# Concise guide for electronic topological transitions

A.A. Varlamov  CNR-SPIN, c/o Dipartimento DICII, Universitá “Tor Vergata”,
Viale del Politecnico 1, I-00133 Rome, Italy  Y.M. Galperin  Department of
Physics, University of Oslo, 0316 Oslo, Norway  A.F. Ioffe Physico-Technical
Institute of Russian Academy of Sciences, Polytekhnicheskaya 26, 194021 St.
Petersburg, Russia  S.G. Sharapov  Bogolyubov Institute for Theoretical
Physics, National Academy of Science of Ukraine, 14-b Metrologicheskaya
Street, Kiev, 03143, Ukraine  Yuriy Yerin  Dipartimento di Fisica e Geologia,
Universitá degli Studi di Perugia, Via Pascoli, 06123 Perugia, Italy

###### Abstract

In this short review we pass through the milestones in the studies of the
electronic topological transitions (ETT) and focus on some recent applications
of the ideas worked out in their classical theory. These are: two-dimensional
electron systems, de Haas–van Alphen effect, classification of ETT in
multidimensional systems, superconductivity in systems close to ETT,
thermoelectricity in heavy-fermion systems, where the cascades of topological
changes of Fermi surface (FS) are generated by magnetic field. The history of
studies of ETT is inextricably linked with Kharkov school of condensed matter
physics, with such names as I.M. Lifshitz, V.G. Bar’yakhtar and many other.
Among them is Moisey Isaakovich Kaganov, who contributed much in studies of
the role of geometry and topology of FS in physical properties of the metals.
Two of the authors (A.V. and Y.G.) had a honor and pleasure to work with
“Musik”, as all friends called Kaganov Blanter1994 ; all of us have been
learning the niceties of science from his books. “The Fermi surface is the
stage on which the drama of the life of the electron is played out” wrote
Kaganov and Lifshitz. We devote this work to their memory.

##  I Origin and adolescence of ETT

The discovery of a surprising new phase transition in liquid helium by W.H.
Keesom and coworkers in Leiden in 1932 who observed a lambda-shaped “jump”
discontinuity in the temperature dependence of the specific heat followed by
the first classification of the general types of transition between phases of
matter, introduced by Paul Ehrenfest in 1933 (see the historical overview
Jaeger1998 ). The proposed classification of phase transitions is based on the
discontinuities of the derivatives of the free energy as a function of the
parameter governing transition. The lowest derivative of the free energy that
is discontinuous at the transition point labels the kind of the transition in
this scheme. First-order phase transitions exhibit a discontinuity in the
first derivative of the free energy with respect to some thermodynamic
variable. Second-order phase transitions are continuous in the first
derivative but exhibit discontinuity in a second derivative of the free
energy. Under the Ehrenfest classification scheme, there could in principle be
third, fourth, and higher-order phase transitions. This classification later
was applied to the study of the transition from the normal to the
superconducting state in metals, discussing liquid-gas, order-disorder, and
paramagnetic-ferromagnetic phase transitions.

As is known, the shape and size of the Fermi surface (FS) may change under the
influence of such factors as uniform compression, changes in the concentration
of ingredients in the alloy and anisotropic deformation Varlamov1989 . In
particular, changes may occur in the topology of the FS: individual voids may
appear and disappear, or a closed FS may transform into an open one. In 50-ies
Mott and Jones Mott1936  made a general observation that singularities in the
electron state density can manifest themselves in certain characteristics of a
metal. Later on, Van Hove VanHove1953  noted that the quasiparticle density of
states in a crystal possesses singularities at the energies corresponding to
change in the FS topology. The anomalies associated with the
$2\frac{1}{2}$-order phase transition were discussed by Ziman Ziman1956  when
he tried to explain the behaviour of the electrical resistance of copper at
low temperature. Finally, in the seminal paper Lifshitz1960  I.M. Lifshitz
consistently formulated the idea of an electron topological transition in
1960, and it was this paper that has stimulated further experimental and
theoretical studies in this direction.

I.M. Lifshitz noticed that changes in topology manifest themselves as certain
anomalies in thermodynamic and galvanomagnetic characteristics of the metal.
He performed a theoretical study of these anomalies and found that, at zero
temperature and in the absence of impurity electron scattering, the
topological changes occur when the Fermi energy reaches a certain critical
value $\epsilon_{c}$. In this case, if the energy of quasi-particles,
$\epsilon(\mathbf{p})$, has an absolute extreme, a void is either formed or
disappears, whereas if $\epsilon_{c}$ is a hyperbolic singular point, then at
$\epsilon=\epsilon_{c}$ there occurs a disruption of the “neck” that connects
two parts of the FS, and an open FS may become closed.

Figure 1:  Panels present two type of Lifshitz transition: neck disruption and
void formation. The plots for conductivity are reproduced from Pfau2017PRL
and are based on Refs. Varlamov1985 ; Varlamov1989  for the clean case and are
presented for three different temperatures (a more smooth curve corresponds to
the higher temperature)

.

Near such transition in 3D metal and at $T=0$ the density of electron states
can be presented as a sum Lifshitz1960

$\nu(\epsilon)=\nu_{0}+\frac{(2m_{1}m_{2}m_{3})^{1/2}}{\pi^{2}\hbar^{3}}|\epsilon-\epsilon_{c}|^{1/2}\Theta\left[\pm(\epsilon-\epsilon_{c})\right]$
(1)

of a smooth function $\nu_{0}$ and an additional term $\delta\nu(\epsilon)$
associated with the transition, which is non-zero only on one side of critical
point $\epsilon_{c}$: on those where either a new void is formed or a neck is
disrupted. Here $m_{i}$ are modules of the principal values of an effective
mass tensor and $\Theta$ is the Heaviside function.

One can easily check that such discontinuity in the derivative of the density
of states results in the appearance at $T=0$ of a non-smooth contribution to
the thermodynamic potential:

$\delta\Omega(z)\sim|z|^{5/2}\Theta\left(\pm z\right)$ (2)

where $z=\epsilon-\epsilon_{c}$ is the energy parameter driving transition. It
characterizes the closeness of the system to the point where the number of
components of topological connectivity of the FS changes. This parameter can
be biased by hydrostatic pressure, anisotropic deformation, doping by some
isovalent impurities, magnetic field etc.

Evidently, the second derivatives of the thermodynamic potential, i.e.
specific heat capacity, diamagnetic susceptibility, have a vertical kink
$\sim|z|^{1/2}\Theta\left(\pm z\right)$. The same non-smoothness appears also
in conductivity Lifshitz1960

$\delta\sigma(z)\sim|z|^{1/2}\Theta\left(\pm z\right).$ (3)

In their turn, the third derivatives (for instance, the coefficient of thermal
expansion) from one side of the transition point demonstrate singularity
$\sim|z|^{-1/2}$. It is from this observation I.M. Lifshitz has named such
electron transitions at $T=0$ and in the absence of impurity scattering as the
$2\frac{1}{2}$-order phase transitions.

At finite temperature however (or when there are impurities that scatter
electrons) the singularity at $z=0$ becomes smeared out, so, strictly
speaking, in this case we cannot call such topological transformation of the
FS a phase transition. To distinguish these cases from the true
$2\frac{1}{2}$-order phase transition, we will call the latter as Lifshitz, or
an electronic topological transition (ETT).

Following the prediction by I.M. Lifshitz of a possibility of changing the FS
topology and related effect on the physical properties of metals had drawn the
attention of both theoreticians and experimentalists (for detailed review see
Varlamov1989 ; Blanter1994  and references therein). Yet, the low temperature
measurements of such tiny changes in heat capacity, or precise study of the
coefficient of thermal expansion were found to be rather complicated, and this
fact decelerated the studies of electronic topological transitions.

As an important milestone in this initial period of the studies of Lifshitz
transitions should be mentioned the paper by Baryakhtar and Makarov BM1965 .
The authors studied the effect of FS topology change under the effect of
pressure on the superconducting transition temperature and found that the
latter exhibits a considerable nonlinearity. Up to the present the discovery
of the $T_{c}(P)$ anomaly (see, e.g., BM1972 ; Higgins1998 ; Lazarev1966 ;
Ignat1968 ) is considered as an experimental proof of the Lifshitz transition
existence.

It is necessary to mention also the paper by M.A. Krivoglaz and Tuo Hao, who
had extended the Lifshitz transition thermodynamics for the case of metal with
impurities Krivoglaz1966 .

##  II Discovery of the giant anomaly in Seebeck coefficient and recursion to
topological transitions

The next stage in the studies of the electronic topological transitions
started at the beginning of 80-ies and was related to the discovery Egorov1983
of the giant peak in Seebeck coefficient of the alloy
$\text{Li}_{1-x}\text{Mg}_{x}$ when the concentration of magnesium approached
$x$=0.2 at. $\%$ (see Fig. 2). In according to the band calculations namely
this value corresponds to the touching of the alloy’s FS the Brillouin zone
edge, i.e. transition from the closed FS to the open one Vaks1981 . It was
observed as a small kink at room temperature, which developed in the well
pronounced peak at nitrogen temperatures with the further growth up to 30
times with respect to the background at the temperature of liquid helium.

Figure 2: The dependence of thermopower on magnesium concentration at
different temperatures $T=4.2$ K (filled circles), $T=78$ K (triangles) and
$T=4.2$ K (blank circles) Egorov1983 .

It is worth to recall that this discovery was preceded by the prediction of
the anomaly in thermoelectric coefficient based on the some naive explanation
Vaks1981 . The authors related the kink feature in the density of states
($\delta\nu\sim\sqrt{z})$ (see Eq. (1)) to the non-smoothness in the mean free
path of the charge carriers,

$l(\bf{p})=l_{0}(\bf{p})+|z|^{1/2}\Theta\left(\pm z\right)l_{1}(\bf{p}),$ (4)

where $l_{0}(\bf{p})$ is the mean free path of electron scattering at the
mainland of the FS with $\bf{p}$ being the quasiparticle momentum and
$l_{1}(\bf{p})$ is the term related to the singular region.

Then they used the Mott’s formula for the Seebeck coefficient
$S=\pi^{2}T/(3e)d(\ln\sigma)/d\epsilon|_{\epsilon=\epsilon_{F}}$ with
$\sigma=\sigma_{0}+\delta\sigma(z)$. Accordingly, $|z|^{-1/2}$ singularity in
thermoelectric coefficient in this approach occurs due to disruption of the
neck. Such additive treatment assumes that the electrons in the both pieces of
the FS are independent. Yet, a more detailed consideration raises questions:
(i) which value of the electron velocity has to be used to calculate
$\delta\sigma(z)$? If one chooses the value of the velocity at the just
disrupted neck, then its smallness would suppress the gained singularity. (ii)
What is the value of the relaxation time, corresponding to the electrons
scattered from the new piece of the FS?

The answers to these questions were given in the papers VP84 ; Varlamov1985 ;
AP86 ; VP88 , where the scattering of the electrons belonging to the FS with
variable number of the components of topological connectivity was studied. The
authors demonstrated that in the case of the ETT of the type of new void
formation three scenarios of the electron scattering on impurities are
possible. The first one is trivial: an electron from the mainland returns to
the mainland after the scattering from impurity. Corresponding probability in
the case of isotropic scattering is determined by the Fermi golden rule:

$\tau_{0}^{-1}=2\pi n_{\mathrm{imp}}\nu_{0}|U|^{2},$ (5)

where $n_{\mathrm{imp}}$ is the impurities concentration and $U$ is the
amplitude of the scattering potential, which we assume independent of
momentum. More precise analysis shows, that this probability depends on the
electron energy $\omega=\epsilon-\epsilon_{F}$, but very weakly: the
correction to Eq. (5) is $\sim\tau_{0}^{-1}\omega/\epsilon_{F}$. Namely this
smallness is responsible for the standard weakness of the Seebeck effect
($S\sim T/\epsilon_{F}$) in normal metals.

The second possibility for electrons after the scattering from impurity is to
travel from the main part of the FS to the small void. Probability of such
process is definitely suppressed by the smallness of the density of states in
the new void (i.e., available states for newcomer in result of scattering
process). However, this smallness for the Seebeck coefficient is over and
above compensated by the strong energy dependence of this contribution. The
last scenario of impurity scattering for electron from the new void is to
return back. This process can be neglected due to smallness of the
corresponding density of states which enters quadratically in the
corresponding probability.

As a result, the total relaxation time can be written as

$\tau_{(3)}^{-1}(\omega,z)\\!=\\!\frac{\tau_{0}^{-1}}{2\sqrt{\epsilon_{F}}}\left[\kappa\left(\pm\omega-\epsilon_{F}\right)\mp\kappa\left(\pm\omega\pm
z\right),\right]$ (6)

where the function

$\kappa(x)=\sqrt{2}\left[\left(\frac{1}{4\tau_{0}^{2}}+x^{2}\right)^{1/2}-x\right]^{1/2}$
(7)

appeared firstly in the paper of Krivoglaz and Tu Hao Krivoglaz1966 . The
choice of signs in Eq. (6) is the following: sign “+” corresponds to the
transition of the neck disruption type (hyperbolic point) while sign “-”
corresponds to the formation of the new void. More explicit form Eq. (6)
acquires in the clean limit, when $\tau_{0}^{-1}\ll T$:

$\tau_{(3)}^{-1}(\omega,z)\\!=\\!\tau_{0}^{-1}\left[1\\!-\\!\frac{\omega}{\epsilon_{F}}\\!-\\!\left(\frac{|z+\omega|}{\epsilon_{F}}\right)^{1/2}\Theta(\\!-\\!z\\!-\\!\omega)\right].$
(8)

Equation (6) allows to present the Seebeck coefficient for the system close to
ETT in the compact form valid for the wide range of impurity concentrations
($\tau_{0}^{-1}\ll\epsilon_{F}$) and temperatures:

$S(T,z)\sim\frac{k_{B}}{eT^{2}\tau_{0}}\int_{-\infty}^{\infty}\frac{\omega
d\omega}{\cosh^{2}\frac{\omega}{2T}}\tau(\omega,z).$ (9)

Figure 3: Panels present theoretical calculations of the signatures of the
Lifshitz transition in thermopower $S$. The plots are reproduced from
Pfau2017PRL  and are based on Refs. Varlamov1985 ; Varlamov1989  for the clean
case and presented for three different temperatures. The sign of the
thermopower (maximum or minimum) is defined by the type of charge carrier (see
Fig. 1).

It is essential that consideration of such problem in the model of neck
disruption reproduces the same result for Seebeck coefficient with replacement
$z\rightarrow-z$ and $\omega\rightarrow-\omega$, the latter means the
replacement of electrons by holes. Therefore the shape of singularity close to
ETT depends only on the character of topological changes of the FS but not on
its geometrical type, viz. void or neck.

The discovery of singularity in Seebeck coefficient in
$\text{Li}_{1-x}\text{Mg}_{x}$ alloy (see Fig. 2) became a trigger for
intensive study of transport phenomena in vicinity of very different
topological transitions, generated by isotropic pressure, anisotropic
deformation, inducing of the isovalent impurities and other methods (see
detailed references in Varlamov1989 ; Blanter1994 ).

##  III ETT in two dimensions

###  III.1 Dissimilarity with the three-dimensional case

End of the XX century was marked by the special interest in low dimensional
systems, this is why let us briefly review now the 2D case (we will follow
Ref. BTV94 ). It turns out that the density of states singularities,
oppositely to the 3D case, are different here depending on the transition
type. In the case of a void formation and an elliptical surface of the
spectrum, the non-smooth part of the density of states (DOS) turns out to be
proportional to
$\Theta\left({\pm\left({\varepsilon-{\varepsilon_{c}}}\right)}\right)$, while
in the case of neck disruption the DOS singularity is proportional to
$\ln\left({\varepsilon-{\varepsilon_{c}}}\right)$ N1966 ; MA1979 ; SK1970 . As
a consequence a singular part of the thermodynamic potential is proportional
either to ${z^{2}}\Theta\left({\pm z}\right)$ (void formation) or
${z^{2}}\ln\left|z\right|$ (neck disruption) at zero temperature and in
absence of a random potential (the role of the impurity scattering was
discussed in Ref. AKR1991 .

One can see that these two expressions are very different from each other. In
the case of neck disruption the singularity is nonzero from both sides of the
transition and it is even on $z$. This fact is a direct consequence of the
above mentioned symmetry of a system corresponding to simultaneous exchange of
holes and electrons branches in the spectrum and a change of $z$ sign (see the
discussion below Eq. (9). In this case the second derivative of the
thermodynamic potential has a logarithmic singularity at the transition point.

In the case of a void formation the DOS becomes nonzero from one side of the
transition only (those corresponding to the new void). First derivatives of
the thermodynamic potential have kinks at $z=0$, and the second ones break.

In accordance with Ehrenfest terminology, both two-dimensional void formation
and neck disruption must be referred as second order phase transitions.
However, it is important to note that this terminology seems to be too general
and can give rise to some misunderstanding. In fact, in the current
literature, the term “second order phase transition” is strictly associated
with the phenomena described by the Landau theory, where it was supposed that
the dependence of the thermodynamic potential on the order parameter is
different, above and below $T_{c}$, due to the special choice of the
temperature dependence of the coefficient in the quadratic term. So, in spite
of the common features between the types of transitions, the 2D void formation
and neck disruption have to be differentiated from the second-order phase
transitions in the Landau sense.

Figure 4: Theoretical calculations of the thermal power $S(z)$ for the case of
the Lifshitz transition of the neck disruption type in two dimensions. The
plot is reproduced from Ref. BPV92 , two curves correspond to different values
of electron concentrations.

###  III.2 Thermopower close to the 2D ETT of the neck disruption type

Slight differences between the two aforementioned ETT lead naturally to some
differences for the thermodynamic and kinetic properties. The two most
remarkable examples are the thermal expansion coefficient and thermoelectric
power. The first one does not exhibit any singularities at all at the point of
void formation and exhibits a sharp peak in the temperature dependence (with
height proportional $\mu/T$) in the case of neck disruption N1966 .
Thermopower dependence versus $z$ exhibits a sharp asymmetric peak with height
$\sim\mu/T$ in the case of void formation (in the 3D case it was
$(\mu/T)^{1/2}$) )AKR1991 ; Z2 .

Special attention in 2D systems requires the case of neck disruption. Here,
unlike the 3D case, the system acquires a new type of symmetry: the relaxation
time remains unchanged when one replaces the electrons by holes and changes
the sign of $z$:

$\tau_{(2)}^{-1}(\omega,z)=\frac{2}{\pi}\tau_{0}^{-1}\ln\frac{4\epsilon_{F}}{|\omega+z|}.$
(10)

Correspondingly, this fact dramatically affects the shape of singularity in
Seebeck coefficient: instead of well pronounced asymmetry with which we are
already familiar for both types of transition in 3D case and void formation in
2D case, in the case of 2D neck disruption the dependence $S(z)$ is even, with
a minimum and maximum of the same height Z2 ; BPV92  (see Fig. 4).

###  III.3 Topogical transitions in a quasi-2DEG: Quantization of entropy

The characteristic example of the ETT in two dimensions is the filling by
electrons of the empty subband formed due to the size quantization in the
quasi-two-dimensional electron gas (quasi-2DEG). Each intersection of the
levels of electron size quantization by the chemical potential (when $\mu$
passes the level $E_{N}$), can be considered as the point of the Lifshitz
phase transition of the void formation type, where the FS acquires a new
component of topological connectivity.

In gated structures, a series of Lifshitz transitions can be controlled by the
gate voltage. Corresponding anomalies in the thermodynamic and transport
characteristics, in particular, thermoelectric coefficient related to the
peculiarities of the energy dependence of the electron momentum relaxation
time have been studied experimentally and theoretically in Refs. Z1 ; Z2  and
BPV92 ; AKR1991 , respectively. In particular, ETTs spectacularly manifest
themselves in behavior of differential entropy (entropy per electron),
$s\equiv V^{-1}(\partial\mathcal{S}/\partial n)_{T}$ ( $\mathcal{S}$ is the
entropy, $V$ is the volume of the system which below we will assume to be
unit). An elegant way to measure this quantity directly was recently
demonstrated Pudalov .

As was shown Varlamov2016 ; Galperin2018 , the quantization of the energy
spectrum of quasi-2DEG into subbands leads to a very specific quantization of
the entropy: $s$ exhibits sharp maxima as the chemical potential $\mu$ passes
through the bottoms of size quantization subbands ($E_{i}$).

Figure 5: Schematic representation of the dependencies of the electronic
density of states (upper panel) and the partial entropy (lower panel) as
functions of the chemical potential. The plot is reproduced from Varlamov2016
.

Indeed, the density of electronic states (DOS) in a non-interacting 2DEG and
in absence of scattering has a staircase-like shape Ando :

$\nu(\mu)=\frac{m^{\ast}}{\pi\hbar^{2}}\sum\limits_{i=1}^{\infty}\Theta\left(\mu-
E_{i}\right),$ (11)

with $m^{\ast}$ being the electron effective mass. The differential entropy,
$s$, can be found using the Maxwell relation

$s=\left(\frac{\partial\mathcal{S}}{\partial
n}\right)_{T}=-\left(\frac{\partial\mu}{\partial
T}\right)_{n}=\left(\frac{\partial n}{\partial
T}\right)_{\mu}\left(\frac{\partial n}{\partial\mu}\right)_{T}^{-1}.$ (12)

The relationship between the electron concentration $n$, chemical potential
$\mu$ and temperature $T$ can be found integrating Eq. (11) over energies with
the Fermi-Dirac distribution:

$n\left(\mu,T\right)=\frac{m^{\ast}}{\pi\hbar^{2}}\sum\limits_{i=1}^{\infty}\int\limits_{0}^{+\infty}\frac{{\Theta}(E-E_{i})}{\exp\left(\frac{E-\mu}{T}\right)+1}\,dE.$
(13)

Using Eq. (13) one can express $s$ in the form of sums over the subband levels
averaged over energy with temperature smearing factor.

In the absence of scattering this result is independent of the shape of the
transverse potential that confines 2DEG and of the material parameters
including the electron effective mass and dielectric constant. The value of
the entropy in the $N$-th maximum depends only on the number of the maximum,
$N$:

$s|_{\mu=E_{n}}\equiv\left(\frac{\partial\mathcal{S}}{\partial
n}\right)_{T,\,\mu=E_{n}}=\frac{\ln 2}{N-1/2}\,.$ (14)

The universality of the above quantization rule can be broken both by disorder
and by electron-electron interactions. Using a simple model of Lorentzian
smearing of the electronic spectrum one can see that it leads to the relative
correction of $\sim 1/T\tau$, where $\tau$ is the electron life time. For the
case of a single-band 2DEG the role of the electron-electron interaction in
the partial entropy was investigated in Pudalov .

The variation of differential entropy $s$ versus chemical potential is
schematically shown in Fig. 5, lower panel. The peaks in it correspond to the
steps of the density of states shown in the upper panel of the same figure.
This dependence can be interpreted in the following way. At low temperatures,
the main contribution to the entropy is provided by the electrons having
energies in the vicinity of the Fermi level, the width of the ‘active’ layer
being $\sim T$. If the electron DOS is constant within the layer then by
adding an electron one does not change the entropy. Hence, the entropy is
independent of the chemical potential, $(\partial S/\partial n)_{T}\rightarrow
0$. However, if the bottom of one of the subbands falls into the active layer,
the number of ‘active’ states becomes strongly dependent on the chemical
potential. In this case, adding an electron to the system, one changes the
number of ‘active’ states in the vicinity of the FS. Consequently, the partial
entropy strongly increases. The peaks of the partial entropy correspond to the
resonances of the chemical potential and electron size quantization levels.
The further increase of the chemical potential brings the system to the region
of the constant density of states, where the partial entropy vanishes again.

At $T\rightarrow 0$ (yet $T\gtrsim\hbar/\tau$ ) the peaks of $s$ are located
at $\mu\rightarrow E_{N}$, $N>1$, the maximal values being $s_{\max}(N)=\ln
2/(N-1/2)$. At finite $T$ the peaks acquire finite widths of the order of $T$
and shift toward negative values of $\mu-E_{N}$ analogously to the shift of
the maximum in the Seebeck coefficient (see Fig. 3).

##  IV  De Haas-van Alphen effect as the 3/2 order phase transition

Let us discuss the situation when the magnetic field $H$ is applied to the
metal and the cross-section of the FS in the direction perpendicular to $H$ is
closed (we will follow Ref. BTV94 ). The motion of the electrons is quantized
in this plane while remains free and effectively one-dimensional in the
magnetic field direction, viz. because he energy spectrum depends on one
momentum projection only. It is important to note that the intersection of the
sequent Landau level and chemical potential level (see Fig. 6) can be treated,
in certain sense, as a $1\frac{1}{2}$ order ETT.

Figure 6: Fermi sphere in magnetic field is transformed in a set of Landau
cylinders, corresponding to the filling by electrons of the degenerated states
at the Landau levels.

Indeed, if one denotes Landau level energies (which can be found from the
Lifshitz-Onsager quantization rule for reasonable fields) as
${E^{\left(n\right)}}$, then the DOS singularity in the vicinity of this
crossing reads LAK73

$\delta\nu\left(\varepsilon\right)\sim{\left({\varepsilon-{E_{n}}}\right)^{-1/2}}\Theta\left({\varepsilon-{E_{n}}}\right).$
(15)

It is clear from the last expression that the number of available for
electrons states for $n$-th level is non-smooth and proportional to
${\left({\varepsilon-{E_{n}}}\right)^{1/2}}\Theta\left({\varepsilon-{E_{n}}}\right)$
and the non-smooth part in the thermodynamic potential turns out to be

$\delta\Omega\left(\varepsilon\right)\sim{\left({\mu-{E_{n}}}\right)^{3/2}}\Theta\left({\mu-{E_{n}}}\right)$
(16)

at zero temperature.

We have to note that the chemical potential depends on the magnetic field due
to the particle number conservation law; nevertheless this dependence becomes
important in the ultra-quantum limit only, which is beyond the scope of our
discussion here LAK73 ; AS64 ; A69 .

It can be concluded that the de Haas-van Alphen oscillations arising when the
next Landau level passes through the chemical potential value, can be treated
as ETT to which the 1½ order has to be attributed. The second derivatives of
the thermodynamic potential tend to infinity at $T=0$ if $z(H)=E_{n}$, and at
finite temperatures exhibit sharp peaks with magnitude
${\left({\mu/T}\right)^{1/2}}$. This singularities are nothing else but
inverse magnetic field oscillations of magnetic susceptibility in the De Haas-
van Alphen effect. It is noteworthy that the kinetic coefficients behave
exactly as ETT in 3D case: conductivity (Shubnikov-de Haas effect),
thermopower VP89 ; PV89 , etc.

##  V Multi-dimensional ETT

After this review of the different phenomena showed by the electronic
topological transitions, we would like to mention some general features
characteristic for such transitions and to classify them formally for
arbitrary space dimensionalities $n$ (we will follow Ref. BTV94 ).

In the neighborhood of any non-degenerate critical point, the spectrum can be
described by a quadratic form and the equation of the Fermi hypersurface can
be expressed as

$\sum\limits_{i=1}^{n}{\frac{{p_{i}^{2}}}{{2{m_{i}}}}}=\epsilon-{\epsilon_{c}}.$
(17)

The shape of this hypersurface is determined by the signature of
$\left\\{{{m_{i}}}\right\\}=1,2,...,n$ and by the sign of
$\epsilon-{\epsilon_{c}}$. The topological type of transition (only the sign
of $\epsilon-{\epsilon_{c}}$ is changed), is established by the signature of
$\left\\{{{m_{i}}}\right\\}$ only.

Let us start the discussion from the 0D case. Here the Fermi sea of electrons
may be formally treated as a point which can be empty
($\epsilon<\epsilon_{c}$) or occupied by two electrons
($\epsilon>\epsilon_{c}$). Therefore, only the transition which can be
formally treated as the void formation type, can occur in this case (the void
does not exist or occupies all 0D space).

In the 1D case, the Fermi sea is presented by the interval
$\left[{-{p_{F}},{p_{F}}}\right]$. These two limiting points play the role of
the Fermi ”surface”. For the surface

$\frac{{p_{x}^{2}}}{{2{m_{x}}}}=\epsilon-{\epsilon_{c}}$ (18)

only two signatures, $[+]$ and $[-]$, are possible and both correspond to void
formation (of electron or hole types). So in this case, like in the 0D case,
only one type of transition, namely void formation, is possible. The same void
formation type transition is typical for the electron system of any
dimensionality and corresponds to the signatures $\left\\{{+++...+}\right\\}$
and $\left\\{{---...-}\right\\}$. But already in the 2D another type of
signature: $\left\\{{+-}\right\\}$ (or $\left\\{{-+}\right\\}$) for the
quadratic form Eq. (17) appears and establishes side by side with the void
formation, another type of ETT - the neck disruption. A similar type of
transition occurs in the 3D space for the signatures $\left\\{{++-}\right\\}$
and $\left\\{{--+}\right\\}$; the same as for higher space dimensions.

In the 4D space, besides the two transitions of the mentioned above nature,
the new transition type corresponding to the symmetrical signature
$\left\\{{++--}\right\\}$ appears.

In the general case of a $n$-dimensional space $n+1$ different signatures are
possible of which $\left\\{{+++...+}\right\\}$ and
$\left\\{{---...-}\right\\}$ describe a new electron and hole void formation,
respectively, $\left\\{{+++...+-}\right\\}$ and $\left\\{{---...-+}\right\\}$
correspond to electron/hole neck disruption and the other are associated with
more complex topological transformations. In general, the surface containing
$p$ pluses and $q$ minuses ($p,q>1$) can be imagined as $(p+q)$ dimensional
hyperboloid and $\frac{1}{2}n+1$ (if $n$ is even) or
$\frac{1}{2}\left({n+1}\right)$ (if $n$ is odd) different kinds of transitions
are possible.

It can be seen that if $n$ is even ($n=2k$) one among these transition types
(which corresponds to the symmetrical signature of $k$ positive and $k$
negative masses) possesses a remarkable symmetry: the system the system does
not vary if all electrons are interchanged with holes and the sign
$\epsilon-{\epsilon_{c}}$ changes also. This symmetry (it was already
described for the 2D system) must imply the DOS singularity dependence to be
even function of $\epsilon-{\epsilon_{c}}$. All other types of transitions do
not exhibit such a symmetry and we cannot expect consequently that the
singular part of the DOS will be even function of energy. It is worthwhile to
stress that such electron-hole symmetry exists for even dimensional spaces
only. For odd dimensions, the two hyperbolic surfaces beyond and above the
transition are not topologically equivalent.

The critical exponent of the DOS non-smoothness may be easily calculated for
the $n$-dimensional case of void formation. Really, the DOS is proportional to
(see LAK73 )

$\nu^{[n]}\left(\epsilon\right)\sim\int\\!{\delta\left[{\varepsilon-\varepsilon({\mathbf{p}})}\right]}{d^{n}}{\mathbf{p}}=\frac{d}{{d\epsilon}}\int\\!{\Theta\left[{\epsilon-\epsilon({\mathbf{p}})}\right]}{d^{n}}{\mathbf{p}},$
(19)

where the index $[n]$ denotes the dimensionality of the space. One can see
that the last integral presents nothing else than a volume inside the equal
energy surface ${\epsilon\left({\mathbf{p}}\right)}=\epsilon$ and is
proportional to the number of electron states. The singular part of it is
proportional to the volume of the $n$-dimensional ellipsoid,

$N^{[n]}(\epsilon)=\frac{\prod\limits_{i=1}^{n}{{p_{i}}}}{2^{n}\pi^{n/2}\hbar^{n}\Gamma(n/2+1)},$
(20)

where $p_{i}$ are the large semi-axis of the ellipsoid in momentum space. Due
to the homogeneity of the quadric form in Eq. (17), the volume of the
$n$-dimensional ellipsoid is proportional to
${\left({\epsilon-{\epsilon_{c}}}\right)^{n/2}}\Theta\left({\epsilon-{\epsilon_{c}}}\right)$.
So the non-smooth part of the DOS is proportional to $dN^{[n]}/d\epsilon$,
i.e.,

$\delta\nu^{[n]}(\epsilon)\sim{\left({\epsilon-\epsilon_{c}}\right)^{n/2-1}}\Theta\left(\epsilon-\epsilon_{c}\right).$
(21)

The contribution to the thermodynamic potential reads

$\delta\Omega^{[n]}\left(\epsilon\right)\sim{\left({\epsilon-\epsilon_{c}}\right)^{n/2+1}}\Theta\left(\epsilon-\epsilon_{c}\right).$
(22)

We can conclude now that $n$-dimensional formation of a void can be considered
as phase transition of $n/2+1$ order at zero temperature, and for high
dimensions ETT-related singularities will be manifested weakly. This
conclusion concerns other types of transitions too.

##  VI Revisited ETT in superconductors

It has been shown that abrupt topological modification of the FS can be
achieved by tuning of different macroscopic parameters such as temperature,
pressure, strain, external magnetic fields and doping.

Figure 7: Dependence of the change of superconducting transition temperature
in thallium and its alloys as a function of the pressure in kilo-atmospheres
for pure Tl (up, curve 1), Tl with 0.45 $\%$ at.(atomic percentage) Hg
(middle, curve 2) and Tl with 0.9 $\%$ at. Hg (bottom, curve 3). Curves $a$
represent linear contribution, while $b_{1}$, $b_{2}$ and $b_{3}$ are
nonlinear components of curves 1, 2, 3, respectively. The dashed lines are the
nonlinear parts of the dependence of $dT_{c}/dP$ on the pressure. The plot was
reproduced from Ref. BM1965 .

A dramatic reshaping of the FS topology viz. Lifshitz transition often leads
to the emergence of the plenty phenomena also in the superconducting phase of
a material and as a result to the unusual behavior and the alteration of many
properties relevant to superconductivity.

On of the first evidence of the Lifshitz transition in a superconductor was
revealed in the above mentioned paper of Baryakhtar and Makarov BM1965  where
the effect of pressure on the superconducting transition temperature $T_{c}$
in various metals was studied both theoretically and experimentally. It was
found that with the increasing of the pressure the critical temperature of
pure thallium and of thallium with Hg impurities undergoes a nonlinear
variation (see Fig. 7.).

Interpretation of such anomalous behaviour can be done within the BCS theory
of superconductivity. The corresponding equation for the temperature dependent
energy gap $\Delta(T)$ can be written in the following form:

$1=\lambda\\!\int\limits_{-{\omega_{D}}}^{{\omega_{D}}}\frac{\nu(\xi)\,d\xi}{2\sqrt{\xi^{2}+{\Delta^{2}(T)}}}\tanh\left({\frac{{\sqrt{{\xi^{2}}+{\Delta^{2}}\left(T\right)}}}{2T}}\right),$
(23)

where $\lambda$ is the constant of electron-phonon interaction, $\omega_{D}$
is the Debye frequency, and $\nu(\xi)$ is the density of states. In the
vicinity of ETT $\nu(\xi)$ is given by Eq. (1).

Theoretical analysis of the Eq. (23) for limiting cases $T=0$ and $T=T_{c}$
allows to interpret qualitatively the experimental data shown in Fig. 7. At
the relatively low pressures the Fermi energy $\epsilon_{F}$ is slightly
higher than $\epsilon_{c}$. Until the difference $\epsilon_{F}-\epsilon_{c}$
remains less than $\omega_{D}$ the critical temperature increases due to the
presence in Eq. (23) of the correction $\delta\nu(\xi)$ which strongly varies
as a function of energy. With the further increase of pressure, corresponding
growth is saturated, yet the other mechanisms enter in interplay: the non-
monotonic behaviour of the critical temperature is connected with the
dependence on pressure of other parameters of $\omega_{D}$, $\nu_{0}$, and
$\lambda$ and the presence of impurities (see Fig. 7, curves 2 and 3)
Lazarev1966 ; Ignat1968 .

Later the same theory was applied successfully for interpretation of the
behavior of the critical temperature as a function of pressure in In-Cd
Higgins1998 , in In-Sn and In-Pb alloys Zavaritskii1971 ; Smith1973 .

The similar qualitative dependence of $T_{c}$ as a function of lattice strain
was detected in strontium ruthenate
$\text{Sr}_{\text{2}}\text{RuO}_{\text{4}}$ that probably exhibits an exotic,
odd-parity kind of superconductivity Steppke2017 . Compressing the $a$ axis of
the $\text{Sr}_{\text{2}}\text{RuO}_{\text{4}}$ lattice drives the critical
temperature through a pronounced maximum, that is a factor of 2.3 higher than
$T_{c}$ of the unstrained material. The combination of experimental data and
theoretical calculations gives evidence that the observed maximum $T_{c}$
occurs at or near a Lifshitz transition when the Fermi level passes through a
Van Hove singularity Steppke2017 ; Sunko2019 ; Barber2019 .

It is worth note that after discovery of the copper-based high-temperature
superconductivity there was an attempt to exploit the theory of ETT for the
elucidation of the order parameter symmetry in such compounds Abrikosov1994 .
The model proposed by A. Abrikosov was based on two pillars: i) existence of a
large dielectric constant associates with the ion cores that leads to a weak
screening of the Coulomb forces and corresponding long range interactions of
conduction electrons; ii) existence of the extended saddle point singularities
and their dominant contribution to formation of the order parameter.
Assumption that the density of states and energy gap $\Delta(p)$ in the
singular region of the momentum space to be much larger than beyond, results
in the modification of the BCS self-consistency Eq. (23). Its solution
demonstrates the possibility of the $s$-wave and $d$-wave order parameter
symmetries coexistence, the absence of the Josephson effect in the
$c$-oriented junctions based on such compounds, their tunneling conductance as
well as the enhancement of the critical temperature.

Despite the fact that this model is rather simple and succeeded to explain
most of the apparently contradictory properties of high-temperature
superconductors at the moment of its proposition, it has not become widely
accepted.

In 2008 high-temperature superconductivity was surprisingly discovered in
iron-based materials. The presence of iron, for long time believed as an enemy
of superconductivity, led to the explosive growth in the study of basic
physical properties of this new family and comprehension of origin of the
unconventional pairing mechanism in these high temperature superconductors. It
was demonstrated by ARPES and other techniques that a common feature of iron-
based superconductors is the multiple-band electronic structure. Some of these
bands are very shallow with Fermi energies of several meV Liu2010 ; Liu2011 ;
Miao2014 ; Okazaki2014 . This circumstance allows to deplete such bands by
means of doping or pressure, and therefore to induce the Lifshitz transition
in such materials. As a result, complex multi-band topology gives rise to a
much richer nomenclature of effects in the vicinity of ETT in these
superconductors in comparison with their conventional counterparts.

One of the such an unusual phenomenon was predicted in a regime of BCS-BEC
(Bose-Einstein) crossover in a two-band superconductor when the chemical
potential $\mu$ is tuned in a narrow energy range around the edge
$E_{\text{edge}}$ of the second band Innocenti2010 .

Shifting the chemical potential for the superconductor one can track the
evolution of FS topology and the emergence of several Lifshitz transitions
(Fig. 8). When $\mu$ is increased above $E_{\text{edge}}$, and the system
undergoes an ETT with the opening of a new 3D FS (see Fig. 8 a,b). When $\mu$
reaches a higher energy threshold, the electronic structure undertakes a
second ETT, the 3D-2D ETT, where one FS changes topology from 3D to 2D, i.e.,
from “spherical” to “cylindrical” or vice versa (see Fig. 8 c,d). This ETT is
a feature of stacks of metallic layers, multilayers, or so-called
superlattices of quantum wells, and therefore it is typical for all existing
high-temperature superconductors and novel materials synthesized by material
design in the search for room temperature superconductivity.

Figure 8: Sketch of the evolution of the FS for a two-band superconductor by
shifting the chemical potential from (a) to (d) so that it crosses two
Lifshitz ETT. The first ETT occurs moving the Fermi level across the band edge
$E_{edge}$ of the second band so that the superconductivity goes from the
single FS in (a) with a single condensate to the two FS in (b) with two
condensates: the first (1) has a 2D topology and the second (2) has a 3D
topology. Changing $E_{F}$ the system crosses the critical energy $E_{3D-2D}$
where the second FS undergoes a 3D-2D ETT shown in panel (b) changing its
topology: the second closed 3D FS (b) becomes the tubular 2D FS in (d). The
first large 2D FS (1) remains nearly constant when the chemical potential is
shifted. Taken from Ref. Innocenti2010, .

In the general case of multi-band metal with deep primary bands and a shallow
one, when the bottom of the shallow band crosses the Fermi level, ETT causes
anomalous density of states with the qualitatively different behaviors on the
two sides of the Lifshitz transition in the superconducting phase Koshelev2014
. The mean-field theory gives the expression for the resulting density of
states

$\displaystyle{\nu_{s}}\left(E\right)$ $\displaystyle=$
$\displaystyle\frac{{{{\left({2m}\right)}^{3/2}}}}{{8{\pi^{2}}}}\operatorname{Re}\left[{\sum\limits_{\delta=\pm
1}{\left({\frac{{\left|E\right|}}{{\sqrt{{E^{2}}-\Delta_{0}^{2}}}}+\delta\operatorname{sgn}\left(E\right)}\right)}}\right.$
(24)
$\displaystyle\left.{\times\sqrt{\mu+\delta\sqrt{{E^{2}}-\Delta_{0}^{2}}}}\right],$

where $\Delta_{0}$ is the pairing amplitude (gap function) induced in the
shallow band by Cooper pair exchange with the deep bands. Eq. (24) was
obtained within conditions that energy gaps in each band are much smaller than
the Fermi energies for the deep bands, while the relation between $\Delta_{0}$
and $\mu$ can be arbitrary. Also, the term corresponding to the contribution
to pairing from the shallow band was excluded in self-consistent equations for
energy gaps.

It is important to note that the term with $\delta=-1$ contributes to Eq. (24)
only if $\mu>0$ and $\left|E\right|<\sqrt{{\mu^{2}}+\Delta_{0}^{2}}$. At
$\Delta_{0}=0$ Eq. (24) yields the normal density of states,

${\nu_{n}}\left(E\right)=\frac{{{{\left({2m}\right)}^{3/2}}}}{{4{\pi^{2}}}}\sqrt{E+\mu}\
\Theta\left({E+\mu}\right),$ (25)

while in the limit $\mu\gg{\Delta_{0}}$ the density of states recovers the
standard symmetric BCS shape

${\nu_{s}}\left(E\right)={\nu_{n}}\left(0\right)\frac{{\left|E\right|}}{{\sqrt{{E^{2}}-\Delta_{0}^{2}}}}.$
(26)

Equation (24) describes main peculiar features of a multiband superconductor
near the Lifshitz transition. On one side of the ETT the density of states
diverges at the energy equal to the induced gap, whereas on the other side it
vanishes.

##  VII Cascade ETT in heavy fermion systems

The specific feature of the extensively studied since 2006 heavy-fermion
systems is that the Lifshitz transitions are driven by a magnetic field (see,
e.g., Ref. Pourret2019JPSJ  for a brief overview). In these compounds, flat
quasiparticle bands with the widths often comparable to the Zeeman splitting
of the energy bands $(g_{\mathrm{eff}}/2)\mu_{B}H$ for field of order
$10\text{\,}\mathrm{T}$ cross the Fermi level. For example, a series of strong
anomalies in the thermoelectric power were recently observed both in the
ferromagnetic Kondo lattice material YbNi4P2 when the magnetic field varies in
the range from $0.4\text{\,}\mathrm{T}$ to $18\text{\,}\mathrm{T}$ Pfau2017PRL
and in the heavy fermion compound YbRh2Si2 when the field varies in the range
from $9.5\text{\,}\mathrm{T}$ to $13\text{\,}\mathrm{T}$ Pourret2019JPSJ .

The experimental findings of Ref. Pfau2017PRL  were theoretically interpreted
in terms of the series of the independent Lifshitz topological transitions as
considered above in Sec. II. The sign of the thermopower signature (maximum or
minimum) allows to determine the type of charge carrier (electrons or holes).
Moreover, resolving the shape of the peaks it is possible to identify the type
of Lifshitz transitions as neck or void type (see Fig. 1).

The theoretical analysis of Ref. Pfau2017PRL  is valid for standing apart
peaks, viz. when the width of the peak is much smaller than the distance
between them. Yet, comparing the experimental situations Ref. Pfau2017PRL  and
that one of Ref. Pourret2019JPSJ,  for YbRh2Si2 one can see that the minimal
distance between the anomalies in the former is $0.75\text{\,}\mathrm{T}$ (for
the most of the peaks it is a few Tesla), while all anomalies in
Pourret2019JPSJ  experiments are separated by the value of
$0.6\text{\,}\mathrm{T}$.

When the thermopower peaks are getting close to each other one should also
include the superposition phenomena between the new pockets occurring in the
narrow energy range. An indication that the the superposition of the different
Lifshitz transitions indeed occurs in the experiments on YbRh2Si2 , is that it
is hard to identify the type of the transition. The asymmetry of the
individual peaks which represents the characteristic feature of the Lifshitz
transition, is blurred in YbRh2Si2 as compared with YbNi4P2 .

To take into account the specificity of a cascade of Lifshitz transitions on
quasiparticle scattering the following model was suggested in Pourret2019JPSJ
. In the review we will not go into specific detail of YbRh2Si2 compound which
was also modelled by ab initio computations and following Ref. Pourret2019JPSJ
represent the general idea how to describe the cascade of Lifshitz
transitions.

We assume that all topological changes of the FS are due to the formation of
new spherical pockets (see Fig. 9). Here the index $i=0$ denotes to regular
part of the FS hereafter referred to as mainland while the critical parts are
labelled by $i=1,...N$. The quasiparticle excitation spectrum for each of them
can be presented as:

$\xi_{i}(\mathbf{p})=\frac{(\mathbf{p}-\mathbf{p}_{ci})^{2}}{2m_{i}}-Z_{i},\qquad
i=0,\ldots N,$ (27)

where $\mathbf{p}_{ci}$ is the position of its center in the Brillouin zone,
$Z_{i}=\mu-\epsilon_{ci}$ is its energetic size, $\epsilon_{ci}=\mu$ is the
critical point of multi-valued function of energy when the i-th voids appears,
$m_{i}$ is the corresponding effective mass.

Figure 9: (Color online) Schematic representation of the various quasiparticle
scattering processes between the pockets with the energy sizes $Z_{i}$.
$\tau_{i,j}$ correspond to the processes from pocket $i$ to pocket $j$
($\tau_{0,j}$ refers to the process from the main FS $Z_{0}$) through
impurities (crosses). Taken from Pourret2019JPSJ .

In Refs. Varlamov1985 ; VP88  it was demonstrated that the anomaly in the
Seebeck coefficient in the vicinity of Lifshitz transition is directly related
to the specific quasiparticle scattering starting from the bulk component of
the mainland FS (the energy size is $Z_{0}$) and ending at the small pocket of
the FS (characterized by the energy size $Z_{i}$). It was shown that the
latter is a trap for quasiparticles, their velocity here is small and they die
away. The next scattering with the dominating probability returns the
quasiparticle back to the mainland FS. Such forth and back scattering give
rise to the singularity of the Seebeck coefficient. At the first glance
generalization of the described scattering mechanism to the case of multiple
Lifshitz transitions seems to be trivial: it is just necessary to account for
the round trips of the quasiparticles from the mainland FS to all other
pockets ($Z_{1},Z_{2},Z_{3}\ldots$), see $\tau_{0,j}$ processes in Fig. 9.
Yet, as will be shown below, this is not enough. An important role is also
played by the “traveling” of the quasiparticles between the newborn FS
elements, $\tau_{1,j}$ and $\tau_{2,j}$ processes in Fig. 9. The expression
for the scattering time for the quasiparticles belonging to the “$l$”-th
pocket accounting for the return to the same pocket and round trips to the
smaller ones with $i>l$, in full analogy with the results of Refs.
Varlamov1985 ; VP88 , is given by:

$\tau^{-1}_{l}\left(\omega,Z_{l},..Z_{N}\right)=\sum_{i=l}^{N}{\tau}_{l,i}^{-1}(\omega,Z_{l}),$
(28)

where

$\tau_{l,i}^{-1}(\omega,Z_{l})=\frac{\kappa_{l}\left(-Z_{i}-\omega\right)}{2{\tau}_{l0}\sqrt{Z_{l}}}.$
(29)

Here the function

$\kappa_{l}(\varepsilon)=\left[{\left(\varepsilon^{2}+\frac{1}{4\tau_{l0}^{2}}\right)^{1/2}-\varepsilon}\right]^{1/2},$
(30)

with $\tau_{l0}^{-1}=\pi^{-1}n_{\mathrm{imp}}|U|^{2}m^{3/2}(2Z_{l})^{1/2}$ as
the relaxation time of the electron initially belonging to the l-th void and
calculated in the golden rule approximation (compare with Eqs. (5),(7). For
the sake of simplicity the effective masses for electrons in all voids are
assumed to be equal, $m_{i}=m$.

Finally, the expression for the Seebeck coefficient (9) becomes now the sum
over involved cascade scattering processes (marked in Fig. 9 by the black, red
and blue arrows, respectively) with the relaxation time $\tau(\omega,z)$
replaced by the corresponding for each term time
$\tau_{l}\left(\omega,Z_{l},..Z_{N}\right)$.

##  VIII Conclusion

As we have shown the field of fermiology, related to study of geometrical and
topological properties of FS, founded by I.M. Lifshitz still remains hot topic
and time to time emerges in different problems of condensed matter: in new
families of superconductors, heavy fermion systems, low dimensional systems,
etc. Considerable merit in it belongs to M.I. Kaganov, both due to his
original contributions and his review article Blanter1994  which became the
handbook for new generations of researchers.

###### Acknowledgements.

A.V. is grateful to V.S. Egorov for valuable discussion and clarification of
the historical circumstances. S.G.Sh. acknowledges a partial support by the
National Academy of Sciences of Ukraine grant “Functional Properties of
Materials Prospective for Nanotechnologies” (project No. 0120U100858). Y.Y.
acknowledges support by the CarESS project.

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