 markup actions for 20k
images) of all training data, which is quite close to the result of a similar
SWAP model (93.1% on $\sim$936000 markup actions); 2) Active learning improves
prediction, AAE+AL model trained on the same amount of labeled images achieves
an accuracy of 93.1%. This result can be easily scaled up on any amount of
images without additional markup cost; 3) If we estimate the accuracy over the
entire Galaxy Zoo 2 dataset (226124 images as in Beck et al. (2018)),
including labeled examples (as the Beck et al. (2018) of SWAP did ), our model
achieves an accuracy of 95.5%, which is quite close to the best SWAP result
(95.7%). At the same time, it requires 2.5 times less markup actions
($\sim$850000 vs. $\sim$2300000 for SWAP).

Figure 2.: Accuracy of different models over active learning process. Table 1.: Accuracy of different models over active learning process. All AAE models are from this work and all SWAP models are from Beck et al. (2018) Model | Acc.(%) | Galaxies | Markup actions | Add markup?
---|---|---|---|---
SWAP | 95.7 | 226124 | 2298772 | Yes
AAE+AL | 95.5 | 226124 | 854310 | Yes
SWAP+RF | 93.1 | 210803 | 936887 | Yes
AAE+AL (20k sample) | 93.1 | Any | 854310 | No

Let $N$ be the size of the entire dataset to be classified, A be the limit of
labeled images used in active learning. Let’s denote by $R$ \- the ratio of
$N$ to $A$: $R=\frac{N}{A}$. We measured how the prediction accuracy of the
model changes with a fixed $A$ and growing $N$. We sampled data with the $N$
range from 30k to 130k with $A$ equal to 7k and 20k. We note that the best
accuracy of classification is achieved with more labeled data. So if we
increase the total amount of data, but keep the value R, we get the best
results (Figures 3).

Figure 3.: Accuracy of different models over active learning process.

##  5 Conclusion

To the best of our knowledge its a first time AAE used in astronomy. Our model
achieves accuracy 93.1% on 20k labeled data without additional markup cost,
this model can easily scale up on any number of images. Our best model with
additional markup achieves accuracy of 95.5%, while using 2.5 times less
markup actions than Beck et al. (2018).

## References

  * Beck et al. (2018) Beck, M. R., Scarlata, C., Fortson, L. F., Lintott, C. J., Simmons, B. D., Galloway, M. A., Willett, K. W., Dickinson, H., Masters, K. L., Marshall, P. J., & et al. 2018, Monthly Notices of the Royal Astronomical Society, 476, 5516–5534. URL http://dx.doi.org/10.1093/mnras/sty503
  * Makhzani et al. (2016) Makhzani, A., Shlens, J., Jaitly, N., Goodfellow, I., & Frey, B. 2016, Adversarial autoencoders. 1511.05644
  * Refregier et al. (2010) Refregier, A., Amara, A., Kitching, T. D., Rassat, A., Scaramella, R., & Weller, J. 2010, Euclid imaging consortium science book. 1001.0061
  * Willett et al. (2013) Willett, K. W., Lintott, C. J., Bamford, S. P., Masters, K. L., Simmons, B. D., Casteels, K. R. V., Edmondson, E. M., Fortson, L. F., Kaviraj, S., Keel, W. C., & et al. 2013, Monthly Notices of the Royal Astronomical Society, 435, 2835–2860. URL http://dx.doi.org/10.1093/mnras/stt1458


AppApp Readings

#  Local magnetic moment formation and Kondo screening
in the presence of Hund exchange: the two-band Hubbard model analysis

T. B. Mazitov  Center for Photonics and 2D Materials, Moscow Institute of
Physics and Technology, Institutsky lane 9, Dolgoprudny, 141700, Moscow
region, Russia  A. A. Katanin  Center for Photonics and 2D Materials, Moscow
Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, 141700,
Moscow region, Russia  M. N. Mikheev Institute of Metal Physics, Kovalevskaya
Street 18, 620219 Ekaterinburg, Russia.

###### Abstract

We study formation and screening of local magnetic moments in the two-band
Hubbard model in the presence of Hund exchange interaction using dynamic mean
field theory approach. The characteristic temperatures of the formation,
beginning and full screening of local magnetic moments are obtained from the
analysis of temperature dependence of the orbital, charge, and spin
susceptibilities, as well as the respective instantaneous correlation
functions. At half filling we find the phase diagram, which is similar to the
single-orbital case with wide region of formation of local magnetic moments
below the orbital Kondo temperature. Similarly to the single-orbital case,
with decreasing temperature the screening of local magnetic moments is
preceded by appearance of fermionic quasiparticles. In the two-band case the
quasiparticles are however present also in some temperature region above the
Mott insulating phase. At finite doping we find broad regime of presence of
local magnetic moments, which coexist with incoherently or coherently moving
holes. We also find, similarly to the half filled case, finite temperature
interval in the doped regime, when the fermionic quasiparticles are formed,
but do not yet screen local magnetic moments.

##  I Introduction

The concept of Hund metals (HM) was introduced Pnictides ; Pnictides1 ; Hund1
; Hund2 ; Hund3 ; Hund4 ; Hund5 ; Freezing ; Delft  to distinguish the
substances where strong electron correlations are induced by Hund exchange
from those where Coulomb interaction yields proximity to Mott insulating (MI)
phase. The Hund metal behavior is obtained in iron pnictides Pnictides ; Hund2
; Pnic1 ; Pnic2 , ruthenates Hund1x ; Hund2x , elemental iron OurIron ;
Hausoel ; OurFe1 ; OurFeGamma , as well as some other transition metals
Hausoel ; OurV , and their compounds OurFeNi . Some aspects of the physics of
high-temperature superconductors can be also described by an effective two-
band model, in which Hund exchange plays an important role High-Tc ; High-Tc1
.

It was suggested in Refs. Hund3 ; Delft  that Hund metals are characterised by
the so called spin-orbital separation, which corresponds to freezing of
orbital degrees of freedom below the orbital Kondo temperature $T_{K,{\rm
orb}}$. The latter temperature is much higher, than the Kondo temperature for
the spin degrees of freedom $T_{K}$. This separation gives a possibility for
the so called spin freezing Freezing  to occur at sufficiently low
temperatures when the orbital degrees of freedom are not active. The spin
freezing yields formation of local magnetic moments, which are important for
magnetic and thermodynamic properties of Hund metals.

The number of previous studies have contrasted the Hund metals to multi-band
systems in the proximity to Mott insulating state at integer fillings,
different from half filling (the so called Hundness vs. Mottness behavior),
see, e.g., Refs. Hund1 ; Hund3 ; Delft ; Deng ; Fanfarillo ; Medici ;
Pnictides1 ; Hund2 . At the same time, one can conjecture that the HM state
away from half filling inherits to some extent its properties from the
proximity the Mott state at half filling. This conjecture is confirmed by
studies of the doping dependence of electronic, charge, and magnetic
properties in broad range of dopings (see, e.g., Refs. Hund1 ; Pnictides1 ;
Hund3 ; Fanfarillo ). In particular, the Hund metal behavior occurs with
doping in the same part of the phase diagram, which is Mott insulating at half
filling, and enhanced on approaching half filling.

Since the properties of Mott transition at half filling in multi-orbital
models are similar to those in the single-orbital model, the HM state finds
some similarities to the proximity of Mott state in the single-band system.
Indeed, the temperature scales of start of the formation of local magnetic
moments and beginning of their screening in single-band systems were actively
discussed recently Our1 ; Our2 ; Toschi ; Katsnelson ; Toschi_new , and it was
found that the temperature of the beginning of formation of local magnetic
moments appears to be much higher, than the temperature of the onset of the
screening. The former temperature is therefore analogous to that for the
orbital freezing in multi-band system.

Finding similarities and differences in the formation of local magnetic
moments in single- and multi-band systems allows therefore for better
understanding of the physical properties of Hund metals, especially in the
most important regime of the unscreened local magnetic moments, which are
expected to form in the temperature range $T_{K}<T<T_{K,{\rm orb}}$. In
particular, this study may reveal peculiarities of the local magnetic moment
formation in multi-band systems at half filling, as well as uncover the
reasons of their stability with doping.

To emphasize various aspects of the similarity and differences of the Mott
transition in the single-orbital model and Hund metal behavior in multi-
orbital model, we consider in the present paper the phase diagrams of the two-
orbital model in the presence of Hund exchange interaction. The important
effect of Hund coupling for reduction of the critical interaction and
formation local magnetic moments was emphasized in this model at half filling
in previous DMFT studies Pruschke ; 2band1 ; Anisimov . Ref. Werner  has
performed investigations of various instabilities at and away from half
filling. The differences of Mottness and Hundness behavior were studied for
this model away from half filling in Refs. 2orb1 ; Fanfarillo .

In the present paper we concentrate on the evolution of local magnetic moments
with temperature, interaction, and doping, starting from the proximity to the
Mott transition at half filling. We show that the effect of Hund exchange is
crucial for the properties of LMM in the considered two-orbital model. By
studying variety of physical observables within the dynamical mean-field
theory (DMFT), we find the temperatures of the start of the formation of local
magnetic moments, appearing well-defined fermionic quasiparticles (QP), and
the onset of screening of local magnetic moments. We show that these scales
are essentially different and comprise the phase diagram at half filling,
which is similar to that obtained previously for the single-orbital case.

Yet, away from half filling we find that in contrast to the single-orbital
model, the local magnetic moments exist also in the unscreened phase in a
broad range of doping levels. This allows us to connect our observations in
the vicinity of half filling to previous studies of the doped multi-band
Hubbard model, see, e.g., Ref. Fanfarillo .

The plan of the paper is the following. In Sect. II we formulate the model and
the physical observables we study; in Sect. III we consider the properties of
the half filled system, and in Sect IV we consider the off half filled case.
Finally, in Sect. V we present Conclusions.

##  II The model and considered observables

We consider the two-band Hubbard model with Ising symmetry of Hund exchange on
the square lattice (the obtained results are however expected to be
qualitatively applicable for an arbitrary density of states)

$\displaystyle H$
$\displaystyle=\sum_{ij,m,\sigma}{(t_{ij}-\mu\delta_{ij})c^{+}_{im\sigma}c_{jm\sigma}}+U\sum_{im}{n_{im\uparrow}n_{im\downarrow}}$
(1)
$\displaystyle+\sum_{i,m<{m^{\prime}},\sigma{\sigma^{\prime}}}{({U^{\prime}}-J\delta_{\sigma{\sigma^{\prime}}})n_{im\sigma}n_{i{m^{\prime}}{\sigma^{\prime}}}},$

where $c^{\dagger}_{im\sigma}(c_{im\sigma})$ is the electron creation
(destruction) operator at site $i$, orbital $m=1,2$ and spin
$\sigma,{\sigma^{\prime}}=\uparrow,\downarrow$;
$n_{im\sigma}=c^{\dagger}_{im\sigma}c_{im\sigma}$ is the electron number
operator, $\mu$ is the chemical potential, and $t_{ij}=-t$ for the nearest
neighbor sites. We assume the interorbital Coulomb interaction
$U^{\prime}=U-2J$. As the unit of energy we use the half bandwidth $D=4t=1$.
In the majour part of calculations (if not stated differently) we fix the
value of Hund exchange interaction $J=U/4$, which roughly corresponds to the
Hund exchange in transition metals.

We use the dynamic mean field theory DMFT , by mapping the lattice model to
the quantum impurity model, subject to the self-consistency condition. To
trace the formation of local moments we calculate in the self-consistent
solution of DMFT the local spin susceptibility,

$\chi_{s}(T)=\int_{0}^{\beta}\left\langle
S_{i}^{z}(\tau)S_{i}^{z}(0)\right\rangle d\tau,$ (2)

where $S^{z}_{i}=S^{z}_{i,1}+S^{z}_{i,2}$,
$S_{i,m}^{z}=(1/2)(c^{+}_{im\uparrow}c_{im\uparrow}-c^{+}_{im\downarrow}c_{im\downarrow})$
is the spin projection at the site i and imaginary time $\tau$, $\beta=1/T$
(Boltzmann’s constant is put to unity). We also consider the local static
charge susceptibility (local charge compressibility)

$\chi_{c}(T)=\frac{dn}{d\mu}=\int_{0}^{\beta}\left(\left\langle
n_{i}(\tau)n_{i}(0)\right\rangle-n^{2}\right)d\tau,$ (3)

where $n_{i}(\tau)=\sum_{m}n_{i,m}(\tau)$,
$n_{i,m}(\tau)=\sum_{\sigma}n_{im\sigma}$, and the change of the chemical
potential $d\mu$ acts only at the impurity site, the orbital susceptibility

$\displaystyle\chi_{\rm orb}(T)=\frac{1}{4}\int_{0}^{\beta}$
$\displaystyle\langle(n_{i,1}(\tau)-n_{i,2}(\tau))$
$\displaystyle\times(n_{i,1}(0)-n_{i,2}(0))\rangle d\tau,$ (4)

as well as the square of the instantaneous local magnetic moment

$\displaystyle\left\langle{\mathbf{S}}_{i}^{2}\right\rangle$
$\displaystyle=\left\langle({\mathbf{S}}_{i,1}+{\mathbf{S}}_{i,2})^{2}\right\rangle$
$\displaystyle=3n/4-3\left\langle
n_{im\uparrow}n_{im\downarrow}\right\rangle+2\left\langle
S_{i,1}^{z}S_{i,2}^{z}\right\rangle.$ (5)

Apart from that, we study the frozen spin ratio Freezing

$R_{s}=\frac{\left\langle
S_{i}^{z}(\beta/2)S_{i}^{z}(0)\right\rangle}{\left\langle(S_{i}^{z})^{2}\right\rangle},$
(6)

which characterizes the degree of the local magnetic moment formation.

For computations, we use the continuous-time quantum Monte Carlo (CT-QMC)
impurity solver, implemented in the iQIST software package iQIST ; iQISTNote .
The analytical continuation of the fermionic spectral functions is performed
using anacont package ana_cont .

##  III Results at half filling

One of the hallmarks of Hund metals is the spin-orbital separation Hund3 ;
Delft , which implies strong difference of spin-and orbital Kondo
temperatures. To obtain an orbital Kondo temperature $T_{\rm K,orb}$, which
sets the upper boundary of the temperature of formation of local magnetic
moments, we consider the orbital susceptibility (II) as an indicator of
changes in the interorbital balance of fillings (see Fig. 1). We identify
$T_{\rm K,orb}$ with the inflection point of the logarithmic temperature
dependence of the squared effective orbital momentum $\mu_{\rm
orb}^{2}=T\chi_{\rm orb}$. One can see that the obtained temperature
dependence implies suppression of the orbital susceptibility in the low
temperature regime $T<T_{\rm K,orb}\sim 0.5t$, which will be mainly considered
below.

Figure 1: (Color online). Temperature dependence of the square of the orbital
moment for different values of the Coulomb interaction $U$. The Kondo
temperature $T_{K,{\rm orb}}$ is obtained from the inflection points of the
$T\chi_{orb}(T)$, which are marked with gray circles. Figure 2: (Color
online). Temperature dependence of the square of the effective local magnetic
moment $\mu^{2}_{eff}=T\chi_{s}$ for different values of the Coulomb
interaction $U$. The Kondo temperatures $T_{K}$ obtained from the inflection
points of the $T\chi_{s}(T)$ dependencies, are marked with gray circles. The
orbital Kondo temperatures $T_{K,{\rm orb}}$ are marked by black crosses. Gray
stars indicate the spin Kondo temperatures calculated with the fit of
$T\chi_{s}(T)$ dependence to the $S=1$ Kondo model.  Figure 3: (Color online).
Temperature dependence of local static charge susceptibility $\chi_{\rm c}$ at
various values of the Coulomb interaction $U$ and Hund coupling constant
$J=U/4$. Gray circles indicate local maxima of $\chi_{\rm c}$. The orbital
Kondo temperatures $T_{\rm K,orb}$ are shown by black crosses. The temperature
dependencies of charge compressibility at $J=0$ and $U=0.6$, $U=2.0$ are shown
by upper and lower solid lines without symbols. The shaded area marks the
range of $\chi_{\rm c}$ values for interactions between $U=0.6$ and $U=2.0$.

On the other hand, the spin Kondo temperature can be determined from the
temperature dependence of the effective local magnetic moment
$\mu^{2}_{eff}(T)=T\chi_{s}(T)$, determined by the spin susceptibility.
Similarly to the orbital Kondo temperatures, we determine $T_{K}$ from the
inflection point of the logarithmic temperature dependence
$\mu^{2}=T\chi_{s}(T)$ (Fig. 2), since the usually used fit to the universal
dependence of $\mu(T)$ for the Kondo model (see, e.g., Refs. ToschiAnd1 ; Our1
; Our2 ; Comment  for the single-band model), is not fully applicable to the
considered two-band model in view of the spin anisotropy, induced by the
density-density form of the interaction in Eq. (1). One can see that for not
too small $U$ the corresponding Kondo temperatures are substantially smaller
than $T_{\rm K,orb}$, but remain substantially higher than in the single-
orbital case Our1 ; Our2  because of the presence of more channels of
screening.

To study the possibility of the formation and subsequent screening of local
magnetic moments in the temperature range $T_{K}<T<T_{\rm K,orb}$, following
previous works Our1 ; Our2 , we consider the temperature dependence of charge
susceptibility (see Fig. 3). One can see that for $U\geq 1.2$, a local maximum
of $\chi_{c}(T)$ is observed, which characterizes the beginning of the
formation of local magnetic moments in this region. With a further decrease in
temperature, for $1.2\leq U<1.4$, a minimum of $\chi_{c}(T)$ is observed with
the subsequent increase, which is associated with the screening of local
moments by conduction electrons, as was previously described for the single-
band model Our1 ; Our2 . For $U\geq 1.4$, it is noticeable that the minimum of
$\chi_{c}(T)$ is absent and the screening does not occur, which is the
signature of an insulating-like behavior in this region. The orbital Kondo
temperatures are also marked in the figure; as one can see, for not too small
$U$ they are in close proximity to the beginning of the formation of local
moments, which indicates a significant influence of interorbital correlations
on the formation of this phase.

Figure 4: (Color online). Temperature dependence of the square of local
magnetic moment $\langle{\mathbf{S}}_{i}^{2}\rangle$ (main plot), the
interorbital spin correlator $\langle S^{z}_{1}S^{z}_{2}\rangle$ (upper right
plot), and the intraorbital (single-orbital) double occupation $-\langle
n_{m\uparrow}n_{m\downarrow}\rangle$ (lower right) at various values of the
Coulomb interaction $U$. Open and filled circles indicate maxima of $\langle
S^{z}_{1}S^{z}_{2}\rangle$ and $-\langle n_{m\uparrow}n_{m\downarrow}\rangle$
respectively. Filled triangles mark local minima of $\chi_{\rm c}$, and
pentagons mark local maxima.

The obtained temperature dependencies of the orbital susceptibility can be
compared to those with “switched off” Hund exchange ($J=0$). In the latter
case the local charge susceptibility predominantly increases with decreasing
temperature, which indicates that the insulator region is located at higher
values of $U$ ($U\sim 2.3$). Therefore, the noticable effect of Hund exchange
is in the lowering of the metal-insulator Coulomb repulsion, which allows for
having $T_{\rm K}\ll T_{\rm K,orb}$

To find the temperatures of (almost) full formation of local magnetic moments,
we consider maxima of the mean square $\langle{\mathbf{S}}_{i}^{2}\rangle$
(see Fig. 4, main plot). This allows us to distinguish the regions of
formation (increase of $\langle{\mathbf{S}}_{i}^{2}\rangle$ with decrease of
$T$), maximal saturation, and screening of the LMM (decrease of
$\langle{\mathbf{S}}_{i}^{2}\rangle$). As one can see, in agreement with the
observations from local charge susceptibility the screening occurs only when
$U<1.4$.

In contrast to the single-orbital case, there are two different contributions
to $\langle{\mathbf{S}}_{i}^{2}\rangle$ according to the Eq. (II). The upper
inset in Fig. 4 shows the interorbital spin correlator $\langle
S^{z}_{1}S^{z}_{2}\rangle$, which originates from Hund exchange, while the
lower inset shows the contribution of the temperature dependence of the
single-orbital double occupation $-\langle
n_{m\uparrow}n_{m\downarrow}\rangle$. The latter quantity reflects the
strength of the intra-orbital correlations. As one can see, the temperature
dependencies of these quantities are similar, but their maxima are achieved at
different temperatures (as noted on the main plot). That is, with decrease of
temperature, correlations first set within each orbital, and then between the
orbitals. One can also see that the minima of the temperature dependence of
charge susceptibility, discussed above, correspond to the maxima of the
intraorbital correlations. While in the single-band model the maxima of the
intra-orbital correlations (which also correspond to the maxima of charge
compressibility) were associated with the onset of the screening, presence of
the interorbital contribution in the two-band model shifts the actual onset of
the screening to lower temperatures. We therefore further use the maximum of
the full squared moment $\langle{\mathbf{S}}_{i}^{2}\rangle$ rather than the
minimum of charge compressibility as a criterion of the onset of the
screening. This criterion also appears applicable in the doped case, in
contrast to the charge compressibility, which is sensitive to the presence of
free charge curriers Our2 , see further discussion below in Sect. IV.

Figure 5: (Color online). Temperature dependence of “frozen spin ratio”
$R_{s}$, Eq. (6) for different values of the Coulomb interaction $U$. Open
circles indicate maxima of $\langle{\mathbf{S}}_{i}^{2}\rangle$ and filled
triangles mark spin Kondo temperatures $T_{K}$. Figure 6: (Color online).
Temperature evolution of local spectral functions $A(\nu)$ at different $U$.
The dotted lines mark the $\nu=0$ boundary, indicating the density of
electronic states at the Fermi level.

Let us compare the obtained temperature dependencies of charge- and spin
susceptibilities to those of the frozen spin ratio (6), see Fig. 5. At the
transition from the metallic-like to insulating-like state, obtained above
from the temperature dependence of charge susceptibility, the temperature
dependence of frozen spin ratio also changes from decreasing to increasing
with the decrease of temperature at low $T$, which corresponds to difference
between screened and unscreened LMM in the metallic-like and insulating-like
regimes. The value $R_{s}>1/2$ was suggested in pioneering study of Ref.
Freezing  as a criterion for the local magnetic moment formation. One can see
that this criterion is fulfilled in the insulating-like regime and in the
metallic regime it is fulfilled above the temperature of full screening of LMM
(i.e. the Kondo temperature $T_{K}$). At the temperature of the beginning of
screening, which was associated above with the maximum of the square of
instantaneous of local magnetic moment $\langle{\mathbf{S}}_{i}^{2}\rangle$,
we observe a rapid decrease in $R_{s}$. Therefore, with the onset of screening
both the amplitude of the total moment and its stability over time $\tau$
decreases. This also confirms the consistency of the methods for determining
the behavior of the LMM. At high temperatures we observe a plateau $R_{s}\sim
0.9$, which appears due to consideration of a short imaginary time interval
$\beta/2$ at large $T$. Therefore, the temperature dependence of the frozen
spin ratio does not reflect the onset of local magnetic moment formation.

Figure 7: (Color online). Frequency dependence of the difference between
spectral functions $A_{U}-A_{U=0.6}$. The dotted lines mark the boundaries of
the excitation energy $\pm(U+J)/2=\pm 5U/8$ (for each $U$), corresponding to
the state $|\sigma,\sigma\rangle$.

In order to see peculiarities of the electronic properties in various regimes,
we also consider the frequency dependencies of the local spectral functions,
which represent the density of states of interacting electrons (see Fig. 6).
At small $U=1.0$ we find van Hove singularity of the density of states at low
temperatures which is smeared by thermal fluctuations, so that the
corresponding evolution of the density of states is only weakly affected by
the interaction effects. However, at somewhat larger $U=1.35$ we find
formation of Hubbard subbands at sufficiently low temperatures. To analyze in
more detail the shape of the spectral functions in the temperature range
$0.2<T<0.4$ we show in Fig. 7 the difference of spectral function $A(\nu)$ at
$T=0.3$ to that at small $U=0.6$ (we have verified that further decrease of
the interaction does not affect the result). One can see that the interaction
$U$ leads to redistribution of the spectral weight from the center of the band
(i.e. from the Fermi level) to the energy $\simeq\pm(U+J)/2=\pm 5U/8$ (at the
corresponding $U$). As we show in Appendix, this excitation energy refers to
the energy of adding (removing) electrons to (from) the state
$|\sigma,\sigma\rangle$, showing that in this region, the formation of local
magnetic moments actually occurs.

At lower temperatures in the metallic region (Fig. 6), both, the quasiparticle
peak and the Hubbard subbands at the energy $\pm(U+J)/2$ exist. Their
simultaneous appearance shows screening of local magnetic moment, cf. Refs.
Our1 ; Our2 . In contrast to the single-orbital model, the narrower side peaks
with the energies $\pm(5J-U)/2=\pm U/8$ appear as well in this regime. The
latter peaks correspond to removing (or adding) electron from (to) the
$\lvert\uparrow\downarrow,0\rangle$ state, which appear as a part of the
processes of local magnetic moment screening. On the other hand, at large $U$
we find the gap of the spectral function, which is continuously filled with
temperature.

Figure 8: (Color online). Interaction dependence of the imaginary part of
zero-frequency self-energy at various values of the temperature $T$. Colored
circles indicate points of rapid growth of $\Gamma=-{\rm Im}\Sigma(0)$,
obtained from largest curvature. Figure 9: (Color online). The obtained phase
diagram at half filling. The orange dotted line (crosses) shows the orbital
Kondo temperatures. The local maxima of $\chi_{c}(T)$ are shown by violet line
(triangles). The preformed local moment (PLM) region, bounded by the increase
of QP damping (open circles) and appearance of QP peak of the spectral
function (stars), is indicated by the purple shaded area. The violet line
(diamonds) marks the suppression of QP peak. The red shaded region (SCR)
corresponds to the LMM screening regime, bounded by the maxima of the
instantaneous magnetic moment (squares). The solid red line (without squares)
indicates the maxima of the intraorbital moment, and the dashed line indicates
the maxima of interorbital correlations. The dark shaded area (FL) indicates
the Fermi liquid phase, bounded by the respective Kondo temperatures
(crosses). The Mott insulator (MI) region is shown by the violet area
determined from the quasiparticle damping. Between the PLM and SCR modes there
is a regime of appearance of quasiparticle peak in the absence of pronounced
screening (QP).

The most complex behavior is observed near the Mott transition (see, e.g., the
result for $U=1.4$). It can be seen that at low temperatures there is a strong
dip at the Fermi level due to insulating behavior, which, however, turns into
a quasiparticle peak with increasing temperature. Next, a region of the formed
local moment is observed, which is also noticeable from the suppression of the
spectral function at the Fermi level.

The transition to the insulating-like state at $U\approx 1.4$ can be further
confirmed by loss of coherence of the quasiparticles as reflected by the
quasiparticle damping, estimated as an imaginary part of the electronic self-
energy $\Gamma=-{\rm Im}\Sigma(0)$ in Fig. 8. Indeed, crossing the border
$U\approx 1.4$, the damping rapidly increases, while to the left of this
border, as can be seen, $\Gamma$ changes only slightly with a change in $U$
and $T$. We associate the transition to the insulating-like behavior with the
increase of damping, characterised by the maximum of the second derivative.
One can see that the obtained boundary between the metallic-like and
insulating-like behavior coincides with that, obtained from the change of the
temperature dependence of local charge compressibility, which is suppressed at
$U\gtrsim 1.4$.

The obtained boundaries of various regimes are shown in the phase diagram for
half filling in Fig. 9. As in earlier works Our1 ; Our2 , at sufficiently
large temperatures and Coulomb interactions we find the preformed local moment
regime (PLM). The upper temperature boundary of PLM regime, determined by the
sharp increase of the quasiparticle damping, is quite close to the line of
charge compressibility maxima and, at sufficiently large $U$, to the orbital
Kondo temperature $T_{\rm K,orb}$, which indicates the correctness of
determining the characteristic temperatures $T_{\rm K,orb}$. This fact also
confirms considering the orbital screening temperature $T_{\rm K,orb}$ as
start of the local magnetic moment formation in the presence of Hund exchange.
With decrease of temperature PLM regime changes first to the quasiparticle
regime (QP), which precedes the screening of LMM (SCR regime) similarly to the
single orbital case Our2 . However, for considered two-band model this regime
“penetrates” into the region above the insulating phase and has the boundary,
which behaves non-monotonically with change of temperature and Coulomb
repulsion. At sufficiently strong interaction (above the mott insulating
phase) the QP peak collapses with decreasing temperature. Finally, in the FL
regime the fully screened local moment Fermi liquid state occurs.

##  IV Results away from half filling

In this section, we consider what changes in the above discussed observables
with doping. In accordance with the consideration of previous Section, we
determine first the orbital Kondo temperatures from the inflection points of
the $T\chi_{\rm orb}(T)$ dependence (Fig. 10). It is noticeable that in
agreement with earlier results Hund3 , $T_{\rm K,orb}$ is almost unchanged
with doping. Therefore, orbital correlations are not strongly dependent on the
concentration of electrons.

Figure 10: (Color online). Temperature dependence of the square of the orbital
moment for various fillings $n$ ($U=1.4$). The Kondo temperature $T_{K,{\rm
orb}}$ is obtained from the inflection points of the $T\chi_{orb}(T)$, which
are marked with filled circles. Figure 11: (Color online). Temperature
dependence of local static charge susceptibility $\chi_{c}$ for various
fillings $n$ (solid lines correspond to $U=1.4$ and dotted lines correspond to
$U=1.6$). Figure 12: (Color online). Main plot: temperature dependence of the
square of local magnetic moment $\langle{\mathbf{S}}_{i}^{2}\rangle$ for
$U=1.4$ (solid lines) and $U=1.6$ (dashed lines). Insets: interorbital spin
correlator $\langle S^{z}_{1}S^{z}_{2}\rangle$ (top) and intraorbital (single-
orbital) double occupation $-\langle n_{m\uparrow}n_{m\downarrow}\rangle$
(bottom) at various fillings $n$ for $U=1.4$. Open and filled circles indicate
maxima of $\langle S^{z}_{1}S^{z}_{2}\rangle$ and $-\langle
n_{m\uparrow}n_{m\downarrow}\rangle$ respectively. Figure 13: (Color online).
Temperature evolution of spectral functions $A(\nu)$ at different interaction
$U$ and concentration $n$. The dotted lines mark the $\nu=0$ boundary,
indicating the density of electronic states at the Fermi level.

We consider in the following the case in the vicinity of the metal-insulator
transition ($U=1.4$) and the insulating phase at half filling ($U=1.6$). The
temperature dependence of charge compressibility (Fig. 11) changes with doping
similarly to the single-band case Our2 . In particular, instead of the maximum
(start of the formation of LMM), followed by the minimum, corresponding to
beginning of screening of LMM, for $n\lesssim 1.8$ it increases with decrease
of temperature due to sufficiently large number of free charge carriers (i.e.
holes), and has a maximum at approximately the same temperature, where the
minimum of the susceptibility was observed for larger fillings. This change of
temperature dependence was identified in Ref. Our2  with the coherent hole
motion (CHM) regime, which is present in the two-orbital model as well.

The temperature dependence of the average square of the local magnetic moment
$\langle{\mathbf{S}}_{i}^{2}\rangle$ is similar to half filling (Fig. 12, main
plot). In particular, near half filling it monotonously increases with
decreasing temperature, which shows the formation of the LMM. With increase of
doping the maximum of the temperature dependence of local magnetic moment is
formed, showing onset of screening. In stark contrast to the single-orbital
case Our2 , the temperature of the maximum of LMM formation only weakly
changes with doping. Resolving the intra- and interorbital contributions (Fig.
12, insets) shows that the maximum of the intraorbital contribution
$-\left\langle n_{im\uparrow}n_{im\downarrow}\right\rangle$ strongly shifts to
larger temperatures similarly to the single-orbital case, but in the two-
orbital model it is compensated by the interorbital contribution $\langle
S_{i,1}^{z}S_{i,2}^{z}\rangle$, which maximum remains unchanged with doping.
Therefore, the interorbital correlations, originating from Hund exchange,
provide a decisive role in preserving LMM with doping.

The evolution of spectral functions in the doped regime is of particular
interest for determining presence of LMM. In the single-band case it was found
Our2  that local magnetic moments at finite doping are present only in
screened form, which corresponds to having narrow quasiparticle peak at the
Fermi level at low temperatures. However, as it is discussed above in this
section, in the two-orbital case the effect of Hund interaction is significant
for providing stability of LMM. The spectral functions $A(\nu)$ are shown in
Fig. 13 for both $U=1.4$ and $U=1.6$. At small doping and temperatures $T\sim
0.2$ – $0.3$ we find a dip of the spectral functions at (or close to the) the
Fermi level, which corresponds to the LMM formation. Thus, despite the
delocalization of charge carriers with decreasing filling, the Hund exchange
between orbitals preserves the LMM. Qualitatively, this can be explained by
the necessity to increase the energy of electrons by Hund exchange when
breaking the LMM, which is comparable to the hopping parameter. Therefore, in
contrast to the single-band model, in multi-band model the hopping becomes
much less effective for breaking the LMM. This is an essence of Hund metal
behavior.

Figure 14:  (Color online). The phase diagram away from half filling for
$U=1.4$. The symbols on the diagram are the same as in the half filled case in
Fig. 9. The PLM+H (PLM+CHM) phases correspond to the preformed local moments,
accompanied by localized (moving) holes. The Mott insulator (MI) region is
shown by the violet area. Between the PLM and SCR modes there is a regime of
appearance of quasiparticles before screening (QP).

To summarize the results for the doped system, we consider the phase diagram
in Fig. 14. We consider the case near the transition ($U=1.4$), for $U=1.6$
the phase diagram looks similar except for the extent of the insulator and
screening regions. As it was noted above, the orbital Kondo temperature
remains almost unchanged with doping (cf. Ref. Hund3 ) and only increases
slightly, which implies an increase of the distribution of electrons in
different orbitals (see Eq. (II)). At small doping the temperatures of charge
compressibility maxima remain close to $T_{K,orb}$, as in the undoped case.
With further decrease of temperature, a wide region of the formation of a
local magnetic moment (PLM) is observed, obtained from observations of
electronic spectral functions. Despite presence of holes, at small doping we
find sufficiently wide PLM+H regime, where owing to the Hund exchange,
delocalized charges do not destroy completely the LMM, which coexists with
free charge carriers (i.e. holes). With significant doping ($n<1.85$),
similarly to the single-orbital case, the maximum of temperature dependence of
charge compressibility is changed by its minimum. The corresponding region
$n<1.85$, which previously was attributed to the coherent hole motion (CHM)
regime, contains however a part with preserved local magnetic moments (PLM+CHM
regime), which finally changes to CHM itself. At low temperatures a screening
region appears. We define the boundary of the screening region by reaching the
maximal value of the LMM, and, as can be seen, the maximum of the intra-
orbital correlations is reached at much larger temperatures, while the effect
of interorbital correlation persists shifts the maximum to much lower
temperatures. Importantly, the temperature of the onset of screening is
suppressed in comparison to the single-orbital model Our2  and separated from
the region of formation of LMM by narrow QP region, which is present in the
single-orbital model only at half filling Our2 . Fully screened local moments
are represented by a Fermi liquid below the Kondo temperature $T_{K}$.

##  V Conclusions

We have considered the effect of local magnetic moments in the presence of
Hund exchange in the two-band Hubbard model. We found that the formation of
local magnetic moments starts below the orbital Kondo temperature $T_{\rm
K,orb}$. In agreement with previous studies, this temperature scale is much
higher than the Kondo temperature $T_{\rm K}$, which opens broad temperature
window for the LMM formation. In essence, it demonstrates the beginning of the
formation of the LMM, conditioned by Hund correlations. At half filling we
find the phase diagram, which looks similarly to the single-orbital case Our1
; Our2 , except that narrow part of the QP region is found also above the
insulating phase.

In conrast to the single-orbital case, we find that preformed local moment
(PLM) regime is preserved with doping. Another important feature obtained is
that the temperature of screening of local magnetic moments and Kondo
temperature $T_{\rm K}$ changes only slightly with doping, as does the orbital
Kondo temperature $T_{K,{\rm orb}}$, discussed previously. The wide region of
formation of local magnetic moments is preserved at finite doping, in contrast
to the single-band model, due to contributions from the inter-orbital
interactions.

The frequency dependence of spectral functions confirms presence of the region
of LMM formation, where the spectral function is suppressed at the Fermi
level, while the spectral weight is redistributed to the energy of
$|\sigma,\sigma\rangle$ states. We also note the presence of the LMM in the
unscreened state even with significant doping, which is absent for the single-
band Hubbard model. With larger doping, the local magnetic moments are
destroyed.

In the present paper, we use the Ising form of Hund exchange, which is often
considered in the first principle DFT+DMFT calculations. Consideration of
SU(2) symmetric Hund exchange has to be performed in future. We also do not
consider magnetic correlations due to the long-range order in the ground
state. In this respect, the results are applicable to frustrated lattices.
Consideration of the effect of local magnetic moments on the possibility of
long-range magnetic order requires considering non-local cluster or
diagrammatic extensions of dynamical mean field theory.

More generally, the results of this paper can be further used for the
description of materials with almost formed local moments, such as Hund’s
metals. The consideration of the effect of local magnetic moments on the
possibility of unconventional superconductivityWernerAndHoshino1 ;
WernerAndHoshino3 ; WernerAndHoshino4 ; High-Tc1  is also of certain interest.

##  VI Acknowledgements

The authors acknowledge the financial support from the BASIS foundation (Grant
No. 21-1-1-9-1) and the Ministry of Science and Higher Education of the
Russian Federation (Agreement No. 075-15-2021-606). A. A. K. also acknowledges
the financial support within the theme “Quant” 122021000038-7 of Ministry of
Science and Higher Education of the Russian Federation.

Configuration |  State energy $E$ |  $E_{1}=E(U^{\prime}=U-2J)$ |  $E_{2}=E_{1}(\mu_{HF})$ |  $E_{3}=E_{2}-E_{2,\lvert\uparrow,\uparrow\rangle}$ |  $E_{3}(J=U/4)$
---|---|---|---|---|---
$\lvert\uparrow,\uparrow\rangle,\lvert\downarrow,\downarrow\rangle$ |  $U^{\prime}-J-2\mu$ |  $U-3J-2\mu$ |  $-2U+2J$ |  $0$ |  $0$
$\lvert\uparrow,\downarrow\rangle,\lvert\downarrow,\uparrow\rangle$ |  $U^{\prime}-2\mu$ |  $U-2J-2\mu$ |  $-2U+3J$ |  $J$ |  $\frac{U}{4}$
$\lvert\uparrow\downarrow,0\rangle,\lvert 0,\uparrow\downarrow\rangle$ |  $U-2\mu$ |  $U-2\mu$ |  $-2U+5J$ |  $3J$ |  $\frac{3U}{4}$
$\lvert\uparrow\downarrow,\uparrow\rangle,\lvert\uparrow\downarrow,\downarrow\rangle,$ |  $U+2U^{\prime}-$ |  $3U-5J-3\mu$ |  |  |
$\lvert\uparrow,\uparrow\downarrow\rangle,\lvert\downarrow,\uparrow\downarrow\rangle$ |  $-J-3\mu$ |  |  $-\frac{3U-5J}{2}$ |  $\frac{U+J}{2}$ |  $\frac{5U}{8}$
$\lvert\uparrow,0\rangle,\lvert\downarrow,0\rangle,$ |  $-\mu$ |  $-\mu$ |  |  |
$\lvert 0,\uparrow\rangle,\lvert 0,\downarrow\rangle$ |  |  |  |  |
$\lvert\uparrow\downarrow,\uparrow\downarrow\rangle$ |  $2U+4U^{\prime}-2J-4\mu$ |  $6U-10J-4\mu$ |  $0$ |  $2U-2J$ |  $\frac{3U}{2}$
$\lvert 0,0\rangle$ |  $0$ |  $0$ |  |  |
Table A1: Energies of various states in the atomic limit; $\mu_{HF}=(3U-5J)/2$
is the chemical potential, corresponding to half filling.

##  Appendix A Green’s function and excitation energies in the atomic limit

We have according to Lehmann’s representation

$\displaystyle G_{l\sigma}$ $\displaystyle(i\nu_{n})=-\int_{0}^{\beta}d\tau
e^{i\nu_{n}\tau}\langle c_{l\sigma}(\tau)c^{\dagger}_{l\sigma}(0)\rangle$
$\displaystyle=-\frac{1}{Z}\sum_{mn}\int_{0}^{\beta}d\tau
e^{(i\nu_{n}+E_{m}-E_{n})\tau-\beta E_{m}}|\langle
m|c_{l\sigma}|n\rangle|^{2}$ $\displaystyle=\frac{1}{Z}\sum_{mn}\frac{|\langle
m|c_{l\sigma}|n\rangle|^{2}}{i\nu_{n}+E_{m}-E_{n}}(e^{-\beta E_{m}}+e^{-\beta
E_{n}})$ (1)

where the summation over $n$ and $m$ is performed over the states listed in
Table A1, $E_{m}$ are the respective energies, and $Z=\sum_{m}\exp({-\beta
E_{m}})$ is the partition function. We therefore obtain

$\displaystyle G_{l\sigma}(i\nu_{n})$ $\displaystyle=$
$\displaystyle\frac{n_{0,\sigma}+n_{0,0}}{i\nu_{n}+\Delta
E_{1}}+\frac{n_{\sigma,\sigma}+n_{\sigma,0}}{i\nu_{n}+\Delta E_{2}}$ (2)
$\displaystyle+$
$\displaystyle\frac{n_{-\sigma,\sigma}+n_{-\sigma,0}}{i\nu_{n}+\Delta
E_{3}}+\frac{n_{\uparrow\downarrow,0}+n_{-\sigma,0}}{i\nu_{n}-\Delta E_{4}}$
$\displaystyle+$
$\displaystyle\frac{n_{\sigma,\uparrow\downarrow}+n_{\sigma,-\sigma}}{i\nu_{n}-\Delta
E_{3}}+\frac{n_{\uparrow\downarrow,-\sigma}+n_{-\sigma,-\sigma}}{i\nu_{n}-\Delta
E_{2}}$ $\displaystyle+$
$\displaystyle\frac{n_{\uparrow\downarrow,\sigma}+n_{\uparrow\downarrow,0}}{i\nu_{n}+\Delta
E_{4}}+\frac{n_{\uparrow\downarrow,\uparrow\downarrow}+n_{\uparrow\downarrow,-\sigma}}{i\nu_{n}-\Delta
E_{1}},$

where $n_{m}=\exp(-\beta E_{m})/Z$ and

$\displaystyle\Delta E_{1}$ $\displaystyle=E_{\lvert
0,0\rangle;\lvert\uparrow\downarrow,\uparrow\downarrow\rangle}-E_{\lvert\sigma,0\rangle;\lvert\sigma,\uparrow\downarrow\rangle}=\frac{3U-5J}{2}=\frac{7}{8}U,$
$\displaystyle\Delta E_{2}$
$\displaystyle=E_{\lvert\sigma,0\rangle;\lvert\sigma,\uparrow\downarrow\rangle}-E_{\lvert\sigma,\sigma\rangle}=\frac{U+J}{2}=\frac{5}{8}U,$
$\displaystyle\Delta E_{3}$
$\displaystyle=E_{\lvert\sigma,0\rangle;\lvert\sigma,\uparrow\downarrow\rangle}-E_{\lvert\sigma,-\sigma\rangle}=\frac{U-J}{2}=\frac{3}{8}U,$
$\displaystyle\Delta E_{4}$
$\displaystyle=E_{\lvert\uparrow\downarrow,0\rangle}-E_{\lvert\sigma,0\rangle;\lvert\sigma,\uparrow\downarrow\rangle}=\frac{5J-U}{2}=\frac{U}{8}.$
(3)

(last equality in each line corresponds to the case $J=U/4$).

In the limit $T\rightarrow 0$ ($\beta\rightarrow\infty$) we have the only non-
zero occupation numbers $n_{\sigma,\sigma}=1/2$ and therefore obtain the
expression for the Green’s function:

$\displaystyle G_{l\sigma}(\nu)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{1}{\nu+\frac{U+J}{2}}+\frac{1}{\nu-\frac{U+J}{2}}\right).$
(4)

Further analysis of spectral functions in the presence of the hopping can be
performed e.g. within the Hubbard-I approximation and broadens peaks,
corresponding to Hubbard subbands. In DMFT analysis the respective peaks are
further broadened by quasiparticle damping, originating from temperature and
interaction effects.

## References

  * (1) P. Werner, E. Gull, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 101, 166405 (2008).
  * (2) Z. P. Yin, K. Haule, and G. Kotliar, Nature Materials 10, 932 (2011).
  * (3) L. de’ Medici, Phys. Rev. B 83, 205112 (2011); L. de’ Medici, J. Mravlje, A. Georges, Phys. Rev. Lett. 107, 256401 (2011).
  * (4) A. Georges, L. de’ Medici, and J. Mravlje, Ann. Rev. Cond. Matt. Phys. 4, 137 (2013).
  * (5) K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum, Phys. Rev. Lett. 115, 136401 (2015).
  * (6) L. de’ Medici in ”The Physics of Correlated Insulators, Metals, and Superconductors Modeling and Simulation” Vol. 7 Forschungszentrum Juelich, 2017, ISBN 978-3-95806-224-5.
  * (7) K. M. Stadler, G. Kotliar, A. Weichselbaum, J. von Delft, Annals of Physics 405, 365 (2019).
  * (8) E. Walter, K. M. Stadler, S. -S. B. Lee, Y. Wang, G. Kotliar, A. Weichselbaum, J. von Delft, Phys. Rev. X 10, 031052 (2020).
  * (9) K. M. Stadler, G. Kotliar, S. -S. B. Lee, A. Weichselbaum, J. von Delft, Phys. Rev. B 104, 115107 (2021).
  * (10) L. de’ Medici, ”Iron-based Superconductivity”, Springer Series in Materials Science, Vol. 211, pp. 409-441 (2015)
  * (11) P. V. Arribi, L. de’ Medici, Phys. Rev. B 104, 125130 (2021).
  * (12) J. Mravlje, M. Aichhorn, T. Miyake, K. Haule, G. Kotliar, and A. Georges, Phys. Rev. Lett. 106, 096401 (2011).
  * (13) J. Mravlje and A. Georges, Phys. Rev. Lett. 117, 036401 (2016).
  * (14) 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).
  * (15) P. A. Igoshev, A. V. Efremov, A. I. Poteryaev, A. A. Katanin, and V. I. Anisimov, Phys. Rev. B 88, 155120 (2013).
  * (16) A. Hausoel, M. Karolak, E. Sasioglu, A. Lichtenstein, K. Held, A. Katanin, A. Toschi, and G. Sangiovanni, Nat. Commun. 8, 16062 (2017).
  * (17) A. S. Belozerov, A. A. Katanin, and V. I. Anisimov, Phys. Rev. B 96, 075108 (2017).
  * (18) A. S. Belozerov, A. A. Katanin, V. I. Anisimov, Phys. Rev. B 107, 035116, (2023).
  * (19) A. S. Belozerov, A. A. Katanin, and V. I. Anisimov, J. Phys.: Condens. Matter 32, 385601 (2020).
  * (20) M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, and A. Georges, Phys. Rev. B 80, 064501 (2009).
  * (21) P. Werner, S. Hoshino, and H. Shinaoka, Phys. Rev. B 94, 245134 (2016).
  * (22) L. de’ Medici, Phys. Rev. B 83, 205112 (2011)
  * (23) L. Fanfarillo and E. Bascones Phys. Rev. B 92, 075136 (2015).
  * (24) X. Deng, K. M. Stadler, K. Haule, A. Weichselbaum, J. von Delft, and G. Kotliar, Nature Communications 10, 2721 (2019); Nat. Commun. 12, 1445 (2021).
  * (25) P. Chalupa, T. Schäfer, M. Reitner, D. Springer, S. Andergassen, and A. Toschi, Phys. Rev. Lett. 126, 056403 (2021).
  * (26) E. A. Stepanov, S. Brener, V. Harkov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 105, 155151 (2022).
  * (27) T. B. Mazitov, A. A. Katanin, Phys. Rev. B 105, L081111 (2022).
  * (28) T. B. Mazitov, A. A. Katanin, Phys. Rev. B 106, 205148 (2022)
  * (29) S. Adler, F. Krien, P. Chalupa-Gantner, G. Sangiovanni, and A. Toschi, SciPost Phys. 16, 054 (2024).
  * (30) Th. Pruschke1 and R. Bulla, Eur. Phys. J. B 44, 217 (2005).
  * (31) J. Steinbauer, L. de’ Medici, and S. Biermann, Phys. Rev. B 100, 085104 (2019).
  * (32) D. Y. Novoselov, D. M. Korotin, A. O. Shorikov, and V. I. Anisimov, J. Phys.: Condens. Matter 32, 235601 (2020).
  * (33) K. Steiner, S. Hoshino, Y. Nomura, and P. Werner, Phys. Rev. B 94, 075107 (2016).
  * (34) S. Ryee, S. Choi, M. J. Han, Phys. Rev. Lett. 126, 206401 (2021); S. Ryee, S. Choi, M. J. Han, Phys. Rev. Research 5, 033134 (2023).
  * (35) A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
  * (36) L. Huang, Y. Wang, Z. Y. Meng, L. Du, P. Werner, and X. Dai, Comput. Phys. Commun. 195, 140 (2015); L. Huang, ibid. 221, 423 (2017).
  * (37) The integrals over $\tau$ in charge correlators and spin correlators are estimated as sums over CT-QMC imaginary time segments.
  * (38) J. Kaufmann and K. Held, Comput. Phys. Commun. 282, 108519 (2023); https://github.com/josefkaufmann/ana_cont.
  * (39) P. Chalupa, P. Gunacker, T. Schäfer, K. Held, and A. Toschi, Phys. Rev. B 97, 245136 (2018).
  * (40) A. A. Katanin, Nat. Commun. 12, 1433 (2021).
  * (41) S. Hoshino and P. Werner, Phys. Rev. Lett. 115, 247001 (2015).
  * (42) P. Werner, A. J. Kim, and S. Hoshino, EPL 124, 57002 (2018).
  * (43) P. Werner and S. Hoshino, Phys. Rev. B 101, 041104(R) (2020).