# Determining the temperature in heavy-ion collisions with multiplicity
distribution

Yi-Dan Song  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),
Institute of Modern Physics, Fudan University, Shanghai $200433$, China  Rui
Wang<EMAIL_ADDRESS>Shanghai Institute of Applied Physics, Chinese
Academy of Sciences, Shanghai $201800$, China  Yu-Gang Ma
<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
$200433$, China  Xian-Gai Deng  Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
$200433$, China  Huan-Ling Liu  Shanghai Institute of Applied Physics, Chinese
Academy of Sciences, Shanghai $201800$, China

###### Abstract

By relating the charge multiplicity distribution and the temperature of a de-
exciting nucleus through a deep neural network, we propose that the charge
multiplicity distribution can be used as a thermometer of heavy-ion
collisions. Based on an isospin-dependent quantum molecular dynamics model, we
study the caloric curve of reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$
with the apparent temperature determined through the charge multiplicity
distribution. The caloric curve shows a characteristic signature of nuclear
liquid-gas phase transition around the apparent temperature $T_{\rm ap}$ $=$
$6.4~{}\rm MeV$, which is consistent with that through a traditional heavy-ion
collision thermometer, and indicates the viability of determining the
temperature in heavy-ion collisions with multiplicity distribution.

heavy-ion collision, temperature, machine learning, multiplicity

###### pacs:

24.10.Ai, 25.70.Gh, 25.70.Mn, 27.60.+j

##  I Introduction

Understanding the properties of nuclear matter is one of the major objectives
in nuclear physics. At zero temperature, the properties of nuclear matter have
been studied extensively, and its equation of state (EOS), including its
isospin dependence, i.e., symmetry energy, has been determined relatively well
Lattimer and Prakash (2007); Baldo and Burgio (2016); Roca-Maza and Paar
(2018), while its properties at finite temperature are relatively little
touched upon. Among these properties, two noticeable examples are the nuclear
liquid-gas phase transition Finn et al. (1982); Siemens (1983); Panagiotou et
al. (1984); Pochodzalla et al. (1995); Ma et al. (1997); Ma (1999); Chomaz et
al. (2000); Richert and Wagner (2001); Natowitz et al. (2002a, b); Ma et al.
(2005); Ma and Ma (2018); Borderie and Frankland (2019) and the temperature
dependence of the ratio of shear viscosity to entropy density ($\eta/s$)
Auerbach and Shlomo (2009); Dang (2011); Fang et al. (2014); Deng et al.
(2016); Mondal et al. (2017); Guo et al. (2017). The latter is also connected
to the nuclear giant dipole resonance at finite temperature Bracco et al.
(1989); Bortignon et al. (1991); Wieland et al. (2006), since both of them are
related to the two-body dissipation of nucleons.

The difficulties of studying the finite temperature properties of nuclear
matter mainly come from the preparation of a finite temperature nuclear
system, as well as the determination of its temperature. Heavy-ion collisions
(HICs) at intermediate-to-low energies provide a possible venue of
investigating the finite temperature properties of nuclear matter Wada et al.
(1989). During the reaction, a transient excited system is formed, and
commonly it can be regarded as a (near)-equilibrium state d’Enterria et al.
(2001); Borderie et al. (1996), since the evolution of its constituent
nucleons is sufficiently short comparing with the global evolution. Its
temperature can be accessed by, e.g., energy spectra through moving source
fitting Wada et al. (1989), excited state populations Chitwood et al. (1986);
Schwarz et al. (1993), (double)-isotope ratios Tsang et al. (1997); Serfling
et al. (1998); Albergo et al. (1985), or quadruple momentum fluctuation
Wuenschel et al. (2010) etc. For a reliable thermometer of HICs, we require it
is insensitive to both the collective effects and the secondary decay of
unstable nuclei after the system disintegrates, which is commonly hard to
achieve. Besides that, because of the difficulty of examining the accuracy of
the apparent temperature obtained through these thermometers, it is not a
trivial task to propose different ways of determining the apparent
temperature, and thus provide more opportunities of crosscheck.

Machine-learning techniques LeCun et al. (2015); Jordan and Mitchell (2015),
which have been applied extensively in physics Carleo et al. (2019) due to
their ability of recognizing and characterizing complex sets of data, provide
an alternative and peculiar way of determining the apparent temperature of
HICs at intermediate-to-low energies. Besides the common uses like particle
identification and tagging in experiments, machine-learning techniques have
various novel applications in physics, e.g., solving quantum many-body problem
Carleo and Troyer (2017); Zhang et al. (2020), analyzing strong gravitational
lenses Hezaveh et al. (2017), classifying phases of matter Carrasquilla and
Melko (2017); van Nieuwenburg et al. (2017); Rodriguez-Nieva and Scheurer
(2019); Rao (2020), extrapolating the cross section of nuclear reactions Ma et
al. (2020), and instructing single crystal growth Yao et al. (2019) and
optimizing experimental control Wu et al. (2020). In a recent work of studying
nuclear liquid-gas phase transition with machine-learning techniques Wang et
al. (2020), it has been shown that machine-learning techniques can capture the
essential features of HICs directly from experimental final-state charge
multiplicity distribution.

In the present letter, we propose that, with the help of machine-learning
technique, the charge multiplicity distribution can be applied to determine
the apparent temperature of HICs at intermediate-to-low energies. Our basic
methodology is as follows. We first obtain the multiplicity of fragments of an
excited nuclear source with given temperature based on a theoretical model,
e.g., transport models Bertsch and Das Gupta (1988); Aichelin (1991),
statistical models Bondorf et al. (1995); Charity et al. (1988) or their
hybrid Gaitanos et al. (2009); Zhang et al. (2018); Ono (2019). We then relate
the final-state charge multiplicity distribution with the temperature of the
source via a deep neural network (DNN). This relation can be employed to
determine the apparent temperature of a certain transient state during HICs at
intermediate-to-low energies through their final-state charge multiplicity
distribution. We use the above method to determine the apparent temperature of
a fragmentation reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$. We further
test the viability of the present method by comparing it with the momentum
fluctuation thermometer Wuenschel et al. (2010), and by studying the
characteristic signature of the caloric curve with the apparent temperature
determined through the above method. To adopt the above method to realistic
HICs analyses with experimental charge multiplicity distribution relies on the
precise determination of the fragment multiplicities from theoretical model.
Nevertheless, as a viability quest, in the present work, we employ an isospin-
dependent quantum molecular dynamics (IQMD) model Ma et al. (2006) to simulate
the de-excitation of the finite temperature nuclear source, and do not require
it to describe precisely the experimental fragment multiplicities. More
accurate description of the experimental fragment multiplicities can be
achieved through, e.g., combining certain transport model with statistical
model of multi-fragmentation Ono (2019), and it is beyond the scope of the
present work.

##  II Methodology

In the present work, we focus on fragmentation reactions Sümmerer and Blank
(2000); Song et al. (2018), i.e., central $\isotope[103]{Pd}$ $+$
$\isotope[9]{Be}$ collision with incident energies ranging from 20 to 400$A$
MeV, and try to determine their apparent temperature through its final-state
charge multiplicity distribution $M_{\rm c}(Z_{\rm cf})$, where $Z_{\rm cf}$
represents the charge number of the final charged fragments, ranging from $1$
to the charge number of the reaction system. In the process of intermediate-
to-low energy HICs, especially for fragmentation reactions, the two incident
nuclei collide and then form a compound-like system. This compound-like system
is regarded as a (near-)equilibrium system, and we can mimic approximately
this transient state of the reaction by a nuclear source or a finite nucleus
at given temperature, with mass number $A$ and proton number $Z$, which are
the same as those in the reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$
Charity (2010). This finite temperature nuclear source has been employed to
study finite-size scaling phenomenon Liu et al. (2019). We first simulate the
evolution of the nuclear source 112Sn ($A=112$ and $Z=50$) at different
initial temperature $T^{model}$, and obtain their $M_{\rm c}(Z_{\rm cf})$
based on the IQMD model. These simulations provide us $M_{\rm c}(Z_{\rm cf})$
from a nuclear source with a given temperature, which is difficult to obtain
directly from HICs. We then establish a relation between the source
temperature $T^{model}$ and $M_{\rm c}(Z_{\rm cf})$ through a DNN, which
recasts the complex relation into a non-linear map through its neurons. Based
on this relation, the apparent temperature of the reaction $\isotope[103]{Pd}$
$+$ $\isotope[9]{Be}$ can be determined through its final-state $M_{\rm
c}(Z_{\rm cf})$. In Fig. 1, we show the basic procedure of the proposed method
to determine apparent temperature of fragmentation reaction
$\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ through the nuclear source 112Sn.

Figure 1: The basic procedure of determining the $T_{ap}$ of
$\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ through its final-state $M_{\rm
c}(Z_{\rm cf})$ with DNN.

###  II.1 IQMD model

The IQMD model, a many-body theory to describe the dynamics of HICs, can be
derived from a time-dependent Hartree theory with Gaussian single-particle
wave function $\phi_{i}(\vec{r},t)$,

$\phi_{i}(\vec{r},t)=\frac{1}{(2{\pi}L)^{3/4}}{\rm
exp}\Big{[}-\frac{[\vec{r}-\vec{r}_{i}(t)]^{2}}{4L}+\frac{i\vec{p}_{i}(t)\cdot\vec{r}}{\hbar}\Big{]},$
(1)

with its spatial center $\vec{r}_{i}(t)$ and momentum center $\vec{p}_{i}(t)$
as variational parameters. Other quantities can be obtained subsequently
through $\vec{r}_{i}(t)$ and $\vec{p}_{i}(t)$. In the above equation, $L$ is
the square of the width of Gaussian wave packet and is set to be 2.18 $\rm
fm^{2}$. Through a product of single-particle wave functions
$\phi_{i}(\vec{r},t)$, we can get the system wave function,

$\psi(\vec{r}_{1},...\vec{r}_{n},t)=\prod_{i=1}^{A}\phi_{i}(\vec{r},t),$ (2)

where $A$ is the mass number of the system. The potential energy $U$ in the
IQMD model includes Skyrme, Yukawa, symmetry, momentum-dependent and Coulomb
terms,

$U=U_{\text{Sky}}+U_{\text{Yuk}}+U_{\text{sym}}+U_{\text{MDI}}+U_{\text{Coul}}.$
(3)

Detailed descriptions of the IQMD model, including the potential and equations
of motion of $\vec{r}_{i}(t)$ and $\vec{p}_{i}(t)$, can be found in Ref.
Hartnack et al. (1998); Ma et al. (2006). Numerous applications of IQMD have
been made for different observable, and some recent ones can be found in Ref.
Li et al. (2018); Feng (2018); Yan et al. (2019); Yan and Li (2019); Yu et al.
(2020).

In the IQMD model, the initial nucleons are generated through the local
density approximation. For a ground-state nucleus, we generate $A$ Gaussian
wave-packets, with their spatial center $\vec{r}_{i}$ $(t=0)$ sampled randomly
within a sphere of radius given by $r_{0}A^{1/3}$ with $r_{0}$ $=$ $1.12~{}\rm
fm$, and their momentum center $\vec{p}_{i}$ $(t=0)$ sampled following the
zero-temperature Fermi-Dirac distribution. A finite nucleus at a given
temperature $T$ can be generated by sampling $\vec{p}_{i}$ $(t=0)$ according
to finite temperature Fermi-Dirac distribution Fang et al. (2014), which is
given by,

$\begin{split}f(\epsilon)=\frac{1}{\exp(\frac{\epsilon-\mu}{T})+1},\end{split}$
(4)

where $\epsilon$ $=$ $\sqrt{p^{2}+m^{2}}$ $+$ $U$ is the single particle
energy, and $\mu$ is the chemical potential. The parameters $p$, $m$ and $U$
are momentum, mass and potential energy, respectively. For simplicity, we have
omitted the contribution from momentum-dependent part in the above equation.

###  II.2 Deep neural network

Figure 2: (a). A single artificial neuron, with $n$ inputs labelled as $x_{1}$
through $x_{n}$ and an output $y$. The output of the neuron is computed by
applying the activation function $f(z)$ to the product of x, weights $W$, and
biases $b$, e.g., z = $\textbf{W}\cdot\textbf{x}$ $+$ b. (b). The feed-forward
neural network used in the present work, consisting of an input layer, an
output layer and four hidden layers. The input of the network is the charge-
weighted charge multiplicity distribution $ZM_{c}(Z_{cf})$.

In the present work, we adopt a feed-forward DNN to establish the relation
between the temperature $T^{\rm Model}$ of the de-exciting nuclear source
generated in IQMD and its final-state charge multiplicity distribution $M_{\rm
c}(Z_{\rm cf})$. We treat $M_{\rm c}(Z_{\rm cf})$ as the input image, while
its corresponding temperature as its label. The DNN contains successively one
input layer, several hidden layers, and one output layer. Each layer generates
its output z through a matrix multiplication of its input x, i.e., z =
$\textbf{W}\cdot\textbf{x}$ $+$ b. The elements in the matrix W are known as
weights and in the vector b as biases. In a normal full-connected neural
network these parameters are single values. The neuron is then followed by an
activation function $f(\textbf{z})$, which turns a linear transform to a non-
linear one. Commonly used activation functions are _sigmoid_ , _tanh_ , and
_ReLU_ (rectified linear unit). $f(\textbf{z})$ is then used as the input of
the next layer. In the present work, the neural network can be treated as a
functional

$T_{\rm ap}^{\rm DNN}=g\big{[}M_{\rm c}(Z_{\rm
cf});\textbf{W},\textbf{b}\big{]},$ (5)

which relates non-linearly a given input $M_{\rm c}(Z_{\rm cf})$, a vector
with $50$ elements in our case, to an output predicted apparent temperature
$T_{\rm ap}^{\rm DNN}$. We employ four hidden layers, with each consists of 32
artificial neurons. The input layer and the first three hidden layers are
followed by _ReLU_ , while the last hidden layer and the output predicted
$T_{\rm ap}^{\rm DNN}$ are connected directly by the matrix multiplication.
The sketch of the DNN used in the present work can be seen in Fig. 2.

We train the network based on a data set $\big{\\{}M_{\rm c}(Z_{\rm
cf}),T^{\rm Model}\big{\\}}$, to minimize the cost function, the difference
between the given temperature and the DNN prediction, i.e., $(T^{\rm
model}-T_{\rm ap}^{\rm DNN})^{2}$, by adjusting its parameters W and b. The
optimization is fulfilled by the $Adam$ Kingma and Ba (2017) package in
_Tensorflow_. During training the network, we use an exponential decreasing
learning rate $\alpha$ $=$ $10^{-6}$ $+$ $(10^{-3}-10^{-6})\exp(-i/10000)$,
with $i$ the training epoch and it is equal to 10,000. To prevent the network
from over fitting the data set, we include a standard $l_{2}$ regularization
term, i.e., a term proportional to the norm of the weight W and the bias b,
$l_{2}(\|\textbf{W}\|^{2}/2+\|\textbf{b}\|^{2}/2)$, with $l_{2}$ a positive
number, in the cost function of the neural network. The $l_{2}$ regularization
prevents the weights and biases from increasing to arbitrary large values
during the optimization.

##  III Results and discussion

###  III.1 Apparent temperature

Figure 3: The charge distribution of fragments from excited nuclear source
$\isotope[112]{Sn}$ at several $T^{\rm model}$. For each $T^{\rm model}$, we
show the simulated data of one run consisting of 2,000 events from the IQMD
model.

Based on the IQMD model, we simulate the de-excitation process of a
$\isotope[112]{Sn}$ nucleus at different $T^{\rm model}$. The [112]Sn nucleus
has the same mass and charge number with the $\isotope[103]{Pd}$ $+$
$\isotope[9]{Be}$ reaction, and it can be regarded as a nuclear source to
mimic the transient excited state of the reaction. We perform $50$ runs for
each $T^{\rm model}$, with each run consists of $2,000$ events. $T^{\rm
model}$ ranges from $0$ to $20~{}\rm MeV$ with $1~{}\rm MeV$ interval. Fig. 3
displays charge distributions of fragments from hot nuclear source [112]Sn at
several $T^{\rm model}$ with the data sample of one run ($2,000$ events) at
each temperature. These fragment charge distributions exhibit typical changes
of disassembly mechanism of hot nuclei with temperature Ma et al. (2005); Ma
(1999), i.e., from evaporation mechanism at lower temperature (e.g. $T^{\rm
model}$= 2 MeV), to multifragmentation at medium temperature (e.g. $T^{\rm
model}$= 8 MeV), till vaporization at higher temperature (e.g. $T^{\rm
model}$= 17 MeV). According to the shapes of these charge distributions versus
temperature, a nuclear liquid-gas phase transition shall begin to happen at a
certain moderate temperature for this system Ma et al. (2005); Liu et al.
(2019); Wang et al. (2020); Wada et al. (2019).

We can train the DNN once we get the above-mentioned charge multiplicity
distributions $M_{\rm c}(Z_{\rm cf})$. In the present work, the total
$50\times 21$ charge multiplicity distributions $M_{\rm c}(Z_{\rm cf})$, and
their corresponding $T^{\rm model}$, are treated as the images and labels,
respectively, and they form the data set $\big{\\{}M_{\rm c}(Z_{\rm
cf}),T^{\rm model}\big{\\}}$ of the DNN. The data set is further divided into
training set and testing set, each contains half of the total data set. After
we trained the DNN with the training set, i.e., determining the parameters W
and b in Eq. (5) to best reproduce the given $T^{\rm model}$, one can predict
the apparent temperature with a given $M_{\rm c}(Z_{\rm cf})$ through Eq. (5).
We show in Fig.4 the histogram of the DNN’s predicting error $\sigma_{\rm T}$,
i.e., the difference between the original $T^{\rm model}$ and its DNN
prediction $T_{\rm ap}^{\rm DNN}$, of the testing set. The standard error of
the DNN prediction is about $0.62~{}\rm MeV$, which is small enough for
further analyses based on the apparent temperature obtained through the
present way.

Figure 4: The histogram of errors $\sigma_{\rm T}$ between the predicted
apparent temperature from DNN $T_{\rm ap}^{\rm DNN}$ and the original $T^{\rm
model}$ given in the IQMD simulations for the excited nuclear source (finite
temperature [112]Sn). The red line is the Gaussian fitting of the histogram.
The inset shows the testing accuracy $\langle\sigma_{\rm T}^{2}\rangle$ for
different charge multiplicity distributions. $X$ represents different ranges
of charge multiplicity distribution.

In the inset of Fig. 4, we show the dependence of the testing accuracy on the
range of charge multiplicity distribution, i.e., only the charge multiplicity
distributions within the given range (represented by $X$ in the inset) are
used in training and testing the DNN. We notice from the inset that the light
fragments, i.e., $Z\in[1,5]$ play major role in determining the apparent
temperature, while including heavier fragments do help to increase the
accuracy. In another perspective, this feature actually indicates the
superiority of the present method to the traditional isotopic ratio
thermometer, since for the later only the information of light fragments is
taken into consideration. At present, we only use the charge multiplicity
distribution to predict the apparent temperature by the DNN. In principle, the
information in momentum space can be included in the present method for better
accuracy.

After establishing the relation between the apparent temperature and $M_{\rm
c}(Z_{\rm cf})$ through training the DNN, we turn to determine the apparent
temperature of $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ fragmentation
reactions. We first examine the reaction dynamics of $\isotope[103]{Pd}$ $+$
$\isotope[9]{Be}$ within the IQMD model. We simulate the reactions
$\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ with incident energy $E_{\rm lab}$
ranging from 20$A$ MeV to 400$A$ MeV, and for each incident energy we employ
$2,000$ events. As the incident energy increases, the reaction becomes more
violent, and the apparent temperature of the reaction increases. In
fragmentation reaction, the projectile and target nuclei initially form a
compound-like system after they collide each other. In the early stage of the
reaction, only a small number of nucleons evaporate or eject, so the mass of
the compound-like system approximately equals to the sum of the projectile and
the target. We exhibit in Fig. 5 the time evolution of the central density of
the heaviest fragment formed in $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$
reaction at different incident energies from the IQMD model. The density in
the IQMD model is obtained through the sum of the single-particle density,

$\rho(\vec{r})=\sum^{A}_{i=1}\rho_{i}=\sum^{A}_{i=1}\frac{1}{(2{\pi}L)^{3/2}}{\rm
exp}\bigg{\\{}-\frac{\big{[}\vec{r}-\vec{r}_{i}(t)\big{]}^{2}}{2L}\bigg{\\}}.$
(6)

We notice from the figure that the central density of the heaviest fragment
exhibits some oscillations at the beginning. This reflects the breathing mode
caused by the initial compression of the system, since before the compound-
like system dismantling, the largest fragment is the compound-like system
itself. The black dotted line represents $\rho$ $=$ $0.156~{}\rm fm^{-3}$,
which is the initial central density of the nuclear source ([112]Sn) at finite
temperature we generate for training the DNN. Therefore, the initial compound-
like system with density around nuclear saturation density can be mimicked
reasonably by an excited nuclear source with certain given temperature.

Figure 5: Time evolution of the central density of the heaviest fragment in
reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ with the IQMD simulation.
Different lines denote the results from different incident energies from 20$A$
MeV to 400$A$ MeV. The black dashed line represents the initial central
density of the nuclear source [112]Sn.

With the final-state charge multiplicity distribution $M_{\rm c}(Z_{\rm cf})$
of the reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ simulated with the
IQMD, we obtain their apparent temperature through the trained DNN in Eq. (5).
We plot the obtained apparent temperature $T_{\rm ap}$ of the reaction [103]Pd
$+$ [9]Be at different incident energies $E_{\rm lab}$ in Fig. 6. Since the
finite temperature nuclear source generated to train the DNN is initialized at
around nuclear saturation density, $T_{\rm ap}$ obtained through the present
method actually reflects the apparent temperature of the early stage of the
compound system in reaction [103]Pd $+$ [9]Be. In Fig. 6, the black symbols
are the predicted $T_{\rm ap}$ of compound system at the early stage of the
reaction by DNN. We further test the effects of imperfect acceptance and
efficiency on the obtained $T_{\rm ap}$, by applying an acceptance and
efficiency cut, respectively, in the IQMD simulations. It shows that these
effects are negligible, which indicates the robustness of the present method
on imperfect experimental acceptance and efficiency.

Figure 6: The apparent temperature $T_{\rm ap}$ of the reaction
$\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ at different $E_{\rm lab}$,
predicted by DNN (black dots) and MFT (red dots). The solid lines are their
fittings to guide for the eye.

In order to provide a crosscheck of the present method, we employ a momentum
fluctuation thermometer (MFT) Wuenschel et al. (2010) to determine the
apparent temperature of the reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$
simulated by IQMD. In MFT, the distribution of a certain species of light
fragment (deuteron in the present work) is assumed to be Maxwellian, and the
temperature of the system is related to the variance of its quadrupole moment
$\sigma$ through

$\sigma^{2}=4A^{2}m^{2}_{0}T^{2},$ (7)

where $m_{0}$ is the mass of a nucleon and $A$ is the mass number of the
fragment. We add in Fig. 6 the apparent temperature predicted by MFT with red
dots. We notice that the $T_{\rm ap}$ of the two methods are very close, which
increased the reliability of the extracted apparent temperature.

###  III.2 Caloric curve

Figure 7: Caloric curve of the reaction $\isotope[103]{Pd}$ $+$
$\isotope[9]{Be}$. Black open squares represent the result based on the IQMD
model, with $T_{\rm ap}$ determined by DNN using $M_{\rm c}(Z_{\rm cf})$. Blue
dashed line is its polynomial fit. The inset shows the specific heat capacity
$\tilde{c}$ derived from the fitted formula.

The caloric curve, i.e., the apparent temperature as a function of excitation
energy per nucleon $E^{*}/A$ of HICs has been considered as an important probe
to the existence of the nuclear liquid-gas phase transition Pochodzalla et al.
(1995); Natowitz et al. (2002a, b); Wada et al. (2019). In order to examine
the validity of $T_{\rm ap}$ determined by DNN using the charge multiplicity
distribution $M_{\rm c}(Z_{\rm cf})$, we study the caloric curve of the
reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$. In the IQMD model, the
excitation energy of the compound system at the early stage of the reaction
$\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ can be obtained by

$E^{*}=U+E_{\text{k}}-E_{\text{0}},$ (8)

where $U$, $E_{\text{k}}$ and $E_{\text{0}}$ are the potential energy, kinetic
energy and experimental binding energy Wang et al. (2017), respectively. To
properly account the energy deposited in the system, the kinetic energy of
emitted or evaporated nucleons, should be counted when calculating the
excitation energy.

We exhibit in Fig. 7 the caloric curve of the $\isotope[103]{Pd}$ $+$
$\isotope[9]{Be}$ reaction from the IQMD simulation with the apparent
temperature determined by DNN using $M_{\rm c}(Z_{\rm cf})$ of the reaction.
As shown in the figure, the increase of $T_{\rm ap}$ slows down when $E^{*}/A$
reaches to about $8~{}\rm MeV$. Traditionally, this characteristic behavior of
the caloric curve is explained that, as the excitation energy increases, the
system is driven to a spinodal region, in which part of the excitation energy
begins to transfer to latent heat. To characterize this feature of caloric
curve quantitatively, the specific heat capacity of the system Chomaz et al.
(2000) is defined

$\tilde{c}\equiv\frac{d(E^{*}/A)}{dT_{\rm ap}}.$ (9)

Note that it is different from $c_{p}$ or $c_{v}$ because it is not
practicable to maintain the external condition on pressure or volume during
the reaction. The apparent temperature corresponding to the maximum of
$\tilde{c}$ is called limiting temperature $T_{\rm lim}$, which can be used to
deduce the critical temperature of nuclear matter Natowitz et al. (2002b). We
further obtain $\tilde{c}$ through a polynomial fit (red dashed line in Fig.
7) of the obtained caloric curve, as shown in the inset of Fig. 7. Based on
$T_{\rm ap}$ obtained by DNN from $M_{\rm c}(Z_{\rm cf})$, the obtained
$T_{\rm lim}$ of the reaction $\isotope[103]{Pd}$ $+$ $\isotope[9]{Be}$ is
about $6.4~{}\rm MeV$. This value of $T_{\rm lim}$ follows the general trend
of the Natowitz’s limiting-temperature dependence to the system size Natowitz
et al. (2002a), and thus indicates the validity of determining the apparent
temperature through charge multiplicity distribution presented in this
article.

##  IV Summary and outlook

In the present work, we have examined the possibility of determining the
apparent temperature $T_{\rm ap}$ of HICs at intermediate-to-low energies
through their final-state charge multiplicity distribution $M_{\rm c}(Z_{\rm
cf})$. Based on the IQMD simulations of de-exciting nuclear sources ([112]Sn)
at given temperatures, we have established a relation between the final-state
$M_{\rm c}(Z_{\rm cf})$ of a nuclear source, and its corresponding
temperature, through training a DNN. The trained DNN can predict the apparent
temperature within an error of $0.62~{}\rm MeV$, which is small enough for
applying it to analyze the reaction dynamics. We have then employed the above
method to obtain the apparent temperature of the $\isotope[103]{Pd}$ $+$
$\isotope[9]{Be}$ reactions at different incident energies simulated by the
IQMD, and subsequently the caloric curve. The caloric curve shows a
characteristic behavior at, i.e., $T_{\rm lim}$ $=$ $6.4~{}\rm MeV$, which
follows the general trend of the limiting temperature’s dependence to system
size and indicates of nuclear liquid-gas phase transition of a given system.
The present method provides an alternative way to determine the apparent
temperature of the HICs at intermediate-to-low energy, and it can be used as a
supplement of the traditional nuclear thermometers. To apply the present
method to the analyses of experimental data relies on the accurate description
of the fragments multiplicity from dynamical model. One of the possible ways
to achieve this goal is using transport model plus statistical
multifragmentation process. Studies following this line are in progress.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

###### Acknowledgements.

This work is partially supported by the National Natural Science Foundation of
China under Contracts No. $11890710$, $11890714$ and No. $11625521$, the Key
Research Program of Frontier Sciences of the CAS under Grant No. QYZDJ-SSW-
SLH$002$, the Strategic Priority Research Program of the CAS under Grants No.
XDB34000000, Guangdong Major Project of Basic and Applied Basic Research No.
2020B0301030008, and the Postdoctoral Innovative Talent Program of China under
Grants No. BX$20200098$.

## References

  * Lattimer and Prakash (2007) J. Lattimer and M. Prakash, Phys. Rep. 442, 109 (2007).
  * Baldo and Burgio (2016) M. Baldo and G. Burgio, Prog. Part. Nucl. Phys. 91, 203 (2016).
  * Roca-Maza and Paar (2018) X. Roca-Maza and N. Paar, Prog. Part. Nucl. Phys. 101, 96 (2018).
  * Finn et al. (1982) J. E. Finn, S. Agarwal, A. Bujak, J. Chuang, L. J. Gutay, A. S. Hirsch, R. W. Minich, N. T. Porile, R. P. Scharenberg, B. C. Stringfellow, et al., Phys. Rev. Lett. 49, 1321 (1982).
  * Siemens (1983) P. J. Siemens, Nature 305, 410 (1983).
  * Panagiotou et al. (1984) A. D. Panagiotou, M. W. Curtin, H. Toki, D. K. Scott, and P. J. Siemens, Phys. Rev. Lett. 52, 496 (1984).
  * Pochodzalla et al. (1995) J. Pochodzalla, T. Möhlenkamp, T. Rubehn, A. Schüttauf, A. Wörner, E. Zude, M. Begemann-Blaich, T. Blaich, H. Emling, A. Ferrero, et al., Phys. Rev. Lett. 75, 1040 (1995).
  * Ma et al. (1997) Y. G. Ma, A. Siwek, J. Péter, F. Gulminelli, R. Dayras, L. Nalpas, B. Tamain, E. Vient, G. Auger, C. O. Bacri, et al., Phys. Lett. B 390, 41 (1997).
  * Ma (1999) Y. G. Ma, Phys. Rev. Lett. 83, 3617 (1999).
  * Chomaz et al. (2000) P. Chomaz, V. Duflot, and F. Gulminelli, Phys. Rev. Lett. 85, 3587 (2000).
  * Richert and Wagner (2001) J. Richert and P. Wagner, Phys. Rep. 350, 1 (2001).
  * Natowitz et al. (2002a) J. B. Natowitz, R. Wada, K. Hagel, T. Keutgen, M. Murray, A. Makeev, L. Qin, P. Smith, and C. Hamilton, Phys. Rev. C 65, 034618 (2002a).
  * Natowitz et al. (2002b) J. B. Natowitz, K. Hagel, Y. Ma, M. Murray, L. Qin, R. Wada, and J. Wang, Phys. Rev. Lett. 89, 212701 (2002b).
  * Ma et al. (2005) Y. G. Ma, J. B. Natowitz, R. Wada, K. Hagel, J. Wang, T. Keutgen, Z. Majka, M. Murray, L. Qin, P. Smith, et al., Phys. Rev. C 71, 054606 (2005).
  * Ma and Ma (2018) C.-W. Ma and Y.-G. Ma, Prog. Part. Nucl. Phys. 99, 120 (2018).
  * Borderie and Frankland (2019) B. Borderie and J. Frankland, Prog. Part. Nucl. Phys. 105, 82 (2019).
  * Auerbach and Shlomo (2009) N. Auerbach and S. Shlomo, Phys. Rev. Lett. 103, 172501 (2009).
  * Dang (2011) N. D. Dang, Phys. Rev. C 84, 034309 (2011).
  * Fang et al. (2014) D. Q. Fang, Y. G. Ma, and C. L. Zhou, Phys. Rev. C 89, 047601 (2014).
  * Deng et al. (2016) X. G. Deng, Y. G. Ma, and M. Veselský, Phys. Rev. C 94, 044622 (2016).
  * Mondal et al. (2017) D. Mondal, D. Pandit, S. Mukhopadhyay, S. Pal, B. Dey, S. Bhattacharya, A. De, S. Bhattacharya, S. Bhattacharyya, P. Roy, et al., Phys. Rev. Lett. 118, 192501 (2017).
  * Guo et al. (2017) C. Q. Guo, Y. G. Ma, W. B. He, X. G. Cao, D. Q. Fang, X. G. Deng, and C. L. Zhou, Phys. Rev. C 95, 054622 (2017).
  * Bracco et al. (1989) A. Bracco, J. J. Gaardhøje, A. M. Bruce, J. D. Garrett, B. Herskind, M. Pignanelli, D. Barnéoud, H. Nifenecker, J. A. Pinston, C. Ristori, et al., Phys. Rev. Lett. 62, 2080 (1989).
  * Bortignon et al. (1991) P. F. Bortignon, A. Bracco, D. Brink, and R. A. Broglia, Phys. Rev. Lett. 67, 3360 (1991).
  * Wieland et al. (2006) O. Wieland, A. Bracco, F. Camera, G. Benzoni, N. Blasi, S. Brambilla, F. Crespi, A. Giussani, S. Leoni, P. Mason, et al., Phys. Rev. Lett. 97, 012501 (2006).
  * Wada et al. (1989) R. Wada, D. Fabris, K. Hagel, G. Nebbia, Y. Lou, M. Gonin, J. B. Natowitz, R. Billerey, B. Cheynis, A. Demeyer, et al., Phys. Rev. C 39, 497 (1989).
  * d’Enterria et al. (2001) D. G. d’Enterria, L. Aphecetche, A. Chbihi, H. Delagrange, J. Díaz, M. J. van Goethem, M. Hoefman, A. Kugler, H. Löhner, G. Martínez, et al., Phys. Rev. Lett. 87, 022701 (2001).
  * Borderie et al. (1996) B. Borderie, D. Durand, F. Gulminelli, M. Parlog, M. F. Rivet, L. Tassan-Got, G. Auger, C. O. Bacri, J. Benlliure, E. Bisquer, et al., Phys. Lett. B 388, 224 (1996).
  * Chitwood et al. (1986) C. B. Chitwood, C. K. Gelbke, J. Pochodzalla, Z. Chen, D. J. Fields, W. G. Lynch, R. Morse, M. B. Tsang, D. H. Boal, and J. C. Shillcock, Phys. Lett. B 172, 27 (1986).
  * Schwarz et al. (1993) C. Schwarz, W. G. Gong, N. Carlin, C. K. Gelbke, Y. D. Kim, W. G. Lynch, T. Murakami, G. Poggi, R. T. de Souza, M. B. Tsang, et al., Phys. Rev. C 48, 676 (1993).
  * Tsang et al. (1997) M. B. Tsang, W. G. Lynch, H. Xi, and W. A. Friedman, Phys. Rev. Lett. 78, 3836 (1997).
  * Serfling et al. (1998) V. Serfling, C. Schwarz, R. Bassini, M. Begemann-Blaich, S. Fritz, S. J. Gaff, C. Groß, G. Immé, I. Iori, U. Kleinevoß, et al., Phys. Rev. Lett. 80, 3928 (1998).
  * Albergo et al. (1985) S. Albergo, S. Costa, E. Costanzo, and A. Rubbino, IL Nuov. Cim. A 89, 1 (1985).
  * Wuenschel et al. (2010) S. Wuenschel, A. Bonasera, L. W. May, G. A. Souliotis, R. Tripathi, S. Galanopoulos, Z. Kohley, K. Hagel, D. V. Shetty, K. Huseman, et al., Nucl. Phys. A 843, 1 (2010).
  * LeCun et al. (2015) Y. LeCun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015).
  * Jordan and Mitchell (2015) M. I. Jordan and T. M. Mitchell, Science 349, 255 (2015).
  * Carleo et al. (2019) G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Rev. Mod. Phys. 91, 045002 (2019).
  * Carleo and Troyer (2017) G. Carleo and M. Troyer, Science 355, 602 (2017).
  * Zhang et al. (2020) Z.-W. Zhang, S. Yang, Y.-H. Wu, C.-X. Liu, Y.-M. Han, C.-H. Lee, Z. Sun, G.-J. Li, and X. Zhang, Chin. Phys. Lett. 37, 018401 (2020).
  * Hezaveh et al. (2017) Y. D. Hezaveh, L. P. Levasseur, and P. J. Marshall, Nature 548, 555 (2017).
  * Carrasquilla and Melko (2017) J. Carrasquilla and R. G. Melko, Nat. Phys. 13, 431 (2017).
  * van Nieuwenburg et al. (2017) E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Nat. Phys. 13, 435 (2017).
  * Rodriguez-Nieva and Scheurer (2019) J. F. Rodriguez-Nieva and M. S. Scheurer, Nat. Phys. 15, 790 (2019).
  * Rao (2020) W.-J. Rao, Chin. Phys. Lett. 37, 080501 (2020).
  * Ma et al. (2020) C.-W. Ma, D. Peng, H.-L. Wei, Z.-M. Niu, Y.-T. Wang, and R. Wada, Chin. Phys. C 44, 014104 (2020).
  * Yao et al. (2019) T.-S. Yao, C.-Y. Tang, M. Yang, K.-J. Zhu, D.-Y. Yan, C.-J. Yi, Z.-L. Feng, H.-C. Lei, C.-H. Li, L. Wang, et al., Chin. Phys. Lett. 36, 068101 (2019).
  * Wu et al. (2020) Y. Wu, Z. Meng, K. Wen, C. Mi, J. Zhang, and H. Zhai, Chin. Phys. Lett. 37, 103201 (2020).
  * Wang et al. (2020) R. Wang, Y.-G. Ma, R. Wada, L.-W. Chen, W.-B. He, H.-L. Liu, and K.-J. Sun, Phys. Rev. Research 2, 043202 (2020).
  * Bertsch and Das Gupta (1988) G. Bertsch and S. Das Gupta, Phys. Rep. 160, 189 (1988).
  * Aichelin (1991) J. Aichelin, Phys. Rep. 202, 233 (1991).
  * Bondorf et al. (1995) J. P. Bondorf, A. Botvina, A. Iljinov, I. Mishustin, and K. Sneppen, Phys. Rep. 257, 133 (1995).
  * Charity et al. (1988) R. Charity, M. McMahan, G. Wozniak, R. McDonald, L. Moretto, D. Sarantites, L. Sobotka, G. Guarino, A. Pantaleo, L. Fiore, et al., Nucl. Phys. A 483, 371 (1988).
  * Gaitanos et al. (2009) T. Gaitanos, H. Lenske, and U. Mosel, Prog. Part. Nucl. Phys. 62, 439 (2009).
  * Zhang et al. (2018) Z.-F. Zhang, D.-Q. Fang, and Y.-G. Ma, Nucl. Sci. Tech. 29, 78 (2018).
  * Ono (2019) A. Ono, Prog. Part. Nucl. Phys. 105, 139 (2019).
  * Ma et al. (2006) Y. G. Ma, Y. B. Wei, W. Q. Shen, X. Z. Cai, J. G. Chen, J. H. Chen, D. Q. Fang, W. Guo, C. W. Ma, G. L. Ma, et al., Phys. Rev. C 73, 014604 (2006).
  * Sümmerer and Blank (2000) K. Sümmerer and B. Blank, Phys. Rev. C 61, 034607 (2000).
  * Song et al. (2018) Y.-D. Song, H.-L. Wei, C.-W. Ma, and J.-H. Chen, Nucl. Sci. Tech. 29, 96 (2018).
  * Charity (2010) R. J. Charity, Phys. Rev. C 82, 014610 (2010).
  * Liu et al. (2019) H. L. Liu, Y. G. Ma, and D. Q. Fang, Phys. Rev. C 99, 054614 (2019).
  * Hartnack et al. (1998) C. Hartnack, R. K. Puri, J. Aichelin, J. Konopka, S. Bass, H. Stöcker, and W. Greiner, Eur. Phys. J. A 1, 151 (1998).
  * Li et al. (2018) P.-C. Li, Y.-J. Wang, Q.-F. Li, and H.-F. Zhang, Nucl. Sci. Tech. 29, 177 (2018).
  * Feng (2018) Z.-Q. Feng, Nucl. Sci. Tech. 29, 40 (2018).
  * Yan et al. (2019) T.-Z. Yan, S. Li, Y.-N. Wang, F. Xie, and T.-F. Yan, Nucl. Sci. Tech. 30, 15 (2019).
  * Yan and Li (2019) T.-Z. Yan and S. Li, Nucl. Sci. Tech. 30, 43 (2019).
  * Yu et al. (2020) H. Yu, D.-Q. Fang, and Y.-G. Ma, Nucl. Sci. Tech. 31, 61 (2020).
  * Kingma and Ba (2017) D. P. Kingma and J. Ba, ArXiv14126980 Cs (2017), eprint 1412.6980.
  * Wada et al. (2019) R. Wada, W. Lin, P. Ren, H. Zheng, X. Liu, M. Huang, K. Yang, and K. Hagel, Phys. Rev. C 99, 024616 (2019).
  * Wang et al. (2017) M. Wang, G. Audi, F. G. Kondev, W. J. Huang, S. Naimi, and X. Xu, Chin. Phys. C 41, 030003 (2017).


# New Approach to Metal–Insulator Phase Transition Kinetics in Magnetic Field.

L.B. Dubovskii  Dubovskiy˙LB@rrcki Theoretical division , NRC”Kurchatov
Institute”, 123182 Moscow, Russia  chair of theoretical physics, MFTI, 141700
Moscow region

###### Abstract

The metal–insulator phase transition is considered on the basis of Ginzburg-
Landau type equations with two different order parameters. An inclusion of
magnetic field in this picture is an important step for understanding of
behavior of the metal–insulator phase transition kinetics. The magnetic field
leads to various singularities of the surface tension and results in drastic
variations of the phase transition kinetics. The strongest singularity is due
to Landau diamagnetism and determines anomalous features of MI transition
kinetics. This singularity supports the well known experimental fact that
almost all semimetals behave like diamagnetic materials.

###### pacs:

75.20.-g, 68.35.Md, 71.30.+h, 73.40.Ns

We investigate phase transitions from an insulator phase to a metal phase (MI
phase transitions). Any MI transition in a crystalline material, at any rate
at zero temperature, must be a transition from a situation in which electronic
bands do not overlap to a situation when they do NFMott . One of the
overlapped bands is initially empty and gets carriers of electron type. The
other one is initially entirely filled with electrons and gets carriers of
hole type due to overlapping of the bands. Small band-crossing leads to two-
band metallic state with small numbers of electrons and holes carriers per
atomic cell. In addition, the numbers of holes and electrons are equal to each
other. The situation is quite typical for common semimetals.

The MI phase transitions are the phase transitions of the first order. The
description of these transitions is usually based on the common approach to
the first order phase transition theory LandLif . The leading parameter which
provides the MI phase transition is the material density $\rho(\vec{r})$ of
the phases. This parameter determines the first order type of the phase
transition at a critical value of $\rho_{c}$. Here $\rho_{c}$ is a function of
pressure $p$ and temperature $T$ ($\rho_{c}=\rho_{c}(p,T)$). Generally, the
pressure $p$ is the leading parameter which governs the MI transition. A
growth of density of the insulator phase leads to overlapping of two
electronic bands. One of these bands is empty and the other one is filled with
electrons. By overlapping of such bands the MI phase transition occurs with an
appearance of a new order parameter which characterizes the density of current
carriers in the system. This parameter equals identically zero in the
insulator phase and grows from zero at the MI interface. This parameter
exhibits itself as the second order phase transition parameter.

So, we can describe the MI phase transition by two order parameters (cf.
Lubensky ; JinwuYe ; dub ; dub1 ). One of the order parameters is the material
density distribution $\rho(\vec{r})$ either of metal phase or of insulating
one. The second parameter is the density of electrons and holes in two bands.
The density of electrons and holes can be described by a vector in the Hilbert
space with electronic and holes components
$\Upsilon^{+}(\vec{r})=(u_{\uparrow}^{*}(\vec{r}),u_{\downarrow}^{*}(\vec{r}),v_{\uparrow}^{*}(\vec{r}),v_{\downarrow}^{*}(\vec{r}))$.
The components of the vector $\Upsilon(\vec{r})$ are functions of electrons
for up and down spins $(u_{\uparrow}(\vec{r}),u_{\downarrow}(\vec{r}))$ and
functions of holes for up and down spins
$(v_{\uparrow}(\vec{r}),v_{\downarrow}(\vec{r}))$. In the absence of magnetic
field all these components of the vector $\Upsilon(\vec{r})$ are real
functions. We can put the following invariant scalar quantity

$n(\vec{r})=n_{e}(\vec{r})+n_{h}(\vec{r})\equiv\left({\Upsilon^{+}\Upsilon}\right)\,,$
(1)

$n_{e}(\vec{r})=|u_{\uparrow}(\vec{r})|^{2}+|u_{\downarrow}(\vec{r})|^{2}\,,\,\,\,n_{h}(\vec{r})=|v_{\uparrow}(\vec{r})|^{2}+|v_{\downarrow}(\vec{r})|^{2}\,.$
It determines densities of electrons and holes in the metal phase and becomes
identically zero in the insulating phase at $T=0$. We take the identity
$n_{h}(\vec{r})=n_{e}(\vec{r})$ due to the local electrical neutrality within
the metal phase.

In the vicinity of MI phase transition, the Ginzburg-Landau (GL) functional
can be assumed as a sum of contributions from order parameters
$\rho(\vec{r})$, $\Upsilon(\vec{r})$:

$\displaystyle\Phi=\Phi\\{\rho,\Upsilon,\Upsilon^{+}\\}=\int
d\vec{r}\emph{F}(\rho,\Upsilon,\Upsilon^{+})\,,$
$\displaystyle\emph{F}=\emph{F}_{\rho}(\rho(\vec{r}))+\emph{F}_{\Upsilon}(\Upsilon(\vec{r}))+\emph{F}_{int}(\rho(\vec{r}),\Upsilon(\vec{r})).$
(2)

The parameter $\rho(\vec{r})$ describes the first order phase transition
between insulating and metal phases whereas the vector function
$\Upsilon(\vec{r})$ of electrons and holes describes the second order phase
transition. We expand the functional $\emph{F}_{\rho}(\rho(\vec{r}))$
according to Cahn – Hillard approximation CahnHillard , see also LifKagan :

$\emph{F}_{\rho}(\rho(\vec{r}))=\varphi(\rho)+\frac{1}{2}\lambda(\rho)(\nabla\rho)^{2}$
(3)

We use the GL expansion LandLif2  for the vector function $\Upsilon(\vec{r})$

$\emph{F}_{\Upsilon}(\Upsilon(\vec{r}))=\alpha\left({\Upsilon^{+}\Upsilon}\right)+\frac{1}{2}\beta{\left({\Upsilon^{+}\Upsilon}\right)}^{2}+\frac{\hbar^{2}}{2m}\left(\nabla\Upsilon^{+}(\vec{r})\nabla\Upsilon(\vec{r})\right);$
$\alpha=\aleph(\rho-\rho_{c})$ (4)

Here we have expanded $\alpha$ linearly over $\rho$ in the vicinity of
$\rho_{c}$. It is in analogy with the well known expansion of the appropriate
parameter in the Landau theory of the second order phase transition on the
temperature $T$ LandLif . The proposed expansion is convenient for an
investigation of the MI phase transitions at a fixed temperature, say at
$T=0$. For simplicity, we have taken the masses of holes and electrons to be
equal to each other. Also the interaction term
$\emph{F}_{int}(\rho(\vec{r}),\Upsilon(\vec{r}))$ is taken as a linear
function of $n(\vec{r})$ (cf. Lubensky ; JinwuYe ; dub ):

$\emph{F}_{int}(\rho(\vec{r}),\Upsilon(\vec{r}))=g(\rho)\left({\Upsilon^{+}\Upsilon}\right)$
(5)

The variation of $\Phi=\Phi\\{\rho,\Upsilon,\Upsilon^{+}\\}$ over
$\rho(\vec{r})$ and $\Upsilon^{+}(\vec{r})$ gives the self consistent
equations of GL type for MI phase transition:

$\displaystyle\lambda^{1/2}(\rho)\nabla\left(\lambda^{1/2}(\rho)\nabla\rho(\vec{r})\right)=\frac{d\varphi}{d\rho}(\rho)+\frac{dg}{d\rho}\left({\Upsilon^{+}\Upsilon}\right)$
$\displaystyle-\frac{\hbar^{2}}{2m}\nabla^{2}\Upsilon+\left(\alpha+g\right)\Upsilon+\beta\left({\Upsilon^{+}\Upsilon}\right)\Upsilon=0\,\,\,\,\,\,\,\,$
(6)

In the case $g\equiv 0$, the parameters $\rho$ and $\Upsilon$ are independent
and the surface tension $\Sigma$ at the interface between the metallic and the
insulating phases can be calculated separately for $\rho$ ($\Sigma_{\rho}$)
and $\Upsilon$ ($\Sigma_{\Upsilon}$). The surface tension $\Sigma_{\rho}$ can
be calculated straightforward (LifKagan ; dub ):

$\Sigma_{\rho}=\int_{\rho_{ins}}^{\rho_{met}}d\rho\sqrt{\varphi(\rho)\lambda(\rho)/2\,}$
(7)

It should be emphasized that the value of $\Sigma_{\rho}$ is of the order of
the value of surface tension at the interface between an insulator and vacuum.
It is well known from experiments kik  that the characteristic values of the
surface tension at the interface between an insulator and vacuum are at least
an order of magnitude lower than those at the interface between a metal and
vacuum. The physical mechanism underlying this difference is related to the
presence of conduction electrons penetrating from the interface to vacuum at a
distance of the order of the separation between atomic layers in the metal
Partenskiy . So,

$\Sigma_{\Upsilon}\gg\Sigma_{\rho}.$

In most cases we can neglect $\Sigma_{\rho}$ in comparison with
$\Sigma_{\Upsilon}$.

Now we consider the behavior of the system in magnetic field. The
investigation of MI interface in the presence of magnetic field is a
remarkably delicate task. The problem is quite similar for introducing of a
magnetic field in Kohn-Sham scheme VRG  based on VR . One can understand the
basic difficulty by considering the expression for the orbital current
density:

$\vec{j}(\vec{r})=\vec{j}_{p}(\vec{r})+\frac{e}{mc}n(\vec{r})\vec{A}(\vec{r})$

For short, we assume the situation with only one conductivity band. The first
term $\vec{j}_{p}(\vec{r})$ is referred to as ”paramagnetic current density,”
and measures the density of canonical momentum in the wave function. It
depends explicitly on the electron density $n(\vec{r})$ and the vector
potential $\vec{A}(\vec{r})$. Here we should emphasize that the situation in
normal metals differs drastically from that in superconductors. In normal
metals the electron density $n(\vec{r})$ is a function of coordinates
$\vec{r}$ whereas in the superconductors the electron density $n(\vec{r})$ is
an integer number of electron charge value GL  and does not depend on
coordinates $\vec{r}$ at all. The integer value is just $2e$ according to the
microscopic theory of superconductivity dG . The brilliant discussion of the
problem in superconductors can be found in MEG . In VRG  it is assumed that
the fields, state and its energy are uniquely determined by the densities, and
the paramagnetic current density. This statement would have not been true if
we had attempted to work with the physical current densities
$\vec{j}(\vec{r})$ as basic variable. In that case one would still have the
freedom of operating a gauge transformation on the vector potential, and hence
multiplying the wave function by a coordinate-dependent phase factor, without
changing the physical current density. Therefore, the state wave function is
not a unique function of $\vec{j}(\vec{r})$. Working with the paramagnetic
current density $\vec{j}_{p}$ eliminates this ambiguity. In the presence of a
magnetic field $\vec{B}$ the mapping between ground-state energy and ground-
state wave function is not one to one. To recover a one to one mapping it is
necessary to supplement the number density distribution by spin- and current-
densities which play the same role with respect to an external field
$\vec{B}$, and vector potential $\vec{A}$, as the density does with respect to
ordinary scalar potential. In summary, the vector potential $\vec{A}$ should
not be involved in the potential $\Phi$ for normal metals because the gauge
invariance is violated.

We assume (cf. RC ; GH ) that the state and its wave function are determined
uniquely by $\vec{B}(\vec{r})$ and $\rho(\vec{r})$. The inclusion of
paramagnetic effects can be carried out straightforward by adding an
additional term into the functional (4):

$\emph{F}_{para}(\vec{B},\Upsilon,\Upsilon^{+})=\mu_{ph}\left(\Upsilon^{+}\vec{B}\vec{\Sigma}\Upsilon\right),\vec{\Sigma}=\begin{pmatrix}\vec{\tau_{u}}\,\,\,\,\,\,\,\,\,\,0\\\
0\,\,\,\,\,\,\,\,\,\,\vec{\tau_{v}}\end{pmatrix}$ (8)

$\mu_{ph}$ is a phenomenological constant, $\vec{\tau_{u}}=\vec{\tau_{v}}$ are
Pauli spinors RC  correspondingly for electrons and holes.

The functional $\Phi\\{\rho,\Upsilon,\Upsilon^{+}\\}$ (New Approach to
Metal–Insulator Phase Transition Kinetics in Magnetic Field.) with the
paramagnetic term (8) gives the energy spectrum $\varepsilon_{k}$ of the
system and the thermodynamical potential $\Omega(T,V,\mu)$ LandLif :

$\Omega=-T\sum_{k}\ln\left(1+\exp\frac{\mu-\varepsilon_{k}}{T}\right)\,,\,\,\,\vec{M}(\vec{r})=-\frac{\partial\Omega}{\partial\vec{B}(\vec{r})}.$

$\vec{M}(\vec{r})$ is the magnetic moment of the metal and we assume that it
depends exclusively on the electron density $n(\vec{r})$ or
$\left({\Upsilon^{+}\Upsilon}\right)$ (see (1)) and does not depend on
$\rho(\vec{r})$, i.e.

$\vec{M}(\vec{r})=\vec{M}(n(\vec{r}))\,\,,\,\,\,n(\vec{r})=\Upsilon^{+}(\vec{r})\Upsilon(\vec{r}).$
(9)

The magnetic field can be introduced as follows:
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec{H}(\vec{r})=\vec{B}(\vec{r})-4\pi\vec{M}(\vec{r}).$

The thermodynamical potential $\Phi\\{\rho,\Upsilon,\Upsilon^{+}\\}$ (New
Approach to Metal–Insulator Phase Transition Kinetics in Magnetic Field.) with
the additional term (8) is a function of the independent variables
$T,V,\vec{B}$. It is inconvenient since $\vec{B}$ means the average of the
microscopic magnetic field in the specimen. The value $\vec{B}$ cannot be
fixed experimentally. Therefore we should fix the external magnetic field
$\vec{H}$. It can be done by adding (see LLe ; AAA ) the following value:

$-\frac{{\vec{H}}{\vec{B}}}{4\pi}+\frac{{\vec{H}}^{2}}{8\pi}\equiv-\vec{H}\vec{M}-\frac{{\vec{H}}^{2}}{8\pi}$

and consider the other thermodynamic potential with the independent variable
$\vec{H}$:

$\displaystyle\Phi_{H}\\{\rho,\Upsilon,\Upsilon^{+}\\}=\int
d\vec{r}\emph{F}_{H}(\rho,\Upsilon,\Upsilon^{+})$
$\displaystyle\emph{F}_{H}(\rho,\Upsilon,\Upsilon^{+})=\emph{F}_{\rho}(\rho(\vec{r}))+\digamma_{\Upsilon,\vec{H}}+\emph{F}_{int}(\rho(\vec{r}),\Upsilon(\vec{r}))$
(10)
$\digamma_{\Upsilon,\vec{H}}=\emph{F}_{\Upsilon}(\Upsilon(\vec{r}))+\emph{F}_{para}(\vec{H}+4\pi\vec{M},\Upsilon,\Upsilon^{+})-\vec{H}\vec{M}-\frac{{\vec{H}}^{2}}{8\pi}.$

The variation of $\Phi=\Phi\\{\rho,\Upsilon,\Upsilon^{+}\\}$ over
$\Upsilon^{+}(\vec{r})$ for the fixed independent variable $\vec{H}$ gives the
equation for $\Upsilon$:

$-\frac{\hbar^{2}}{2m}\left(\frac{d^{2}\Upsilon}{dx^{2}}\right)+\left(\alpha+g\rho\right)\Upsilon+\beta\left({\Upsilon^{+}\Upsilon}\right)\Upsilon+\mathbb{H}\Upsilon=0\,;$
(11) $\mathbb{H}\Upsilon=\mu_{ph}\begin{pmatrix}H_{z}&H_{x}&0&0\\\
H_{x}&-H_{z}&0&0\\\ 0&0&H_{z}&H_{x}\\\
0&0&H_{x}&-H_{z}\end{pmatrix}\Upsilon-\vec{H}\frac{\partial\vec{M}}{\partial(\Upsilon^{+}\Upsilon)}\Upsilon\,.$

Now we take $g\equiv 0$ and calculate the surface tension $\Sigma_{\Upsilon}$.
One by one, we shall consider two cases: 1) $\vec{M}\equiv 0$ and 2)
$\mu_{ph}\equiv 0$. In the first case components of $\Upsilon(x)$ can be taken
as real functions and an exact solution can be found. The first two matrix
rows of Eq. (11) are not connected with the other two rows and they can be
rewritten as the following set of two coupled equations for the components
$\Upsilon$ :

$\displaystyle\hat{L}(u_{\uparrow},u_{\downarrow})u_{\uparrow}+\mu_{ph}(H_{x}u_{\downarrow}+H_{z}u_{\uparrow})=0$
$\displaystyle\hat{L}(u_{\uparrow},u_{\downarrow})u_{\downarrow}+\mu_{ph}(H_{x}u_{\uparrow}-H_{z}u_{\downarrow})=0\,.$
(12)

Here $\hat{L}(u,v)$ is the following linear differential operator:

$\hat{L}(u,v)\varphi=-\frac{\hbar^{2}}{2m}\frac{d^{2}\varphi}{dx^{2}}+\alpha\varphi+\beta\left(|u|^{2}+|v|^{2}\right)\varphi\,.$

Solution of the system (New Approach to Metal–Insulator Phase Transition
Kinetics in Magnetic Field.) can be sought in the form:

$u_{\downarrow}(x)=qu_{\uparrow}(x)\,\,,$

where $q$ depends on $H_{x}$ and $H_{z}$. The both equations of the system
$(\ref{glh3x})$ become identical to each other under the following condition:
$qH_{x}+H_{z}={H_{x}}/{q}-H_{z}\,.$

So, we have $q=q_{\pm}$,

$q_{+}=\frac{|H_{x}|}{\sqrt{H_{x}^{2}+H_{z}^{2}}+H_{z}sgn{H_{x}}}\,\,\,\,\,,\,\,\,\,\,\,\,q_{-}=-\frac{1}{q_{+}}\,.$

The first equation of the system $(\ref{glh3x})$ can be rewritten:

$\displaystyle-\frac{\hbar^{2}}{2m}\frac{d^{2}u_{\uparrow}}{dx^{2}}+\tilde{\alpha}u_{\uparrow}+\tilde{\beta}(1+q^{2}){|u_{\uparrow}|}^{2}u_{\uparrow}=0\,\,;\,\,\,\,$
$\displaystyle\tilde{\alpha}=\alpha+\mu_{p}(qH_{x}+H_{z})\,,\,\,\tilde{\beta}=\beta(1+q^{2})\,;\,\,q=q_{\pm}\,\,.$
(13)

We can also $\tilde{\alpha}$ represent as follows

$\tilde{\alpha}=a(\rho-\rho_{c\vec{H}})\,,\,\,\,\,\,\,\,\rho_{c\vec{H}}=\rho_{c0}-\mu_{ph}Hsgn{H_{x}}a^{-1}\,\,,\,\,$

$H=\sqrt{H_{x}^{2}+H_{z}^{2}}\,.$

Equation (New Approach to Metal–Insulator Phase Transition Kinetics in
Magnetic Field.) has two solutions: (i) a homogeneous solution
$u_{\uparrow}\equiv 0$ corresponding to the insulating state and (ii) the
solution $(u_{\uparrow}^{0},u_{\downarrow}^{0})$ having the absolute values

$|u_{\uparrow}^{0}|^{2}=-\frac{\tilde{\alpha}}{\tilde{\beta}(1+q^{2})}\,;\,\,|u_{\downarrow}^{0}|^{2}=q^{2}|u_{\uparrow}^{0}|^{2}.$
(14)

The sum of these two terms equals the total electron density $n_{e0}$ at
infinity, $|u_{\uparrow}^{0}|^{2}+|u_{\downarrow}^{0}|^{2}=n_{e0}\,\,.$

We introduce the dimensionless function $f(x)$ as
$u_{\uparrow}(x)={u_{\uparrow}^{0}}f(x)$ and Eq. (New Approach to
Metal–Insulator Phase Transition Kinetics in Magnetic Field.) has the form

$-\xi_{\vec{H}}{{}^{2}}(T)\frac{d^{2}f}{dx^{2}}-f+f^{3}=0\,\,,\,\,\,\xi_{\vec{H}}(T)=\sqrt{\frac{\hbar^{2}}{2m|\tilde{\alpha}|}}$
(15)

with the solution $f(x)=\tanh\left[\frac{x}{\sqrt{2}\xi_{\vec{H}}(T)}\right]$.
$\xi_{\vec{H}}$ and $f(x)$ are the functions of the magnetic field $\vec{H}$.

The next two matrix rows of Eq. (11) can be examined in the same way. So, we
get a solution of Eq. (11) in the explicit form
$\Upsilon^{+}=(u_{\uparrow}\,,\,u_{\downarrow}\,,\,v_{\uparrow}\,,\,v_{\downarrow})$.
The expression enables to analyze the surface tension due to the order
parameter $\Upsilon$ in the magnetic field $\vec{H}$. The free energy
$\emph{F}_{\Upsilon,\vec{H}}$ due to the order parameter $\Upsilon$ and
$\vec{H}$ per unit area of the interface is

$\displaystyle\emph{F}_{\Upsilon,\vec{H}}=\int_{0}^{\infty}dx\,\digamma_{\Upsilon,\vec{H}}\equiv\int_{0}^{\infty}dx\emph{F}_{\Upsilon}\Upsilon(\vec{r}))+$
$\displaystyle\int_{0}^{\infty}dx\left[\emph{F}_{para}(\vec{H}+4\pi\vec{M},\Upsilon,\Upsilon^{+})-\vec{H}\vec{M}-\frac{{\vec{H}}^{2}}{8\pi}\right]$
(16)

The surface energy $\Sigma_{\Upsilon,\vec{H}}$ represents the difference
between $\emph{F}_{\Upsilon,\vec{H}}$ and the free energy of the uniform metal
phase filling the half-space of the specimen. In the paramagnetic case (New
Approach to Metal–Insulator Phase Transition Kinetics in Magnetic Field.) with
$\vec{M}(\vec{r})\equiv 0$, the two last terms in the integrand do not give
contribution into $\Sigma_{\Upsilon,\vec{H}}$. The calculation is
straightforward (cf.dG ) and gives $\Sigma_{\Upsilon,\vec{H}}$:

$\Sigma_{\Upsilon,\vec{H}}=\frac{2\sqrt{2}}{3}\xi_{\vec{H}}(T)\tilde{\beta}n_{0}^{2}$
(17)

Here $n_{0}$ is the bulk density of electrons in metal. And
$n_{0}=\left({\Upsilon^{+}\Upsilon}\right)|_{|\vec{r}|\rightarrow\infty}$ ,
see (1). If we put $n_{0}$ to be of the order of an unit per crystal cell we
get for $\Sigma_{\Upsilon,\vec{H}}$ (17) the typical value for metals. In
addition, $\Sigma_{\Upsilon,\vec{H}}$ is a singular function in the limit
$n_{0}\Rightarrow 0$, namely $\Sigma_{\Upsilon,\vec{H}}\sim n_{0}^{3/2}$ as
$n_{0}\Rightarrow 0$. And also
$\Sigma_{\Upsilon,\vec{H}}\sim(\rho-\rho_{c\vec{H}})^{3/2}$ in the limit
$\rho\Rightarrow\rho_{c\vec{H}},\,\,\,\,\,\,\rho_{c\vec{H}}=\rho_{c0}-\mu_{ph}Hsgn{H_{x}}a^{-1}$
.

Now we consider the second situation

$-\frac{\hbar^{2}}{2m}\left(\frac{d^{2}\Upsilon}{dx^{2}}\right)+\alpha\Upsilon+\beta\left({\Upsilon^{+}\Upsilon}\right)\Upsilon-\vec{H}\frac{\partial\vec{M}}{\partial(\Upsilon^{+}\Upsilon)}\Upsilon=0$

We assume that $\vec{M}(\vec{r})$ depends on electron density $n(\vec{r})$ or
$\left({\Upsilon^{+}\Upsilon}\right)$ and does not depend on $\rho(\vec{r})$,
i.e.

$\vec{M}(\vec{r})=\vec{M}(n(\vec{r}))\,\,,\,\,\,n(\vec{r})=\Upsilon^{+}(\vec{r})\Upsilon(\vec{r})$

In this case we can put components of $\Upsilon(x)$ and $\Upsilon^{+}(x)$ in
the form of real functions which do not vary in spin space and depend on $x$:
$\Upsilon^{+}(x)=\chi(x)(s_{u\uparrow},s_{u\downarrow},s_{v\uparrow},s_{v\downarrow})$,
$s_{u\uparrow},s_{u\downarrow}$ and $s_{v\uparrow},s_{v\downarrow}$ are
components of two vectors in spin space and are normalized by the condition

$\Upsilon^{+}(x)\Upsilon(x)=\chi^{2}(x)=n(x)\,.$

We arrive at the equation for the scalar function $\chi(x)$:

$\frac{\hbar^{2}}{2m}\frac{d^{2}\chi}{dx^{2}}=\alpha\chi+\beta\chi^{3}-\vec{H}\frac{\partial\vec{M}}{\partial(n(x))}\chi$
(18)

If we assume $\lim|_{\chi\Rightarrow
0}\,\,\,\left\\{\chi*\frac{\partial\vec{M}}{\partial(n(x)}\right\\}\Rightarrow
0$, the Eq.(18) has two uniform solutions: 1) the solution $\chi\equiv 0$ for
the insulating phase, 2) the solution $\chi_{0}$ for the metallic phase:
$\chi_{0}^{2}=|\alpha|/\beta\,.$ The function $\chi(x)$ satisfies the
following boundary conditions: $\chi=0,\,$ at $\,x=0$ and
$\chi\Rightarrow\chi_{0}\,,\,\,$ at $x\Rightarrow\infty$. We multiply (18) by
${d\chi}/{dx}$ and find the first integral of it:

$-\frac{\hbar^{2}}{2m}\left(\frac{d\chi}{dx}\right)^{2}-|\alpha|\chi^{2}+\frac{\beta}{2}\chi^{4}-\vec{H}\vec{M}(\chi^{2})=C_{0}$
(19)

$C_{0}$ can be found from the boundary conditions. So,

$C_{0}=-|\alpha|\chi_{0}^{2}+\frac{\beta}{2}\chi_{0}^{4}-\vec{H}\vec{M}(\chi_{0}^{2})\,;\,\,\chi^{2}(x)\equiv
n(x)\,.$
$\displaystyle\frac{\hbar^{2}}{2m}\left(\frac{d\chi}{dx}\right)^{2}=-|\alpha|(\chi^{2}-\chi_{0}^{2})+\frac{\beta}{2}(\chi^{4}-\chi_{0}^{4})-$
$\displaystyle-\vec{H}(\vec{M}(\chi^{2})-\vec{M}(\chi_{0}^{2})).$ (20)

From the last equation the function $\chi(x)$ can be extracted in the implicit
form. And the surface tension due to the order parameter $\chi$ can be written
as follows:

$\displaystyle\Sigma_{\Upsilon}=\hbar\sqrt{\frac{\beta}{m}}\int_{0}^{\chi_{0}}d\chi\Gamma^{1/2}(\chi)\theta(\Gamma(\chi))\,\,,\,\,\,\,$
$\displaystyle\Gamma(\chi)=\left[(\chi_{0}^{2}-\chi^{2})^{2}+2\beta^{-1}\vec{H}(\vec{M}(\chi_{0}^{2})-\vec{M}(\chi^{2}))\right]$
(21)

Here $\theta(\upsilon)=0\,\,$ for $\upsilon<0\,\,$ ;
$\,\,\theta(\upsilon)=1\,\,$ for $\,\,\upsilon>0\,.$ The presence of
$\theta(\Gamma(\chi))$ in the integrand of (New Approach to Metal–Insulator
Phase Transition Kinetics in Magnetic Field.) leads to the appearance of a
singularity of the value $\Sigma_{\Upsilon}$ for $H=0$. It leads also to
singularities in the points where $\vec{M}$ becomes zero. If $\vec{H}\equiv
0$,

$\Sigma_{\Upsilon 0}=\Sigma_{\Upsilon}|_{H\Rightarrow
0}=\frac{2\hbar}{3\beta\sqrt{m}}|\alpha|^{3/2}$ (22)

Below we consider $\Sigma_{\Upsilon}$ for Pauli paramagnetism and Landau
diamagnetism LandLif . In both cases the magnetic moment $\vec{M}$ can be
represented as $\vec{M}=\mu_{H}\vec{B}$. In the case of the Pauli
paramagnetism $\mu_{H}=\mu_{para}=1/2*\left(1/2\pi\right)^{2}\left(e^{2}/\hbar
c\right)\left(p_{F}/mc\right)$. In the case of the Landau diamagnetism
$\mu_{H}=\mu_{dia}=-\frac{1}{3}\mu_{para}$; $p_{F}=(3\pi^{2})^{1/3}\hbar
n^{1/3}$ is Fermi momentum. $\Sigma_{\Upsilon}$ (New Approach to
Metal–Insulator Phase Transition Kinetics in Magnetic Field.) equals:

$\displaystyle\Sigma_{\Upsilon}=1.5*\Sigma_{\Upsilon
0}\int_{0}^{1}dx\Gamma_{1}^{1/2}(x)\theta(\Gamma_{1}(x))\,\,,$
$\displaystyle\Gamma_{1}(x)=\left[(1-x^{2})^{2}\pm
DG_{\pm}(1-x^{2/3})\right]^{1/2}$ (23)

$D=2\beta\mu_{eff}H^{2}/|\alpha|,\,\mu_{eff}=1/(2\pi)^{2})(e^{2}/\hbar
c)(p_{eff}/mc),$

$p_{eff}=\hbar\left(3\pi^{2}\aleph\rho_{c}/\beta\right)^{1/3}\,.$

Here $\mu_{eff}$ is the effective susceptibility and $p_{eff}$ is the
effective Fermi momentum. In the low magnetic field ($D\ll 1$) we represent
$\Sigma_{\Upsilon}$ as follows:

$\Sigma_{\Upsilon}=\Sigma_{\Upsilon 0}+\Delta\Sigma_{\Upsilon}\,.$ (24)

In the cases of Pauli paramagnetism and Landau diamagnetism we have

$\displaystyle\Delta\Sigma_{\Upsilon,Pauli}\cong
0.536H^{2}\frac{3^{1/3}}{4\pi^{2/3}}\frac{\hbar
e^{2}}{cm^{3/2}}\frac{1}{|\alpha|^{1/6}\beta^{1/3}}$
$\displaystyle\Delta\Sigma_{\Upsilon,Landau}\cong-0.290H^{2}\frac{3^{1/3}}{4\pi^{2/3}}\frac{\hbar
e^{2}}{cm^{3/2}}\frac{1}{|\alpha|^{1/6}\beta^{1/3}}$ (25)

When $T$ goes to $T_{c}$, $|\alpha|\Rightarrow 0$ and
$|\Delta\Sigma_{\Upsilon}|$ rises. The value is restricted by the condition
$D\ll 1$. It means a reduction of surface tension $\Sigma_{\Upsilon}$ in the
case of Landau diamagnetism and an increase of it in the case of Pauli
paramagnetism. The asymmetry of behavior of the value of $\Sigma_{\Upsilon}$
between Landau diamagnetism and Pauli paramagnetism increases for larger
values of $D$. In the limit $D\gg 1$ we have for Pauli paramagnetism
$\Sigma_{\Upsilon}\gg\Sigma_{\Upsilon 0}$:

$\Sigma_{\Upsilon;Pauli}\cong
G_{+}^{1/2}|\alpha|^{-1}\mu_{eff}^{1/2}H\Sigma_{\Upsilon
0}\sim|\alpha|^{2/3}\,\,,$ (26)

For Landau diamagnetism:

$\Sigma_{\Upsilon;Landau}\equiv 0\,,\,\,\,\,D\succeq
4.645\,\,.\,\,\,\,\,\,\,\,\,$ (27)

It is seen from (New Approach to Metal–Insulator Phase Transition Kinetics in
Magnetic Field.) that $\Sigma_{\Upsilon}\geq 0$. It should be emphasized that
the value of surface tension in magnetic field $\Sigma_{\Upsilon}$ can
strongly differ from the value of $\Sigma_{\Upsilon 0}$ in the absence of
magnetic field. In particularly, $\Sigma_{\Upsilon}$ decreases for the case of
Landau diamagnetism and can be done even zero. In the opposite case of Pauli
paramagnetism $\Sigma_{\Upsilon}$ can be done substantially more than
$\Sigma_{\Upsilon 0}$. This effect leads to a strong asymmetry of the phase
transition kinetics for diamagnetic and paramagnetic materials in the magnetic
field. This effect is very important even in low magnetic fields. It is
connected with the exponential dependence of probability for thermal
nucleation as a function of the value of surface tension. The probability in
the process of phase transitionLandLif  equals approximately:

$W=\omega_{th}\exp{\left(-\frac{U_{0}}{T}\right)}s^{-1}\,;\,\,\,\,\,\omega_{th}=\frac{\omega_{0}V}{4\pi
R_{c}^{3}/3}\,.$ (28)

Here $V$ is the volume of the system, $R_{c}$ is the critical radius of
thermal fluctuations and $\omega_{0}$ is the frequency of heterophase
fluctuations in the metastable phase, i.e. the characteristic frequency of
small oscillations in the potential well of a nucleus of the stable phase.
Usually, $\omega_{0}$ can be estimated numerically by the order of the
magnitude as about the Debye frequency $\omega_{D}$. It should be emphasized
that the value of $\omega_{th}$ is really great (cf.LifKagan ; budu2 ):

$\omega_{th}\simeq\exp{(N_{0})}\,s^{-1}\,,\,\,\,N_{0}\gg 1\,.$ (29)

For the system with the volume $V\approx 1cm^{3}\,,\,\,\,\omega_{D}\approx
10^{11}s^{-1}$, and the critical radius $R_{c}$ of about several interatomic
distances, we have $N_{0}\approx 80$. We should note that in real experiment
the critical $R_{c}$ should be not more than several interatomic distances. In
opposite case (see e.g.budu2 ), the time of decay of the metastable system
($\sim 1/W$) will exceed the time of existence of our Universe. The same
dramatic situation takes place also for quantum nucleationLifKagan ; budu2 .
Small corrections of $R_{c}$ of the order of $N_{0}^{-1}$ lead to modulation
of the value $W$ of the order of unity (see (29)). For example, if a variation
of the surface tension is of the order ten percent, the modification of $W$ is
about $\sim 2500$ or $\sim 1/2500$. So, if $D\sim 0.1$ in Eq (New Approach to
Metal–Insulator Phase Transition Kinetics in Magnetic Field.), $W$ increases
in $\sim 2500$ times for the case of Landau diamagnetism and decreases in
$\sim 2500$ times for the case of Pauli paramagnetism. In the vicinity of
phase transition the value $D\sim 1$ can be reached even for extremely small
magnetic fields because $|\alpha|\Rightarrow 0$. So, the probability for
thermal nucleation $W$ in the presence of magnetic field for diamagnetic
substances is much more than that for paramagnetic substances. Probably, this
fact accounts for the well known diamagnetic behavior of all semimetals kik .
The possible reason of this behavior is the kinetic nature of stable phase
nucleation in the metastable environment with a small barrier due to the small
surface tension. So, the growth of diamagnetic semimetals is much more
probable than of paramagnetic ones. To verify this assumption experimentally
it is necessary to investigate the phase transition kinetics by shielding the
magnetic field. The other possible verification can be done by carrying out
the experiment in an altering in time magnetic field.

In summary, it is built a new approach to MI phase transition kinetics. It is
based on two order parameters of sufficiently different nature. The equations
of GL type are formulated for these two order parameters. Effects of magnetic
field are included in the kinetics of MI phase transition kinetics. The exact
solution of equations is obtained for paramagnetic case (11) with nontrivial
behavior as a function of the uniform magnetic field $\vec{H}$. It is shown a
radically different behavior of MI phase transition for Landau diamagnetism
and Pauli paramagnetism cases.

The author thanks S.N.Burmistrov and Yu.Kagan for valuable critical comments.
The work is supported in part by the Grant of RFBR No 16-02-00382.

## References

  * (1) N. F. Mott, ”Metal–Insulator” Second Edition Taylor@Francis London–New York–Philadelphia,1990.
  * (2) L.D. Landau, E.M. Lifshitz . Statistical Physics. Part 1: Course of Theoretical Physics, Vol. 5 (3rd ed.) Publisher: Butterworth-Heinemann; 3 edition, 1980.
  * (3) P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge university press, 1995.
  * (4) Jinwu Ye, Phys. Rev. Lett., 97, 125302 (2006)
  * (5) L. Dubovskii, JETP Letters 99, No. 1, pp. 22-26 (2014).
  * (6) L. Dubovskii, J. Low Temp. Phys. 182, 192-205 (2016).
  * (7) J. W. Cahn and J. E. Hillard, J. Chem. Phys. 28 ,258 (1958); ibid. 42 , 93, (1964).
  * (8) I. M. Lifshits and Yu. Kagan. Sov. Phys. JETP 35, 206 (1972)
  * (9) L. P. Pitaevskii, E. M. Lifshitz. Statistical Physics, Part 2. Vol. 9 (1st ed.), 1980.
  * (10) Tables of physical quantities. Editor I. K. Kikoin. M., Atomizdat, 1976 (in Russian).
  * (11) M.B. Partensky UFN, 128, 69 (1979).
  * (12) G. Vignale, M. Rasolt, D. J. W. Geldart, Adv. Quantum Chem. 21, 235 (1990).
  * (13) G. Vignale and M. Rasolt, Phys. Rev. Lett. 59, 2360 (1987); G. Vignale, PRB 37 10865 (1988).
  * (14) V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546
  * (15) P. G. De Gennes. Superconductivity of Metals and Alloys, W. A. Benjamin, INC. New York, 1966.
  * (16) E. G. Maksimov. Phys. Usp. 53, 1185 1190 (2010).
  * (17) A. K. Rajagopal and J. Callaway, Phys. Rev. B7, 1912 (1973).
  * (18) C. J. Grayce, R. A. Harris, Phys. Rev. A50, 3089 (1994).
  * (19) L.D. Landau, E.M. Lifshitz. Vol. 8. Electrodynamics of Continuous Media (2ed., Pergamon), 1984.
  * (20) A.A. Abrikosov. Fundamentals of the theory of metals, NY Butterworth-Heinemann, 1988.
  * (21) S. N. Burmistrov and L. B. Dubovskii, Sov. Phys. JETP 66, 414 (1987).