# Inverse magnetocaloric effect and phase separation induced by giant van Hove
singularity in itinerant ferromagnetic metal

P. A. Igoshev E-mail<EMAIL_ADDRESS>M. N. Mikheev Institute of Metal
Physics, 620041, S. Kovalevskoj Str. 19, Ekaterinburg, Russia  Amirkhanov
Institute of Physics of DFRC of RAS, Makhachkala 367003, Russia  I. A.
Nekrasov E-mail<EMAIL_ADDRESS>Institute of Electrophysics UB RAS,
620016, Amundsena Str. 106, Ekaterinburg, Russia

###### Abstract

Magnetic phase separation and magnetocaloric effect is considered within the
Hubbard model for face-centered cubic (fcc) lattice with giant van Hove
singularity of electron spectrum at band bottom. Within the Hartree-Fock
approximation it is shown that such model of itinerant magnet exhibits the
first-order ferromagnet-paramagnet phase transition (FOPT) with phase
separation and inverse magnetocaloric effect deep inside the phase separation
region. The thermodynamic theory based on Landau expansion for grand potential
for the ferromagnetic-paramagnetic phase transition is developed for phase-
separated state. It is rigorously shown that ferromagnetic phase involved in
the phase separated state exhibits negative magnetic susceptibility in the
vicinity of tricritical point. Thus, an entropy of the magnetically ordered
phase may increase when the magnetic field is applied which implies inverse
character of the total magnetic entropy change $\Delta S$. Temperature
dependence of $\Delta S$ for the mean-field solution of the non-degenerate
Hubbard model is analyzed in detail for different band filling values. The
possibility to control $\Delta S$ sign by changing both the temperature and
the band filling of magnetocaloric materials is demonstrated and this seems to
be promising from the point of view of interpretation a lot of experimental
data and possible technological applications. It is shown that giant van Hove
singularity, being a main cause of phase separation, strongly affects
temperature dependence of $\Delta S$ in the vicinity of tricritical point.

##  I Introduction

The magnetocaloric effect (MCE), i.e. a change of a sample temperature
$T$(entropy $S$) when the external magnetic field is applied under the
condition of fixed entropy (temperature), is observed in a lot of magnetically
ordered systems. The strength of MCE typically has maximal value close to
critical temperature of magnetic phase transition. Over the past 40 years, a
huge amount of experimental data on MCE has been accumulated, see reviews
Franco et al., 2012; Li, 2016; Franco et al., 2018. The current state of
affairs in the field of materials used for magnetic cooling is described in
Ref. Tishin and Spichkin, 2014, whereas medical applications are reviewed in
Ref. Tishin et al., 2016. Details of standard theoretical approaches to the
study of MCE are described in the review de Oliveira and von Ranke, 2010.

The temperature profile of isothermal entropy change $\Delta S(T)$ and its
maximum absolute value $\Delta S_{\rm max}$ substantially depend on the order
of magnetic phase transition as well as on the nature of the electronic states
forming magnetic order. Most important quantity for applications
characterizing the magnetic cooling efficiency is relative cooling power (RCP)
proportional to $\Delta S_{\rm max}$, and its peak width. Thus, an ability to
calculate $\Delta S(T)$ profile for real materials based on characteristics of
their electronic structure is of a great importance for practical
applications. In addition, not only a large value $\Delta S$ at peak
temperature, but also the table-like temperature dependence of $\Delta S$ can
be used to construct an ideal Ericsson cycle of magnetic refrigeration.
Recently the compounds exhibiting such behaviour attracted a lot of attention
Fu et al. (2014); Agurgo Balfour et al. (2015); Liu et al. (2016); Zheng et
al. (2017); Zhong et al. (2018).

Perhaps the most known material being examined for practical use is rare-earth
system Gd5Si2Ge2 exhibiting so-called giant MCE ($\Delta S_{\rm max}\sim 19$
J/kg/K under the magnetic field 5 T) at room temperature Pecharsky and
Gschneidner (1997a). The magnetic properties of this system are mainly driven
by localized magnetic moments of Gd $f$-shell. For this compound, the position
of $\Delta S(T)$ maximum is close to the Curie temperature $T_{\rm C}\sim 273$
K of the first-order ferromagnetic (FM)–paramagnetic (PM) magnetic phase
transition.

Rare-earth compounds are not only materials with remarkable MCE properties.
There are some examples of transition metal compounds which also exhibit large
and unexplained MCE potential values. The MnAs, MnFeP0.45As0.55 and
La(FexS1-x)13 systems demonstrate $\Delta S_{\rm max}$ value close to that of
the Gd5Si2Ge2 system. Note that, first-order FM-PM phase transition in the
following compounds occurs at temperature close to room temperature: 318 K in
MnAs Wada et al. (2002), about 300 K in MnFeP0.45As0.55 Tegus et al. (2002).
First-order FM-PM phase transition occurs about 195 K in La(Fe0.88Si0.12)13
and its origin is likely the peak of electron density of states in the
vicinity of Fermi level in paramagnetic phase Fujita et al. (2003).
Interestingly, the order of FM–PM magnetic phase transition in the system
La(FexSi1-x)13 depends on the concentration of Si: at low concentrations there
is a first-order phase transition (FOPT), while at larger concentrations –
second-order phase transition (SOPT) occurs Franco et al. (2017). Magnetism in
the systems MnFeP0.45As0.55 and La(FexSi1-x)13 has pronounced itinerant
character Yamada and Goto (2003).

All above mentioned compounds demonstrate so called direct MCE: an increase of
the magnetic field within an isothermal process results in the entropy
decrease ($\Delta S<0$), and the temperature increase within the adiabatic
process ($\Delta T>0$). However, there is another class of compounds
exhibiting opposite response to an applied magnetic field ($\Delta T<0$,
$\Delta S>0$), which is named inverse MCE.

Typical classes of systems with inverse MCE are Heusler alloys (e.g. NiMnIn
Zhang et al. (2007), NiMnSn Krenke et al. (2005), Ni2Mn1-xCuxGa Wang and Liu
(2013) with martensitic antiferromagnetic (AFM)–FM phase transition), some
manganites (e.g. Pr0.5Sr0.5MnO3 Hsini et al. (2018), MnFeP0.8Ge0.2Caron et al.
(2009) (with PM$+$FM phase separation), La1-xCaxMnO3 Restrepo-Parra et al.
(2014)), intermetallic compounds (e.g. PrNi5 von Ranke et al. (1998), ErRu2Si2
Samanta et al. (2007), CoMnSi1-xGex Sandeman et al. (2006)), for more details
see the reviews Gschneidner et al. (2005); Franco et al. (2012); Li (2016); de
Oliveira and von Ranke (2010). In the Heusler alloys, the FM–PM magnetic phase
transitions typically give direct MCE while inverse MCE occurs under FM-AFM
transition.

Besides observed above ordinary ferromagnetic systems, there are also systems
where the ferromagnetic order coexists with some other ordered phases in terms
of phase separation (PS): La0.27Nd0.40Ca0.33MnO3Liu et al. (2006)
(FM$+$charge-ordered phase separation), Mn0.99Cu0.01As, Gd5Ge2.3Si1.7Phan et
al. (2010)(FM$+$AFM phase separation). The compound (alloy) FeRh exhibits
large value of $\Delta S_{\rm max}>0$ (inverse MCE) under corresponding FM-AFM
FOPT upon temperature decrease with critical temperature 270 K Annaorazov et
al. (1992); Gimaev et al. (2020). This phase transition goes through FM+AFM
phase separation region.

There are also some materials that exhibit inverse MCE: MnRhAs Balli et al.
(2014), Co and Mn Co-Doped Ni2MnGa Wang and Liu (2013), Er2Fe17 Khedr et al.
(2019); Álvarez-Alonso et al. (2012), DyAl2von Ranke et al. (2000),
Pr0.46Sr0.54MnO3Naik et al. (2011) in the vicinity of FOPT temperature.
Typically, there are two different temperatures for inverse and direct MCE.
Moreover, an inverse MCE provides an additional peak in $\Delta S$ temperature
profile (see e.g. Eu0.55Sr0.45MnO3 Liu et al. (2006), Ni50Mn33.13In13.90 Zhang
et al. (2007),ErGa2, HoGa2 dos Reis et al. (2014), HoFeSi Zhang et al.
(2014)).

Previously some theoretical studies of MCE in materials and corresponding
models with FOPT were performed. A Monte-Carlo-based solution of the Potts-
Blume-Emery-Griffiths model was used to investigate a series of
Ni45Co5Mn37In13 Comtesse et al. (2014) and Ni-Co-Mn-(Sn,Al) alloys Sokolovskiy
et al. (2018), which have two phase transitions with direct and inverse MCE.
Within the Potts-Blume-Emery-Griffiths model, the inhomogeneity arising during
the FOPT was taken into account by introducing an additional term to the
Hamiltonian responsible for a disorder, which simulates the mixing of
martensitic and austenitic phases. Investigation of MCE in the case of SOPT
for the mean-field treated Hubbard model for different bare densities of
states was performed in Refs. de Oliveira and von Ranke, 2010; Igoshev et al.,
2017. Theoretical study of MCE for FM-AFM FOPT including also ferrimagnetic
states, taking into account the possibility of phase separated state formation
within the Hubbard model was performed for Bethe Ivchenko and Igoshev (2021)
and square Igoshev et al. (2023) lattices.

Another direction of theoretical studies is based on phenomenological Landau
theory. The latter was designed to describe the FM-PM SOPT using a simple free
energy functional Landau and Lifshitz (2013). However, this theory exhibits
wide opportunities to consider more general cases, e.g. in the case of FOPT.
The Landau theory equally well can be applied to both itinerant and localized
electron systems. For the itinerant systems a direct derivation of Landau
series coefficients within the Stoner approximation can be done
straightforward starting from bare electronic density of states Moriya (2012).
However, the phenomenological approach (corresponding to paramagnon
renormalization of Landau series coefficients) proved itself to be more
successful Murata and Doniach (1972); Moriya and Kawabata (1973a, b); Moriya
(2012). Some examples of the Landau theory application for explanation of FOPT
and MCE in itinerant systems Co(S,Se)2, Lu(Co,Al)2, and Lu(Co,Ga)2 can be
found in Ref. Yamada and Goto, 2003. Necessary for FOPT sign changes of Landau
series coefficients may be provided due to the presence of peculiarities of
density of states in the vicinity of the Fermi level Fujita et al. (2003) or
due to an introduction of additional interactions (e. g. AB2 compounds Yamada
and Goto (2003), MnAs de Oliveira (2004); de Medeiros et al. (2013); de
Oliveira and von Ranke (2010), MnFeP0.45As0.55 de Oliveira and von Ranke
(2005), YCo2Bloch et al. (1975); Duc et al. (1992)).

However, phase separation accompanying FOPT in metal systems stays beyond this
approach and is somehow unavailable without an appropriate extension of Landau
theory.

Thus, there is a problem of theoretical modeling and explanation of MCE in
itinerant systems exhibiting FOPT FM-PM. Another complexity comes from
necessary presence of phase-separated state in itinerant systems near FOPT
region, and pure localized electron model (without itinerant component) can
not consistently treat appearing inhomogeneities Balli et al. (2009);
Pecharsky et al. (2009); Xu et al. (2015); de Medeiros et al. (2013); Casanova
et al. (2003); Jia et al. (2006); Matsumoto et al. (2004). At the moment there
is a lack of theoretical studies of MCE for variety of experimental data in
systems with FOPT and PS. Additional issue is that conventional Maxwell
relations for experimental data treatment in a case of FOPT should be applied
with a great caution. One should also take into account that phase volumes of
PS state depend on temperature and magnetic field Liu et al. (2006).
Neglecting this dependency leads to spurious results Pecharsky and Gschneidner
(1997b); Balli et al. (2009), which were discussed in Refs. Comtesse et al.,
2014; Liu et al., 2007.

The plan of the presentation is the following. An extension of the Landau
theory for an account of PS state during FM-PM FOPT is presented in Section
II. Section III contains the example of MCE within the mean-field solution of
the non-degenerate Hubbard model for an face-centered cubic lattice with the
parameter set corresponding to the position of Fermi level in the vicinity of
giant van Hove singularity of the electronic density of states. The paper is
ended by concluding remarks in Sec. IV.

##  II FM-PM FOPT thermodynamics: the extension of Landau theory

Conventional Landau functional (free energy) does not allow to describe phase
separation which can accompany any FOPT in itinerant magnets Igoshev et al.
(2009, 2010, 2015). The Landau functional typically is considered as a
functional of the magnetization only, and besides the latter plays a role of
an order parameter. From a general point of view, in itinerant magnet order
parameter jump at FOPT point should result in band filling jump which in turn
may result in phase separation formation. Instead of introducing band filling
$n$ as an inner argument of free energy, chemical potential $\mu$ should be
introduced as an argument of the Landau functional and grand potential
$\Omega$ — instead of free energy $F$ Landau and Lifshitz (2013); Igoshev
(2023). In this section we present general thermodynamical approach to FOPT
including PS state as well as to standard SOPT within the Landau theory.

Let us write Landau expansion for grand potential $\Omega$ with respect to the
order parameter $m$ (the coefficients $A^{\Omega}_{\alpha}$ depend on $T$ and
$\mu$)

$\Omega(T,\mu,h|m)=A^{\Omega}_{0}(T,\mu)+A^{\Omega}_{2}(T,\mu)m^{2}+A^{\Omega}_{4}(T,\mu)m^{4}+A^{\Omega}_{6}(T,\mu)m^{6}-hm,$
(1)

where $A^{\Omega}_{6}>0$ to provide stable solution. The coefficients
$A^{\Omega}_{\alpha}$ up to sixth order are hold to supply not only SOPT but
also the possibility of FOPT.

External arguments (i.e. set up by environment such as $\mu,T$ and $h$) are
written before the vertical line ‘$|$’, internal argument, magnetization $m$,
which can be found by minimization of $\Omega$ under given external
parameters, is written after ‘$|$’. Note that the internal parameter specifies
particular phase for the system. We consider phase transitions between phases
differing by a value of $m$ only, e. g. paramagnetic-ferromagnetic or
antiferromagnetic-ferrimagnetic phase transitions.

The equation of state defining magnetization $m=m(T,\mu,h)$ for given $T,\mu$
and $h$ is determined by extremum condition of Eq. (1)
$\partial\Omega/\partial m=0$, and has a form

$f(T,\mu|m)=h,$ (2)

where

$f(T,\mu|m)=2\left(A^{\Omega}_{2}(T,\mu)m+2A^{\Omega}_{4}(T,\mu)m^{3}+3A^{\Omega}_{6}(T,\mu)m^{5}\right)$
(3)

and $\partial f/\partial m>0$ (the criterion of $\Omega$ minimum at $m$
point).

We are interested in the calculation of zero-field magnetic susceptibility

$\left(\frac{dm}{dh}\right)_{h=0}=\left(\frac{\partial m}{\partial
h}\right)_{h=0}+\eta_{\rm env}(h=0)\left(\frac{\partial
m}{\partial\mu}\right)_{h=0},$ (4)

where the derivative

$\eta_{\rm env}=d\mu/dh$ (5)

originates from the dependence $\mu=\mu(h)$, which in turn should be setup to
provide constraints determined by physical environment.

Using Eq. (2) we get $\mu$-fixed (as denoted by bar) zero-field magnetic
susceptibility $\bar{\chi}\equiv\left({\partial m}/{\partial
h}\right)_{h=0}=\left({\partial f}/{\partial m}\right)^{-1}_{h=0}$, i. e.

$\bar{\chi}=\frac{1}{2}(A^{\Omega}_{2}+6A^{\Omega}_{4}m^{2}+15A^{\Omega}_{6}m^{4})^{-1},$
(6)

where $m$ satisfies Eq. (2) at $h=0$. At $h=0$ the Eq. (2) may have up to two
solutions $m\geqslant 0$. While in the case $A^{\Omega}_{2}<0$ unique minimum
exists at arbitrary values of $A^{\Omega}_{4}$ and positive $A^{\Omega}_{6}$,
at $A^{\Omega}_{2}>0$, $m=0$ $\Omega$ minimum exists (corresponding to PM
state) and in the case
$\left(A^{\Omega}_{4}\right)^{2}>3A^{\Omega}_{2}A^{\Omega}_{6}$,
$A^{\Omega}_{4}<0$, additional finite-$m$ minimum of $\Omega$ exists,
$m=m_{\rm FM}$,

$m^{2}_{\rm
FM}=\frac{-A^{\Omega}_{4}+\sqrt{\left(A^{\Omega}_{4}\right)^{2}-3A^{\Omega}_{2}A^{\Omega}_{6}}}{3A^{\Omega}_{6}}.$
(7)

corresponding to FM state. Then the Eq. (7) can be written as

${m^{2}_{\rm FM}}=-\frac{2A^{\Omega}_{2}}{A^{\Omega}_{4}}.$ (8)

For magnetically ordered phase, we calculate $\bar{\chi}$, taking into account
Eqs. (2) and (6),

$\bar{\chi}_{\rm FM}=\frac{1}{8m^{2}_{\rm
FM}(A^{\Omega}_{4}+3A^{\Omega}_{6}m^{2}_{\rm FM})}.$ (9)

For PM solution $m=m_{\rm PM}=0$, from Eq. (6) we have

$\bar{\chi}_{\rm PM}=\frac{1}{2A^{\Omega}_{2}}.$ (10)

Further we consider two different cases: (i) single-phase (homogeneous) and
(ii) phase-separated state (see next subsection).

###  II.1 Magnetic susceptibility at fixed band filling

For a single-phase case (i) band filling defined as
$n=-\partial\Omega/\partial\mu=\mathfrak{n}(\mu,T|m)$, where

$\mathfrak{n}(T,\mu|m)=-(\dot{A}^{\Omega}_{0}+\dot{A}^{\Omega}_{2}m^{2}+\dot{A}^{\Omega}_{4}m^{4}+\dot{A}^{\Omega}_{6}m^{6}),$
(11)

where $\dot{A}^{\Omega}_{\alpha}=\partial{A^{\Omega}_{\alpha}}/\partial\mu$.
Therefore for the single-phase state there comes additional $h$-independent
condition to determine $\mu$

$\mathfrak{n}(T,\mu|m)=n$ (12)

as a function of $T$ and $m$. Magnetic field $h$ affects the system only
through $m$ and $\mu$.

The fixed band filling magnetic susceptibility (as denoted by lower index
“$n$”) reads

$\chi_{n}\equiv\left(\frac{dm}{dh}\right)_{n,h=0}$ (13)

is obtained directly from Eq. (4), where $\mu$ changes consistently with $m$
to satisfy Eq. (12), so that $\eta_{n}\equiv\left({d\mu}/{dh}\right)_{n,h=0}$
and

$\eta_{n}=-\left(\frac{{\partial\mathfrak{n}}/{\partial
m}}{{\partial\mathfrak{n}}/{\partial\mu}}\right)_{h=0}\chi_{n}.$ (14)

Therefore we get

$\chi_{n}=\left(\frac{\partial f}{\partial m}-\frac{\partial
f}{\partial\mu}\frac{{\partial\mathfrak{n}}/{\partial
m}}{{\partial\mathfrak{n}}/{\partial\mu}}\right)^{-1}$ (15)

which is always positive for any single-phase metallic electron system.

###  II.2 Magnetic susceptibility of phases for phase-separated state

Now we consider the case (ii) for phase-separated state caused by FOPT with PM
and FM phases (instead of these phases there can be any other pair of phases
differing from each other only by the presence of non-zero component of total
magnetization).

Figure 1:  Schematic plot of $\Omega(m)$ for $A^{\Omega}_{4}=-0.1$,
$A^{\Omega}_{6}=1$, and different values of $A^{\Omega}_{2}$ in units of
$A^{\Omega}_{2,\rm FOPT}$ (shown in legend). Arrows show the positions of
minimum of $\Omega(m)$, corresponding to minimum position $m=m_{\rm FM}(m_{\rm
PM})$ for FM (PM) phase. Solid (dashed) lines show the result at
$h=0~{}(h=10^{-4})$.

Main difference of FOPT in itinerant systems from the FOPT in localized
systems is that in the former case, a jump of the order parameter $m$
(magnetization) at the transition point leads to a band filling jump. For the
phase-separated state total band filling is a weighted sum of both phase band
fillings, see Eq. (11), satisfying Eq. (2) at the thermodynamic equilibrium.
At given $T$ and $h$ at FOPT line the criterion of PS appearance is a falling
of $n$ into the band filling jump interval, see details in Refs. Igoshev et
al., 2010; Ivchenko and Igoshev, 2021; Igoshev, 2023; Igoshev et al., 2015.
Therefore all extensive quantities describing PS state acquire additional
linear dependence on band filling $n$. Thus, for $\mu$ fixed at FOPT line band
filling $n$ plays a role of additional degree of freedom.

At FOPT line, grand potential $\Omega(\mu,T,h|m_{\Phi})$ for different phases
$\Phi=\text{FM},\text{PM}$ involved in PS should be equal, where $m_{\rm FM}$
(FM phase) and $m_{\rm PM}$ (PM phase) are the roots of Eq. (2). The
definition of phase separation boundary line $\mu={\mu_{\rm PS}}(T,h)$ reads

$\Omega(T,{\mu_{\rm PS}}(T,h),h|m_{\rm FM})=\Omega(T,{\mu_{\rm
PS}}(T,h),h|m_{\rm PM}).$ (16)

In the case of a zero magnetic field $h=0$, Eq. (16) becomes
$A_{4}+2A_{6}{m^{2}_{\rm FM}}=0$. Using the Eq. (7) one can get zero-field
FOPT line equation in terms of $T$ and $\mu$ variables

$A^{\Omega}_{2}(T,\mu)=A^{\Omega}_{2,\rm FOPT}(T,\mu),$ (17)

where

$A^{\Omega}_{2,\rm
FOPT}(T,\mu)=\frac{\left(A^{\Omega}_{4}(T,\mu)\right)^{2}}{4A^{\Omega}_{6}(T,\mu)}$
(18)

Fig. 1 shows a schematic example of $m$ dependence of $\Omega(T,\mu,h|m)$ at
zero and finite magnetic field, see Eq. (1), when coefficient $A^{\Omega}_{2}$
is increases through $A^{\Omega}_{2,\rm FOPT}$ value. One can see that at
$h=0$ global minumum position of $m$ dependence of $\Omega(T,\mu,h|m)$ changes
from $m=m_{\rm FM}$ to $m=m_{\rm PM}$ as $A^{\Omega}_{2}(T,\mu)$ becomes
larger than $A^{\Omega}_{2,\rm FOPT}(T,\mu)$. At
$A^{\Omega}_{2}(T,\mu)>(4/3)A^{\Omega}_{2,\rm FOPT}(T,\mu)$ local FM minimum
disappears.

Let ${m^{2}_{\rm FM}},A^{\Omega}_{4}\rightarrow 0$ at FOPT line, so from Eq.
(8) $A^{\Omega}_{2}\rightarrow 0$ and we arrive at tricritical point
($T=T^{\ast}_{0}$, $\mu=\mu^{\ast}$) for which
$A^{\Omega}_{2}(T,\mu)=A^{\Omega}_{4}(T,\mu)=0$. The continuation of FOPT line
corresponding to equation $A^{\Omega}_{2}(T,\mu)=0$ in
$A^{\Omega}_{4}(T,\mu)>0$ region yields SOPT line. Thereby, at tricritical
point phase transition changes its order.

Using the Eqs. (17) and (8), we simplify the expression (9) for $\mu$-fixed FM
phase magnetic susceptibility

$\bar{\chi}_{\rm FM}=\frac{1}{8A^{\Omega}_{2}}>0,$ (19)

which is essentially positive in case of FOPT. Obtained results are applicable
for both itinerant and localized (with replacements $\mu\rightarrow n$ and
$\Omega\rightarrow F$) systems undergoing FOPT.

Finally, we should obtain the dependence of chemical potential $\mu={\mu_{\rm
PS}}(T,h)$ to keep the system in the PS regime at finite $h$. The condition
that the system remains in PS region when the magnetic field is applied is
given by Eq. (16). Differentiating Eq. (16) with respect to $h$ and taking
into account the relation $\partial\Omega/\partial m=0$ at $m=m_{\rm PM,FM}$,
we get

$\frac{\partial\Omega}{\partial h}(T,{\mu_{\rm PS}}(T,h),h|m_{\rm
PM})+\frac{\partial\Omega}{\partial\mu}(T,{\mu_{\rm PS}}(T,h),h|m_{\rm
PM})\left(\frac{d\mu}{dh}\right)_{\rm PS}\\\ =\frac{\partial\Omega}{\partial
h}(T,{\mu_{\rm PS}}(T,h),h|m_{\rm
FM})+\frac{\partial\Omega}{\partial\mu}(T,{\mu_{\rm PS}}(T,h),h|m_{\rm
FM})\left(\frac{d\mu}{dh}\right)_{\rm PS}.$ (20)

We obtain an analogue of Clausius–Clapeyron relation determining the chemical
potential derivative in Eq. (4) in the PS state as

$\eta_{\rm PS}\equiv\left(\frac{d\mu}{dh}\right)_{{\rm PS},h=0}=-\frac{m_{\rm
FM}-m_{\rm PM}}{n_{\rm FM}-n_{\rm PM}},$ (21)

instead of $\eta_{\rm n}$ for single-phase case, see Eq. (14). Substituting
the Eq. (1) into the last equation one obtains

$\eta_{\rm PS}=(1/{m_{\rm
FM}})(\dot{A}^{\Omega}_{2}+\dot{A}^{\Omega}_{4}{m^{2}_{\rm
FM}}+\dot{A}^{\Omega}_{6}m^{4}_{\rm FM})^{-1}.$ (22)

We have common $\eta=\eta_{\rm PS}$ for both phases involved in PS differing
by $m$ value only. Then, using the equation of state (2), we get phase
susceptibility

$\chi^{\rm PS}_{\Phi}=\bar{\chi}_{\Phi}\left(1-\eta_{\rm
PS}\left(\frac{\partial f}{\partial\mu}\right)_{\Phi}\right).$ (23)

So, we get directly

$\displaystyle\chi^{\rm PS}_{\rm PM}$ $\displaystyle=$
$\displaystyle\bar{\chi}_{\rm PM},$ (24) $\displaystyle\chi^{\rm PS}_{\rm FM}$
$\displaystyle=$ $\displaystyle-\bar{\chi}_{\rm FM}\left(1+\frac{2{m^{2}_{\rm
FM}}(\dot{A}^{\Omega}_{4}+2\dot{A}^{\Omega}_{6}{m^{2}_{\rm
FM}})}{\dot{A}^{\Omega}_{2}+\dot{A}^{\Omega}_{4}{m^{2}_{\rm
FM}}+\dot{A}^{\Omega}_{6}m^{4}_{\rm FM}}\right).$ (25)

If ${m_{\rm FM}}$ is small (the vicinity of tricritical point) then the second
term in brackets can be neglected and one get in the leading order with
respect to $m$

$\chi^{\rm PS}_{\rm FM}\simeq-\frac{1}{8A^{\Omega}_{2}}<0.$ (26)

One can see that in the vicinity of tricritical point $\chi^{\rm PS}_{\rm FM}$
is essentially negative (inverse sign with respect to Eq. (19)) and equal by
absolute value to $\bar{\chi}_{\rm FM}$ from Eq. (19).

Above we discussed magnetic susceptibilities of PM and FM phases involved in
PS. We have shown that these two phases behave differently under the change of
the magnetic field. There is significant contribution to magnetic
susceptibility of FM phase involved in PS state that comes from a change in
chemical potential as a function of the magnetic field (see Eqs. (21) and
(23)), while for PM phase there is no such contribution since $\left(\partial
f/\partial\mu\right)_{\rm PM}=0$. Let us stress here that total response for a
system in the PS state should be obtained as a phase weighted average (see
details in the model example below). To our knowledge this result is new,
although some similar results (for example, a similar expression to Eq. (21)
was obtained earlier for the alloys theory Krivoglaz and Sadovskii (1964) for
another problem).

###  II.3 Entropy calculation

We are mainly interested in the calculation of $(\Delta S)_{\rm env}$ which is
dependent on the physical environment (we mark this by the index “env”, (i)
“env”$=$ “n” and (ii) “env” $=$ “PS”)

$(\Delta S)_{\rm
env}=\int_{0}^{h}dh^{\prime}\left(\frac{dS(h^{\prime})}{dh^{\prime}}\right)_{\rm
env},$ (27)

thus the sign of $(\Delta S)_{\rm env}$ is defined by the sign of
$(dS/dh)_{\rm env}$ at small enough magnetic field.

By looking at the entropy definition $S=-{\partial\Omega(T,\mu|m)}/{\partial
T}=S(T,\mu|m)$ one can see that $S$ does not depend on magnetic field $h$
explicitly, but only through chemical potential $\mu$ and magnetization $m$.
So we have

$\left(\frac{dS}{dh}\right)_{\rm env}=\frac{\partial S}{\partial m}\chi_{\rm
env}+\frac{\partial S}{\partial\mu}\eta_{\rm env},$ (28)

where $\eta_{\rm env}$ is determined by (5).

Now we discuss and compare the computation of the $({dS}/{dh})_{\rm env}$ in
both single-phase (i) and phase-separated (ii) cases. For the case (i) one
gets

$\left(\frac{\partial S}{\partial
h}\right)_{n}=\chi_{n}\left(\frac{dS}{dm}\right)_{n},$ (29)

where

$\left(\frac{\partial S}{\partial m}\right)_{n}=\left(\frac{\partial
S}{\partial m}-\frac{\partial
S}{\partial\mu}\frac{{\partial\mathfrak{n}}/{\partial
m}}{{\partial\mathfrak{n}}/{\partial\mu}}\right),$ (30)

with $\chi_{n}$ defined by Eq. (15).

Consider how the presence of PS and the negative sign of the magnetic response
can affect the change in entropy when the magnetic field is applied. Let
$S_{\Phi}(T,h)=S(T,\mu_{\rm PS}(T,m_{\Phi},h)|m_{\Phi})$ be the entropy of
phase $\Phi$ involved in PS.

From Eq. (28) one gets the entropy change derivative for the case (ii) for the
phase $\Phi$ involved in PS

$\left(\frac{dS_{\Phi}}{dh}\right)_{\rm PS,\Phi}=\frac{\partial S}{\partial
m}\chi^{\rm PS}_{\Phi}+\frac{\partial S}{\partial\mu}\eta_{\rm PS},$ (31)

where $\eta_{\rm PS}$ is determined by Eq. (21). Here it is taken into account
that when the magnetic field is applied, not only the magnetization of the
ferromagnetic phase $m$ changes, but also $\mu$. In the vicinity of
tricritical point one can expect that the first ($\chi$) term dominates over
the second ($\eta$) term. Thus, we have

$\left(\frac{dS_{\Phi}}{dh}\right)_{\rm PS}=\frac{\chi_{\rm
PS}}{\chi_{n}}\left(\left(\frac{dS}{dh}\right)_{n}+\frac{\partial
S}{\partial\mu}\left(\frac{\chi_{n}}{\chi_{\rm PS}}\eta_{\rm
PS}-\eta_{n}\right)\right).$ (32)

Thus, we have the contributions to the infinitesimal entropy change in FM
phase: the $\chi$ term is opposite to the usual sign due to the negative phase
susceptibility and $\eta$ term has an uncertain sign. Therefore one can expect
that an inverse MCE occurs as a direct consequence of PS for the parameter
region for which direct MCE occurs away of PS region:

$\left(\frac{dS_{\rm FM}}{dh}\right)_{\rm PS}>0.$ (33)

We state that the participation of FM phase in PS substantially can generally
change both magnetic and entropy response. Furthermore, PS presence can change
direct MCE away of PS region to the inverse MCE. For the paramagnetic phase
involved in PS, the phase susceptibility is positive (see Eq. (6)). Thus, in
the phase-separated (mixed) state, the magnetic response of the paramagnetic
and ferromagnetic phases are opposite in sign, which can lead to the fact that
the ferromagnetic phase contributes an inverse sign to the isothermal change
of the entropy $\Delta S$ when the magnetic field is applied.

##  III Entropy change for a first-order FM-PM transition induced by giant an
Hove singularity: the case of fcc lattice

In this Section we present an example of calculation of generalized Landau
theory coefficients $A^{\Omega}_{\alpha}$ (see previous Section) and the MCE
isothermal change of the entropy $\Delta S(T)$ for the case of non-degenerate
Hubbard model face-centered cubic lattice

$\mathcal{H}=\sum_{ij\sigma}t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}-h\sum_{i\sigma}\sigma
n_{i\sigma},$ (34)

where $t_{ij}$ is non-zero and equal to $t$ ($t^{\prime}$), when sites $i$ and
$j$ are nearest (next nearest) neighbours, $h=\mu_{\rm B}H$ with $H$ being a
magnetic field, $c^{\dagger}_{i\sigma}/c_{i\sigma}$ is a creation/annihilation
electron operator and $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$ is an
operator of number of electrons with spin projection $\sigma=\pm 1$ at the
site $i$, $U$ is matrix element of local Coulomb interaction.

To cast giant van Hove singularity at the band bottom, we consider the case
$t^{\prime}=-t/2$, so the density of states for bare electron spectrum
$\epsilon^{\rm fcc}_{\mathbf{k}}=(1/N)\sum_{ij}t_{ij}\exp[{\rm
i}\mathbf{k}(\mathbf{R}_{i}-\mathbf{R}_{j})]$ can be written

$\tilde{\rho}_{\rm
fcc}(\epsilon)=\frac{1}{t}\sqrt{\frac{2}{3+\epsilon/t}}\rho_{\rm
sc}\left(\sqrt{2}\sqrt{3+\epsilon/t}\right),$ (35)

where $\rho_{\rm sc}(\epsilon)$ is the density of state (DOS) of simple cubic
lattice spectrum within nearest neighbour approximation with unit hopping
integral $t_{\rm sc}=1$ Vollhardt et al. (1996); Ulmke (1998); Igoshev and
Irkhin (2022). This expression has giant (square-root) van Hove singularity at
the band bottom, $\epsilon=-3t$. To simplify calculations we replace
$\rho_{\rm sc}(\epsilon)$ by infinite-dimensional DOS for hypercubic lattice
(hc)

$\rho^{\infty}_{\rm
hc}(\epsilon)=\frac{1}{\sqrt{12\pi}}\exp[-\epsilon^{2}/12]$ (36)

with the same $\left({\epsilon^{\rm sc}_{\mathbf{k}}}\right)^{2}$ averaged
over Brillouin zone as analogous $\left({\epsilon^{\rm
hc}_{\mathbf{k}}}\right)^{2}$ in the case of lattice dimension $d=\infty$.
From Eqs. (35) and (36) we get the expression

$\rho_{\rm fcc}^{\infty}(E)=\frac{\exp(-E/2)}{\sqrt{2\pi E}},$ (37)

where dimensionless energy variable $E=1+\epsilon/(3t)$ is introduced,
$\rho_{\rm fcc}^{\infty}(E)dE=\tilde{\rho}_{\rm
fcc}^{\infty}(\epsilon)d\epsilon$, see Fig. 2. DOS for the case $d=\infty$
only weakly deviates from three-dimensional case in the vicinity of the band
bottom, see Fig. 2. $\rho_{\rm fcc}(\epsilon)$ only weakly deviates from
$\rho^{\infty}_{\rm fcc}(E)$ in the vicinity of band bottom. Below we take
$3t$ as energy unit.

Figure 2: Density of electronic states for a next nearest neighbor infinite-
dimensional ($d=\infty$) and three-dimensional ($d=3$) fcc lattice with
$t^{\prime}=-t/2$, see Eq. (37) and Refs. Ulmke, 1998; Igoshev and Irkhin,
2022. The energy is calculated from the band bottom, $E=1+\epsilon/(3t)$.

Further the Hubbard model is treated within the mean-field (Hartree-Fock, HFA)
approximation under the assumption that we have ferromagnetic ordering induced
by giant van Hove singularity at the band bottom. The application of HFA here
is justified by the dominance of the crossed particle-hole electron-electron
interaction channel Metzner et al. (2012); Igoshev and Katanin (2023) with
renormalized (screened) interaction parameter $U$. Due to the presence of a
van Hove singularity in the vicinity of the Fermi level the Stoner criterion
may be fulfilled with small enough $U$ parameter.

A magnetization $m$ (in units of $\mu_{\rm B}$) and a chemical potential $\mu$
are determined by following HFA equations

$n=\frac{1}{N}\sum_{\mathbf{k}\sigma}f_{\mu_{0}}(\varepsilon_{\mathbf{k}\sigma}),$
(38)

and

$m=\frac{1}{N}\sum_{\mathbf{k}\sigma}\sigma
f_{\mu_{0}}(\varepsilon_{\mathbf{k}\sigma}),$ (39)

where $f_{\mu}(\varepsilon_{\mathbf{k}\sigma})$ is the Fermi function with
$\varepsilon_{\mathbf{k}\sigma}=\epsilon^{\rm
fcc}_{\mathbf{k}}-\sigma(Um/2+h)$ is HFA electronic spectrum. Within the HFA
effective chemical potential $\mu_{0}=\mu-Un/2$, subband spin splitting is
$Um/2+h$. At low $T$ $\mu_{0}\simeq E_{\rm f}$.

At first, we discuss conventional approach allowing to describe the properties
of paramagnetic and ferromagnetic phases for single-phase state. The
corresponding thermodynamical potential is free energy Moriya (2012)

$F_{\rm HFA}(T,n,h|m)=\Omega_{\rm HFA}(T,\mu,h|m)+\mu n,$ (40)

where $\mu$ is determined by $\partial\Omega/\partial\mu=0$ and HFA grand
potential of a single-phase state reads

$\Omega_{\rm HFA}(T,\mu,h|m)=\Omega_{0}(T,\mu_{0}(T,n_{\rm
HFA}(T,\mu,m),m),Um/2+h)\\\ -(U/4)(n^{2}_{\rm HFA}(T,\mu,m)-m^{2}),$ (41)

with grand potential of band (free) electrons

$\Omega_{0}(T,\mu,h)=2\mu-\frac{T}{N}\sum_{\mathbf{k}\sigma}\ln\left[2\cosh\frac{\epsilon^{\rm
fcc}_{\mathbf{k}}-\sigma h-\mu}{2T}\right],$ (42)

where $n=n_{\rm HFA}(T,\mu,m)$ is a solution of equation

$\mu_{0}(T,n,m)=\mu-Un/2.$ (43)

Here, auxiliary magnetic field (exchange splitting for free electronic
subbands) $h$ and charge field $\mu=\mu_{0}(T,n,m)$ are required to setup the
magnetization and band filling values $n$, $m$.

The expansion of free energy defined by Eq. (40) is

$F_{\rm
HFA}(T,n,h|m)=A^{F}_{0}(T,n)+A^{F}_{2}(T,n)m^{2}+A^{F}_{4}(T,n)m^{4}+A^{F}_{6}(T,n)m^{4}-hm,$
(44)

where $A^{F}_{0}(T,n)$ is a free energy at $m=0$ and $h=0$. Also,

$\displaystyle A^{F}_{2}(T,n)$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\Pi^{-1}-U),$ (45) $\displaystyle A^{F}_{4}(T,n)$
$\displaystyle=$
$\displaystyle\frac{1}{64\Pi^{5}}\left({{\Pi^{\prime}}^{2}}-\frac{1}{3}{\Pi\Pi^{\prime\prime}}\right),$
(46) $\displaystyle A^{F}_{6}(T,n)$ $\displaystyle=$
$\displaystyle\dfrac{1}{1536\Pi^{7}}\left(\frac{7{\Pi^{\prime}}^{2}}{\Pi^{2}}\left({\Pi^{\prime}}^{2}-\Pi\Pi^{\prime\prime}\right)+\dfrac{2}{3}{\Pi^{\prime\prime}}^{2}+\Pi^{\prime}\Pi^{{}^{\prime\prime\prime}}-\dfrac{1}{15}{\Pi\Pi^{\prime\prime\prime\prime}}\right),$
(47)

where $\Pi=\Pi(T,E_{\rm F})$ with Fermi level corresponding to given band
filling $n$, where

$\Pi(T,E_{\rm F})=-\frac{1}{N}\sum_{\mathbf{k}}f^{\prime}_{E_{\rm
F}}(\epsilon^{\rm fcc}_{\mathbf{k}})$ (48)

is up to constant factor magnetic susceptibility in the paramagnetic phase.
Prime above $\Pi$ denotes $E_{\rm F}$ derivative.

The calculation of the dependence $A^{F}_{4}$ on $T$ and $E_{\rm F}$ using Eq.
(46) allows to analyze the applicability of conventional approach. In the case
$A^{F}_{4}>0$ convensional Landau theory yields the phase transition line
$A^{F}_{2}=0$, separating FM region at $A^{F}_{2}<0$, or equivalently

$U\Pi(T,E_{\rm F})>1,$ (49)

from paramagnetic region at $A^{F}_{2}>0$. In the case $T\rightarrow 0$ we get
$\Pi(T,E_{\rm F})\rightarrow\rho(E_{\rm F})$ which transforms Eq. (49) into
conventional Stoner criterion. Numerical calculation shows that $A^{F}_{4}$ is
negative in the small region in the vicinity of van Hove singularity position
at small temperatures, see Fig. 3. We find that singular behaviour of
$\rho(\epsilon)$ can change the sign of $A^{F}_{4}$ and this makes the
conventional Landau theory not applicable. In this situation it is necessary
to consider accurately the possibility of FOPT FM-PM.

Figure 3:  (a) Phase diagram at $U=1$ in terms $T$, $E_{\rm F}$ calculated
within the generalized Landau theory, see Sec. II using coefficients
$A^{\Omega}_{2}$, $A^{\Omega}_{4}$, $A^{\Omega}_{6}$, see Eq. (50-53).
$A^{\Omega}_{2}=0$ corresponds to SOPT line when
$A^{\Omega}_{4},A^{\Omega}_{6}>0$, FOPT line is set by
$A^{\Omega}_{2}=A^{\Omega}_{2,\rm FOPT}$. Dashed black line indicates line
$S_{0}^{\prime}(T,E_{\rm F})=0$. Violet region indicates the region
$A^{F}_{4}(T,E_{\rm F})<0$ inside. Tricritical point (TCP) position is shown
by an arrow. Red region indicates the region $A^{\Omega}_{2}<0$, blue region
indicates the region $0<A^{\Omega}_{2}<A^{\Omega}_{2,\rm FOPT}$,
$A^{\Omega}_{4}<0$. (b) The same as (a) in the vicinity of TCP.

For itinerant electron system the FOPT results in phase separation which is
convenient to investigate using $\Omega_{\rm HFA}(T,\mu,h|m)$ instead of
$F_{\rm HFA}(T,n,h|m)$ using $\mu$ as main variable (see discussion in
Refs.Igoshev et al., 2009, 2010, 2015). Such transition is accompanied not
only by a jump of order parameter but also by jump of band filling.

Instead of expansion (44) one can use the expansion of $\Omega_{\rm HFA}$, see
Eq. (41), cf. Sec. II, with explicit expressions for coefficients

$\displaystyle A^{\Omega}_{2}(T,\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\Pi^{-1}-U),$ (50) $\displaystyle
A^{\Omega}_{4}(T,\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{{192f\Pi^{4}}}(3U{\Pi^{\prime}}^{2}-f\Pi^{\prime\prime}),$
(51) $\displaystyle A^{\Omega}_{6}(T,\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{720}\frac{1}{8f^{3}\Pi^{7}}\left(9(11f-1)U^{2}{\Pi^{\prime}}^{4}-3f(31f-11)U\Pi^{\prime\prime}{\Pi^{\prime}_{0}}^{2}\right.$
(53)
$\displaystyle\left.+2f^{2}(2f+3){\Pi^{\prime\prime}}^{2}+15f^{2}(f-1)\Pi^{\prime\prime\prime}\Pi^{\prime}-f^{3}\Pi^{\prime\prime\prime\prime}\Pi\right),$

where $f=1+U\Pi$, arguments $T,\mu$ of $\Pi$ are omitted for brevity. Note
that $A^{\Omega}_{2}(T,\mu)=A^{F}_{2}(T,n)$ with $n$ and $\mu$ related by Eq.
(38), but $A^{\Omega}_{4}\neq A^{F}_{4}$, $A^{\Omega}_{6}\neq A^{F}_{6}$. The
derivation of Eqs. (50-53) will be published later Igoshev (2024).

The calculation of $A^{\Omega}_{\alpha}(T,\mu)$, $\alpha=2,4,6$ using Eqs.
(50-53) allows to calculate preliminary phase diagram within the generalized
Landau theory, see Fig. 3. Accordingly to the discussion of Sec. II, in the
case $A^{\Omega}_{6}>0$ FM region corresponds to $A^{\Omega}_{2}<0$ at
$A^{\Omega}_{4}>0$ and $A^{\Omega}_{2}<A^{\Omega}_{2,\rm FOPT}$ for
$A^{\Omega}_{4}<0$. Otherwise PM phase is established. In the case
$A^{\Omega}_{4}<0$ the equation $A^{\Omega}_{2}=A^{\Omega}_{2,\rm FOPT}$
yields FOPT line with corresponding PS region (see below); in the case
$A^{\Omega}_{4}>0$ the equation $A^{\Omega}_{2}=0$ yields SOPT line. While
using $\Omega$ expansion allows to make qualitative conclusions about phase
diagram, it is very inconvenient for detailed calculations since necessary
number of terms in the series to be hold is a priori unknown. To this end a
better way is straightforward numerical minimization of $\Omega_{\rm HFA}$
given by Eq. (41).

For definiteness we suppose that two possible phases are characterized by
$n_{\rm FM}(\mu,T,h),m_{\rm FM}(T,\mu,h)$ and $n_{\rm PM}(T,\mu,h),m_{\rm
PM}(T,\mu,h)$ and each of these sets of parameters provides the local minimum
of $\Omega_{\rm HFA}$. In our consideration chemical potential $\mu$ governs
the position of global minimum of $\Omega_{\rm HFA}$. So one can assume that
there exists some critical value $\mu=\mu_{\rm PS}$ so that $\Omega_{\rm
HFA}(\mu,T,h|m_{\rm FM})<\Omega_{\rm HFA}(\mu,T,h|m_{\rm PM})$ for
$\mu<\mu_{\rm PS}(T,h)$ and $\Omega_{\rm HFA}(\mu,T|m_{\rm FM})>\Omega_{\rm
HFA}(\mu,T,h|m_{\rm FM})$ for $\mu>\mu_{\rm PS}(T,h)$ (arguments $\mu$, $T$
and $h$ of $m_{{\rm FM},{\rm PM}}$ are omitted for brevity). Thus, as $\mu$
increases, both $m$ and $n$ jump at $\mu=\mu_{\rm PS}$, and a state with $n$
satisfying $n_{\rm FM}<n<n_{\rm PM}$ corresponds to phase-separated state. The
case of the second-order magnetic phase transition is reproduced within this
scheme when $m_{\rm FM}=m_{\rm PM}$ (and therefore $n_{\rm FM}=n_{\rm PM}$):
in this case phase separation is absent and treatments using $F_{\rm HFA}$ and
$\Omega_{\rm HFA}$ potentials are equivalent. An application of $F_{\rm
HFA}$-based scheme in the case $n_{\rm FM}<n<n_{\rm PM}$ results in
thermodynamical unstable single-phase state. This is fully analogous to first-
order transition water-vapour with pressure being analogous to $(-\mu)$
parameter, while H2O density being analogous to $n$.

Using rather dense grid by $\mu$ and $T$ variables allows to localize the
position $\mu=\mu_{\rm PS}$ where $\Omega_{\rm HFA}(T,\mu,h|m_{\rm
FM})=\Omega_{\rm HFA}(T,\mu,h|m_{\rm PM})$ for numerically calculated
$\Omega_{\rm HFA}$. For given $T$, $\mu$ phase characteristics $n_{\rm
FM},m_{\rm FM}$ and $n_{\rm PM},m_{\rm PM}$ are obtained by numerical solution
of Eqs. (38-39). Fig. 4 presents the $T-n$ phase diagram at $U=1.0$ including
FM-PM phase separation region due to first-order transition at low
temperatures for $n\simeq 0.6$ which provides the closeness of Fermi level to
the band bottom (van Hove singularity). While in terms of $T-E_{\rm F}$, see
Fig. 3, FOPT line is very close to SOPT line, corresponding to FOPT line PS
region is very wide. Phase boundaries are shown in two variants: in a zero and
finite magnetic fields $h=10^{-4}$.

Figure 4: $T-n$ phase diagram of the Hubbard model for an infinite-dimensional
fcc lattice with $U=1.0$. PS region FM + PM at $h=0$ is filled by red color,
black solid line indicates the corresponding PS boundary at finite $h$. Purple
line is PM-FM phase boundary obtained from the Stoner criterion $U\Pi(T,E_{\rm
F})=1$. Its solid part corresponds to second-order phase FM-PM transition,
whereas dotted part has no sense since it falls into the region of phase
separation. The vertical dashed lines show the temperature boundaries of the
phase separation in zero $T^{\ast}_{0}$ (right) and finite $T^{\ast}(h)$
(left) magnetic field. Colors of horizontal dashed lines correspond to the
values of $n$ presented in Fig. 5.

Curves denoting phase boundaries, $n=n_{\rm FM,PM}(T,h)$, merge at
$T=T^{\ast}(h)$ which corresponds to the tricritical point
$(T^{\ast}(h),n^{\ast}(h))$, whose position depends on $h$. At $h=0$ the
position of tricritical point is $T^{\ast}_{0}\equiv T^{\ast}(h=0)=0.04645$,
$n^{\ast}_{0}\equiv n^{\ast}(h=0)=0.6066$. At this point, a change in the
order of phase transition occurs at $h=0$: for $T>T^{\ast}_{0}\equiv
T^{\ast}(h=0)$, the phase transition is a second-order phase transition and
the transition point is determined by the criterion $U\Pi(T,E_{\rm F})=1$; for
$T<T^{\ast}$ the curve determined by this criterion is inside the PS region
and the criterion is inapplicable. We denote the temperature boundary of the
PS region as $T_{\rm PS}(n,h)$. From Fig. 4 it can be seen that when $n\simeq
n^{\ast}$ the difference is $T_{\rm PS}(n,h)$ from $T_{\rm PS}(n,h=0)$ can be
significant.

Below we study $\Delta S(T)$ near temperature boundary $T_{\rm PS}(n,h)$.
Similar investigations of $\Delta S(T)$ for Bethe lattice (second order phase
transition) and square lattice (first order phase transition) are presented in
Refs. Igoshev et al., 2017; Ivchenko and Igoshev, 2021; Igoshev et al., 2023.
Here we calculate the entropy per site within the HFA as

$S(\mu,T,h)=\sum_{\sigma}S_{0}(T,E^{\rm HFA}_{\rm F,\sigma}(\mu,n,m)),$ (54)

where

$E^{\rm HFA}_{\rm F,\sigma}(\mu,n,m)=\mu-Un/2+\sigma(Um/2+h)$ (55)

being the effective Fermi level of $\sigma$ spin subband for HFA solution for
$n$ and $m$, see Eqs. (38–39). An entropy per site and one spin subband for
non-interacting electrons with spectrum $\epsilon_{\mathbf{k}}$ and Fermi
level $E_{\rm F}$ is

$S_{0}(T,E_{\rm
F})=\frac{1}{N}\sum_{\mathbf{k}\sigma}\kappa\left(\frac{\varepsilon_{\mathbf{k}}-E_{\rm
F}}{T}\right)$ (56)

with auxiliary function

$\kappa(x)=\ln\left[2\cosh(x/2)\right]-(x/2)\tanh(x/2).$ (57)

To calculate volume-averaged thermodynamical quantities such as $m$, $S$ for
the case of phase-separated state one needs to introduce the phase volume
fractions $0<x_{\rm FM},x_{\rm PM}<1$ for given $n$ by the equation

$n=x_{\rm FM}n_{\rm FM}(T,\mu_{\rm PS}(T,h),h)+x_{\rm PM}n_{\rm PM}(T,\mu_{\rm
PS}(T,h),h),$ (58)

together with $x_{\rm FM}+x_{\rm PM}=1$. Then any extensive quantity is just a
corresponding phase-weighted average. For example, total magnetization and
entropy then are

$m=x_{\rm FM}m_{\rm FM}(T,\mu_{\rm PS}(T,h),h)+x_{\rm PM}m_{\rm PM}(T,\mu_{\rm
PS}(T,h),h),$ (59) $S=x_{\rm FM}S_{\rm FM}(T,\mu_{\rm PS}(T,h),h)+x_{\rm
PM}S_{\rm PM}(T,\mu_{\rm PS}(T,h),h),$ (60)

where $m_{\rm FM,PM}$ ($S_{\rm FM,PM}$) is the phase magnetization (entropy)
for both phases involved in PS.

Fig. 5 shows the temperature dependence of total $m$ and $\Delta S$ (see Eqs.
(59) and (60)) for the case $U=1.0$ at various band fillings from the interval
of PS existence $[0.50,0.63]$ (these values are shown by horizontal lines on
$T-n$ phase diagram, see Fig. 4). To illustrate the effect of PS on $m$ and
$\Delta S$ in Fig. 5 the temperature dependence of these quantities is also
calculated within the assumption that the system is single-phase.

Figure 5:  Temperature dependence of (a) $m$, (b) $\Delta S$ for fillings in
the vicinity of $n=n^{*}_{0}=0.06066$ in the $[0.54,0.62]$ filling interval.
$U=1.0,h=10^{-4}$. Particular filling is shown by color, see the legend of
Fig. 5. Dashed lines show the result of calculation by conventional way not
taking into account PS possibility.

The temperature behavior of $m$ and $\Delta S$ dramatically depends on $n$
value. There is substantial effect of PS on $m$ temperature dependence: at the
$T=T_{\rm PS}(n,0)$ $m$ dependence exhibit a valuable kink. The height of
$m(T)$ kink decreases when $n$ increases towards $n^{\ast}$. For the
calculation not accounting PS formation, no kinks in $m(T)$ dependence is
found, instead there is pyromagnetic behavior, which is an artifact of
inconsistent treatment. When $T$ approaches finite-field PS boundary $T_{\rm
PS}(n,h)$ with $n<n^{\ast}_{0}$ $\Delta S$ has a sharp peak and is positive
(inverse MCE). With further increase in temperature in a narrow interval
$[T_{\rm PS}(n,h),T_{\rm PS}(n,0)]$ $\Delta S$ is almost linearly dependent on
temperature, rapidly changing from positive peak to moderate negative value.
With a further increase in $T$, a typical negative peak for the second-kind
transition with $\Delta S<0$ at $T=T_{\rm C}$ takes place.

For $n>n^{\ast}_{0}$ there is a weak feature (kink) of the dependence $\Delta
S$ associated with leaving the region of the phase separation, which height
increases as $n$ approaches $n^{\ast}_{0}$, but there is no second-order
transition.

For $n\approx n^{\ast}_{0}$, the behavior of $\Delta S$ is the most
interesting. The peculiarity is formed by an inverse MCE peak at the point
$T=T^{\rm PS}(n^{\ast}(h),h)$ and by a another direct MCE peak-like feature
not corresponding to SOPT in the vicinity to $T=T^{\ast}_{0}$, between which
there is a linear growth interval $[T^{\ast}(h),T^{\ast}_{0}]$ (for $n\neq
n^{\ast}$ this temperature range has a smaller width).

The change of PS boundaries $n_{\rm FM}$, $n_{\rm PM}$ under magnetic field
switching is maximal at $T=T^{\ast}_{0}$.

Figure 6:  (a) Temperature dependence of phase magnetization for ferromagnetic
$m_{\rm FM}$ and paramagnetic $m_{\rm PM}$ phases involved in PS in zero
magnetic field (solid lines) and finite magnetic field (dashed lines). (b)
Temperature dependence of phase entropy change $\Delta S_{\rm FM}$ and $\Delta
S_{\rm PM}$. The temperature is taken in units of $T^{\ast}_{0}$. The vertical
dashed line shows the position $T^{*}(h)$. The sample average value of $\Delta
S$ and $m$ for $n=n^{\ast}_{0}$ is shown for comparison. The parameters are
the same as in Fig. 4.

A remarkable consequence of this is a perspective of a control the sign and
magnitude of $\Delta$ by changing the temperature in a narrow interval
$[T^{\ast}(h),T^{\ast}]$, when $n\approx n^{\ast}_{0}$. When $n$ significantly
differs from $n^{\ast}_{0}$ as the temperature increases, the system leaves PS
area until a significant magnetic response is formed in the system.

Interestingly, the temperature dependencies $\Delta S(T)$ and $m(T)$ for small
$T$ are similar to that found for DyAl2 Lima et al. (2005). Authors suppose
that this anomaly is a consequence of an anisotropy and explains it within the
Heisenberg model. The region of such dependence is limited by the temperature
of the spin-flop transition. Here one can see an analogy with the results
obtained, since the presence of two projections of magnetization along and
perpendicular to the magnetic field in anisotropic systems is in a manner
similar to the presence of two phases in phase-separated state of metallic
systems.

Figure 7:  Temperature dependence of phase FM susceptibility $\chi^{\rm
PS}_{\rm FM}$ and $\eta_{\rm PS}$ at FOPT line. The temperature is taken in
units of $T^{\ast}_{0}$. The parameters are the same as in Fig. 4.

Fig. 6 shows the temperature dependence of $m$ and $\Delta S_{\Phi}$
calculated separately for the phases $\Phi$ involved in PS in the case
$U=1.0$. Note that within the PS region, these quantities by definition do not
depend on $n$, whereas $n$ dependence of total $\Delta S$ originates from
volume fraction $n$ dependence only, see Eqs. (59-60). For the case
$n=n^{\ast}_{0}$, $T$ dependencies of total $m$ and $\Delta S$ (averaged over
the sample volume) are also shown. For the ferromagnetic phase, the magnetic
response (susceptibility) to the magnetic field is negative at
$T>0.02=0.43T^{\ast}_{0}$ (see also Fig. 7 and derivation of this fact within
the Landau theory in Sec. II) and we have inverse MCE effect, $\Delta S_{\rm
FM}>0$, for almost all $T<T^{*}(h)$. For paramagnetic phase we have direct MCE
effect $\Delta S_{\rm PM}<0$ up to $T<0.0393=0.85T^{\ast}_{0}$. For the case
$T>0.85T^{\ast}_{0}$ we have rapid change of both phase $\Delta S_{\rm FM}$
and $\Delta S_{\rm PM}$ which is related with convergence $m_{\rm FM}$ and
$m_{\rm PM}$ for phases involved in PS, see Eq. (6).

Effective Fermi level of both spin subbands $E^{\rm HFA}_{\rm F,\sigma}$ at
FOPT line, see Eq. (55), strongly depends on $T$. Fig. 8 shows $E^{\rm
HFA}_{\rm F,\sigma}$ in zero and finite for both phases involved in PS. One
can see that effective Fermi level for $\downarrow$-subband $E^{\rm HFA}_{\rm
F,\downarrow}$ lies in the vicinity of van Hove singularity position $E=0$.

Figure 8:  Subband effective Fermi level $E^{\rm HFA}_{\rm F,\sigma}$ of FM
and PM phase participating in PS in zero $h=0$ (solid lines) and finite
$h=10^{-4}$ (dashed lines) magnetic field. Red (violet) lines correspond to FM
(PM) phase $E^{\rm HFA}_{\rm F,\uparrow}$, blue (green) lines correspond to FM
(PM) phase $E^{\rm HFA}_{\rm F,\downarrow}$. The parameters are the same as in
Fig. 4. Black dotted line indicates the line $S_{0}^{\prime}(T,E_{\rm F})=0$,
see Fig. 3. Vertical dot-dashed line indicates the temperature bound of PS
region in finite $T^{*}(h)$ magnetic field. The temperature is taken in units
of $T^{\ast}_{0}$. The parameters are the same as in Fig. 4.

A cause of inverse MCE for FM phase at FOPT line requires an explanation. In
general, the reason for this fact is that when the magnetic field is applied,
the chemical potential changes, adjusting $n$ to into the phase separation
region, cf. discussion in Sec. II. To reveal the origin of MCE sign dependence
at FOPT line we calculate the magnetic field derivative of phase entropy for
the phase $\Phi$

$\left(\frac{dS_{\Phi}}{dh}\right)_{\rm
PS}=\frac{1}{\Pi^{\Phi}_{\uparrow}+\Pi^{\Phi}_{\downarrow}+2U\Pi^{\Phi}_{\uparrow}\Pi^{\Phi}_{\downarrow}}\sum_{\sigma}\left(\sigma(1+U\Pi^{\Phi}_{\bar{\sigma}})\chi^{\rm
PS}_{\Phi}+2\eta_{\rm
PS}\Pi^{\Phi}_{\bar{\sigma}}\right)S^{\prime}_{0}(T,E^{\rm HFA}_{\rm
F,\sigma}(T,n_{\Phi},m_{\Phi})),$ (61)

where $\eta_{\rm PS}$ is given by Eq. (21) and

$\chi^{\rm
PS}_{\Phi}=\frac{\Pi^{\Phi}_{\uparrow}+\Pi^{\Phi}_{\downarrow}+2U\Pi^{\Phi}_{\uparrow}\Pi^{\Phi}_{\downarrow}}{1-U^{2}\Pi^{\Phi}_{\uparrow}\Pi^{\Phi}_{\downarrow}}+\frac{\Pi^{\Phi}_{\uparrow}-\Pi^{\Phi}_{\downarrow}}{1-U^{2}\Pi^{\Phi}_{\uparrow}\Pi^{\Phi}_{\downarrow}}\eta_{\rm
PS},$ (62)

where $\Phi=$ FM, PM enumerates phases involved in PS and
$\Pi^{\Phi}_{\sigma}=\Pi(T,E^{\rm HFA}_{\rm
F,\sigma}(\mu,n_{\Phi},m_{\Phi}))$. $T$ dependencies of
$\left({dS_{\Phi}}/{dh}\right)_{\rm PS}$ and $S^{\prime}_{0}(T,E^{\rm
HFA}_{\rm F,\sigma})$ for both spin projection in FM phase involved in PS are
shown in Fig. 9. Subband contributions are proportional to
$S_{0}^{\prime}(E^{\rm HFA}_{\rm F,\sigma})$ which is shown in Fig. 9(b). One
can see that $S_{0}^{\prime}$ dominates for $\sigma=\downarrow$ and positive.
The strong dependence of $\left({dS_{\rm FM}}/{dh}\right)_{\rm PS}$ on $T$ and
the change in its sign are due to several facts: (i)
$|S^{\prime}_{0}\left(T,E^{\rm HFA}_{\rm F,\uparrow}\right)|\ll
S^{\prime}_{0}\left(T,E^{\rm HFA}_{\rm F,\downarrow}\right)$, (ii) the sign
change of $S^{\prime}_{0}(T,E^{\rm HFA}_{\rm F,\downarrow})$ at $T\approx
0.85T^{\ast}_{0}$, (iii) the sign change of $\chi^{\rm FM}_{\rm PS}$ at
$T\approx 0.02=0.43T^{\ast}_{0}$, see Fig. 7, cf. the discussion of negative
susceptibility of FM phase in the vicinity of tricritical point.

At low temperature $\sigma=\downarrow$ contribution into $(dS_{\rm
FM}/dh)_{\rm PS}$ dominates but $\chi$\- and $\eta$-terms have opposite signs
and almost cancel each other. $\downarrow$ $\chi$-term is negative since in
this $T$ region both $S^{\prime}_{0}(E_{\rm F,\downarrow}),\chi^{\rm PS}_{\rm
FM}>0$. At $T=0.02=0.43T^{\ast}_{0}$ $\chi^{\rm PS}_{\rm FM}$ and, in turn,
$\chi$-term change their sign and both $\chi$\- and $\eta$-terms are positive
which establishes inverse MCE for FM phase. $\eta_{\rm PS}$ holds its sign and
the sign of $\eta$-term is fully determined by the sign of
$S_{0}^{\prime}(E_{\rm F,\downarrow})$. Further $T$ increase results in
dominating of $\chi$-term over $\eta$-term. In the vicinity of $T^{*}_{0}$
$\downarrow$-term $\chi$ and $\eta$-term as well as $S_{0}^{\prime}(E^{\rm
HFA}_{\rm F,\downarrow})$ change their sign which results in sing change of
$(dS_{\rm FM}/dh)_{\rm PS}$. The sign change of $S^{\prime}_{0}(E_{\rm F})$ is
associated with giant van Hove singularity of the density of states at band
bottom.

Now we see the reasons for the unusual behavior of $\Delta S$ in Fig. 4: the
entropy of the paramagnetic phase decreases when the magnetic field is applied
(excepting for $T$ in a close vicinity of $T^{\ast}_{0}$), while the entropy
of the ferromagnetic phase increases, it is directly related to a decrease in
magnetization when applying the magnetic field (see Fig. 6) and impact of
giant van Hove singularity of the density of states.

Thus, in the presence of a first-order phase transition sample-averaged
inverse MCE effect can take place: when the magnetic field is applied, the FM-
phase part of the sample is disordered and its entropy increases. There are
two reasons for this: the presence of phase separation, which generally
results in the sum of two phase contributions to $\Delta S$, one of which is
negative and the other is positive (the latter corresponds to inverse MCE);
the second is the influence of giant van Hove singularity $\rho(\varepsilon)$,
leading to the inverse sign $\Delta S$ already for the paramagnetic phase.

Figure 9:  Temperature dependence of properties for FM phase involved in PS.
(a) The derivative $dS_{\rm FM}/dh$ see Eq. (61) (black line) and separate
contributions into this: violet (green) lines indicate first $\chi$-term
(second $\eta$-term) in Eq. (61). Solid lines indicate sum of both spin
projections contributions, dashed (dotted) line indicates spin up (down)
projection. (b) $S_{0}^{\prime}(E^{\rm HFA}_{\rm F,\sigma})$ for both spin
projections. The temperature is taken in units of $T^{\ast}_{0}$. The
parameters are the same as in Fig. 4.

##  IV Conclusions

This presentation models the first-order paramagnetic–ferromagnetic transition
in the vicinity of the tricritical point for an itinerant system on fcc
lattice for a small coupling constant which can simulate the formation of
ferromagnetic order in itinerant compounds with giant van Hove singularity of
the electron spectrum. While the Hartree-Fock approximation used is valid only
qualitatively and it can be in principle replaced some better approximation,
the above qualitative conclusions are valid for any itinerant system
undergoing a first-order phase transition, which always leads to an unusual
behavior of $\Delta S$ for the case when the electron filling lies in the
corresponding filling interval. Independently on the approximation used FOPT
results in phase separation which characteristics depend on approximation. The
latter is thereby only quantitative effect. Note that the Coulomb long-range
energy not accounted for in the employed approximation cannot qualitatively
change the results: as shown by the experiments of scanning tunnel microscopy
Battisti et al. (2017), in such cases the system undergoes nanoscale phase
separation (a specific implementation of the heterogeneity configuration can
be fixed by the location of impurities and defects), which provides an energy
minimum taking into account the energy of the Coulomb long-range action.

The role of van Hove singularity is manyfold: (i) vHS is in general necessary
origin of ferromagnetic ordering in itinerant electron systems, (ii) vHS
results in first-order phase transition between paramagnetic and ferromagnetic
phases with the corresponding phase separation, (iii) non-standard electronic
properties caused by vHS result in turn inverse entropy response on the
application of magnetic field.

It is shown that the magnetocalorical effect in metallic systems undergoing a
first-order phase transition to the ferromagnetic state can have significant
features due to the sample splitting into two phases having different values
of band filling. To observe this effect, it is sufficient only to provide the
special behavior of the density of states, no additional interactions should
be considered. In particular, it was found that the contribution to $\Delta S$
from the ferromagnetic phase in the low-temperature region has an inverse
sign. With increasing temperature, this contribution has pronounced peak at
the temperature of the leaving PS region in a finite field and has a strong
almost linear dependence inside the PS region in zero magnetic field and away
PS PS region in finite magnetic field. This feature can find technical
application in devices based on the control of the $\Delta S$ sign by changing
the temperature or other parameters.

Perfect relation between nearest and next-nearest neighbour hopping producing
giant van Hove singularity appears to be not absolutely necessary. Earlier
investigations on the problem yields that some devitions from this relation
results in the replacement of divergence of DOS by a wide plateu with high
value of $\rho(\epsilon)$ Igoshev and Irkhin (2019a, b); Igoshev and Katanin
(2023); Igoshev and Irkhin (2022). This does not completely breaks anomalous
thermodynamical properties which allows to expect the PS existence and inverse
MCE in this case.

The description of MCE within a novel generalized Landau theory can acquire
perspective extensions taking into account additional interactions like
magnetoelastic coupling Podgornykh and Shcherbakova (2006) and spin-
fluctuations Igoshev et al. (2007) etc.

##  V Acknowledgments

We are grateful to V. Yu. Irkhin, A. N. Ignatenko, N. S. Pavlov for fruitful
discussions. The theoretical studies in Section 2 is supported by the state
assignment of the Ministry of Science and Higher Education of the Russian
Federation (theme “Quant” No. 122021000038-7) for development of generalized
Landau functional approach. The theoretical studies in Section 3 are in part
supported by the Russian Science Foundation (Project No. 24-43-00156) for
entropy calculations within Hartree-Fock approximation for the Hubbard model.

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