# Lifshitz transitions and angular conductivity diagrams in metals with
complex Fermi surfaces

A.Ya. Maltsev  V.A. Steklov Mathematical Institute of Russian Academy of
Sciences
119991 Moscow, Gubkina str. 8

###### Abstract

We consider the Lifshitz topological transitions and the corresponding changes
in the galvanomagnetic properties of a metal from the point of view of the
general classification of open electron trajectories arising on Fermi surfaces
of arbitrary complexity in the presence of magnetic field. The construction of
such a classification is the content of the Novikov problem and is based on
the division of non-closed electron trajectories into topologically regular
and chaotic trajectories. The description of stable topologically regular
trajectories gives a basis for a complete classification of non-closed
trajectories on arbitrary Fermi surfaces and is connected with special
topological structures on these surfaces. Using this description, we describe
here the distinctive features of possible changes in the picture of electron
trajectories during the Lifshitz transitions, as well as changes in the
conductivity behavior in the presence of a strong magnetic field. As it turns
out, the use of such an approach makes it possible to describe not only the
changes associated with stable electron trajectories, but also the most
general changes of the conductivity diagram in strong magnetic fields.

##  I Introduction

In this paper, we will try to describe the most general relationship between
the Lifshitz transitions (see Lifshits1960 ; etm ), leading to a change in the
topology of the Fermi surface, and angular diagrams that describe the behavior
of the magnetic conductivity of a metal in strong magnetic fields. It must be
said that at present topological Lifshitz transitions are actually represented
by a very wide range of phenomena associated with topological properties of a
Fermi surface and their changes, and the study of the variety of such
phenomena is an interesting and rapidly developing area of condensed matter
physics (see, for example, Volovik2017 ; Volovik2018 ).

Here, however, we will consider the most classical definition of the Lifshitz
transitions (Lifshits1960 ), namely, a change in the topology of the Fermi
surface when passing the critical points of the dispersion relation
$\epsilon({\bf p})$ (see Fig. 1).

Figure 1: Reconstruction of the Fermi surface and arising of new components
when passing through the critical points of the relation $\epsilon({\bf p})$
(Lifshits1960 )

As is well known, the dispersion relation $\epsilon({\bf p})$ can be
considered either as a periodic function in the quasi-momentum space
$\mathbb{R}^{3}$, or as a smooth function on the three-dimensional torus
$\mathbb{T}^{3}$ obtained from $\mathbb{R}^{3}$ by factorization with respect
to the reciprocal lattice vectors. The singular points of the function
$\epsilon({\bf p})$ are defined by the condition $\,\nabla\epsilon({\bf
p})=0\,$, and the corresponding energy levels, as is known, correspond to
arising of the Van Hove singularities in the density of electron states.

The singularities of the function $\epsilon({\bf p})$ include the points of
its local minima and maxima, as well as saddle singular points (assuming that
all singular points of $\epsilon({\bf p})$ are non-degenerate).

Saddle points of a function in three-dimensional space, as is well known, can
have index 1 or 2, depending on whether the increment of the function near
this point can be represented in the form

$d\epsilon({\bf p})\,=\,a^{2}dp_{1}^{2}+b^{2}dp_{2}^{2}-c^{2}dp_{3}^{2}$

or

$d\epsilon({\bf p})\,=\,a^{2}dp_{1}^{2}-b^{2}dp_{2}^{2}-c^{2}dp_{3}^{2}$

in some local Euclidean coordinate system. As is well known from the Morse
theory, the number of saddle singular points of both types for a function on a
three-dimensional torus is always at least three. In fact, for real dispersion
laws, it is often larger, in particular, whenever Fermi surfaces of genus
greater than 3 arise. Here, we are interested precisely in the saddle
singularities of the relation $\epsilon({\bf p})$.

As was shown in Lifshits1960 , the passage of the Fermi level through critical
points of $\epsilon({\bf p})$ (for example, under strong external pressure)
leads to singularities in the thermodynamic quantities of the electron gas in
the crystal (the Lifshitz transitions), as well as possible abrupt changes in
the behavior of the magnetic conductivity in strong magnetic fields. The
latter circumstance is associated with a possible significant change in the
geometry of the trajectories of the system

${\dot{\bf p}}\,\,\,\,=\,\,\,\,{e\over c}\,\,\left[{\bf v}_{\rm gr}({\bf
p})\,\times\,{\bf B}\right]\,\,\,\,=\,\,\,\,{e\over
c}\,\,\left[\nabla\epsilon({\bf p})\,\times\,{\bf B}\right]\,\,\,,$ (I.1)

describing the semiclassical dynamics of electrons in an external magnetic
field, on the Fermi surface. The main effect here is a sharp change in the
behavior of the magnetic conductivity due to arising (or disappearance) of
open trajectories of system (I.1) during topological reconstructions of the
Fermi surface.

The important role of open trajectories of system (I.1) in description of the
conductivity of metals in strong magnetic fields was also first revealed by
the school of I.M. Lifshitz (see lifazkag ; lifpes1 ; lifpes2 ; etm ). Since
the trajectories of system (I.1) are defined by the intersections of the
surfaces $\,\epsilon({\bf p})={\rm const}\,$ by planes orthogonal to the
magnetic field, the geometry of such trajectories is essentially determined by
the geometry and the topology of the Fermi surface. In particular, the
question of whether the Fermi surface is bounded or unbounded in the ${\bf p}$
\- space is of great importance.

As was shown in lifazkag , the contributions of closed and open periodic
trajectories to the conductivity tensor differ significantly in the limit
$\,\omega_{B}\tau\rightarrow\infty\,$ (i.e., in the limit of strong magnetic
fields). In particular, if there are only closed trajectories on the Fermi
surface, the conductivity decreases in all directions in the plane orthogonal
to ${\bf B}$ in the specified limit. The asymptotic behavior of the total
conductivity tensor can then be represented in the form

$\sigma^{kl}\,\,\,\,\simeq\,\,\,\,{ne^{2}\tau\over
m^{*}}\,\left(\begin{array}[]{ccc}(\omega_{B}\tau)^{-2}&(\omega_{B}\tau)^{-1}&(\omega_{B}\tau)^{-1}\cr(\omega_{B}\tau)^{-1}&(\omega_{B}\tau)^{-2}&(\omega_{B}\tau)^{-1}\cr(\omega_{B}\tau)^{-1}&(\omega_{B}\tau)^{-1}&*\end{array}\right)\quad,$
(I.2)

($\omega_{B}\tau\,\rightarrow\,\infty$).

At the same time, the contribution of open periodic trajectories to the
conductivity tensor is strongly anisotropic in the plane orthogonal to ${\bf
B}$ and can be represented in the leading order as

$\sigma^{kl}\,\,\,\,\simeq\,\,\,\,{ne^{2}\tau\over
m^{*}}\,\left(\begin{array}[]{ccc}(\omega_{B}\tau)^{-2}&(\omega_{B}\tau)^{-1}&(\omega_{B}\tau)^{-1}\cr(\omega_{B}\tau)^{-1}&*&*\cr(\omega_{B}\tau)^{-1}&*&*\end{array}\right)\quad,$
(I.3)

($\omega_{B}\tau\,\rightarrow\,\infty$).

In the formulas (I.2) - (I.3), as everywhere further, it is assumed that the
$z$ axis is directed along the magnetic field. In the relation (I.3), it is
also assumed that the direction of the axis $x$ coincides with the mean
direction of the open trajectories in the ${\bf p}$ -space. The sign $\simeq$
in both formulas means asymptotic behavior, i.e. each of the components
actually contains some dimensionless factor of order 1. The quantity
$\omega_{B}$ plays the role of the electron cyclotron frequency in the metal,
while the quantity $\tau$ represents the mean free time of electrons. The
quantity $m^{*}$ has the meaning of the effective mass of an electron in a
crystal. The relation $\,\omega_{B}\tau\gg 1\,$, as is also well known,
requires the use of sufficiently pure single-crystal samples at very low
temperatures ($T\leq 1K$) and sufficiently strong magnetic fields ($B\geq
1Tl$).

The value $n$ usually plays the role of the concentration of current carriers
in the metal. In the formulas (I.2) - (I.3), however, it is also proportional
to the measure of the corresponding trajectories (closed or periodic) on the
Fermi surface. The latter circumstance is especially important in the
situation we are considering, since the measure of open trajectories can be
determined by the proximity to the Lifshitz transition point $\epsilon_{0}$.
It is this situation that occurs, for example, in Lifshits1960 , where the
arising and disappearance of periodic open trajectories on a ‘‘warped
cylinder’’ surface is considered. In this situation, the leading term of the
conductivity tensor in the presence of open trajectories on the Fermi surface
was represented in Lifshits1960  in the form

$\sigma^{kl}=\left(\begin{array}[]{ccc}\gamma^{2}a_{xx}&\gamma a_{xy}&\gamma
a_{xz}\cr\gamma a_{yx}&\gamma^{2}a_{yy}+\beta^{1/2}b_{yy}&\gamma
a_{yz}+\beta^{1/2}b_{yz}\cr\gamma a_{zx}&\gamma
a_{zy}+\beta^{1/2}b_{zy}&a_{zz}\end{array}\right)$ (I.4)

($\omega_{B}\tau\,\rightarrow\,\infty$), where
$\,\gamma=(\omega_{B}\tau)^{-1}\,$,
$\,\beta=|(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}|\,$, the quantities
$a_{kl}$ represent some constants, and the quantities $b_{kl}$ have a weak
(logarithmic) dependence on $\beta$.

As we can see, the formula (I.4) allows not only to observe the Lifshitz
transition in the described situation, but also to determine the proximity to
this transition when changing the parameters of influence on a sample.

In the general situation, the Fermi surface is an arbitrary 3-periodic surface
in ${\bf p}$ \- space (Fig. 2), and the problem of describing the trajectories
of the system (I.1) is quite difficult. For the first time, the problem of
complete classification of open trajectories of system (I.1) was set by S.P.
Novikov in MultValAnMorseTheory  and then intensively studied in his
topological school (see zorich1 ; dynn1992 ; Tsarev ; dynn1 ; zorich2 ;
DynnBuDA ; dynn2 ; dynn3 ). As a result of studying the Novikov problem, a
number of deep topological results have been obtained, and by now a complete
classification of the open trajectories of system (I.1) for arbitrary periodic
dispersion relations $\epsilon({\bf p})$ has been obtained. Consequences from
the topological theorems obtained in the study of the Novikov problem also led
to a description of a number of physical effects associated with the behavior
of open trajectories of (I.1) and, in addition, made it possible to give a
complete classification of possible asymptotic behavior of conductivity in
strong magnetic fields for metals with arbitrarily complex Fermi surfaces (see
e.g. PismaZhETF ; ZhETF1997 ; UFN ; JournStatPhys ; TrMian ; ObzorJetp ).

Figure 2: Trajectories of system (I.1) on a general periodic Fermi surface

Here we are interested in changes in the open trajectories of system (I.1)
under the Lifshitz transitions, i.e. changes in the topology of the Fermi
surface when passing singular points of the dispersion relation $\epsilon({\bf
p})$. We will assume here that the Fermi surface has the most general form,
and in describing the trajectories we will use the general classification
obtained in the study of the Novikov problem. To describe the situation, we
will use angular diagrams that specify the type of trajectories of system
(I.1) on the Fermi surface depending on the direction of the magnetic field.
The angular diagram is thus the unit sphere $\mathbb{S}^{2}$, which
parametrizes the directions of ${\bf B}$ and determines the type of
trajectories on the Fermi surface for each direction. Since the type of
trajectories of system (I.1) determines the asymptotic behavior of the
conductivity tensor in the limit of strong magnetic fields, it is natural to
also call such diagrams conductivity (magnetic conductivity) diagrams for a
given Fermi surface. We are mainly interested here in the changes in such
diagrams that accompany the Lifshitz transitions. Experimental observation of
changes (sharp jumps) in such diagrams can generally serve as one of the tools
for studying the Lifshitz transitions in metals with complex Fermi surfaces.

Here we present the general picture of changes in the conductivity diagrams
when passing the singularities of the relation $\epsilon({\bf p})$, based on
the general theory of such diagrams, constructed in the study of the Novikov
problem. In the next section, we present the general classification of the
diagrams corresponding to various Fermi surfaces and describe their connection
with the angular diagrams for the entire dispersion relation $\epsilon({\bf
p})$. In section 3, we will describe typical changes in angular diagrams
corresponding to topological transitions of various types on Fermi surfaces of
arbitrary complexity.

##  II General Facts about Angular Conductivity Diagrams in Metals

The basis for classifying the open trajectories of system (I.1) is the
description of its stable open trajectories. Here we call open trajectories of
(I.1) stable if they do not vanish and retain their global geometry under
small variations of all problem parameters, in particular, small variations of
the level $\epsilon_{F}$ and rotations of the direction of ${\bf B}$. As
follows from the results of zorich1 ; dynn1992 ; dynn1 , stable open
trajectories of system (I.1) have the following remarkable properties.

1) Each stable open trajectory of system (I.1) lies in a straight strip of
finite width in a plane orthogonal to ${\bf B}$ and passes through it (Fig.
3).

2) The mean direction of all stable open trajectories for a given direction of
the magnetic field is given by the intersection of the plane orthogonal to
${\bf B}$ and some integral (generated by two reciprocal lattice vectors)
plane $\Gamma$, the direction of which is invariable for small variations of
the problem parameters.

Figure 3: The form of a stable open trajectory of system (I.1) in a plane
orthogonal to ${\bf B}$ (schematically)

Property (1) of stable open trajectories manifests itself directly in the
behavior of the magnetic conductivity in strong magnetic fields. Namely, here,
as in the case of periodic open trajectories, there is a strong anisotropy of
the conductivity in the plane orthogonal to ${\bf B}$, and the main term in
the asymptotics of the conductivity tensor is also given by the formula (I.3).
For special directions of ${\bf B}$, stable open trajectories (I.1) can be
periodic. In the generic case, however, they are quasi-periodic and have no
periods in the ${\bf p}$ -space. The direction of maximum suppression of
conductivity belongs to the corresponding plane $\Gamma$, which makes it
experimentally observable (PismaZhETF ; UFN ). An integral plane in ${\bf p}$
\- space can also be defined as a plane orthogonal to some integer direction
of the original crystal lattice. The plane $\Gamma$ can thus be defined by
some irreducible integer triple $(m^{1},m^{2},m^{3})$. The numbers
$(m^{1},m^{2},m^{3})$ were defined in PismaZhETF  as topological numbers
observed in the conductivity of normal metals.

Each family of stable open trajectories is defined by some region (stability
zone) $\Omega$ on the angular diagram corresponding to the same values
$(m^{1},m^{2},m^{3})$. In the general case, an angular diagram may contain
some (finite or infinite) number of stability zones $\Omega_{\alpha}$
corresponding to different values of
$(m^{1}_{\alpha},m^{2}_{\alpha},m^{3}_{\alpha})$. The presence of stability
zones and their location on the unit sphere $\mathbb{S}^{2}$ is an important
component of the diagram of conductivty of a metal in strong magnetic fields.

In addition to stable open trajectories of system (I.1), there are also
unstable open trajectories. First of all, they may include periodic
trajectories, for example, those considered above. Periodic trajectories are,
in a sense, semi-stable, namely, they are preserved under rotations of ${\bf
B}$ orthogonal to their mean direction, and collapse under all other
rotations. These trajectories correspond to one-dimensional arcs on the
angular diagram, which mark the presence of such trajectories on the Fermi
surface for the corresponding directions of ${\bf B}$. The set of
corresponding arcs on the sphere $\mathbb{S}^{2}$ is also an important part of
a metal magnetic conductivity diagram.

Periodic open trajectories, however, are not the only type of unstable open
trajectories of (I.1). Namely, there are open trajectories of system (I.1) of
much more complex geometry, which are unstable both with respect to small
rotations of ${\bf B}$ and small variations of the Fermi level $\epsilon_{F}$
(Tsarev ; DynnBuDA ; dynn2 ). Such trajectories can be conditionally divided
into two main types, namely, Tsarev-type trajectories and Dynnikov-type
trajectories. Tsarev-type trajectories can only be observed for partially
irrational directions of ${\bf B}$, when the plane orthogonal to ${\bf B}$
contains one (up to proportionality) reciprocal lattice vector. On the
contrary, Dynnikov-type trajectories can arise only for directions of ${\bf
B}$ of complete irrationality (the plane orthogonal to ${\bf B}$ does not
contain reciprocal lattice vectors).

Unstable trajectories of both types have rather complex behavior on the Fermi
surface, which in this case should itself have sufficient complexity. However,
the behavior of Tsarev-type trajectories in planes orthogonal to ${\bf B}$ is
much simpler than that of Dynnikov-type trajectories. Namely, Tsarev-type
trajectories have an asymptotic direction that is the same in all planes
orthogonal to ${\bf B}$ for a given direction of ${\bf B}$. As a consequence,
the contribution of Tsarev-type trajectories to the conductivity tensor also
has strong anisotropy in the plane orthogonal to ${\bf B}$ and is close in
form to the contribution (I.3), although it differs from it in some details.

Dynnikov-type trajectories have much more complex behavior in planes
orthogonal to ${\bf B}$, wandering along them in a rather chaotic manner (Fig.
4). Among the features of the contribution of such trajectories to the
magnetic conductivity tensor, one can distinguish the suppression of
conductivity along the direction of the magnetic field (see ZhETF1997 ), as
well as the arising of fractional powers of the parameter $\omega_{B}\tau$ in
the asymptotics of the tensor components in the limit
$\,\omega_{B}\tau\rightarrow\infty\,$ (ZhETF1997 ; TrMian ). We note here that
the study of arising and geometric properties of Dynnikov-type trajectories is
an actively developing area at the present time (see, for example, Tsarev ;
DynnBuDA ; dynn2 ; zorich2 ; Zorich1996 ; ZorichAMS1997 ; ZhETF1997 ; zorich3
; DeLeo1 ; DeLeo2 ; ArxivObzor ; BullBrazMathSoc ; JournStatPhys ;
ZorichLesHouches ; DeLeoDynnikov1 ; DeLeoDynnikov2 ; Dynnikov2008 ;
Skripchenko1 ; Skripchenko2 ; DynnSkrip1 ; DynnSkrip2 ; AvilaHubSkrip1 ;
AvilaHubSkrip2 ; TrMian ; DeLeoObzor ; UMNObzor ).

In view of the particular complexity of the geometry of trajectories of the
Tsarev or Dynnikov type, such trajectories are usually called chaotic. Stable
open trajectories of the system (I.1), as well as periodic trajectories, are
called topologically regular.

Figure 4: Form of the Dynnikov chaotic trajectory in a plane orthogonal to
${\bf B}$ (schematically)

The arising of unstable trajectories of the Tsarev or Dynnikov type on the
Fermi surface is associated with particularly complex angular diagrams, which
we describe below. The location of the corresponding directions of ${\bf B}$
in such diagrams is perhaps the most interesting information about the Fermi
surface.

Before describing the types of diagrams corresponding to fixed Fermi surfaces,
it is convenient to describe the angular diagrams corresponding to the entire
dispersion relation $\epsilon({\bf p})$. Such diagrams were introduced in
dynn3  and are based on an important property of open trajectories of (I.1),
namely, the type of open trajectories of system (I.1) for a given direction of
${\bf B}$ is the same for all energy levels $\,\epsilon({\bf p})={\rm
const}\,$ at which they appear. Moreover, the situation in the general case
can be described as follows (dynn3 ).

Consider a smooth 3-periodic function $\epsilon({\bf p})$ whose values lie in
the interval $[\epsilon_{\min},\epsilon_{\max}]$. Consider some fixed
direction of ${\bf B}$ and the corresponding system (I.1). Let us assume for
simplicity that the direction of ${\bf B}$ is not rational. Then the following
assertions can be formulated.

1) Open trajectories of system (I.1) appear either in some closed energy
interval

$\epsilon_{\min}\,\,\,<\,\,\,\epsilon_{1}({\bf
B})\,\,\,\leq\,\,\,\epsilon({\bf p})\,\,\,\leq\,\,\,\epsilon_{2}({\bf
B})\,\,\,<\,\,\,\epsilon_{\max}$

or only at one energy level $\,\epsilon_{0}\,=\,\epsilon_{1}({\bf
B})\,=\,\epsilon_{2}({\bf B})\,$.

2) In the case $\epsilon_{1}({\bf B})<\epsilon_{2}({\bf B})$, all nonsingular
open trajectories in the interval $\left[\epsilon_{1}({\bf
B}),\epsilon_{2}({\bf B})\right]$ lie in straight strips of finite width in
planes orthogonal to ${\bf B}$ and pass through them (Fig. 3). All of them
have the same mean direction given by the intersection of the plane orthogonal
to ${\bf B}$ with some integral plane $\Gamma$ in the ${\bf p}$-space.

3) For generic directions of ${\bf B}$ the values $\epsilon_{1}({\bf B})$ and
$\epsilon_{2}({\bf B})$ coincide with the values of some continuous functions
$\tilde{\epsilon}_{1}({\bf B})$ and $\tilde{\epsilon}_{2}({\bf B})$ defined
everywhere on $\mathbb{S}^{2}$. However, for directions of ${\bf B}$
corresponding to arising of periodic open trajectories, the values of
$\epsilon_{1}({\bf B})$ and $\epsilon_{2}({\bf B})$ have ‘‘jumps’’ with the
following inequalities

$\epsilon_{1}({\bf B})\,\leq\,\tilde{\epsilon}_{1}({\bf
B})\,\leq\,\tilde{\epsilon}_{2}({\bf B})\,\leq\,\epsilon_{2}({\bf B})$

4) The property $\tilde{\epsilon}_{1}({\bf B})<\tilde{\epsilon}_{2}({\bf B})$,
and the integral plane $\Gamma$ are stable with respect to small rotations of
${\bf B}$, so that each of the planes $\Gamma_{\alpha}$ corresponds to a
certain ‘‘stability zone’’ $\widehat{\Omega}_{\alpha}$ in the space of
directions of ${\bf B}$.

According to dynn3 , the picture of stability zones for an arbitrary
dispersion relation $\epsilon({\bf p})$ can correspond to only one of the
following situations.

1) The entire unit sphere is the only stability zone $\widehat{\Omega}$
corresponding to some integral plane $\Gamma$.

2) The angular diagram contains an infinite number of stability zones whose
union is everywhere dense on the sphere $\mathbb{S}^{2}$ (see, for example,
Fig. 5).

Figure 5: Stability zones for the dispersion relation $\epsilon({\bf p})=\cos
p_{x}\cos p_{y}+\cos p_{y}\cos p_{z}+\cos p_{z}\cos p_{x}$ (DeLeoObzor )

Case (1), as a rule, is observed for dispersion relations of a rather special
form, close to the dispersion relations in quasi-one-dimensional conductors.
For the vast majority of real dispersion relations, however, case (2) takes
place. We will call here the dispersion relations corresponding to case (1)
dispersion relations with simple angular diagram. Similarly, the dispersion
relations corresponding to case (2) will be called relations with a complex
angular diagram. Here we are primarily interested in dispersion relations with
complex angular diagrams.

The complement ${\cal M}$ to the union of stability zones is a rather complex
set of fractal type on the sphere $\mathbb{S}^{2}$. According to the
conjecture of S.P. Novikov (ArxivObzor ), this set has measure zero and the
fractal dimension strictly less than 2. The first part of the Novikov
conjecture was recently proved for dispersion relations satisfying the
additional condition $\epsilon(-{\bf p})=\epsilon({\bf p})$ (I.A. Dynnikov, P.
Hubert, P. Mercat, and A.S. Skripchenko, in the process of publication). The
second part of the conjecture is confirmed by serious numerical studies, but
has not yet been proved rigorously.

The points of the set ${\cal M}$ represent accumulation points of decreasing
stability zones. Moreover, the set ${\cal M}$ can contain special rational
directions of ${\bf B}$ (see DynMalNovUMN ), as well as directions of ${\bf
B}$ corresponding to arising of trajectories of the Tsarev or Dynnikov type.
The set of special rational directions of ${\bf B}$, however, is only a
countable subset of the set ${\cal M}$, so that ‘‘almost all’’ points of the
set ${\bf M}$ represent, in fact, ‘‘chaotic’’ directions of these two types.
The values of the functions $\tilde{\epsilon}_{1}({\bf B})$ and
$\tilde{\epsilon}_{2}({\bf B})$ coincide on the set ${\cal M}$, as well as on
the boundaries of all the zones $\widehat{\Omega}_{\alpha}$.

It is easy to see that open trajectories appear on a fixed Fermi surface for a
given direction of ${\bf B}$ only if the Fermi level falls within the
corresponding interval $[\epsilon_{1}({\bf B}),\epsilon_{2}({\bf B})]$. In
particular, each stability zone $\Omega_{\alpha}$ at the conductivity diagram
is a subdomain of some zone $\widehat{\Omega}_{\alpha}$ defined for the entire
dispersion relation. As a rule, most of the conductivity diagram of a metal is
in fact the region corresponding to the presence of only closed trajectories
on the Fermi surface.

It can also be seen that when determining the zones $\Omega_{\alpha}$, as well
as the Tsarev and Dynnikov directions for a fixed Fermi surface, one can use
the functions $\tilde{\epsilon}_{1}({\bf B})$ and $\tilde{\epsilon}_{2}({\bf
B})$, while to determine the directions corresponding to arising of unstable
periodic trajectories, it is necessary to know the functions
$\epsilon_{1}({\bf B})$ and $\epsilon_{2}({\bf B})$. The latter circumstance
manifests itself, in particular, in a certain difference in the shape of zones
$\Omega_{\alpha}$ from the zones $\widehat{\Omega}_{\alpha}$. Namely, the set
of directions of ${\bf B}$ corresponding to arising of open trajectories
associated with the zone $\Omega_{\alpha}$ is somewhat larger than the zone
itself and also contains an infinite set of segments adjacent to the boundary
of $\Omega_{\alpha}$ and corresponding to arising of periodic trajectories on
the Fermi surface (Fig. 6). The periodic trajectories can then be considered
stable for directions of ${\bf B}$ inside $\Omega_{\alpha}$ and unstable on
additional segments.

Figure 6: The zones $\widehat{\Omega}_{\alpha}$ (top) and the zones
$\Omega_{\alpha}$ (bottom) with adjoining segments corresponding to arising of
unstable periodic trajectories on the Fermi surface (schematically, nets of
directions of ${\bf B}$ inside the zones corresponding to stable periodic
trajectories are also indicated).

Such an arrangement of the zones $\Omega_{\alpha}$ actually leads to a rather
complicated analytical behavior of the conductivity tensor near their
boundaries, which makes it difficult to determine the shape of these zones in
direct measurements of the conductivity (see, for example, AnalProp ). At the
same time, however, there are methods for experimentally determining the exact
mathematical boundaries of the zones $\Omega_{\alpha}$, which makes it
possible to experimentally determine their exact shape (see ExactBound ).

The zones $\Omega_{\alpha}$ represent regions with piecewise smooth boundaries
on the sphere (see e.g. dynn3 ). Other than that, we are not aware of any
restrictions on their shape. In particular, there may be unconnected stability
regions corresponding to the same values of $(m^{1},m^{2},m^{3})$. For
simplicity, we agree here to consider the union of such domains as one
disconnected stability zone on $\mathbb{S}^{2}$. In this sense, each region
$\Omega_{\alpha}$ and its diametrically opposite one form the same stability
zone. In addition, stability zones can also be non-simply connected (see e.g.
DynMalNovUMN ). The latter, however, takes place for very specific Fermi
surfaces, which are special mathematical examples. For real dispersion laws,
we will assume here that all the zones $\Omega_{\alpha}$ are simply connected
domains with piecewise smooth boundaries on $\mathbb{S}^{2}$.

Below we give a brief description of various types of angular conductivity
diagrams in metals, as well as their changes with a change in the value of
$\epsilon_{F}$ in the interval $[\epsilon_{\min},\epsilon_{\max}]$ (UltraCompl
), which we will need later. Here, we will be primarily interested in
conductivity diagrams that correspond to dispersion laws with complex angular
diagrams, i.e., diagrams containing an infinite number of zones
$\widehat{\Omega}_{\alpha}$.

It is easy to see that if the value of $\epsilon_{F}$ is sufficiently close to
the value $\epsilon_{\min}$ or $\epsilon_{\max}$, the Fermi surfaces are small
ellipsoids, and open trajectories of system (I.1) are absent on them for any
direction of ${\bf B}$. It can also be noted that the Hall conductivity is of
the electronic type in the first case and of the hole type in the second one.
The corresponding conductivity diagrams can be called zero-type diagrams and
denoted by $0_{-}$ or $0_{+}$, depending on the sign of the Hall conductivity.
In the general case, for a fixed dispersion relation $\epsilon({\bf p})$, we
can single out some values $\epsilon_{1}^{{\cal A}\prime}$ and
$\epsilon_{2}^{{\cal A}\prime}$ such that the angular diagrams of the types
$0_{-}$ and $0_{+}$ correspond to situations

$\epsilon_{F}\,\in\,(\epsilon_{\min},\epsilon_{1}^{{\cal
A}\prime})\quad\text{\rm and}\quad\epsilon_{F}\,\in\,(\epsilon_{2}^{{\cal
A}\prime},\epsilon_{\max})$

respectively.

For generic dispersion relations, we can also single out the values
$\epsilon_{1}^{\cal A}$ and $\epsilon_{2}^{\cal A}$, such that the situations

$\epsilon_{F}\,\in\,(\epsilon_{1}^{{\cal A}\prime},\epsilon_{1}^{\cal
A})\quad\text{\rm and}\quad\epsilon_{F}\,\in\,(\epsilon_{2}^{\cal
A},\epsilon_{2}^{{\cal A}\prime})$

correspond to conductivity diagrams containing only one-dimensional arcs
corresponding to arising of unstable periodic trajectories on the Fermi
surface. Diagrams of this type can be denoted by the symbols $1_{-}$ and
$1_{+}$ depending on the type of the Hall conductivity observed for directions
of ${\bf B}$ corresponding to the presence of only closed trajectories on the
Fermi surface.

The interval $(\epsilon_{1}^{\cal A},\epsilon_{2}^{\cal A})$ corresponds to
conductivity diagrams containing stability zones $\Omega_{\alpha}$. We can say
that such diagrams have a sufficient level of complexity, and it is they that
will be mainly of interest to us here. For generic dispersion relations with
complex angular diagrams (with an infinite number of zones
$\widehat{\Omega}_{\alpha}$), however, it is natural to divide this interval
into three intervals (see UltraCompl )

$\epsilon_{1}^{\cal A}\,\,\,<\,\,\,\epsilon_{1}^{\cal
B}\,\,\,<\,\,\,\epsilon_{2}^{\cal B}\,\,\,<\,\,\,\epsilon_{2}^{\cal A}$

Conductivity diagrams corresponding to the situations

$\epsilon_{F}\,\in\,(\epsilon_{1}^{\cal A},\epsilon_{1}^{\cal
B})\quad\text{\rm and}\quad\epsilon_{F}\,\in\,(\epsilon_{2}^{\cal
B},\epsilon_{2}^{\cal A})\,\,\,,$

can be called diagrams of the $A_{-}$ and $A_{+}$ types, respectively. For
diagrams of this type, in all regions of directions of ${\bf B}$ corresponding
to the presence of only closed trajectories on the Fermi surface, the Hall
conductivity has the same type (electronic and hole, respectively).

Conductivity diagrams corresponding to the situation

$\epsilon_{F}\,\in\,(\epsilon_{1}^{\cal B},\epsilon_{2}^{\cal B})\,\,\,,$

can be called diagrams of type $B$. These diagrams are specified by the fact
that in the space of directions of ${\bf B}$ (on the unit sphere
$\mathbb{S}^{2}$), among the regions corresponding to the presence of only
closed trajectories on the Fermi surface, there are both regions of the
electronic Hall conductivity, and regions of the hole Hall conductivity.

In fact, there are also two additional important differences between diagrams
of type $A$ and diagrams of type $B$ (see UltraCompl ).

1) Generic diagrams of type $A$ contain a finite number of zones
$\Omega_{\alpha}$, while generic diagrams of type $B$ contain an infinite
number of stability zones.

2) Generic diagrams of type $A$ do not contain directions of ${\bf B}$
corresponding to arising of Tsarev or Dynnikov trajectories, while generic
diagrams of type $B$ contain such directions.

Fig. 7 (schematically) shows a possible evolution of the conductivity diagram
in the situation we describe when $\epsilon_{F}$ changes from
$\epsilon_{1}^{\cal A}$ to $\epsilon_{2}^{\cal A}$.

Figure 7: Possible evolution of the conductivity diagram when $\epsilon_{F}$
changes from $\epsilon_{1}^{\cal A}$ to $\epsilon_{2}^{\cal A}$ and passes
through $\epsilon_{1}^{\cal B}$ and $\epsilon_{2}^{\cal B}$. The zones
$\Omega_{\alpha}$ and the electronic (e) and hole (h) Hall conductivity
regions are shown schematically. We can distinguish diagrams of type $A_{-}$
((a) and (b)), diagrams of type $B$ ((c) and (d)) and diagrams of type $A_{+}$
((e) and (f)).

Diagrams in the interval $(\epsilon_{1}^{\cal B},\epsilon_{2}^{\cal B})$
contain also stability zones with a more complex boundary than in the
intervals $(\epsilon_{1}^{\cal A},\epsilon_{1}^{\cal B})$ and
$(\epsilon_{2}^{\cal B},\epsilon_{2}^{\cal A})$. Namely, here we have zones,
part of the boundary of which is adjacent to the regions of the electronic
Hall conductivity, and part to the regions of the hole Hall conductivity.

We also note here that the parts of the boundaries of $\Omega_{\alpha}$
adjacent to the electronic Hall conductivity regions are determined by the
relation $\tilde{\epsilon}_{1}({\bf B})=\epsilon_{F}$, and the parts of the
boundaries adjacent to the hole Hall conduction regions are determined by the
relation $\tilde{\epsilon}_{2}({\bf B})=\epsilon_{F}$.

The above picture corresponds to generic dispersion relations with angular
diagrams containing an infinite number of zones $\widehat{\Omega}_{\alpha}$.
This picture, in fact, may have the following degenerations.

1) $\epsilon_{1}^{{\cal A}\prime}=\epsilon_{1}^{\cal A}$ or
$\epsilon_{2}^{\cal A}=\epsilon_{2}^{{\cal A}\prime}$, such that the
corresponding interval $(\epsilon_{1}^{{\cal A}\prime},\epsilon_{1}^{\cal A})$
or $(\epsilon_{2}^{\cal A},\epsilon_{2}^{{\cal A}\prime})$ shrinks to a point.
In the above picture, in this case, there are no diagrams of the type $1_{-}$
or $1_{+}$, such that diagrams of the type $0_{-}$ and $A_{-}$ or $A_{+}$ and
$0_{+}$ (or both) immediately pass into each other. Such degeneracies often
arise for dispersion relations with sufficiently high symmetry (for example,
cubic).

2) Degeneration $\epsilon_{1}^{\cal B}=\epsilon_{2}^{\cal B}$. In this case,
there are no diagrams of type $B$ in the above picture, and the diagram
arising at the level $\epsilon_{1}^{\cal B}=\epsilon_{2}^{\cal B}$ coincides
with the angular diagram for the entire dispersion relation. The corresponding
dispersion relations form a very special class (of infinite codimension in the
space of periodic $\epsilon({\bf p})$) and we will not consider them here. We
emphasize here only that in this case we have in mind dispersion relations
whose diagrams remain complex (contain an infinite number of stability zones),
despite the presence of degeneracy. In addition to such cases, there are also
deformations of the dispersion relations, under which the interval
$(\epsilon_{1}^{\cal B},\epsilon_{2}^{\cal B})$ is infinitely narrowed, and
the corresponding angular diagrams are simplified and become diagrams with one
stability zone at the degeneracy point. Such degenerations, in a sense,
correspond to the boundary between the dispersion relations of the two types
described above and are observed in a much more general situation.

Although we are primarily interested here in dispersion relations with an
infinite number of stability zones, let us also present here a typical picture
of the change in the conductivity diagram for relations corresponding to the
presence of only one zone $\widehat{\Omega}$. As before, we will assume here
that all stability zones are simply connected, which corresponds to realistic
dispersion relations that arise in real conductors.

As in the previous case, for generic dispersion relations, here we can also
introduce a set of reference points

$\epsilon_{\min}\,<\,\hat{\epsilon}^{{\cal
A}\prime}_{1}\,<\,\hat{\epsilon}^{\cal A}_{1}\,<\,\hat{\epsilon}^{\cal
B}_{1}\,<\,\hat{\epsilon}^{\cal B}_{2}\,<\,\hat{\epsilon}^{\cal
A}_{2}\,<\,\hat{\epsilon}^{{\cal A}\prime}_{2}\,<\,\epsilon_{\max}\,\,\,,$

separating intervals with diagrams of different types.

The intervals $[\epsilon_{\min},\hat{\epsilon}^{{\cal A}\prime}_{1})$ and
$(\hat{\epsilon}^{{\cal A}\prime}_{2},\epsilon_{\max}]$, and also
$(\hat{\epsilon}^{{\cal A}\prime}_{1},\hat{\epsilon}^{\cal A}_{1})$ and
$(\hat{\epsilon}^{\cal A}_{2},\hat{\epsilon}^{{\cal A}\prime}_{2})$, as
before, correspond here to diagrams of the types $0_{-}$, $0_{+}$, $1_{-}$,
and $1_{+}$, respectively.

The intervals $(\hat{\epsilon}^{\cal A}_{1},\hat{\epsilon}^{\cal B}_{1})$ and
$(\hat{\epsilon}^{\cal B}_{2},\hat{\epsilon}^{\cal A}_{2})$ correspond to the
diagrams $A_{-}$ and $A_{+}$ respectively. The only peculiarity here is that
on such diagrams there is only one stability zone corresponding to a single
set $(m^{1},m^{2},m^{3})$. The region that does not belong to the stability
zone corresponds to the electronic Hall conductivity for diagrams of the type
$A_{-}$, and to the hole conductivity for diagrams of the type $A_{+}$.

The diagram appearing in the interval $(\hat{\epsilon}^{\cal
B}_{1},\hat{\epsilon}^{\cal B}_{2})$ will be called here a diagram of type
$\widehat{B}$. It contains a single stability zone covering the entire unit
sphere $\mathbb{S}^{2}$.

As in the previous case, the above picture admits degenerations. In
particular, the situations $\,\hat{\epsilon}^{{\cal
A}\prime}_{1}=\hat{\epsilon}^{\cal A}_{1}\,$ and $\,\hat{\epsilon}^{\cal
A}_{2}=\hat{\epsilon}^{{\cal A}\prime}_{2}\,$ correspond here to the same
types of degeneracy as in the case of complex angular diagrams. The
degeneration $\,\hat{\epsilon}^{\cal B}_{1}=\hat{\epsilon}^{\cal B}_{2}\,$
corresponds to the ‘‘boundary’’ between dispersion relations with complex
angular diagrams and those with simple angular diagrams.

##  III Lifshitz transitions and general principles of changing of
conductivity diagrams

Changing of the picture of open trajectories of system (I.1) can be quite
simple and visual for fairly simple Fermi surfaces. An illustrative example is
the classical reconstruction considered in Lifshits1960  (Fig. 1), where the
compact Fermi surface takes the form of a warped cylinder. It is easy to see
that open trajectories arise in this case only for directions of ${\bf B}$
orthogonal to the cylinder axis and are periodic. In the general case,
however, the description of open trajectories on the Fermi surface is a rather
complicated problem and often requires serious numerical studies (see e.g.
DynnBuDA ; DeLeo1 ; DeLeoObzor ). In this paper, we will try to describe the
most general features of the changes in angular diagrams during the Lifshitz
transitions, based on the general topological results obtained in the study of
the Novikov problem. As we will see below, such features can lead in this case
to a number of very special regimes in the conduction behavior, which are
observed experimentally and are inherent precisely to situations close to
topological transitions.

A natural indicator of the topological complexity of a Fermi surface is its
rank, namely, the number of independent directions in which the surface
extends in ${\bf p}$ \- space. It is easy to see that the rank of the Fermi
surface can take on the values 0, 1, 2 and 3 (Fig. 8).

Figure 8: Examples of Fermi surfaces of rank 0, 1, 2 and 3

Moreover, since the Fermi surface can also be considered as a compact surface
in a three-dimensional torus, it also has topological genus $g$, which can
take the values 0, 1, 2, 3, 4, … and so on. (Fig. 9). For topological reasons,
the rank of a Fermi surface cannot exceed its genus. It is also important
that, in addition to the topological complexity, the Fermi surface can have a
very complex geometric shape in the ${\bf p}$ \- space, which also has a
significant effect on the shape of the trajectories of system (I.1).

Figure 9: Topological surfaces of genus 0, 1, 2 and 3

The passage of singular points of the relation $\epsilon({\bf p})$ with
increasing energy $\epsilon_{F}$ changes the topology of the Fermi surface. It
is easy to see that the passage of the minima and maxima of $\epsilon({\bf
p})$ leads to arising and disappearance of (small) components of the Fermi
surface, while the passage of saddle singular points leads to the merging or
decay of individual components, or to a change in their genus. If we talk
about a reconstruction of a connected component of the Fermi surface, then
passing a saddle singular point of index 1 increases its genus by one, while
passing a singular point of index 2 decreases its genus by one. More
generally, passing a singular point of index 1 can also lead to a merging of
individual components, while passing a point of index 2 can lead to a
splitting of one component into two.

According to the Morse theory, a smooth function $\epsilon({\bf p})$ on the
torus $\mathbb{T}^{3}$ has at least three saddle singular points of both index
1 and index 2 (and, of course, at least one minimum and maximum). Quite often,
however, the number of saddle singular points of $\epsilon({\bf p})$ exceeds
the lower estimates given, and the genus of the Fermi surface is 4 or more.

Certainly, a change in the topology of the Fermi surface often gives obvious
indications of a possibility of arising of non-closed trajectories of system
(I.1) on it. This is especially true for changes in the rank of the Fermi
surface, as well as arising of periodic open trajectories. Usually, in these
cases, the Fermi surfaces have a fairly simple shape and correspond to fairly
simple angular conductivity diagrams.

It will be of interest to us here to consider the situation when the Lifshitz
transitions occur on fairly complex Fermi surfaces, which also correspond to
fairly complex conductivity diagrams. Since the structure of such diagrams is
formed mainly by the pattern of stability zones on them, it will be of
interest to us, first of all, to trace the changes in this pattern during
topological transitions.

Above, we described the evolution of the picture of stability zones on a
complex conductivity diagram, starting from the moment they appear on the
diagram until they completely disappear (Fig. 7). As the value of
$\epsilon_{F}$ increases, the diagram changes monotonically, such that the
region corresponding to the presence of only closed trajectories on the Fermi
surface and the electron Hall conductivity decreases monotonically (until it
disappears), and the analogous region corresponding to the hole Hall
conductivity increases monotonically (since its emerging). In particular,
sections of the boundaries of $\Omega_{\alpha}$ adjacent to the first region
move outside the stability zones, and sections adjacent to the second region
move inside the zones. As we said above, segments of the first type are
defined by the relation $\tilde{\epsilon}_{1}({\bf B})=\epsilon_{F}$, and
segments of the second type are determined by the relation
$\tilde{\epsilon}_{2}({\bf B})=\epsilon_{F}$.

In energy intervals that do not contain reconstructions of the Fermi surface,
the evolution of the conductivity diagram is continuous. At the same time, as
was pointed out in dynn3 , the functions $\tilde{\epsilon}_{1}({\bf B})$ and
$\tilde{\epsilon}_{2}({\bf B})$ can be locally constant in some domains on the
unit sphere. This phenomenon is associated precisely with topological
reconstructions of the surface $S_{F}$, and the values of these functions on
such ‘‘plateaus’’ coincide with the energies at which the corresponding
reconstructions (the Lifshitz transitions) are observed. As can be seen, the
picture of stability zones can in this case ‘‘jump’’ in some region of
$\mathbb{S}^{2}$ corresponding to a ‘‘plateau’’ of the function
$\tilde{\epsilon}_{1}({\bf B})$ or $\tilde{\epsilon}_{2}({\bf B})$.

We will try to consider here in most detail possible changes in the
conductivity diagram during the Lifshitz transitions, including a description
of the regimes of behavior of the tensor $\sigma^{kl}({\bf B})$ corresponding
to such changes. As we said above, we will start with the cases corresponding
to arising or disappearance of stable open trajectories on the Fermi surface.

In connection with the study of open trajectories of system (I.1), we will be
interested in the structure of the Lifshitz transitions associated with the
passage of saddle singular points of the relation $\epsilon({\bf p})$. Fig. 10
shows the reconstructions of the Fermi surface when passing singular points of
index 1 and 2 with increasing Fermi energy. Reconstructions in Fig. 10 look
like mutually inverse, we must remember, however, that in both cases, for
increasing $\epsilon_{F}$, the region $\,\epsilon({\bf p})<\epsilon_{F}\,$
increases and the region $\,\epsilon({\bf p})>\epsilon_{F}\,$ decreases.

Figure 10: Reconstructions of the Fermi surface when passing saddle singular
points of $\epsilon({\bf p})$ of index 1 (top) and index 2 (bottom)

All changes in the picture of open trajectories on the Fermi surface with
increasing $\epsilon_{F}$ can be associated with two processes in planes
orthogonal to ${\bf B}$, namely, the formation of open trajectories from
closed electron-type trajectories and the decay of open trajectories into
closed hole-type trajectories (Fig. 11). Similarly, as the value of
$\epsilon_{F}$ decreases, these processes go in the opposite direction.

Figure 11: Formation of open trajectories and their decay with increasing
value of $\epsilon_{F}$

It is easy to see that for directions of ${\bf B}$ close to the axis of the
cone

$a^{2}dp_{1}^{2}+b^{2}dp_{2}^{2}-c^{2}dp_{3}^{2}\,=\,0$ (III.1)

(in the coordinate system corresponding to the given saddle singular point)
or, respectively,

$a^{2}dp_{1}^{2}-b^{2}dp_{2}^{2}-c^{2}dp_{3}^{2}\,=\,0$ (III.2)

no changes in the picture of open trajectories of system (I.1) can occur. The
equation (III.1) or (III.2) thus selects two ellipsoidal regions on the unit
sphere, in which the conductivity diagram (in our sense) certainly does not
change when passing through the corresponding singular point. A change in the
picture of open trajectories on the Fermi surface can thus occur only in the
circular region separating opposite ellipsoidal regions on $\mathbb{S}^{2}$
(Fig. 12). It can also be noted that in the case of observing of sharp changes
along the boundary of this region (or part of it), it is not difficult to
determine the parameters of the corresponding singular point (more precisely,
the quantities $b/a$ and $c/a$).

Figure 12: Changes in trajectories when passing through a singular point for
different directions of ${\bf B}$ and the area of possible changes in the
conductivity diagram (shaded)

For further consideration, we need a brief description of the structure of
system (I.1) on the Fermi surface in the presence of stable open trajectories
on it (see zorich1 ; dynn1 ; dynn3 ). We give here this description using a
model Fermi surface.

Consider in ${\bf p}$ \- space a periodic family of integral planes connected
by cylinders (Fig. 13). As before, we call a plane in the ${\bf p}$ \- space
integral if it is generated by two reciprocal lattice vectors.

Figure 13: Model Fermi surface carrying stable open trajectories of system
(I.1)

We assume that the surface under consideration is periodic with periods equal
to the reciprocal lattice vectors. In addition, we assume that all planes are
divided into even and odd ones, so that the even planes remain even, and the
odd planes remain odd, when shifted by any period.

It is easy to see that for directions of ${\bf B}$ almost orthogonal to the
direction of the planes, our cylinders contain closed trajectories, separating
the planes from each other (Fig. 13). In this case, our planes contain stable
open trajectories of system (I.1) with the mean direction given by the
intersection of the plane orthogonal to ${\bf B}$ and the integral direction
of the planes. The carriers of open trajectories are in this case
(periodically deformed) planes with holes formed after the removing of closed
trajectories (Fig. 14). It is also easy to see that the directions of open
trajectories are opposite to each other on even and odd planes.

Figure 14: Carrier of stable open trajectories of system (I.1) on the Fermi
surface

The above picture is stable with respect to small rotations of ${\bf B}$ and
is preserved as long as there are closed trajectories separating integral
planes on the cylinders connecting these planes. The corresponding stability
zone $\Omega$ is obviously the larger, the higher and narrower the cylinders
connecting the planes, and vice versa, is small for wide cylinders of low
height. It can also be noted that the disappearance of a cylinder of closed
electron-type trajectories corresponds to sections of the boundary of $\Omega$
adjacent to the hole Hall conductivity regions, while the disappearance of a
cylinder of closed hole-type trajectories corresponds to sections of the
boundary adjacent to the regions of the electronic Hall conductivity.

The presented picture is topological and geometrically it can look much more
complicated. In particular, carriers of open trajectories can be deformed much
more strongly, and the cylinders connecting them can have a very small height
and a rather complex shape. Nevertheless, the described topological
representation of the Fermi surface always arises when it contains stable open
trajectories of system (I.1) (zorich1 ; dynn1 ; dynn3 ). This representation
is not unique for a given Fermi surface; in particular, different such
representations for the same surface arise in different stability zones.

Let us now describe the possible changes in the picture of stability zones
when passing a saddle point of index 1 (see Fig. 10), using the above
structure. Let us first consider the case when the reconstruction of the Fermi
surface leads to the formation of stable open trajectories (Fig. 11) for some
direction of ${\bf B}$. Using the structure described above, we will show here
that the stability zones $\Omega_{\alpha}$ arising as a result of such a
reconstruction have, in a certain sense, a special shape, and also a specific
contribution to the conductivity tensor in the limit $\,B\rightarrow\infty\,$.

Since the arising of open trajectories occurs due to the reconstruction of the
Fermi surface, all such trajectories must pass through a narrow neck that
appears after the passage of the saddle singular point (Fig. 15). This means,
in particular, that the cycle $c$ shown in Fig. 15, must pass both through the
carrier of open trajectories running in one direction and through the carrier
of open trajectories running in the opposite direction. From this it follows
then that it also passes from one base of a cylinder of closed trajectories
separating these carriers to its other base. It can be seen, therefore, that
the height of at least one cylinder of closed trajectories connecting two
carriers of open trajectories is very small and tends to zero when approaching
the topological transition point. In addition, this cylinder has a saddle
singular point at each of its bases, which are adjacent to two different
carriers of open trajectories. It is not dificult to show then that such
points must lie on different necks in the ${\bf p}$ \- space, and the cylinder
itself, thus, passes through both these necks. It is easy to see then that
quite small rotations of ${\bf B}$, except for those orthogonal to the vector
connecting the indicated necks, will lead to the disappearance of the
indicated cylinder of closed trajectories and, thus, to going beyond the zone
$\Omega_{\alpha}$. It can be seen, therefore, that the stability zone formed
as a result of the reconstruction must be a very narrow region on the sphere
$\mathbb{S}^{2}$ (Fig. 16).

Figure 15: A neck formed after a topological reconstruction and a cycle
intersecting the resulting open trajectories  Figure 16: A stability zone
formed as a result of passing a singular point of index 1 with an increase in
the value of $\epsilon_{F}$ (schematically)

As the transition point is approached, the width of the region
$\Omega_{\alpha}$ tends to zero, so that $\Omega_{\alpha}$ tends to a one-
dimensional arc on the unit sphere (Fig. 16). It is easy to see that this arc
is a segment of a great circle orthogonal to some integer direction in the
${\bf p}$ \- space (namely, to the vector connecting the two necks considered
above). At the points of this segment, therefore, the plane orthogonal to
${\bf B}$ always contains some fixed reciprocal lattice vector. It is also not
difficult to show that the width of the region $\Omega_{\alpha}$ tends to zero
according to the law $\,\sim\sqrt{(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}}\,$
when approaching the transition point.

The passage of a saddle point of index 1 can lead to arising of both a finite
and an infinite number of narrow stability zones on the angular diagram. In
the latter case, as can be seen, we should expect the arising of an angular
diagram of type $B$ described above.

It can also be noted that the total area of the stability zones arising as a
result of the topological reconstruction tends to zero when approaching the
transition point. As a consequence, the passage of a singular point of index 1
does not cause an abrupt decrease in the area of the regions corresponding to
the Hall conductivity of the electronic type (and the presence of only closed
trajectories on the Fermi surface).

The formation of stability zones when passing through a singular point of
index 1 can occur on diagrams of the $0_{-}$, $1_{-}$, $A_{-}$, and $B$ types.
In the very first case, as is easy to see, this leads to a change of the type
$0_{-}$ immediately to the type $A_{-}$. As we have already said, such
degeneracies are typical for dispersion relations of sufficiently high
symmetry and arising of several singular points of the same type at one energy
level.

To describe the behavior of the conductivity tensor in our situation, we also
need to discuss the measure of open trajectories that appear on the Fermi
surface. In fact, for generic directions of ${\bf B}$ (maximal irrationality),
the measure of such trajectories is small in the parameter of proximity to the
transition point. To show this, consider the Fermi surface immediately before
passing through (one or several) singular points of index 1. For generic
directions of ${\bf B}$, there are only closed trajectories on it in this
case, and the Fermi surface itself represents a set of a finite number of
(non-equivalent) cylinders of closed trajectories separated by singular
trajectories (Fig. 17).

Figure 17: An example of a Fermi surface consisting of cylinders of closed
trajectories of the system (I.1)

Near the transition point at $\,\epsilon_{F}>\epsilon_{0}\,$, thin necks
appear on the Fermi surface, connecting its various parts. Considering such
necks on each of the cylinders of closed trajectories, we can see that with a
strong decrease in their diameter, most of the closed trajectories do not
undergo any changes (Fig. 18). As a result, in the limit
$\,\epsilon_{F}\rightarrow\epsilon_{0}\,$ almost all trajectories for such
directions of ${\bf B}$ remain closed. It is also easy to see that the
fraction of trajectories changed on each of the cylinders is proportional to
the ratio of the neck width to the height of the cylinder, i.e.
$\sqrt{(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}}$. The corresponding factor
also arises for the contribution (I.3) of open trajectories to the
conductivity tensor in the limit $\,\omega_{B}\tau\rightarrow\infty\,$ (as
well as a factor containing a weak logarithmic dependence on
$(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}$ due to the proximity of the
trajectories to singular points of system (I.1) in the narrow necks).

Figure 18: Thin necks adjacent to a cylinder of closed trajectories of finite
height and reconstruction of trajectories on the Fermi surface (cylinders of
conserved trajectories are shaded)

The above discussion, however, needs one important addition. Namely, in
addition to the proximity to a topological transition point, in the situation
under consideration, the proximity of the direction of ${\bf B}$ to the
directions corresponding to arising of periodic open trajectories on the Fermi
surface (even before the transition point) can also play an important role.
Such trajectories can exist on both sides of the topological transition and
occupy a finite area on the Fermi surface. For directions of ${\bf B}$ close
to such directions, at least some of the cylinders of closed trajectories
described above (Fig. 17) have small heights and large ‘‘transverse’’ sizes in
${\bf p}$ \- space. In this case, the ratio of the neck diameter to the
cylinder height can remain finite.

The proximity of generic directions of ${\bf B}$ to the directions described
above may be caused by the specific geometry of the stability zones (Fig. 16).
This applies, first of all, to the limit segment inside the zone which can be
a set of directions corresponding to arising of periodic trajectories even
before the transition point. As we have seen above, the width of the region
$\Omega_{\alpha}$ is also proportional to
$\sqrt{(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}}$ and the same we can also say
about the height of a part of the cylinders of closed trajectories that arise
for our directions of $\,{\bf B}\in\Omega_{\alpha}\,$ before the topological
transition (when there are no open trajectories for these directions yet). As
a consequence, the measure of open trajectories on the Fermi surface in the
zone $\Omega_{\alpha}$ can remain finite near the transition. In this case,
however, the emerging stable open trajectories have a specific geometry.
Namely, they are limited by straight strips of rather large width and repeat
the geometry of periodic trajectories on small scales (see Fig. 19).

Figure 19: Stable open trajectories arising for directions of ${\bf B}$ close
to directions of arising of periodic trajectories (schematically)

For the described directions of ${\bf B}$, we can introduce a function
$\mu({\bf B})$, which determines the ratio of the minimal width of strips
containing open trajectories to the size of the Brillouin zone. The
contribution of open trajectories to the conductivity at $\,\omega_{B}\tau\gg
1\,$ is different in the intervals $\,1\ll\omega_{B}\tau\leq\mu({\bf B})\,$
and $\,\omega_{B}\tau\gg\mu({\bf B})\,$. In the first case, this contribution
can be approximated by the formula (I.3), provided that the direction of the
$x$ axis coincides with the direction of periodic open trajectories. In the
second case, the direction of the $x$ axis must coincide with the mean
direction of stable open trajectories in ${\bf p}$ \- space, and the total
contribution of such trajectories to the conductivity tensor can be
represented as

$\sigma^{kl}\,\,\,\simeq\,\,\,{ne^{2}\tau\over
m^{*}}\left(\begin{array}[]{ccc}\mu^{2}(\omega_{B}\tau)^{-2}&(\omega_{B}\tau)^{-1}&\mu(\omega_{B}\tau)^{-1}\\\
(\omega_{B}\tau)^{-1}&\mu^{-2}&\mu^{-1}\\\
\mu(\omega_{B}\tau)^{-1}&\mu^{-1}&*\end{array}\right)$ (III.3)

($\omega_{B}\tau\,\rightarrow\,\infty$).

The function $\mu({\bf B})$ has a strong dependence on the direction of ${\bf
B}$ and goes to infinity for directions corresponding to the arising of
periodic trajectories that exist on both sides of the transition.

The zones $\Omega_{\alpha}$, which have additional (rotational) symmetry and
appear as a result of emerging of several singular points of index 1 at the
same level $\epsilon_{0}$, also deserve special mention. The sizes of such
zones tend to zero in all directions as the transition point is approached,
and the measure of open trajectories on the Fermi surface at $\,{\bf
B}\in\Omega_{\alpha}\,$ is proportional to
$\sqrt{(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}}$.

Symmetric stability zones, however, have one more peculiarity. Namely, the
vector $(m^{1},m^{2},m^{3})$ for such zones coincides with the direction
passing through the center of the zone (see dynn3 ), and for this direction of
${\bf B}$ there are no open trajectories on the Fermi surface. For directions
of ${\bf B}$ lying in a symmetric zone of small sizes, stable open
trajectories lie in straight strips of large width and have rather complex
behavior on small scales (see Fig. 20).

Figure 20: Stable open trajectories arising for directions of ${\bf B}$ lying
in a symmetric stability zone of small sizes

To describe the contribution to the conductivity given by open trajectories in
symmetric stability zones arising during the Lifshitz transitions, it is also
natural to consider the intervals $\,1\ll\omega_{B}\tau\leq\mu({\bf B})\,$ and
$\,\omega_{B}\tau\gg\mu({\bf B})\,$ for the function $\mu({\bf B})$, which has
the same meaning as above. In the first interval, the behavior of the
conductivity is more complicated (the arising of intermediate fractional
powers of $\omega_{B}\tau$ is possible), while the total contribution of such
trajectories to the conductivity tensor also contains a small factor of the
order of $\sqrt{(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}}$. In the second
interval, the contribution of open trajectories to the conductivity is similar
to the contribution (III.3), and also multiplied by a factor of the order of
$\sqrt{(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}}$. It is likely that the
observation also of a weak logarithmic dependence on
$(\epsilon_{F}-\epsilon_{0})/\epsilon_{F}$, due to the proximity to singular
points of $\epsilon({\bf p})$, is somewhat complicated here from the
experimental point of view.

It must be said that the conductivity in the region of the small stability
zones, arising as a result of the Lifshitz transitions, as a whole, has a
rather complex behavior, and it is probably more convenient to study the
geometry of such zones using methods that differ from methods of direct
measurements of conductivity (see, for example, ExactBound ). At the same
time, the arising of such zones plays a very important role in changing the
structure of a general conductivity diagram, especially in the case of arising
of complex diagrams of the type $B$.

Let us now consider the second possible situation, namely, the disappearance
of stable open trajectories of system (I.1) when passing through a singular
point of index 1 (Fig. 11). In this situation, therefore, we will talk about
the disappearance of a stability zone or a part of it. The corresponding
changes, obviously, can occur only on diagrams of the types $A_{-}$, $B$, and
$A_{+}$.

As we have already seen, the decay of stable open trajectories of system (I.1)
becomes possible when the carriers of such trajectories are no longer
separated from each other and the possibility of ‘‘jumping’’ between such
carriers appears. Thus, a topological reconstruction of the Fermi surface
leads to the decay of stable open trajectories if, as a result of the
reconstruction, a new cylinder is formed that connects two carriers and makes
it possible to ‘‘jump’’ between them for a given direction of ${\bf B}$. This
is exactly the situation that leads to the formation of plateaus in the values
of the functions $\tilde{\epsilon}_{1}({\bf B})$ and
$\tilde{\epsilon}_{2}({\bf B})$ (in our case, of $\tilde{\epsilon}_{2}({\bf
B})$) (see dynn3 ), which, in turn, should lead to jumps in the conductivity
diagrams, considered by us here.

As we have already noted, a change in the conductivity diagram during our
reconstructions can occur only in a special circular region (Fig. 12). With a
change of considered type, we can observe an instantaneous disappearance of
zones of finite sizes (or their parts) at the moment of the topological
transition. In addition, the measure of open trajectories on the Fermi surface
here also remains finite up to the moment of transition and instantly vanishes
(for generic directions of ${\bf B}$) when the surface is reconstructed.

At the same time, the proximity to the Lifshitz transition affects the
specifics of the trajectories of system (I.1), as well as the behavior of the
conductivity tensor, also in the described situation. In this case, it is
expressed in arising of very long closed trajectories of system (I.1) on the
Fermi surface for generic directions of ${\bf B}$ lying in the disappeared
stability zones or their disappeared parts (see Fig. 21). The average length
of such trajectories tends to infinity near the transition and decreases away
from it. In addition, as we noted above, the values of the functions
$\epsilon_{1}({\bf B})$ and $\epsilon_{2}({\bf B})$ differ from the values of
$\tilde{\epsilon}_{1}({\bf B})$ and $\tilde{\epsilon}_{2}({\bf B})$ for
directions of ${\bf B}$ corresponding to arising of periodic open
trajectories. As a consequence of this, the periodic trajectories of system
(I.1) on the Fermi surface do not disappear immediately at the moment of
transition, but persist for some time. As a result, the region of a vanished
stability zone (or its vanished part) is covered by a net of one-dimensional
arcs corresponding to the presence of periodic trajectories on the Fermi
surface. The net of corresponding arcs becomes denser when approaching a
topological transition and becomes infinitely dense at the moment of
transition.

Figure 21: Long closed trajectories arising on the Fermi surface near a
topological transition (schematically)

The described features of the trajectories of (I.1) near the Lifshitz
transition also lead to a rather complicated behavior of the conductivity
tensor in strong magnetic fields. The corresponding behavior of
$\sigma^{kl}({\bf B})$ was described in AnalProp , where it appeared in very
narrow regions near the boundaries of the zones $\Omega_{\alpha}$ in the
conductivity diagrams. Here, however, such behavior occurs in rather large
areas, namely, in the place of the disappeared zones $\Omega_{\alpha}$ or
their parts. In such domains, for generic directions of ${\bf B}$, it is
natural to introduce an (approximate) function $\lambda({\bf B})$, which
determines the ratio of the average size of long closed trajectories to the
size of the Brillouin zone. In addition, when considering the conductivity in
these regions, it is natural to keep the coordinate system corresponding to
the disappeared open trajectories, namely, choosing the mean direction of the
former open trajectories in the ${\bf p}$ \- space as the $x$ axis.

A natural consequence of the geometry of trajectories in the regions under
consideration is that their contribution to the conductivity manifests itself
as the contribution of closed trajectories under the much stronger condition
$\,\omega_{B}\tau\gg\lambda({\bf B})\,$, while in the interval
$\,1\ll\omega_{B}\tau\leq\lambda({\bf B})\,$ their contribution rather
corresponds to that of open trajectories. However, even in the limit
$\,\omega_{B}\tau\gg\lambda({\bf B})\,$, the contribution of the resulting
closed trajectories preserves anisotropy in the plane orthogonal to ${\bf B}$.

In addition, it can be seen that the long closed trajectories are formed from
open trajectories located on two different carriers. For dispersion relations
satisfying the condition $\,\epsilon({\bf p})=\epsilon(-{\bf p})\,$ (and Fermi
surfaces of not too large genus), this actually implies the relation
$\,\langle v^{z}_{gr}\rangle\rightarrow 0\,$ at $\,\lambda({\bf
B})\rightarrow\infty\,$ for the trajectory-averaged electron group velocity
along the direction of magnetic field. As a consequence, the contribution of
such trajectories to the conductivity along the magnetic field actually tends
to zero in the limit $\,\omega_{B}\tau\gg\lambda({\bf B})\gg 1\,$. In the
latter, such a contribution is similar to the contribution to the conductivity
given by unstable Dynnikov’s trajectories in the limit $\,\omega_{B}\tau\gg
1\,$.

In general, the total contribution of the long closed trajectories to the
symmetric $s^{kl}$ and antisymmetric $a^{kl}$ parts of the conductivity tensor
in the limit $\,\omega_{B}\tau\gg\lambda({\bf B})\,$ can be represented as
(AnalProp )

$s^{kl}(B)\,\,\,\simeq\,\,\,\left(\begin{array}[]{ccc}0&0&\,\,0\cr
0&0&\,\,0\cr 0&0&\sigma^{zz}(\lambda)\end{array}\right)\,\,\,+\\\
+\,\,\,{ne^{2}\tau\over
m^{*}}\left(\begin{array}[]{ccc}(\omega_{B}\tau)^{-2}&\lambda(\omega_{B}\tau)^{-2}&\lambda(\omega_{B}\tau)^{-2}\cr\lambda(\omega_{B}\tau)^{-2}&\lambda^{2}(\omega_{B}\tau)^{-2}&\lambda^{2}(\omega_{B}\tau)^{-2}\cr\lambda(\omega_{B}\tau)^{-2}&\lambda^{2}(\omega_{B}\tau)^{-2}&\lambda^{2}(\omega_{B}\tau)^{-2}\end{array}\right)$

(where $\,\sigma^{zz}(\lambda)\rightarrow 0\,$ at $\lambda\rightarrow\infty$),

$a^{kl}(B)\,\,\,\simeq\,\,\,{ne^{2}\tau\over
m^{*}}\left(\begin{array}[]{ccc}0&(\omega_{B}\tau)^{-1}&(\omega_{B}\tau)^{-1}\cr(\omega_{B}\tau)^{-1}&0&(\omega_{B}\tau)^{-1}\cr(\omega_{B}\tau)^{-1}&(\omega_{B}\tau)^{-1}&0\end{array}\right)$

It must be said that the condition $\,\omega_{B}\tau\gg\lambda({\bf B})\,$ can
be quite strong, and in many cases some intermediate regime between the regime
(I.3) and the dependence described above can be observed. We also note that,
in the general case, in addition to the contribution described above, we must
also add the contribution (I.2) from ‘‘ordinary’’ closed trajectories, which
are also present on the Fermi surface in the described situation.

The value $\lambda({\bf B})$ goes to infinity at the topological transition
point. For a fixed generic direction of ${\bf B}$, its behavior near
$\epsilon_{0}$ can be (approximately) described by the dependence

$\lambda\,\sim\,\sqrt{\epsilon_{F}/|\epsilon_{F}-\epsilon_{0}|}$

At the same time, the dependence of $\lambda$ on the direction of ${\bf B}$ is
quite complicated, in particular, $\lambda$ goes to infinity on the
(preserved) arcs corresponding to arising of periodic trajectories of (I.1).
In general, vanishing stability zones (or parts of them) can be called regions
of complex conductivity behavior in strong magnetic fields.

In fact, both described effects (arising and disappearance of stability zones)
can be observed simultaneously (in different parts of an angular diagram) when
passing through a saddle singular point of index 1. Fig. 22 shows an example
of one of such reconstructions of the Fermi surface. It is easy to see that
before the reconstruction the angular diagram is rather simple (of the type
$A_{-}$) and contains one stability zone (with a diametrically opposite one).
After the reconstruction, a part of the stability zone disappears being
replaced by a zone of complex conductivity behavior. In addition, many small
zones arise that separate the region of the electron Hall conductivity from
the region of the hole Hall conduction that appears in the diagram. It can be
shown that, in the immediate vicinity of the topological transition, chains of
small stability zones are located very close to zones of complex conductivity
behavior, while further away from the transition they shift towards the
‘‘equator’’. In general, the conductivity diagram after the transition is of
type $B$ and, in the generic case, must also contain directions of ${\bf B}$
corresponding to arising of unstable trajectories of the Tsarev or Dynnikov
type.

Figure 22: An example of a passage of a saddle singular point of index 1 and
the corresponding change in the angular conductivity diagram (schematically,
the change in the picture of stability zones and the formation of a region of
complex conductivity behavior are shown)

We note here that the passage of singular points with an increase in the value
of $\epsilon_{F}$ can cause abrupt changes in the conductivity diagrams,
however, preserving the general direction of their evolution, shown in Fig. 7.
As a consequence, the disappearance of a stability zone or part of it leads in
this situation to an abrupt increase in the area corresponding to the hole
Hall conductivity (and presence of only closed trajectories on the Fermi
surface). At the same time, the area of the region corresponding to the
electronic Hall conductivity (and presence of only closed trajectories on the
Fermi surface) immediately near the transition remains unchanged. It can be
seen, therefore, that if the described effect (disappearance of stable open
trajectories when passing through a singular point of index 1) takes place for
a diagram of type $A_{-}$, the type of the diagram changes to type $B$. From
the same considerations, we can conclude that the observation of the described
effect on diagrams of type $B$ does not change the type of a diagram.
Observation of the described effect on a diagram of type $A_{+}$ preserves the
type of the diagram or changes it to type $1_{+}$.

In connection with what has been said above, we would like to note here
another important circumstance. Namely, diagrams of type $B$ are generic
diagrams and should, generally speaking, be observed in the study of
sufficiently rich families of Fermi surfaces of sufficiently complex shape.
However, they have not yet been experimentally discovered. The same is also
true for unstable trajectories of the Tsarev or Dynnikov type, which must
accompany such diagrams. One of the reasons for this, in our opinion, may be a
rather small width of the interval $(\epsilon^{\cal B}_{1},\epsilon^{\cal
B}_{2})$ and, accordingly, a low probability of the value $\epsilon_{F}$
falling into it for real dispersion relations. In this regard, it can be
expected that the use of the Lifshitz transitions can provide good
opportunities for observing such diagrams, as well as nontrivial regimes of
conductivity behavior corresponding to arising of Tsarev or Dynnikov
trajectories on the Fermi surface.

In general, the effects we have discussed can produce changes in the
conductivity diagrams listed below, with the following possible changes in the
diagram type

$\begin{array}[]{c}0_{-}\rightarrow A_{-}\\\ 1_{-}\rightarrow A_{-}\\\
A_{-}\rightarrow\\{A_{-},B\\}\\\ B\rightarrow B\\\
A_{+}\rightarrow\\{1_{+},A_{+}\\}\end{array}$

The effects described above correspond to the passage of a saddle singular
point of index 1 in the ‘‘forward’’ direction, namely, as the Fermi energy
$\epsilon_{F}$ increases. Certainly, as a result of an external influence, the
Lifshitz transitions can be performed both in the ‘‘forward’’ and ‘‘backward’’
directions. Moreover, an external action does not necessarily simply change
the Fermi level, but generally changes the entire dispersion relation
$\epsilon({\bf p})$. Obviously, in the general case, the topology of a
reconstruction of the Fermi surface is actually determined by changes in the
relation $\epsilon({\bf p})$ near the corresponding singular point. It is
natural, however, to keep the terms ‘‘the passage of a singular point in the
forward or backward direction’’, based on the coincidence of the topology of
the corresponding transition with the topology of the transition with
increasing or decreasing $\epsilon_{F}$.

Considering the change in the relation $\epsilon({\bf p})$ to be continuous,
in a rather narrow neighborhood of a topological transition, the influence of
the general change in $\epsilon({\bf p})$ can be neglected and it can be
assumed that the main changes in the picture of stability zones are caused
precisely by the reconstruction of the Fermi surface. In order to describe the
corresponding changes in the angular diagram, we can use the picture obtained
in the consideration given above. As we have already said, the above picture
refers to the passage of a singular point of index 1 in the forward direction.
In the general case, we are interested here in a similar description for
passing a point of index 1 in the backward direction, as well as passing a
point of index 2 in the forward and backward directions.

The passage of a singular point of index 1 in the backward direction naturally
leads to effects opposite to those described above. Namely, changing of stable
open trajectories when passing a saddle point of index 1 in the backward
direction can produce changes in the conductivity diagrams of the types listed
below, with the following possible changes in the type of diagram:

$\begin{array}[]{c}A_{-}\rightarrow\\{0_{-},1_{-},A_{-}\\}\\\
B\rightarrow\\{A_{-},B\\}\\\ A_{+}\rightarrow A_{+}\\\ 1_{+}\rightarrow
A_{+}\end{array}$

In this case, the picture of stability zones in the conductivity diagram can
undergo the following specific changes

1) Reducing the size of a certain (finite or infinite) number of stability
zones and their disappearance directly at the topological transition point.

2) Formation of regions of complex conductivity behavior of finite size in the
region of the hole Hall conductivity when approaching a topological transition
and their transformation into stability zones at the transition point (or
their attachment to already existing zones).

The passage of a saddle singular point of index 2 in the forward direction is
in fact similar to the passage of a singular point of index 1 in the backward
direction, however, with the replacement of the electronic Hall conductivity
regions by the hole Hall conductivity regions, and vice versa. Thus, a change
in stable open trajectories when passing a saddle singular point of index 2 in
the forward direction can produce changes in the conductivity diagrams of the
types listed below with the following possible changes in the diagram type:

$\begin{array}[]{c}1_{-}\rightarrow A_{-}\\\ A_{-}\rightarrow A_{-}\\\
B\rightarrow\\{A_{+},B\\}\\\
A_{+}\rightarrow\\{0_{+},1_{+},A_{+}\\}\end{array}$

In this case, the picture of stability zones in the conductivity diagram can
undergo the following specific changes.

1) Reducing the size of a certain (finite or infinite) number of stability
zones and their disappearance directly at the topological transition point.

2) Formation of regions of complex conductivity behavior of finite size in the
region of the electronic Hall conductivity when approaching a topological
transition and their transformation into stability zones at the transition
point (or their attachment to already existing zones).

Similarly, changing of stable open trajectories when passing a saddle singular
point of index 2 in the backward direction can produce changes in the
conductivity diagrams of the types listed below, with the following possible
changes to the diagram type:

$\begin{array}[]{c}A_{-}\rightarrow\\{1_{-},A_{-}\\}\\\ B\rightarrow B\\\
A_{+}\rightarrow\\{A_{+},B\\}\\\ 1_{+}\rightarrow A_{+}\\\ 0_{+}\rightarrow
A_{+}\end{array}$

In this case, the picture of stability zones in the conductivity diagram can
undergo the following specific changes.

1) Arising (of a finite or infinite number) of new stability zones of zero
size at the transition point and a gradual increase in their size with further
distance from it.

2) Disappearance of some stability zones of finite size (or their parts) with
the formation of regions of complex conductivity behavior with the electronic
type of Hall conductivity. If this effect is observed on a $A_{+}$ diagram, it
turns into a $B$ type diagram, with the arising of an infinite number of
stability zones, as well as directions of ${\bf B}$ corresponding to the
occurrence of Tsarev’s or Dynnikov’s trajectories on the Fermi surface.

It can also be noted here that each of the above descriptions can also be used
in the case of passing several singular points of the same type at once. Such
a situation, in fact, can arise quite often for dispersion relations with
additional (rotational) symmetry.

As for the simultaneous passage of singular points of different types, such a
situation is a nongeneric situation and is observed only for special
dispersion relations. In particular, it may refer to relations separating
relations with simple angular diagrams (with a single stability zone) and
relations with complex angular diagrams.

Each of the above descriptions of the changes in conductivity diagrams,
certainly, can also be used for dispersion relations with simple angular
diagrams, taking into account the peculiarities of the evolution of
conductivity diagrams for such relations.

In conclusion, we would like to note that although the above picture refers
primarily to changes in the structure of stability zones in conductivity
diagrams, it also, in fact, describes many features of arising and
disappearance of unstable open trajectories of system (I.1) on the Fermi
surface under the described reconstructions. Moreover, if we talk about
trajectories of the Tsarev or Dynnikov type, their arising is uniquely related
with diagrams of the type $B$ and, thus, is completely determined by the
picture of stability zones on the unit sphere.

If we consider unstable periodic open trajectories of system (I.1), then, as
can be seen, most of them are also associated with stability zones and appear
either near their edges or in regions of complex conductivity behavior. As can
also be seen, most of these trajectories do not change directly at the
topological transition point, but undergo changes at some distance from it,
following the changes in the corresponding stability zone.

Among the unstable periodic trajectories of system (I.1), however, we should
also especially note the trajectories that are not tied to any specific
stability zone on the conductivity diagram. In fact, the corresponding
directions of ${\bf B}$ almost always belong to some zones
$\widehat{\Omega}_{\alpha}$ for the entire dispersion relation, and the
corresponding trajectories are consistent with stable open trajectories,
occurring in these zones. However, when they appear far from the corresponding
interval $(\tilde{\epsilon}_{1}({\bf B}),\tilde{\epsilon}_{2}({\bf B}))$, they
usually are not very interesting from this point of view. Instead, however,
they are usually closely related to the topology of a given Fermi surface and
exhibit its geometric properties well. A representative example of such
trajectories are the periodic trajectories considered in Lifshits1960 . It is
easy to see that changes associated with the arising or disappearance of such
trajectories correspond to transitions between diagrams of type 0 and 1, or do
not change the diagram type.

##  IV Conclusion

The paper considers the topological Lifshitz transitions in metals and related
changes in galvanomagnetic phenomena from the point of view of the general
Novikov problem. Namely, the picture of possible changes in electron
trajectories in a metal during topological reconstructions of the Fermi
surface and the corresponding changes in the behavior of electrical
conductivity in the presence of strong magnetic fields is considered. The
consideration is based on the classification of non-closed electron
trajectories arising on Fermi surfaces of arbitrary complexity, and of
corresponding behavior of conductivity in strong magnetic fields. The main
analysis is based on the description of possible changes in the picture of
stable open trajectories on the Fermi surface during topological transitions.
As shown in the paper, the Lifshitz transitions are accompanied by a certain
number of such changes, which make it possible to determine the features of
the transition topology based on the observation of the conductivity in strong
magnetic fields. The results obtained in the work can serve as one of the
tools for studying topological Lifshitz transitions in metals with complex
Fermi surfaces.

The study was supported by the grant of the Russian Science Foundation №
21-11-00331, ‘‘Geometric methods in the Hamiltonian theory of integrable and
almost integrable systems’’

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