Topological integrability, classical and quantum chaos, and the theory of dynamical systems in the physics of condensed matter
aa r X i v : . [ m a t h - ph ] A p r Topological integrability, classical andquantum chaos, and the theory of dynamicalsystems in the physics of condensed matter.
A.Ya. Maltsev , S.P. Novikov , L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences142432 Chernogolovka, pr. Ak. Semenova 1A V.A. Steklov Mathematical Institute of Russian Academy of Sciences119991 Moscow, Gubkina str. 8
Abstract
The paper is devoted to the questions connected with the investi-gation of the S.P. Novikov problem of the description of the geometryof level lines of quasiperiodic functions on a plane with different num-bers of quasiperiods. We consider here the history of the question, thecurrent state of research in this field, and a number of applications ofthis problem to various physical problems. The main attention is paidto the applications of the results obtained in the field under consider-ation to the theory of transport phenomena in electron systems.
In this paper we will mainly consider the formulation of problems, researchand results related to the S.P. Novikov problem, set in the early 1980s (see[42]) and having relation in fact to many areas of research, in particular, suchas the theory of dynamical systems, the theory of quasiperiodic functions,the theory of topological phenomena in the physics of condensed matter, thetheory of transport phenomena in systems of various dimensions, etc. In thevery first formulation, the problem of S.P. Novikov was connected with thedescription of the levels of multivalued functions on manifolds, but it has thesame natural formulation also in the language of the theory of dynamicalsystems. Most of our paper will be connected with the investigations of1he Novikov problem which relate to dynamical systems arising on complexFermi surfaces in metals in the presence of an external magnetic field. Asis well known, the description of the trajectories of such systems can beeffectively reduced to the description of the level lines of height functionson periodic two-dimensional surfaces embedded in three-dimensional space.More precisely, the dynamics of electron states in a crystal in the presenceof an external magnetic field is described by the system (see, e.g. [1, 25, 49])˙ p = ec [ v gr × B ] = ec [ ∇ ǫ ( p ) × B ] , (1.1)where p is the quasimomentum of the electron state. It is extremely im-portant here that the space of quasimomenta represents actually a three-dimensional torus T , and not the Euclidean space R , as in the case of freeelectrons.The space of electron states can also be represented as the space R pro-vided that any two values of p that differ by a reciprocal lattice vector, definethe same quantum-mechanical state. The function ǫ ( p ) represents in thiscase a 3-periodic function in R , and its level surfaces are two-dimensional3-periodic surfaces in the same space. The reciprocal lattice L ∗ in the quasi-momentum space can be defined with the aid of its basis vectors a , a , a ,which are connected by the relations a = 2 π ~ l × l ( l , l , l ) , a = 2 π ~ l × l ( l , l , l ) , a = 2 π ~ l × l ( l , l , l )with the basis vectors ( l , l , l ) of the direct lattice of the crystal. As is nothard to see, the exact phase space then represents a torus T = R /L ∗ , obtained from the complete p - space with the aid of the factorization bythe vectors m a + m a + m a , m , m , m ∈ Z The system (1.1) is a Hamiltonian system with the Hamiltonian H = ǫ ( p ) and the Poisson bracket { p , p } = ec B , { p , p } = ec B , { p , p } = ec B It is easy to see that the consequence of this fact is, in particular, theconservation of the quantity ǫ ( p ) , as well as the projection of the quasimo-mentum on the direction B , along trajectories of the system. As a conse-quence, geometrically the trajectories of the system (1.1) in p - space are2ctually given by the intersections of surfaces of constant energy ǫ ( p ) = constwith planes orthogonal to the magnetic field. It is also easy to see that dueto this property the problem under consideration has two natural interpre-tations, namely, the problem of describing the trajectories of a dynamicalsystem with an ambiguous conservation law on a compact surface and theproblem of describing the level lines of a quasiperiodic function on a planewith three quasiperiods. It must be said that both these interpretations playan extremely important role in the study of the problem.The important role of the geometry of the trajectories of the system (1.1)in the behavior of the electrical conductivity of metals in strong magneticfields (magnetoconductivity) was first discovered by the school of I.M. Lifshits(I.M. Lifshits, M.Ya. Azbel, M.I. Kaganov, V.G. Peschanskii) in the late1950s - early 1960s (see [26, 27, 28, 29, 30, 31, 32, 24]). We note at oncethat in the theory of normal metals usually only the single energy level ǫ F (the Fermi level) is important for most processes occurring in metal, while alllevels far from the Fermi level are always either completely filled, or empty,and do not affect the ongoing processes. As a consequence, in the theory ofgalvanomagnetic phenomena, only the trajectories of the system (1.1), whichlie on the Fermi surface S F : ǫ ( p ) = ǫ F , are interesting, and it is the complexity of the Fermi surface that determinesthe features of electron transport phenomena in strong magnetic fields.Thus, in the paper [26] there was indicated the principal difference inthe behavior of the magnetoconductivity in strong magnetic fields in thepresence of only closed trajectories of the system (1.1) on the Fermi surfacein p - space and in the presence of periodic open trajectories in p - spaceon this surface. In both cases, the asymptotic behavior of the conductivitytensor can be represented as a regular series in inverse powers of B , and, inparticular, has the following form in the presence of only closed trajectorieson the Fermi surface in p - space: σ ik ≃ ne τm ∗ ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ∗ , ω B τ → ∞ (1.2)In the formula (1.2) the value n represents the concentration of chargecarriers in the metal, and the value m ∗ determines the order of the effective3ass of the electron in the crystal. The time τ plays the role of the meanfree time of an electron and depends on the purity as well as the temperatureof the crystal. The value ω B = eB/m ∗ c has the meaning of the cyclotronfrequency of the electron in the crystal, it should be noted here that, unlikethe case of the free electron gas, the cyclotron frequency is defined here onlyfor closed trajectories of the system (1.1) and coincides with the parameter ω B only in order of magnitude. The sign ≃ expresses here the equality up toa dimensionless coefficient (of order 1) and the notation ∗ means also someconstant. Here and in what follows we will always assume that the axis z in our considerations is chosen along the direction of the magnetic field. Itis easy to see that the formula (1.2) coincides essentially with the analogousformula for the free electron gas, differing from it only by possible numericalparameters.A completely different situation arises in the presence of periodic opentrajectories on the Fermi surface in p - space. As was shown in [26], afterthe choice of the axis x along the average direction of the periodic trajectoryin p - space, the principal term of the asymptotic expansion of the tensor σ ik ( B ) can be represented in the form σ ik ≃ ne τm ∗ ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ( ω B τ ) − ∗ ∗ ( ω B τ ) − ∗ ∗ , ω B τ → ∞ (1.3)It can be seen that the electric conductivity tensor here has a stronganisotropy in the plane orthogonal to B in strong magnetic fields, whichmakes its behavior fundamentally different from the case of the free electrongas. It is easy to see that the formula (1.3) allows also to measure the averagedirection of periodic trajectories in p - space as the direction of the greatestsuppression of conductivity in the plane orthogonal to B . Let us note herethat both the closed and periodic trajectories in p - space are represented byclosed trajectories of the system (1.1) in the torus T . It can thus be seenthat for the description of galvanomagnetic phenomena in metals, not onlythe shape of the trajectories of (1.1) on the Fermi surface is important, butalso their homology classes under the embedding in the torus T .Quasiclassical trajectories of the system (1.1) correspond also to quasi-classical trajectories (electron packets) in the x - space, which are determinedfrom the system ˙ x = v gr ( p ) = ∇ ǫ ( p )The electron trajectories in x - space do not coincide in common with thetrajectories of the system (1.1), in particular, they are not plane. However,their shape correlates quite strongly with the shape of the trajectories of41.1). For example, the projections of the electron trajectories in the x -space onto the plane, orthogonal to B , are similar to the correspondingtrajectories of the system (1.1), rotated by 90 ◦ . As already noted above, theshape of the trajectories of the system (1.1) becomes important in the limit ω B τ ≫ p - space between two scattering acts.In the papers [27, 28] examples of open trajectories of a more generalform on Fermi surfaces of different shapes were considered. The trajectoriesconsidered in [27, 28] are not periodic in general case, however, they also havean average direction in the plane orthogonal to B , which also leads to a sharpanisotropy of the electric conductivity tensor in the same plane. It must besaid, however, that the analytic properties of the electric conductivity tensorare generally more complex here in comparison with the case of periodictrajectories (see e.g. [24, 38]).The problem of the complete classification of various types of trajectoriesof system (1.1) was first set by S.P. Novikov in the work [42] and was activelyinvestigated in his topological school during the last decades. At present, theNovikov problem of the classification of trajectories of (1.1) (with an arbitrarydispersion law) has been studied in many details, and in particular, ratherprofound results have been obtained, which have already found applicationsin the theory of solids.Let us note at once that the most important part of the results obtainedin the investigation of Novikov’s problem was the description of stable opentrajectories of the system (1.1) for an arbitrary dispersion law (A.V. Zorich,I.A. Dynnikov). The remarkable geometric properties of such trajectoriesmade it possible to define important topological characteristics (topologicalquantum numbers) observable in studies of conductivity of normal metals,which, in particular, provide a convenient tool for determining the orientationof the crystal lattice in such studies (S.P. Novikov, A.Ya. Maltsev). We alsonote here that the description of the geometric properties of stable opentrajectories of system (1.1) allows, in addition, to approach more strictlythe description of the analytic properties of conductivity in strong magneticfields in the presence of trajectories of this type.At the same time, a detailed study of the system (1.1) has also led tothe discovery of new, rather nontrivial, types of trajectories of such systemswhose properties are the subject of active study at the present time. Let usnote here that these trajectories exhibit very interesting (chaotic) propertiesboth from the geometric point of view and from the point of view of thedescription of transport phenomena in normal metals when they appear.We must now say that the applications of the problem of describing the5rajectories of the system (1.1) are not really limited to transport phenom-ena in normal metals in strong magnetic fields. For example, the problem ofdescribing the trajectories of systems similar to (1.1) also arises in the de-scription of transport phenomena in two-dimensional electron systems placedin artificially created quasiperiodic (super)potentials in the presence of an ex-ternal magnetic field. The Novikov problem is formulated here as the problemof describing the geometry of the level lines of a quasiperiodic function on aplane with a fixed number of quasiperiods. It is not difficult to see here thatthe problem of describing the level curves of a function with three quasiperi-ods (the first case following the periodic one) coincides in reality with theNovikov problem in its formulation given above. Thus, all the results ob-tained in the study of the trajectories of the system (1.1) are also transferredto transport phenomena in two-dimensional electron systems in superpoten-tials with three quasiperiods. We must note here that, unlike the situationwith normal metals, the parameters of such systems are controlled in thiscase, which makes it possible to implement any of the cases of behavior ofthe system (1.1) which is interesting to us.The problem of describing the geometry of the level lines of functionswith a larger number of quasiperiods is actually much more complicated.For example, for the case of functions with four quasiperiods, a rather se-rious result has now been obtained (S.P. Novikov, I.A. Dynnikov), whichdistinguishes an important class of potentials that have topologically regu-lar open level lines analogous to stable open trajectories of the system (1.1)in the three-dimensional case. But on the whole, the Novikov problem forfour quasiperiods has been studied much less compared to the case of threequasiperiods.Here we try to give the most complete overview of the results obtainedto date and describe, as much as possible, the range of possible applicationsof the Novikov problem in physical problems. In this chapter we will consider stable open trajectories of the system (1.1)and describe the main properties of transport phenomena in metals in strongmagnetic fields in the presence of such trajectories on the Fermi surface. Wesay at once that under the stability of the open trajectories of the system (1.1)we mean here the preservation of such trajectories, as well as their geometricproperties, for all small variations in the direction of B or the energy level6igure 1: The form of a stable open trajectory in the plane, orthogonal to B , in the space of quasimomenta. ǫ ( p ) = ǫ . We also note that we consider the system (1.1) in the coveringspace of quasimomenta, where geometrically its trajectories are given by theintersections of surfaces of constant energy and planes orthogonal to themagnetic field. As we said above, the dispersion relation ǫ ( p ) is assumedhere to be an arbitrary smooth 3-periodic function in the p - space, withperiods equal to the vectors of the reciprocal lattice.Let us describe here two important properties of stable open trajectoriesof the system (1.1), which follow from the results obtained in the papers[50, 14, 15].1) All stable open trajectories of the system (1.1) for a fixed direction of B lie in straight strips of finite width in planes orthogonal to B , passingthrough them (Fig. 1).2) All stable open trajectories of the system (1.1) for a fixed direction of B have the same mean direction in p - space defined by the intersection ofthe plane orthogonal to B with some integral (generated by two vectors ofthe reciprocal lattice) plane Γ , unchanged for all close directions of B .In a more precise formulation, it follows from the papers [50, 14] that theproperties (1)-(2) hold for open trajectories of (1.1) for the directions of B ,sufficiently close to rational directions, while it follows from the paper [15]that properties (1)-(2) hold for open trajectories that are stable with respectto variations of the energy level ǫ ( p ) = ǫ . It is not difficult to see here thatto satisfy conditions (1)-(2) it is sufficient to require either the stability oftrajectories with respect to small rotations of the direction of B , or stabilitywith respect to variations of the energy level. Let us also note here thatproperty (1) was first expressed by S.P. Novikov in the form of a conjecture,and was thus proved later for stable open trajectories of (1.1).Conditions (1) and (2) play an important role for transport phenomenain metals and served as the basis for introducing in [43] (see also [44]) impor-7ant topological characteristics (topological quantum numbers) observablein magnetic conductivity in the presence of stable open trajectories on theFermi surface. Thus, the fulfillment of conditions (1) and (2) leads to astrong anisotropy of the conductivity in the plane orthogonal to B in thelimit ω B τ ≫ σ ik ≃ ne τm ∗ o (1) o (1) o (1) o (1) ∗ ∗ o (1) ∗ ∗ , ω B τ → ∞ (2.1)provided that the axis x coincides with the mean direction of the open trajec-tories in the p - space. Thus, the mean direction of stable open trajectoriesin p - space is directly observable as the direction of the largest suppres-sion of conductivity in the plane orthogonal to B in strong magnetic fields.Variations of the direction of the magnetic field within the stability zone forthe given family of open trajectories determine, in this case, the direction ofthe integral plane Γ in p - space associated with this family according tocondition (2).We note here that the plane Γ is integral in the space of quasimomenta,which means that it is generated by some two vectors of the reciprocal lattice.In particular, it does not have to coincide in the general case with any of thecrystallographic planes in the coordinate space. Instead, it can be given incoordinate space by the relation M ( x , l ) + M ( x , l ) + M ( x , l ) = 0 , M , M , M ∈ Z and, thus, is orthogonal to one of the integer crystallographic directions.The irreducible integer triple ( M , M , M ) represents here a topologicalcharacteristic of the corresponding family of stable open trajectories that isdirectly observable in conductivity measurements in strong magnetic fields.Integral triples ( M α , M α , M α ) for the complete set of all different StabilityZones Ω α in the space of directions of B were called in [43] the topologicalquantum characteristics (topological quantum numbers) observable in theconductivity of normal metals.It can thus be seen that the measurement of the conductivity for differentdirections of B within the same Stability Zone Ω α allows us to determinequite accurately some (known) crystallographic direction in the coordinatespace. The same conductivity measurement for the directions of B lying in8wo different Stability Zones (with different topological quantum numbers)makes it possible to completely determine the orientation of the crystal latticeof a single crystal. Let us note here that such a method of determining theorientation of a single crystal sample is significantly more convenient than,for example, the measurement of the exact angular diagram of conductivity,since (as we shall see below) the exact boundaries of the Stability Zones areactually difficult to observe in direct conductivity measurements (see, e.g.[38, 39]), and also because precise determination of the theoretical boundariesof the Stability Zones for a given dispersion relation is also a serious task inthe general case (see [7]).Stability Zones represent domains with piecewise smooth boundaries inthe space of directions of B (on the unit sphere S ). The full Stability ZoneΩ α can be defined as a complete domain on the sphere S such that for anydirection B ∈ Ω α there are stable open trajectories on the Fermi surfacethat correspond to the same topological quantum numbers ( M α , M α , M α ).It is easy to see that the full Stability Zone is invariant under the substitution B → − B and often consists of two opposite connected components on theunit sphere.The addition to the union of the bands Ω α on the unit sphereˆ S = S \ ∪ Ω α is by definition the set of directions of B for which stable open trajectories arenot present on the Fermi surface. At the same time, however, the set ˆ S cancontain directions of B , corresponding to the appearance of unstable opentrajectories of the system (1.1) on the Fermi surface. Among such directions,we should especially note the directions of B , which lead to the appearance ofperiodic open trajectories on the Fermi surface that are unstable for B ∈ ˆ S .The presence of a large number of such directions of B on the set ˆ S is in facta consequence of the topological structure of the system (1.1) for B ∈ Ω α ,which we briefly describe below.The topological structure of the system (1.1) in the presence of stableopen trajectories was described in the papers [50, 15] and is based on aproperty that can be called “topological integrability” for systems of thistype. Namely, consider the system (1.1) on a fixed surface ǫ ( p ) = const .Let us assume that the direction of B has maximal irrationality and removefrom this surface all closed trajectories of the system (1.1) (homologous tozero in the torus T ). It is not difficult to see that the remainder of thesurface will be the carrier of the open trajectories of the system (1.1). Asfollows from the works [50, 15], in the presence of stable open trajectories ofthe system (1.1) on the surface, each connected component of such a carrier9 . . B Figure 2: Structure of a connected component of the Fermi surface carryingstable open trajectories of the system (1.1) in the covering p - space.represents a two-dimensional torus T with contractible holes embedded inthe torus T .Returning to the covering p - space, we can thus say that any connectedcomponent of the energy (Fermi) surface carrying stable open trajectoriesof the system (1.1) has a well-defined topological structure, defined by thetrajectories of the system (1.1). Namely, each such connected componentrepresents a union of integral (periodically deformed) planes in the p - spaceconnected by parts formed by cylinders of closed trajectories of the system(1.1). It can be noted here also that for almost any real dispersion law thecorresponding parts will in fact be simple cylinders of closed trajectoriesbounded by singular trajectories on their bases (Fig. 2).The results of investigations of the Novikov problem in solid state theoryare most significant in cases when the dispersion relation for the electron ina crystal has a rather complex form. In particular, this refers to the shape ofthe Fermi surface, which is also assumed here to be quite complicated. Wecan say, for example, that a metal has a complex Fermi surface S F , if it hasrank 3, where the rank of the surface is determined by the dimension of theimage of the mapping H ( S F ) → H ( T )It can also be seen that a connected Fermi surface of rank 3 must neces-sarily have genus g ≥ p - space. It can also be noted that for the overwhelmingnumber of real crystals, the number of nonequivalent integral planes in thedescribed representation is exactly two, since a larger number of such planescorresponds to rather large genera of the Fermi surface. Based on the formof real dispersion relations, it is this situation, therefore, that should be con-sidered typical when stable open trajectories of the system (1.1) appear onthe Fermi surface. We also note here that the structure shown at Fig. 2 hasa purely topological character and can be visually much more complicatedfor real Fermi surfaces.It is not difficult to see that the presence of at least one pair of carri-ers of stable open trajectories excludes the appearance of open trajectoriesof the system (1.1) having a different form, and that all the present opentrajectories have the same direction in the p - space in this case. We notethat this property is also true for Fermi surfaces consisting of several con-nected components (in the torus T ), provided that these components donot intersect each other. This circumstance is especially important for Fermisurfaces of real crystals, which usually consist of several components, de-termined by different dispersion relations (the property of non-intersectionof different components of the Fermi surface is, as a rule, preserved in thiscase). This property of stable open trajectories for the full Fermi surface wascalled the Topological Resonance in [34, 35] and plays an important role indescribing angular diagrams for conductivity in real conductors. In partic-ular, this property excludes the possibility of crossing two Stability Zoneswith different topological quantum numbers on the angular diagram.It is easy to see also, that in the situation described above the opentrajectories of the system (1.1) exist on all the presented integral planes forall directions of B , except possibly for the direction orthogonal to the planeΓ α (if this direction belongs to the Zone Ω α ). It can also be seen that thedescribed structure of the Fermi surface is preserved under rotations of thedirection of B as long as all the cylinders of closed trajectories connectingthe integral planes remain unbroken. The boundary of the correspondingStability Zone on the angular diagram is thus determined by the conditionthat the height of one of the cylinders of closed trajectories described above(Fig. 3) goes to zero. 11 B B
Figure 3: The vanishing of the height of one of the cylinders of closed tra-jectories of the system (1.1), followed by a “jump” of trajectories from oneintegral plane to another when crossing the boundary of a Stability Zone onthe angular diagram. B Ω α Figure 4: The Fermi surface having a Stability Zone Ω α with a simple bound-ary.It is not difficult to see that the total boundary of a Stability Zone on theangular diagram can be either “simple” and correspond to the disappearanceof only one cylinder of closed trajectories (Fig. 4), or “compound” and bedetermined by the disappearance of different cylinders of closed trajectoriesin its different parts (Fig. 5). Besides that, it is also easy to see that in thecase of a compound boundary of a Stability Zone we can have cases when thecylinders of different (electron and hole) types disappear in different partsof the boundary (Fig. 5), and when different cylinders of the same typedisappear in its different parts (Fig. 6).What also can be noted in the situation described above is that the rep-resentation of the Fermi surface, shown at Fig. 2, allows actually to give aneffective description of the trajectories of the system (1.1) also immediatelyafter crossing the boundary of the Stability Zone Ω α on the angular dia-gram. Indeed, after the intersection of the boundary of Ω α the trajectoriescan jump between the previous carriers of open trajectories, however, theFermi surface is still divided into pairs of such carriers separated from eachother by the remaining cylinders of closed trajectories. It can be seen in thiscase that the corresponding component is completely divided into closed tra-12 Ω α Figure 5: The Fermi surface having a Stability Zone Ω α with a compoundboundary. Ω α B Figure 6: The Fermi surface having a Stability Zone Ω α with a compoundboundary, determined by the disappearance of different cylinders of the sametype in its different parts.jectories if the intersection of the plane orthogonal to B with the plane Γ α has an irrational direction, and can contain open periodic trajectories if thisintersection has a rational direction in the p - space. It is also easy to seethat the resulting closed trajectories are strongly elongated in one directionand have very long length, so that in the immediate vicinity of the boundaryof Ω α we have the relation T ≥ τ , where T is the typical time of a turnalong such trajectories. As a consequence, trajectories of this type are indis-tinguishable from open trajectories from the experimental point of view, andthe exact boundary of the Zone Ω α is actually unobservable in direct con-ductivity measurements even in rather strong magnetic fields. Besides that,when approaching the boundary of the Zone Ω α from the outside, there willbe more and more directions of B , corresponding to the presence of periodicopen trajectories on pairs of former carriers of stable open trajectories of sys-tem (1.1). It can thus be stated that for direct conductivity measurementsthe “experimentally observable” Stability Zone ˆΩ α does not coincide with13 α ^ Ω α ^ Ω α Ω α Figure 7: A schematic view of the experimental Stability Zones ˆΩ α observ-able in direct measurements of conductivity in strong magnetic fields (the“nets” of special directions of B , corresponding to the appearance of peri-odic trajectories on the Fermi surface, are also shown).the exact mathematical Stability Zone Ω α and contains it as a subset (Fig.7). It must be said that the analytic dependence of the tensor σ ik ( B ) bothon the direction and on the value of B inside the Zone ˆΩ α is generally quitecomplicated and can be approximated by different regimes in its differentparts (see [38]). Let us also note here that the analytic properties of thetensor σ ik ( B ) in the Zones ˆΩ α can play, presumably, a certain role in con-sidering the conductivity of polycrystals in strong magnetic fields (see, e.g.[12, 13]), which also possesses rather non-trivial properties in metals withcomplex Fermi surfaces. At the same time, in spite of the unobservability ofthe exact boundaries of the Stability Zones in direct conductivity measure-ments, it is nevertheless possible to indicate other experimental methods thatallow one to determine the exact mathematical Stability Zones for real ma-terials. In particular, as can be shown, this problem can be solved using thestudy of classical or quantum oscillation phenomena (cyclotron resonance,de Haas - van Alphen effect, Shubnikov - de Haas effect, etc.) in strongmagnetic fields (see [39]).As also follows from the arguments of the previous paragraph, each Stabil-ity Zone Ω α is in fact surrounded by some additional domain Ω ′ α , restrictedby some “second boundary” of the Stability Zone, determined by the disap-pearance of at least one more cylinder of closed trajectories presented at Fig.2. Everywhere in the region Ω ′ α the Fermi surface can be represented as aunion of pairs of former carriers of stable open trajectories admitting jumpsof the trajectories of the system (1.1), separated by the remaining cylindersof closed trajectories. It is easy to see that in the most common case we14re considering (exactly two carriers of open trajectories in the Zone Ω α ), inthe domain Ω ′ α there can be no open trajectories other than (unstable) peri-odic trajectories whose direction is determined by the same rule as the meandirection of the open trajectories in the Zone Ω α . It must be said that forFermi surfaces of very high genus (more than two carriers of open trajectoriesin the Zone Ω α ), of course, we can have the situation when the “merged”carriers of open trajectories coexist with unbroken carriers having the samedirection in p - space. In this case, one can speak of a Stability Zone with acomplex boundary or of the imposition of Stability Zones corresponding tothe same topological quantum numbers. The region Ω ′ α can be called the“derivative” of the Stability Zone Ω α , since the structure of the system (1.1)in this region is closely related to the structure of the same system in theZone Ω α . It is easy to see that, like the first boundary of the Stability Zone,its second boundary can also be either simple or compound in the sense indi-cated above (Fig. 8). It can also be seen that for any connected part of theregion Ω ′ α its inner (bordering Ω α ) and outer boundaries correspond to thedisappearance of the cylinders of closed trajectories of the opposite (electronand hole) types at Fig. 2. We also note here that for a precise determinationof the second boundaries of the Stability Zones in real crystals, we can alsopropose a number of experimental methods, including the study of classicalor quantum oscillations in strong magnetic fields mentioned earlier (see [40]).It should also be noted that the type of cylinder of closed trajectories dis-appearing at any of the sections of the (first) boundary of a Stability Zone isalso observable experimentally and, moreover, plays an important role in theconductivity behavior outside of Ω α . Namely, it can be shown that the typeof the vanishing cylinder determines the behavior of the Hall conductivityin strong magnetic fields in that part of the region Ω ′ α , where the condition τ ≫ T starts to be fulfilled. Let us note at once that the behavior of theHall conductivity in this limit is usually associated with the concentration ofelectron and hole carriers in a conductor, however, the type of current carri-ers is not determined in the presence of open trajectories of the system (1.1)on the Fermi surface. Let us consider here for simplicity (the most realistic)the case when the connected component of the Fermi surface carrying stableopen trajectories has exactly two nonequivalent carriers of open trajectories.In this case it can be shown (see [40]) that in calculating the Hall conduc-tivity in the indicated part of Ω ′ α one can assume that such a component isbounding either the carriers of the electron type, or carriers of the hole type,depending on the type of the vanishing cylinder of closed trajectories on thecorresponding part of the boundary of Ω α . More precisely, the connectedcomponent must be regarded as bounding carriers of the electron type if onthe corresponding part of the boundary of Ω ′ α and Ω α we observe disap-15 α Ω α / B B Ω α / Ω α / Ω α / Ω α Ω α / B Ω α Ω α / Figure 8: The second boundaries of the Stability Zones, bounding the do-mains Ω ′ α .pearance of a cylinder of closed hole-type trajectories, and bounding carriersof the hole type in the opposite case. To calculate the contribution of such acomponent to the Hall conductivity in the limit τ ≫ T , one can use one ofthe formulas σ = 2 ec (2 π ~ ) B V − (the first case) (2.2) σ = − ec (2 π ~ ) B V + (the second case) (2.3)where V − and V + represent the volumes of the regions bounded by thecomponent under consideration (in the Brillouin zone) and determined byconditions ǫ ( p ) < ǫ F and ǫ ( p ) > ǫ F respectively. (To calculate the total16all conductivity, it is necessary to carry out summation over all connectedcomponents of the Fermi surface).Let us recall here that the Hall conductivity represents a “transverse”conductivity in the plane orthogonal to the magnetic field, and traditionallyhas a positive sign for positively charged current carriers and negative fornegatively charged ones in strong magnetic fields (electron charge e is as-sumed to be negative here). As we have already mentioned, for conductorswith Fermi surfaces of general form, the carrier charge must be effectivelyconsidered positive or negative, depending on the trajectories of the system(1.1). The type of current carriers is usually well defined for metals withsimple Fermi surfaces (of rank zero) that admit only closed trajectories ofthe system (1.1), and does not depend in this case on the direction of themagnetic field. At the same time, for metals with complex Fermi surfaces,the carrier type is not defined for the directions of B lying in the StabilityZones, as mentioned above. At the same time, the type of carriers can bedefined for directions of B lying outside any of the Stability Zones, whenthe Fermi surface contains only closed trajectories of the system (1.1), andthis property is stable with respect to small rotations of B . (Let us notehere that this fact does not actually mean the existence of closed trajec-tories of the system (1.1) of a fixed type (electron or hole) on each of thecomponents of the Fermi surface. Nevertheless, although on any part of theFermi surface in this case there can be closed trajectories of both types, itis possible to effectively assign to it a fixed number of carriers of a certaintype (in the Brillouin zone) by relating the number of carriers to one of thevolumes bounded by the corresponding component of the Fermi surface.) Asfollows from the formulas (2.2), (2.3), the value of the Hall conductivity isthen locally constant (for a fixed value of B ) under the condition τ ≫ T .It can be seen that in this case it is natural to divide the angular diagramsfor conductivity into two classes, namely, to diagrams for which the carriershave the same charge (type A) everywhere outside the Stability Zones, anddiagrams for which in different regions outside the Stability Zones carriershave different charge (type B).We must at once say that diagrams of type A are a priori simpler and,moreover, appear, apparently, in most of the cases when studying the con-ductivity of real crystalline conductors. In particular, such diagrams includeall diagrams containing only a finite (or infinite) number of Stability Zonesthat do not divide the sphere S into unrelated domains. As for the B-typediagrams, it is easy to see that the Stability Zones should form a rather com-plex structure here, splitting S into regions with different types of currentcarriers. Nevertheless, from a theoretical point of view, B-type diagrams alsorepresent general diagrams, and in particular arise each time when there is17t least one Stability Zone with a compound boundary defined by the disap-pearance of cylinders of closed trajectories of different types on its differentparts (Fig. 5). As already noted above, in this case we necessarily have asituation when in different parts of the region Ω ′ α the corresponding compo-nent of the Fermi surface corresponds to carriers of different types for genericdirections of B . It can also be noted that the type of carriers does not changein generic case also after crossing the second boundary of a Stability Zone(see [40]), so that the regions of different types of charge carriers that arise inthis case have a rather complex structure, separated by “chains” of StabilityZones on S (Fig. 9). Note also that the “chains” contain generically aninfinite number of Stability Zones. Thus, for example, the “corner” pointof the Stability Zone at Fig. 5 can be adjoined by another Stability Zoneonly if the corresponding direction of B is associated with the appearanceof periodic open trajectories on the Fermi surface. In the case of a generalposition the corner point of the Zone at Fig. 5 must be adjoined by a chainof an infinite number of decreasing Stability Zones.As we have already said, diagrams of type A are, apparently, the maintype that arises for real crystal conductors. At the same time, the detec-tion of a B-type diagram for some real material would allow us to observea wide variety of different behavior regimes for the magnetoconductivity fordifferent directions of B . It can be noted at the same time that despitethe possible experimental difficulties in observing the entire complex pictureof the Stability Zones arising in this case, the identification of the type ofthe diagram can be easily established by the behavior of the Hall conduc-tivity outside the (experimentally observable) Stability Zones. We note herethat, since the total Fermi surface can consist of several components thatcontribute to the Hall conductivity, a B-type diagram differs in the generalcase by the presence of at least two different values of Hall conductivity (nomatter what sign) in different domains outside the experimentally observableStability Zones at a given value of B .We can also define extended Stability ZonesΣ α = Ω α ∪ Ω ′ α , defined by the union of domains Ω α and Ω ′ α . It is not difficult to see thenthat the open trajectories of the system (1.1), other than those consideredabove, can arise only on the set S \ ∪ α ¯Σ α of directions of B on the angular diagram. We note here that the sizeand shape of the extended Stability Zones are not related in general to the18 (cid:2) IIIIII
Figure 9: The angular diagram of type B and the Stability Zones separat-ing regions corresponding to different values of the Hall conductivity on S (schematically, only the mathematical boundaries of a finite number of exactStability Zones are shown).size of the experimentally observable Stability Zones ˆΩ α , so that the ZonesΣ α can be both subsets of ˆΩ α , or contain ˆΩ α as subsets. Note also that,unlike Zones Ω α , the Zones Σ α can overlap with each other. In particular,if the Zones Σ α cover the entire sphere S , this means that on the givenFermi surface only stable or periodic open trajectories of the system (1.1)can appear.Thus, in the most general case, on the angular diagram of conductivity wecan have a finite or infinite number of disjoint mathematical Stability ZonesΩ α corresponding to different topological numbers ( M α , M α , M α ) and cov-ering some part of the sphere S . On the remaining part of S for almostall directions of B the Fermi surface contains only closed trajectories of thesystem (1.1). For special directions of B , nevertheless, unstable periodictrajectories of the system (1.1) or chaotic trajectories of Tsarev or Dynnikovtype (they will be considered in the next chapter) may appear on the Fermisurface. As we have already said, theoretically, all the angular diagrams of19onductivity can be divided into two types A and B. The diagrams of thesecond type differ in this case by the presence of regions outside the StabilityZones with different values of the Hall conductivity in strong magnetic fieldsand an infinite number of Stability Zones Ω α (generically) which divide theseregions in the general case. As we have already noted, in the experimentson direct measurement of the magnetoconductivity, the exact mathematicalStability Zones are usually included in the wider “experimentally observable”Stability Zones ˆΩ α . In addition, as was also noted above, for each StabilityZone Ω α we can indicate an adjoining domain Ω ′ α , where the description oftrajectories of the system (1.1) is actually closely related to the structure of(1.1) in the Zone Ω α and does not allow the appearance of open trajectoriesof (1.1), other than periodic. We note here again that the described struc-ture of the angular diagram of conductivity holds for all Fermi surfaces (notnecessarily defined by a single dispersion relation) consisting of componentsthat do not intersect each other.Returning to the general problem of describing the geometry of the trajec-tories of the system (1.1) with an arbitrary dispersion law, it is also necessaryto give a description of the angular diagrams for the total dispersion relation ǫ ( p ) , introduced by I.A. Dynnikov in the work [18]. Let us formulate herethe main statements given in [18], which represent a basis for describing suchdiagrams.Consider an arbitrary dispersion relation given by a smooth 3-periodicfunction ǫ ( p ) , such that ǫ min ≤ ǫ ( p ) ≤ ǫ max . Let us fix an arbitrarydirection of B and consider the energy levels ǫ ( p ) = const containing opentrajectories of the system (1.1) in the extended p - space.Then:Energy levels containing open trajectories of (1.1) represent either a con-nected interval ǫ ( B /B ) ≤ ǫ ≤ ǫ ( B /B )or only one isolated point ǫ = ǫ ( B /B ).For generic directions of B the boundaries of the interval of open trajec-tories ǫ ( B /B ) and ǫ ( B /B ) are determined by the values of some globallydefined continuous functions ˜ ǫ ( B /B ) and ˜ ǫ ( B /B ) . At the same time,for special directions of B , corresponding to appearance of periodic opentrajectories of system (1.1), we can write the relations ǫ ( B /B ) ≤ ˜ ǫ ( B /B ) , ǫ ( B /B ) ≥ ˜ ǫ ( B /B ) (2.4)Every time when we have the situation ˜ ǫ ( B /B ) > ˜ ǫ ( B /B ) , all the(non-singular) open trajectories of system (1.1), arising for generic directions20f B , have a regular shape, represented at Fig. 1, and the same meandirection given by the intersection of the plane orthogonal to B and someintegral plane Γ in the p - space.All such trajectories, arising for generic directions of B , and also theintegral plane Γ are stable with respect to small rotations of B , and thecomplete set of directions of B , corresponding to the same plane Γ , rep-resents a finite domain with piecewise smooth boundary on the angular di-agram S . The regions Ω ∗ α , corresponding to the presence of stable opentrajectories with the same integral plane Γ α on any of the energy levels,represent in this case the Stability Zones for the entire dispersion relation ǫ ( p ) . On the boundaries of the Zones Ω ∗ α we always have the relation˜ ǫ ( B /B ) = ˜ ǫ ( B /B ) . Let us also note here that the boundaries of the Sta-bility Zones Ω ∗ α have in this case both the “electron” and the “hole” type,and are determined by the disappearance of simultaneously two cylinders ofclosed trajectories of opposite types when crossing them. The complete setof Stability Zones Ω ∗ α forms an everywhere dense set on the angular diagram(Fig. 10), which can generally contain either one or infinitely many differentStability Zones.Let us also note that on the boundaries of the Zones Ω ∗ α we have opentrajectories of the system (1.1) having a regular form, shown at Fig. 1, whichin this case are not stable with respect to all small rotations of B , as wellas energy level variations.It is not difficult to see that the angular diagram of a Fermi surface, deter-mined by one dispersion relation ǫ ( p ) = ǫ F , can be “nested” in the diagramof the total dispersion relation, so that the Stability Zones, defined for afixed Fermi surface, represent subsets of a part of the Zones Ω ∗ α . For Fermisurfaces defined by several dispersion relations, in general, such a naturalembedding can be absent. It should be noted here that the latter situationarises actually only for sufficiently complex Fermi surfaces containing severalcomplex components.The complement to the set (cid:8) ∪ Ω ∗ α (cid:9) on the sphere S forms a rather com-plex set (of Cantor type) and represents the directions of B correspondingto the appearance of chaotic open trajectories at one energy level ǫ ( B /B ) = ˜ ǫ ( B /B ) = ˜ ǫ ( B /B )According to the conjecture of S.P. Novikov (see [36]), this set has measurezero and the Hausdorff dimension strictly less than 2 on the unit sphere. Wecan note here that the structure of the set of such special directions and theproperties of chaotic trajectories are actively investigated at the present time21 .. . ... . . ... ..... . .... . . .. .... ... ... . ..... ... .. . Figure 10: The angular diagram for a general dispersion relation ǫ ( p )(schematically, only a finite number of decreasing Stability Zones Ω ∗ α andspecial “chaotic” directions is shown).(see, e.g. [16, 18, 33, 51, 52, 53, 55, 5, 6, 8, 56, 9, 19, 10, 46, 47, 20, 21, 2, 3,11]).In conclusion of this chapter, it can be noted that although angular di-agrams of conductivity for the complete dispersion relation are not yet ob-served in the experiment, it is possible that they will still be observed in themeasurement of (photoinduced) conductivity in semiconductors in extremelystrong magnetic fields (see [17]). In this chapter we will consider the trajectories of the system (1.1), whichhave chaotic properties. We will start with a simpler example, constructedby S.P. Tsarev (private communication, 1992-1993). The general idea of22 C +−+ B C C . a δ . α ^ a Figure 11: The Brillouin zone and the behavior of the trajectories of thesystem (1.1) for a Fermi surface carrying chaotic trajectories of Tsarev type.constructing chaotic trajectories, suggested by Tsarev, can be expressed bythe following scheme:Consider a family of identical integral (horizontal) planes connected byidentical cylinders, as shown at Fig. 11. Let us assume that the centersof all the bases of the cylinders shown at Fig. 11 lie in one (for simplicity,vertical) plane intersecting the horizontal planes in some direction ˆ α . Wealso assume that the constructed cylinders are repeated periodically, so thatthe constructed surface is periodic with some periods a , a , lying in thehorizontal plane, and also a vertical period a . It is not difficult to see thatthe constructed surface can be regarded as a periodic Fermi surface, and wemust divide all the integral planes (even and odd), and the cylinders ( C and C ) into two different classes. Let us consider a horizontal magneticfield B , orthogonal to the direction ˆ α , and the trajectories of the system(1.1) corresponding to this direction. The trajectories of the system (1.1)can be considered as trajectories on integral planes (of two different types),sometimes jumping from one plane to another. Under the above conditions,it is not difficult to see that any trajectory that jumps from a plane to anadjacent plane inevitably jumps to the next plane and continues in the samedirection as on the original plane (Fig. 11). It is easy to see here that all the(non-singular) trajectories of the system (1.1) have an asymptotic directionin the p - space defined by the relations between the effective radius of thecylinders and the periods a , a , a . At the same time, for any irrationaldirection ˆ α (with respect to the given crystal lattice), no regular trajectoryof (1.1) can lie in a straight line of finite width in the corresponding plane,orthogonal to B .It is easy to see, that in the above construction each regular trajectoryof the system (1.1) sweeps out half the surface of genus 3 and, in this sense,has a chaotic behavior on the Fermi surface. In the extended p - space,23owever, the trajectories of Tsarev are more like the trajectories described inthe previous chapter, having asymptotic directions in the plane orthogonalto B . As was shown in [16], the last property is actually observed for allchaotic trajectories arising for directions of B of irrationality 2 (the planeorthogonal to B contains a reciprocal lattice vector). As in the case ofstable open trajectories, the contribution of Tsarev’s chaotic trajectories tomagnetoconductivity has a more complex analytic behavior than that givenby the formula (1.3), but has similar geometric properties. In particular,here also the general formula (2.1) takes place with a proper choice of thecoordinate system.More complex examples of chaotic trajectories of the system (1.1) werefirst constructed by I.A. Dynnikov in the work [16]. The Dynnikov trajec-tories arise for directions of B of maximal irrationality and have complexchaotic behavior both on the Fermi surface and in the extended p - space.As a rule, the Dynnikov trajectories everywhere densely sweep componentsof genus 3 (or more) with contractible holes in the Brillouin zone, having anobvious chaotic behavior on these components. In the extended p - space,the behavior of such trajectories in planes orthogonal to B , resembles diffu-sional motion to some extent, although, of course, it is not diffusion in thestrict sense of the word (Fig. 12). In general, the behavior of trajectoriesof this type on the Fermi surface and in the extended p - space representsan important example of an emergence of classical chaos in the condensedmatter physics.The behavior of the magnetoconductivity (and other transport phenom-ena) in strong magnetic fields in the presence of chaotic trajectories of Dyn-nikov type on the Fermi surface has the most complicated form. The mostinteresting phenomenon arising in this case is the suppression of the conduc-tivity along the direction of the magnetic field in the limit ω B τ → ∞ ([33]).Thus, because the chaotic trajectory sweeps out a part of the Fermi surface,which is invariant under the transformation p → − p , the contribution ofthis part to the longitudinal conductivity disappears in the above limit. Inthis situation, the longitudinal conductivity is created only by the remainingpart of the Fermi surface filled with closed trajectories of the system (1.1).As a consequence of this fact, the longitudinal conductivity must have sharpminima for the directions of B corresponding to the appearance of suchtrajectories on the Fermi surface.Another interesting circumstance arising in the description of transportphenomena in the presence of Dynnikov-type trajectories on the Fermi sur-face is the appearance of fractional powers of the parameter ω B τ in theasymptotic behavior of the components of the tensor σ ik ( B ) as ω B τ → ∞ ([33, 41]). Let us note here that the first indication of this circumstance24 + ++ + ++++++ − − − − − −−−−− − − Figure 12: Dynnikov chaotic trajectory in a plane orthogonal to B .in [33] was actually based on some additional property of the trajectories(self-similarity) constructed in [16]. It must be said that this property isnot, generally speaking, common for trajectories of the Dynnikov type. Nev-ertheless, as can be shown (see [41]), the appearance of such powers is infact a more general fact and is associated with important characteristics (theZorich - Kontsevich - Forni indices) of dynamical systems on surfaces. TheZorich - Kontsevich - Forni indices play an important role in describing thebehavior of the trajectories of dynamical systems and can be determinedfor a fairly wide class of dynamical systems and foliations on surfaces (see[51, 52, 53, 54, 55, 56]). Let us also note here that for the dynamical system(1.1) the existence of the Zorich - Kontsevich - Forni indices, strictly speak-ing, requires additional justification and does not follow automatically fromthe general theory based on certain generic requirements. As an example ofsuch a justification, we can indicate the work [2], in which the constructionand investigation of Dynnikov type trajectories was carried out, and the ex-istence of the indicated indices for the Fermi surface of a rather general formwas established.We give here a general description of the Zorich - Kontsevich - Forniindices defined in the general case for foliations generated by closed 1-forms25n compact surfaces M g . We will follow here the work [55], where for “almostall” foliations of this type the following properties were indicated:For a foliation generated by a closed 1-form on a surface of genus g , weconsider a layer (level line) in general position and fix an initial point P onit. On the same layer, we fix another point P , in which this layer approachesclose to P after passing a sufficiently large path along the surface M g . Wejoin the points P and P by a short segment and define a closed cycle onthe surface M g . Let us denote the homology class of the resulting cycle by c P ( l ) , where l represents the length of the corresponding section of the layerin some metric. Then:There is a flag of subspaces V ⊂ V ⊆ . . . ⊆ V g ⊆ V ⊂ H ( M g ; R ) , such that:1) For any such layer γ and any point P ∈ γ lim l →∞ c P ( l ) l = c , where the non-zero asymptotic cycle c ∈ H ( M g ; R ) is proportional to thePoincare cycle and generates the subspace V .2) For any linear form φ ∈ Ann ( V j ) ⊂ H ( M g ; R ) , φ / ∈ Ann ( V j +1 )lim sup l →∞ log |h φ, c P ( l ) i| log l = ν j +1 , j = 1 , . . . , g − φ ∈ Ann ( V ) ⊂ H ( M g ; R ) , || φ || = 1 |h φ, c P ( l ) i| ≤ const , where the constant is determined only by foliation.4) The subspace V ⊂ H ( M g ; R ) is Lagrangian in homology, where thesymplectic structure is determined by the intersection form.5) Convergence to all the above limits is uniform in γ and P ∈ γ , i.e.depends only on l .It can be seen that the above statements give extremely important in-formation about the behavior of level lines of a foliation (or trajectories ofa dynamical system) on the manifold M g . It can also be shown that theproperties described above also significantly affect the behavior of chaotictrajectories in the extended p - space under the condition that the Zorich- Kontsevich - Forni indices exist for the system (1.1) on the Fermi surface.Indeed, suppose that for some system (1.1) chaotic Dynnikov trajectories26rise on a complex Fermi surface and fill a part of the Fermi surface boundedby closed singular trajectories. For greater rigorousness, we can use the pro-cedure of gluing the corresponding holes in the p - space and define a newsmooth Fermi surface carrying chaotic trajectories of the same global geom-etry as the system (1.1) (see [18]). We also put for simplicity (as is the casefor the most realistic situations), that the corresponding carriers of chaotictrajectories have genus 3.Under the condition that the Zorich - Kontsevich - Forni indices exist forthe system under consideration, we can in this case speak of the presence ofa flag of subspaces V ⊂ V ⊆ V ⊆ V ⊂ H ( M ; R ) , possessing the properties listed above.We note that in the generic case we assume that 1 > ν > ν > dim V = 2 , dim V = dim V = 3 .To describe the properties of the trajectories of the system (1.1) in theextended p - space which we need, let us now consider the map in homology H ( M g ; R ) → H ( T ; R ) , induced by the embedding M ⊂ T . It is not difficult to see that theimages of all spaces V j must belong to a two-dimensional subspace defined bythe plane orthogonal to B . In addition, from the absence of a linear growthof the deviation from the point P with increasing length of a trajectory inthe plane orthogonal to B in examples of chaotic trajectories of Dynnikovtype we get that the image of the asymptotic cycle c is equal to zero underthis mapping. The image of the subspace V in the generic case is one-dimensional and determines the selected direction in the plane orthogonal to B , along which the average deviation of the trajectory grows faster with itslength ( ∼ l ν ) , than in the direction orthogonal to it. In the general case,we must also assume that the image of the space V is two-dimensional andcoincides with the plane orthogonal to B . Considering the 1-forms dp x and dp y as a basis of 1-forms, subject to the above condition (2), we can in oursituation choose the coordinate system ( x, y, z ) in such a way that for somereference sequences of values l we will have the relations | ∆ p x ( l ) | ≃ p F (cid:18) lp F (cid:19) ν , | ∆ p y ( l ) | ≃ p F (cid:18) lp F (cid:19) ν for the deviations of the trajectory along the coordinates p x and p y whenpassing a part of the approximate cycles on the Fermi surface describedabove. 27t can thus be seen that the existence of the Zorich - Kontsevich - Forniindices predetermines certain properties of “wandering” of electron trajecto-ries in the extended p - space, and, consequently, in the coordinate space,according to the specifics of the electron motion in magnetic fields. In turn,as well as for closed or stable open trajectories, the geometric properties ofchaotic trajectories have a decisive influence on the characteristics of electrontransport phenomena in strong magnetic fields. In particular, we can expecthere the manifestation of the Zorich - Kontsevich - Forni indices in the studyof the magnetoconductivity in crystals in the limit ω B τ → ∞ .Indeed, as a more detailed analysis of the kinetic equations in this situa-tion shows (see [41]), the existence of the Zorich - Kontsevich - Forni indicesis manifested directly in the behavior of the components of the conductivitytensor in the plane, orthogonal to B , and, in particular, leads to the followingdependencies of the components σ xx and σ yy on the magnetic field: σ xx ( B ) ≃ ne τm ∗ ( ω B τ ) ν − , σ yy ( B ) ≃ ne τm ∗ ( ω B τ ) ν − It should be noted here that the above relations do not in fact representthe principal term of any asymptotic expansion of the quantities σ xx ( B )and σ yy ( B ) and define more likely some common “trend” in their behavior.Strictly speaking, it is also more correct here to write the relationslim sup ω B τ →∞ log σ xx ( B )log ω B τ = 2 ν − , lim sup ω B τ →∞ log σ yy ( B )log ω B τ = 2 ν − σ xx ( B ) and σ yy ( B ) , since their exact dependence satisfies actually a number of importantrestrictions.We note here, in addition, that the presence of the Zorich - Kontsevich -Forni indices for chaotic trajectories of Dynnikov type also allows us to writethe following relationlim sup ω B τ →∞ log | ∆ s xy ( B ) | log ω B τ ≤ ν + ν − B , whichcan also serve as an estimate for the overall trend of the decrease of thisquantity (see [41]).We would now like to note that all the above relations have been obtainedwithin the framework of the kinetic theory on the basis of a purely quasiclas-sical analysis of the evolution of electron states in crystalline conductors. At28 + ++ + ++++++ − − − − − −−−−− − − Figure 13: The phenomenon of magnetic breakdown on the Dynnikov chaotictrajectory in strong magnetic fields.the same time, as is well known, electron transport phenomena in sufficientlystrong magnetic fields also have observable quantum corrections caused byquantum phenomena in electron systems (see e.g. [1, 25, 32, 49]). Here wewould like to note that for Dynnikov chaotic trajectories (unlike most othertrajectories of other types) the most significant of the quantum effects is thephenomenon of magnetic breakdown that arises in sufficiently strong mag-netic fields. The phenomenon of (intraband) magnetic breakdown here isclosely related actually to the presence of saddle singular points inside car-riers of chaotic trajectories and consists in the possibility of jumps from onesection of the trajectory to another (at a fixed p z ) for a sufficiently closeapproach of such sections to each other (Fig. 13).The jump probability for two given sections increases with increasingof B and tends to 1 / B → ∞ . It is not difficult to seehere that the mean time τ (1) ( B ) of the electron motion between such jumpsis a sufficiently rapidly decreasing function of B . The effect of magneticbreakdown becomes significant when τ (1) ( B ) turns out to be of order of τ ,and the problem represents one of the possible models of quantum chaos inthe limit τ (1) ( B ) ≪ τ . In the general case, the main effect of magnetic29reakdown on transport phenomena in magnetic fields can be expressed byintroducing the effective mean free time τ eff ( B ) , determined by the formula τ − eff ( B ) = τ − + τ − ( B )It is not difficult to show (see [41]) that the introduction of the time τ eff ( B ) leads to a faster decreasing of all components of the conductivitytensor σ ik ( B ) (including the conductivity along the direction of B ) withincreasing of B .As already noted, the phenomenon of magnetic breakdown represents apurely quantum phenomenon in the electron system. For a more detaileddescription of the quantum picture arising in the presence of chaotic tra-jectories on the Fermi surface, it is first of all necessary to consider thespectrum of the one-electron states near the Fermi energy in this situation.Here it is most convenient to start the analysis of the one-electron spectrumfrom a purely quasiclassical spectrum, i.e. from the spectrum, described bythe “quantization” of quasiclassical orbits in the p - space. As we told al-ready, the Dynnikov chaotic trajectories exist only on one energy level ǫ (see [15, 16, 18]), which should be close enough to the Fermi energy for thepossibility of experimental observation of the regimes described above. Atthe energy levels ǫ < ǫ all such trajectories break up into long closed trajec-tories of the electron type, and at the levels ǫ > ǫ - into closed trajectoriesof the hole type. It can thus be seen that in the purely quasiclassical approx-imation (implying also the limit τ → ∞ ), we should expect the appearanceof delocalized electron states only at the level ǫ = ǫ , while electron statesnear the level ǫ must be localized and determined by the quantization onthe long closed trajectories.In accordance with the rules of the quasiclassical quantization (see [1,25, 32, 49], the closed trajectories of the system (1.1) for each p z should beselected according to the rule S ( ǫ, p z ) = 2 πe ~ Bc (cid:18) n + 12 (cid:19) , n ≫ , where S ( ǫ, p z ) represents the area bounded by a closed trajectory in theplane orthogonal to B in the p - space.It is easy to see that the areas of long closed trajectories S ( ǫ, p z ) tend toinfinity as ǫ → ǫ , which leads to a rapid decrease in the distance betweenelectron levels in the same limit (Fig. 14). Such a behavior of the arisingquasiclassical levels makes it possible to easily distinguish their contributionto a number of quantum corrections on the background of the contributionof electron levels arising on “short” closed trajectories of the system (1.1),30 ε Figure 14: Quasiclassical electron levels near a special energy level carryingchaotic trajectories of the system (1.1).the distance between which considerably exceeds the distance between levelsarising on long closed trajectories of (1.1). Let us note here also that instudying the phenomena associated with the quantization of electron levels,usually the levels associated with the extremal trajectories on the Fermisurface, satisfying the condition ∂S/∂p z = 0 , are revealed.In the absence of the magnetic breakdown, the level structure shown atFig. 14, is destroyed by scattering on impurities under the condition T ≥ τ near the level ǫ . Scattering on impurities is associated with the electronmotion in the coordinate space and “mixes” electron levels with different p z , making the problem of determining of the new level structure essentiallythree-dimensional. It can be seen, however, that for a sufficiently large valueof the magnetic field, we can have a much wider region near the value ǫ ,where the conditions T ≥ τ (1) ( B ) and τ (1) ( B ) ≪ τ are fulfilled. In thecorresponding region, the change in the structure of the electron levels will becompletely determined by the phenomenon of the magnetic breakdown andwill be described by the behavior of the levels of two-dimensional systemsfor fixed values of p z . Systems of this type can be considered as one ofthe important models of quantum chaos, where a two-dimensional electronsystem has special quasiperiodic properties. The study of the structure ofelectron levels, as well as the transport properties of such systems, representsan important problem here both from the point of view of the mathematicaltheory of quantum chaos and from the point of view of the condensed matterphysics.As already noted, all the above regimes are observed for special directionsof B , corresponding to the appearance of chaotic trajectories of Dynnikovtype on the Fermi surface. It must be said that for many of the real sub-stances, such directions may in fact not be present on the angular diagram. Inthe general case, as was shown by I.A. Dynnikov (see [16, 18]), the Lebesguemeasure of the corresponding directions on the angular diagram for genericFermi surface is equal to zero. According to the conjecture of S.P. Novikov([34, 35]), the fractal dimension of the set of such directions on the angulardiagram for generic Fermi surface is strictly less than 1 (but may be larger for31 + ++ + ++ ++ +−− − − −− − Figure 15: A stable open trajectory in the Zone Ω α of small sizes.special Fermi surfaces). Nevertheless, for substances with a rather complexFermi surface, it is quite possible to expect an experimental observation ofthe described regimes for specially selected directions of B . In particular,the appearance of chaotic trajectories at levels arbitrarily close to the Fermilevel should always be observed on the angular diagrams of type B describedin the previous chapter. It should also be noted that stable open trajectories,corresponding to sufficiently small Stability Zones Ω α , can also have a rathercomplex shape (Fig. 15) and have the features of both regular and chaoticbehavior depending on the interval of the values of the parameter ω B τ .In conclusion of this chapter we consider briefly one more applicationof the Novikov problem to two-dimensional electron systems, which is ac-tively studied in modern experiments. Namely, we consider the problem ofa two-dimensional electron gas with a high mobility of carriers placed in theartificially created potential V ( r ) . We note at once that at the present timethere are many different methods of creating potentials of this type, many ofthem, in fact, are based on the use of superposition of certain periodic struc-tures formed in the plane of the electron system. One of the most commonmethods, in particular, is the use of superposition of (one-dimensional) in-terference patterns of laser radiation, which causes the polarization of atomsforming the sample (see e.g. [48]). 32 a a ηη η a a a a η η η η Figure 16: Quasiperiodic potentials with three and four quasiperiods, ob-tained by superposition of three and four one-dimensional potentials withdifferent directions and period values, respectively.The Novikov problem in these systems arises in the description of electrontransport phenomena in the presence of a sufficiently strong magnetic field B , orthogonal to the sample plane. The quasiclassical consideration of theelectron motion (see e.g. [23, 4]) allows one to use here the approximationof quasiclassical cyclotron orbits whose centers drift in the presence of thepotential V ( r ) (Fig. 17). As can be shown, the drift of cyclotron orbitcenters occurs along the level lines of the potential V ( r ) averaged over thecorresponding cyclotron orbits. As in the case of normal metals, the mainrole here is played by electrons with energy close to the Fermi energy, so thatfor the electron motion one can also introduce a fixed cyclotron radius r B corresponding to the given problem. It is not difficult to see that the averagedpotential ¯ V B ( r ) has then the same quasiperiodic properties as the potential V ( r ) , and the description of the motion of orbits along the level lines of sucha potential represents the Novikov problem with the corresponding numberof quasiperiods.The great similarity between the two problems presented above makesit possible to transfer many results from the theory of the conductivity ofnormal metals to the theory of transport phenomena in the described two-dimensional systems (see e.g. [37]). We would like to note here that in thesituation of two-dimensional systems, a much larger number of parametersof the system is controlled, which makes it possible to actually implementpractically any of the interesting cases that arise in the investigation of theNovikov problem. In particular, two-dimensional electron systems can be aconvenient experimental tool for studying chaotic regimes for specially se-lected parameters of such systems. It can also be noted here that by specif-ically choosing the parameters of the system, one can implement here botha purely quasiclassical case of chaotic behavior and a magnetic breakdownmode on chaotic trajectories. The latter circumstance is actually due to the33 B r B V(r)
Figure 17: The cyclotron orbit in the presence of a quasiperiodic potential V ( r ) .fact that the semiclassical description here is also approximate, and here alsojumps of cyclotron orbits between level lines of ¯ V B ( r ) , close enough to eachother, are possible (see [23]). As can be shown, different behavior modes (inthe limit τ → ∞ ) are determined here by external parameters of the sys-tem. In general, it can be noted that the described two-dimensional electronsystems can represent a very convenient experimental base for studying bothregular regimes in magnetotransport phenomena and the models of classicaland quantum chaos described above.In conclusion, let us formulate a theorem that defines an important classof quasiperiodic functions on a plane with four quasiperiods whose level lineshave properties analogous to the properties of stable open trajectories of thesystem (1.1) (see [45, 22]). We give here, in fact, only the main corollaries ofthe results obtained in [45, 22], a more detailed description of the resultingtopological picture can be found in the papers [45, 22]. It can be recalledhere that a quasiperiodic function on a plane with four quasiperiods f isdefined as the restriction to the plane of some 4-periodic function F in four-dimensional space under a linear embedding R ⊂ R . According to [45, 22],the following assertion can be stated:1) All non-singular open level lines of the corresponding functions f in R lie in straight strips of finite width, passing through them;2) The mean direction of all non-singular open level lines of the functions f is given by the intersection of the corresponding plane R with someintegral three-dimensional plane Γ α (Π) in the space R , which is locallyconstant in the space G , . 34t is not difficult to see that the statements formulated above correlatein a certain sense with the results obtained in the paper [50] for the case ofquasiperiodic functions with three quasiperiods. It should also be noted thatthe proof of this theorem in the case of four quasiperiods requires, in fact,much greater effort. References [1] A.A. Abrikosov., Fundamentals of the Theory of Metals., Elsevier Sci-ence & Technology, Oxford, United Kingdom, 1988.[2] A. Avila, P. Hubert, A. Skripchenko., Diffusion for chaotic plane sec-tions of 3-periodic surfaces.,
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