Lifshitz transitions, type-II Dirac and Weyl fermions, event horizon and all that
aa r X i v : . [ c ond - m a t . o t h e r] M a y Lifshitz transitions, type-II Dirac and Weyl fermions, event horizon and all that
G.E. Volovik
1, 2 and K. Zhang
1, 3 Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland Landau Institute for Theoretical Physics, acad. Semyonov av., 1a, 142432, Chernogolovka, Russia State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Laser spectroscopy, Shanxi University, Taiyuan 030006, P. R. China (Dated: October 16, 2018)The type-II Weyl and type-II Dirac points emerge in semimetals and also in relativistic systems.In particular, the type-II Weyl fermions may emerge behind the event horizon of black holes. Inthis case the horizon with Painlev´e-Gullstrand metric serves as the surface of the Lifshitz transition.This relativistic analogy allows us to simulate the black hole horizon and Hawking radiation usingthe fermionic superfluid with supercritical velocity, and the Dirac and Weyl semimetals with theinterface separating the type-I and type-II states. The difference between such type of the artificialevent horizon and that which arises in acoustic metric is discussed. At the Lifshitz transitionbetween type-I and type-II fermions the Dirac lines may also emerge, which are supported by thecombined action of topology and symmetry. The type-II Weyl and Dirac points also emerge asthe intermediate states of the topological Lifshitz transitions. Different configurations of the Fermisurfaces, involved in such Lifshitz transition, are discussed. In one case the type-II Weyl pointconnects the Fermi pockets, and the Lifshitz transition corresponds to the transfer of the Berry fluxbetween the Fermi pockets. In the other case the type-II Weyl point connects the outer and innerFermi surfaces. At the Lifshitz transition the Weyl point is released from both Fermi surfaces. Theyloose their Berry flux, which guarantees the global stability, and without the topological support theinner surface disappears after shrinking to a point at the second Lifshitz transition. These examplesreveal the complexity and universality of topological Lifshitz transitions, which originate from theubiquitous interplay of a variety of topological characters of the momentum-space manifolds. For theinteracting electrons, the Lifshitz transitions may lead to the formation of the dispersionless (flat)band with zero energy and singular density of states, which opens the route to room-temperaturesuperconductivity. Originally the idea of the ehancement of T c due to flat band has been put forwardby the nuclear physics community, and this also demonstrates the close connections between differentareas of physics. PACS numbers:
I. INTRODUCTION
Massless Weyl fermions are the building blocks of Standard Model. In the chiral gauge theory of weak interactions,the fundamental elementary particles are Weyl fermions with a pronounced asymmetry between the SU (2) doubletof left-handed Weyl fermions and the SU (2) singlet of right-handed Weyl ferrmions. The masslessness of the Weylfermions is topologically protected. The corresponding topological invariant – the Chern number – has values N = − N = +1 for the left and right particles respectively. The gapless Weyl fermions are at the origin of theanomalies in quantum field theories, such as chiral anomaly, and the coresponding symmetry protected Chern numberscharacterize the anomalous action. The Dirac particles, which emerge below the symmetry breaking electroweaktransition, are the composite objects obtained by the doublet-singlet mixing of Weyl fermions with opposite chirality.The topological invariants N = ± Such diabolical (conical)point represents the monopole in the Berry phase flux, and it is described by the same momentum-space Chernnumber N . Weyl fermionic excitations are known to exist in the chiral superfluid He-A, where the related effects– chiral anomaly and chiral magnetic effect – have been experimentally observed, and in electronic topologicalsemimetals.
The Weyl points supported by the higher values of the Chern number, | N | >
1, are also possible. In this case instead of the conical point with linear spectrum of fermions, one has the higher order band touchingpoint, when for example the spectrum is linear in one direction and quadratic in the other directions. These are theso-called semi-Dirac or semi-Weyl semimetals.
Recently the attention is attracted to the so-called type-II Weyl points.
A remarkable property of this type ofWeyl point is that it is the node of co-dimension 3 in the 3D momentum space, which is accompanied by the nodes ofthe co-dimension less than three: the nodes of co-dimension 1 (Fermi surfaces) or nodes of co-dimension 2 (Dirac lines).The transition between the type I and type II Weyl points is the quantum phase transition, while the symmetry doesnot necessarily change at this transition. The quantum phase transitions with the rearrangement of the topology ofthe energy spectrum, at which the symmetry remains the same, are called Lifshitz transitions. Originally I.M. Lifshitzintroduced the topological transitions in metals, at which the connectedness of the Fermi surface changes. Manynew types of Lifshitz transition become possible, where the topologically protected nodes of other co-dimensions areinvolved. . There is a variety of topological numbers, which characterize the momentum space manifolds of zeroes.Together with the geometry of the shapes of the manifolds, this makes the Lifshitz transitions widespread in fermionicsystem.In relativistic theories there are several scenarios of emerging of the type-II Weyl points. In particular, the transitionfrom the type-I to the type-II Weyl points occurs at the black hole event horizon. The type II Weyl point mayalso emerge as the intermediate state of the topological Lifshitz transition, at which the Fermi surfaces exchangetheir global topological charge N . This Weyl point also naturally appears if the relativistic Weyl fermions arenot fundamental, but emerge in the low energy sector of the fermionic quantum vacuum, for example, in the vacuumof the real (Majorana) fermions. These scenarios will be discussed here in connection to the topological materials.Some of these considerations suggest that the inhomogeneous Weyl semimetal can serve as a platform for simulatingthe black hole with stationary metric and Hawking radiation before the equilibrium is reached. Situations when thetopological invariants are transported between the Fermi surfaces through type II Weyl point will be considered.The plan of rest of the paper is as follows. Sec. II describes the transformation of the type I to type II Weyl fermionsthrough the intermediate Dirac line. Such transition may occur not only in semimteals, but also in chiral superfluid,where the transition is regulated by superflow due to Doppler effect experienced by Weyl excitations. The symmetryprotected topological number of Dirac line appearing at Lifshithz transition is discussed. In Sec. III, we considerthe behavior of the spectrum of Weyl fermions across the event horizon using the Painlev´e-Gullstrand space-time.Behind the horizon the Weyl fermions with type II spectrum emerge. The Fermi surfaces, which touch each other atthe type-II Weyl point, become closed when the Planck scale physics is involved. Simulation of the event horizon andHawking radiation in Weyl and Dirac semimetals is dicussed. In Sec. IV we consider Lifshitz transitions, which aregoverned by the interplay of different topological invariants, on example of the transfer of global topological invariantsbetween the Fermi surfaces. In Sec. V the formation of the flat band in the vicinity of the topological transtion isconsidered. Finally in Sec. VI we review our results and discuss some open questions, in particular in relation to thepossibility of room-temperature superconductivity in exotic topological materials.
II. DIRAC LINE AT THE TRANSITION BETWEEN TYPE-I AND TYPE-II WEYL POINTS
A particular example of emergence of the type-II Weyl fermions in relativistic theories is when the relativisticWeyl fermions are not fundamental, but represent the fermionic excitations in the low energy sector of the fermionicquantum vacuum.
The type-I and type-II Weyl fermions may emerge, for example, in the vacuum of the real(Majorana) fermions. The general form of the relativistic Hamiltonian for the emergent Weyl fermions is obtainedby the linear expansion in the vicinity of the topologically protected Weyl point p (0) with Chern number N = ± H = e jk ( p j − p (0) j )ˆ σ k + e j ( p j − p (0) j ) . (1)This expansion suggests that the position p (0) of the Weyl point, when it depends on coordinates, serves as the U (1)gauge field, A ( r , t ) ≡ p (0) ( r , t ), acting on relativistic fermions. The parameters e jk ( r , t ) and e j ( r , t ) play the role ofthe emergent tetrad fields, describing the gravity experienced by Weyl fermions.The energy spectrum of the Weyl fermions depends on the ratio between the two terms in the rhs of Eq.(1 ), i.e.on the parameter | e j [ e − ] kj | . When | e j [ e − ] kj | < e j = 0. At | e j [ e − ] kj | > The Lifshitz transition between the two regimes occurs at | e j [ e − ] kj | = 1. In the relativisticregime, the spectrum of Weyl fermions at the transition contains zeroes of co-dimension 2 – the Dirac line. In general,the existence of the nodal lines requires the special symmetry: they are protected by topology in combination withsymmetry.There are indications that in some materials the maximum of the superconducting transition temperature occursjust in the vicinity of the Lifshitz transitions (see also Sec. V). In particular, the enhancement of T c at the type-I to-type-II topological transition in Weyl semimetals has been discussed in Ref. . A. Relativistic system
To reveal properties of this Lifshits transition, let us start with considering the topological charge of the nodal lineusing a simple choice of the tetrads for the relativistic Weyl fermions in the gravitational field: H = c σ · ˆ p − f cp z . (2)For f = 0 the Weyl cone is tilted, and for f > f = 1, the Hamiltonian has the form H = (cid:18) c ( p x + ip y ) c ( p x − ip y ) − cp z (cid:19) , (3)and the energy spectrum has the nodal line on the p z -axis, i.e. E ( p ⊥ = 0 , p z ) = 0 for all p z . We consider severalapproaches to characterize stability of the nodal Dirac lines in relativistic systems, which could be extended tocondensed matter systems.In the first approach we take into account that the matrix in Eq.(3) belongs to the class of the 2 n × n matrices ofthe type: H = (cid:18) B ( p ) B + ( p ) C ( p ) (cid:19) , (4)and the topological properties of the considered nodes in the spectrum are characteristics of this class. Of course, itis difficult to expect such matrices in real physical systems, except for the case of n = 1, which naturally emerges atLifshitz transition. But it is instructive to consider the general n case. The determinant of such matrix is the productof the determinants of matrices B and B + : D ( H ) = − D ( B ) D ∗ ( B ) . (5)The nodal lines – zeroes of co-dimension 2 – are zeroes of D ( B ) and are described by the winding number of the phaseΦ of the determinant D ( B ) = | D ( B ) | e i Φ : N = I C dl πi D − ( B ) ∂ l D ( B ) = tr I C dl πi B − ( p ) ∂ l B ( p ) , (6)where C is the closed loop in momentum space around the line. The line in momentum space with the non-zerowinding number of the phase Φ is the momentum-space analog of the vortex line in superfluids and superconductors,which is characterized by the winding number of the phase of the order parameter.For the particular case of 2 × D ( B ) = B = c ( p X + ip y ), the invariant can be written as N = tr I C dl πi · [ σ z H − f =1 ( p ) ∂ l H f =1 ( p )] , (7)where the Dirac line corresponds to the p z -axis.The form (7) of invariant N is somewhat counterintuitive, since the integral of this type represents the trueinteger-valued invariant only if σ z commutes or anticommutes with the Hamiltonian. The latter does not happenhere, nevertheless the integral is stiil integer-valued, which can be shown in a straightforward way. For p z = 0 theHamiltonian anticommutes with σ z , and the integral is the well defined topological invariant with N = 1 for any f . At p z = 0 the Hamiltonian does not anticommute with σ z . However, Eq.(7) remains integer for the general p z if f = 1.To see that we apply the second approach. Let us consider p z as parameter and the arbitrary loops around the line p ⊥ = 0 with fixed p z . Taking into account that H − ( f = 1) = 1 p ⊥ ( c σ · ˆ p + cp z ) , (8)one obtains that the variation of N over p z is zero: dN ( p z ) dp z = 0 , N ( p z ) = tr I C ( p z ) dl πi · [ σ z H − f =1 ( p ⊥ , p z ) ∂ l H f =1 ( p ⊥ , p z )] . (9) FIG. 1:
Type-I and type-II Weyl points (black dots) across the Lifshitz transition.
When the superfluid velocityexceeds the pair-breaking velocity(”speed of light”), the type-I Weyl points in the original chiral superfluid are converted to thetype-II Weyl points. The process of this Lifshitz transition is shown for Eq.(12) describing quasiparticles in the chiral superfluid He-A in the presence of superfluid current with velocity v = v ˆ x . Green arrows depict the vector configurations in the p y = 0plane of the momentum space monopole located at Weyl points.( top left ): Two original type-I Weyl points at v < c .( top right ): At the critical speed v = c , two Dirac lines are formed, by which two Weyl points are connected. The red and bluelines correspond to the Dirac nodes in the hole and particle spectrum of Eq.(12) respectively.( bottom ): Particle and hole Fermi pockets connected via the type-II Weyl points appear when v > c . Thus the integral N ( p z ) = 1 for any p z at f = 1.Finally, the stability of the vortex line in momentum space can be understood through consideration in terms ofthe determinant of the Hamiltonian matrix, i.e. D ( H ), in a way somewhat similar to that in Ref. D ( H ) = c p z ( f − − c p ⊥ . (10) D ( H ) is nonzero for 0 < f <
1, is zero on line at f = 1, and has zeros on the conical Fermi surface at f >
1. For f = 1one can define the generalized root q ( H ) of det H – a polynomial function of the matrix elements of the Hamiltonian– in such a way that | q ( H ) | = | D ( H ) | . So q ( H ) is our D ( B ) in Eq.(5). The corresponding polinomial is q ( H f =1 ) = D ( B ) = c ( p x + ip y ) . (11)It has zero on the line p ⊥ = 0 which is protected by 2 π winding of the phase of q around the line. This gives riseto the topologically stable zero in the determinant D ( H ) and thus to topologically stable zero in the quasiparticlespectrum. For f = 1, the integral Eq.(7) depends on p z and on the radius of the closed loop C . B. Chiral superfluid
It should be pointed out that the Dirac line emerges for the relativistic fermions mainly due to the Lorentz invarianceof the linear spectrum. In principle, the nodal line may disappear when the higher order nonrelativistic corrections aretaken into account, such as the Planckian quadratic term of momentum discussed in Sec.III A, if there is no additionalsymmetry, which could support the stability of the nodal line. The nonlinear terms are natural in condensed mattersystems, and we consider the energence of the Dirac line at Lifshitz transition, which can be realized in chiral superfluidsystem, such as superfluid He-A. The simple model Hamiltonian with the Dirac lines existing at Lifshitz transitionis: H = p x v + τ p − p F m + τ cp x + τ cp y . (12)Here the Weyl points are in positions ± p F ˆ z ; the superfluid velocity with respect to the heat bath v = v ˆ x is transverseto the direction towards the Weyl points. The first term in the rhs of Eq. (12) comes from the Doppler shift producedby superflow; τ i are the Pauli matrices in the Bogoliubov-Nambu space; in He-A v F = p F /m ≫ c . Here c is themaximum speed of quasiparticle propagating in the plane ( p x , p y ) in vicinity of the Weyl points, where the spectrumis relativistic in the linear expansion of the Hamiltonian.The transition between the type-I Weyl fermions and the type-II Weyl fermions tales place, when the flow velocity v reaches the ”speed of light” c , see Fig. 1. For v < c there are two Weyl points at p (0) = ± p F ˆ z with oppositetopological charges N = ±
1, and thus with opposite chiralities of the relativistic Weyl fermions living near the Weylpoints, Fig. 1 ( top left ). At v > c there are two banana shape particle and hole Fermi surfaces, which contact eachother at the type-II Weyl points, see Fig. 1 ( bottom ). Exactly at the Lifshitz transition, at v = c , one has Dirac lines,which connect the Weyl points in Fig. 1 ( top right ). At v = c the matrix Hamiltonian belongs to the class of matricesin Eq.(4). The corresponding determinant D ( B ) describing the topology of the line at f = 1 in Eq.(5) is: D ( B ) = p − p F m + icp y . (13)It has the topologically protected lines of zeroes at p y = 0, p x + p z = p F in Fig. 1(b). These are the vortex linesin momentum space with the winding numbers N = ± D ( B ) in Eq.(6), where thecontour C is along closed loop surrounding the Dirac lines. C. Lifshitz transition with crossing of Dirac lines and Hopf linking
This type of transtion can be seen on example of the modification of the model describing the rhombohedralgraphite , with B = ( p x + ip y ) (cid:0) p x + ip y + t + e ip z a + t − e − ip z a (cid:1) . (14)This model has two nodal lines – the straight one along the z direction and the spiral around the straight one. Dueto the lattice periodicity originating from the layers type construction along z direction, the spectrum of quasiparticlealong p z direction can be described in terms of the one dimensional Brillouin zone. As a result the nodes in thespectrum are the closed loops. From the viewpoint of knot theory, these two nodal lines form a Hopf link – thesimplest nontrivial link consisting of two unknots. . The Hopf link with t + > t − is the mirror image of the onewith t + < t − , and these two configurations cannot be connected with combination of Reidemeister moves. Thismeans that they are not ambient isotopic, and the may transform to each other only via the special type of Lifshitztransition, which in our case occurs at t + = t − . To characterize the difference between these kinds of two Hopf linkednodal lines, we assign a fixed direction and calculate the linking number via N l = ( P p ǫ p ) /
2, where p is the crossingin the diagram of Hopf link of nobal lines, and ǫ p is the sign of the oriented crossing. From Fig.(2), we can find that N l = 1 for t + > t − , while N l = − t + < t − respectively.On other examples of knotted nodal lines see e.g. in Ref. . III. TRANSITION BETWEEN TYPE-I AND TYPE-II DIRAC/WEYL VACUA AND EVENT HORIZON
The section II B demonstrated the scenario of Lifshitz transition between type-I and type-II Weyl points, whenthe flow velocity of the chiral superfluid liquid exceeds the “light speed” of an emergent Weyl quasiparticles. In Sec. (a) t + > (cid:0)(cid:1) (b) (cid:2) + = (cid:3)(cid:4) (c) (cid:5) + < t (cid:6) FIG. 2:
Topological Lifshitz transition of Hopf linked nodal lines. (a) Hopf link with linking number N l = 1. (b)Lifshitz transition. (c) Hopf link with linking number N l = − III A we shall see that analagous transition occurs for the relativistic fermions when the event horizon of the blackhole is crossed and the frame drag velocity exceeds the speed of light. This analogy suggests a route for simulationof an event horizon in inhomogeneous condensed matter systems, which is accompanied by the analog of Hawkingradiation. This will be discussed in Sec. III B.
A. Type-II Weyl fermions behind the black hole horizon
In general relativity the convenient stationary metric for the black hole both outside and inside the horizon isprovided in the Painlev´e-Gullstrand spacetime with the line element: ds = − c dt + ( d r − v dt ) = − ( c − v ) dt − v d r dt + d r . (15)This is stationary but not static metric, which is expressed in terms of the velocity field v ( r ) describing the framedrag in the gravitational field. The Painlev´e-Gullstrand is equivalent to the so-called acoustic metric, where v ( r )is the velocity of the normal or superfluid liquid.For the spherical black hole the frame drag velocity field (the velocity of the free-falling observer) is radial: v ( r ) = − ˆ r c r r h r , r h = 2 M Gc . (16)Here M is the mass of the black hole; r h is the radius of the horizon; G is the Newton gravitational constant. Theminus sign in Eq.(16) gives the metric in case of the black hole, while the plus sign would characterize the gravity ofa white hole.Let’s us consider a Weyl particle in the Painlev´e-Gullstrand space-time. The tetrad field corresponding to themetric in Eq.(15) has the form: e jk = cδ jk and e j = v j , (17)which leads to the following Hamitonian: H = ± c σ · p − p r v ( r ) + c p E UV , v ( r ) = c r r h r . (18)Here the plus and minus signs correspond to the right handed and left handed fermions respectively; p r is the radialmomentum of fermions. The second term in the rhs of (18) is the Doppler shift p · v ( r ) caused by the frame dragvelocity (compare with Eq.(12) for chiral superfluid).The third term in Eq.(18) is the added nonlinear dispersion to take into account the Planckian physics, whichbecomes important inside the horizon. The parameter E UV in the third term is the ultraviolet (UV) energy scale,at which the Lorentz invariance is violated. The UV scale is typically associated with but does not necessarilycorrespond to the Planck energy scale. For the interacting fermions such term can arise in effective Hamiltonian H = G − ( ω = 0 , p ) even without violation of Lorentz invariance on the fundamental level: the Green’s function G ( ω, p ) may still be relativistic invariant, while the Lorentz invariance of the Hamiltonian is violated due to theexistence of the heat bath reference frame. In this case the UV scale is below the Planck scale.In Fig. 3, we present the Fermi surfaces, which appear behind the horizon at different positions r < r h . Behindthe black hole horizon, the Weyl point in the spectrum transforms to the pair of the closed Fermi surfaces with thetouching point: the type-II Weyl point. The p term in Eq.(18) makes the Fermi surfaces attached to the type-IIWeyl point closed, while it provides only a small correction when cp ≪ E UV . The latter is valid if the spectrum insidethe black hole is considered in the vicinity of the horizon, where r h − r ≪ r h , see Fig. 3, at positions r = 0 . r h and r = 0 . r h . Similar o the situation in section II, near the horizon the Dirac(Weyl) cone is tilted, and behind thehorizon it crosses zero energy and forms the Fermi surfaces corresponding to the type-II Weyl points. But there isno Dirac line at the horizon, where Lifshitz transition occurs: as we mentioned in section II B this is because of thequadratic term, which violates the Lorentz invariance. Instead, the Fermi pockets start to grow from the Weyl pointwith | p r | < p UV ( v ( r ) − c ) /c ≪ p UV , when the horizon is crossed.In the full equilibrium the Fermi pockets must be occupied by particles and ”holes”. One of the mechanisms of thefilling of the Fermi pockets in the process of equilibration will be observed by external observer as Hawking radiation. The Hawking temperature is determined by effective gravitational field at the horizon: T H = ~ π (cid:18) dvdr (cid:19) r = r h (19)If the Hawking radiation is the dominating process of the black hole evaporation, the lifetime of the black hole isastronomical. However, the other much faster mechanisms involving the trans-Planckian physics are not excluded. B. Artificial black hole and Hawking radiation from Lifshitz transition
Based on the discussion in Sections II B and III A, one can suggest a new route through which the black hole horizonand ergosurface can be simulated using the inhomogeneous condensed systems with emergent type-I and type-II Welyfermions. The interface which separates the regions of type-I and type-II Weyl points may serve as the event horizon,on which the Lifshitz transition takes place. In general case, such an artificial horizon may have the shape differentfrom the spherical surface. The shape of the horizon is not important if we are interested in the local temperature ofHawking radiation, which is determined by the local effective gravity at the horizon.Let us consider the completely flat artificial event horizon on example of Eq.(2). We assume that the parameter f depends on z , and f ( z ) crosses unity at z = z hor . The plane z = z hor separates the region with type-I Weyl fermions( f ( z ) <
1) from the region with type-II Weyl fermions ( f ( z ) > z (review onartificial horizons and ergoregions in acoustic metric see in Ref. 54). Fig. 4 demonstrates the Weyl cones in theenergy-momentum space ( bottom ) and the analogues of the light cone ( top ) for quasiparticles on two sides of the eventhorizon. Behind the horizon the Weyl cone is overtilted so that the upper cone crosses the zero energy level, andthe Fermi surfaces (Fermi pockets) are formed, which are connected by the type-II Weyl point. Correspondingly thefuture light cone is overtilted behind the horizon so that all the paths fall further into the black hole region.The filling of the originally empty states inside the Fermi surfaces causes the Hawking radiation. For the flat horizonthe Hawking temperature determined by the effective gravitational field at the horizon is: T H = ~ c π (cid:18) dfdz (cid:19) z = z hor (20)Note that in the Weyl semimetals the mechanism of formation of the artificial event horizon (and its behaviorafter formation) is different from the traditional mechanism, which is based on the supercritical flow of the liquid orBose-superfluids. In the latter case the effective metric (the so-called acoustic metric) is produced by the flowof the liquid and thus represents the non-static state. Due to the dissipation (caused, say, by the analogue of Hawkingradiation) the flow relaxes and reaches the sub-critical level, below which the horizon disappears. On the contrary,
FIG. 3:
Type II Weyl point behind the Black hole event horizon.
Contours of Fermi surfaces attached to the type-IIWeyl fermions of Standard Model at different radial positions r inside the black hole horizon. Here p r is the radial componentof the momentum p ; p ⊥ = p p − p r and p = ~ /r h , where r h is the radius of spherical black holes. The contours with p r > p r < r only one of the two Fermi surfaces is shown. The process of the filling of particle and hole Fermi pocketsinside the horizon is observed as the Hawking radiation outside the horizon. in semimetals the tilting of the Weyl cone occurs without the flow of the electronic liquid, and thus the state withthe horizon is fully static. The dissipation after the formation of the horizon (caused, say, by analogue of Hawkingradiation) leads to the filling of the electron and hole Fermi pockets. After the Fermi pockets are fully occupiedthe final state is reached, but it still contains the event horizon, though the Hawking radiation is absent. Similarmechanism takes place in the fermionic superfluids, such as superfluid He, where depending on the parameters ofthe system the flow may or may not remain supercritical after the Fermi pockets are occupied, see Fig. 26.1 in Ref.3.
IV. LIFSHITZ TRANSITIONS WITH TYPE-II WEYL AND DIRAC POINTS AT THE TRANSITION
As we mentioned in Sec. I, with the multiplicity of topological invariants for the manifolds of nodes in the fermionicspectrum, Lifshitz transitions become diverse and complex. This can be seen on examples of the Lifshitz transitionswith the reconstruction of the Fermi surfaces, where several topological invariants may interplay. In Sec. II and Sec.III we discussed how the Dirac lines and Fermi surfaces emerge in the Lifshitz transition between two types of theWeyl point. Here we discuss the opposite case, when the type-II Weyl and type-II Dirac points emerge during theFermi surface Lifshitz transitions. By the type-II Weyl point the topological invariant N is transported between theFermi surfaces.In general, topological invariants which are involved in the complex topological Lifsihitz transitions are: (i) theinvariant N , which is responsible for the local stability of the Fermi surface; (ii) the invariant N , which is the globalinvariant describing the closed Fermi surface: when the Fermi surfaces collapse to a point, it becomes the type-I Weylpoint with the topological charge N ; and (iii) the N invariant in Eq.(7) which characterizes the Dirac line. All threetopological invariants are involved in the complex Lifshitz transition. For Fermi surfaces with non-vanishing N , thereis the type-II point attached to the Fermi surfaces at the critical point of Lifshitz transition. This type-II point hasalso the nontrivial N , with the contour C chosen as the infinitesimal loop around the cone, see also reference [22]. This is the consequence of the π Berry phase along the infinitesimal loop around the Weyl point. And of course, theinvariant N supports the local stability of the Fermi surface and does not allow to make a hole in the Fermi surfaceand disrupt it.Here, we present three models, each with its own characterstics, which exhibit complex topological Lifshitz transitioninduced by the interplay between N , N and N invariants. A. Lifshitz transitions via marginal Dirac point
Fig. 5 demonstrates the Lifshitz transition, where the intermediate state represents the type-II Dirac point. Suchtransition has been discussed in relativistic theory with the CPT-violating perturbation.
The corresponding z < z hor z > z hor z = z hor
FIG. 4:
Weyl cone and light cone of artificial black hole.
The artificial event horizon can be simulated in Weyl semimetalsusing the interface between type-I Weyl material ( z > z hor ) and type-II Weyl material ( z < z hor ). With decreasing z , the Weylcones (lower row) and the corresponding ”light cones” for Weyl quasiparticle (upper row) are gradually titled. In the upperrow the light cone is overtilted behind the horizon, so that quasiparticle can move only away from the horizon into the blackhole region. The lower row demonstrates the process of Lifshitz transition at the horizon. Behind the horizon the Weyl coneis overtilted and two Fermi surfaces appear (red lines correspond to zero energy), connected by type-II Weyl point. Filling ofthe Fermi surfaces by particles and holes behind the horizon corresponds to the Hawking radiation, if the electrons and holescome from the region outside the horizon. The process of tunneling of quasiparticles from outside the horizon to inside thehorizon is seen as Hawking radiation with temperature in Eq.(19) for the black hole or by Eq.(20) for the flat horizon.After the particle and hole states in the Fermi surfaces are fully occupied, the Hawking radiation stops. While in the blackhole the shapes of the horizon and ergosurface are determined by Einstein equations, in semimetals they can be designed. FIG. 5:
Illustration of the process of topological Lifshits transition via type-II marginal Dirac point.
RelativisticWeyl fermions in Eq.(21) are considered. ( right ): Fermi surfaces of the right and left Weyl fermions. They enclose the Berry phase monopoles with topological charge N = +1 and N = − left ): Fermi surfaces are topologically trivial.( middle ): At the border between the two regimes the Fermi surfaces are attached to the marginal Dirac point, which is formedby merging of two Weyl points. Hamiltonian for a massive Dirac particle with mass M has the form: H = (cid:18) σ · ( c p − b ) − b MM − σ · ( c p + b ) + b (cid:19) . (21)Here the 4-vector b µ = ( b , b ) causes the shift of the positions of the Berry phase monopoles in opposite direction andformation of two Fermi surfaces with the global charge N = ± b > b + M as is shown in Fig.5(c). One Fermisurface enclosing the Berry phase monopole with topological charge N = +1 is formed by the right-handed Weylfermions; while the other one, which encloses the Berry phase monopole with topological charge N = −
1, comes fromthe left-handed Weyl fermions. Positions of the monopoles are at p ± = ± b (cid:18) b − b − M b − b (cid:19) / . (22)At critical point of Lifshitz transition, b = b + M , two Berry phase monopoles with opposite chirality mergeforming the Dirac point with trivial topological charge N = 0. In contrast, the non-vanishing N locally preservesFermi surfaces at this critical point. As a result of this interplay between N and N , the Fermi surfaces are attachedto the Dirac point forming the type-II Dirac point, see Fig. 5(b). Note that as different from the Weyl point, theDirac point is marginal: it has trivial global topology and is not stable, if there is no special symmetry which canstabilize the node. Here the type-II Dirac point appears exactly at the Lifshitz transition, similar to the appearanceof the Dirac nodal line in Fig. 1.
B. Lifshitz transitions via Weyl points
For the type of transition discussed in Sec. IV A, the topological index N is not involved, because the intermediatestate is the Dirac point. To obtain the type-II fermions with non-vanishing N at the Lifshitz transition, one should1 FIG. 6:
Lifshitz transition and the transport of topological charge between two Fermi surfaces through thetype-II Weyl point in chiral superlfuid. (a) θ < π/
2. In this case, the red Fermi surface and the blue Fermi surface areboth globally non-trivial, with N = − N = 1 respectively. Green arrows represent the configuration of correspondingBerry phase monopoles in the p y = 0 plane. When θ is increased to the case of (b) θ = π/
2, one can find that both Fermisurfaces are globally trivial. Equivalently, two Berry phase monopoles are pushed out from Fermi surfaces and form two type-IIWeyl points as the intermediate states of Lifshitz transition. (c ) θ > π/
2. Two Berry phase monopoles are pulled into the Fermisurfaces again and the globally non-trivial property of Fermi surfaces are revived. The only difference is that the N of everyFermi surface with θ > π/ N = +11 and N = − consider the system in which the Fermi surfaces are connected not by the type-II marginal Dirac point but by thetopologically stable type-II Weyl point.The corresponding Hamiltonian is obtained by the natural extension of Eq.(12) for quasiparticles in chiral superfluid He-A in the presence of the superfluid current. If the current is chosen perpendicular to the directions towards theWeyl nodes, then at v > c we obtain two type-II Weyl points, which connect two banana shape Fermi surfaces in Fig.6(b). These type-II Weyl points, in addition to the Berry monopole invariant N = ±
1, have the nonzero value of thetopological charge | N | calculated for the infinitesimal closed loop C around the cones. If the closed loop C is withinthe symmetry plane of two Fermi surfaces, the integral is independent on the shape and the radius of the contour C of integration. In general, however, when the symmetry is violated, only the integration over the infinitesimal loopgives the integer value of the invariant, see Sec.IV C.Let us now change the direction of the current. If θ is the angle between the current and the directions to thenodes, the Hamiltonian in the laboratory frame becomes: H = p x v + τ p − p F m + τ c ( p x sin θ − p z cos θ ) + τ cp y . (23)Eq.(23) is identical to Eq.(12) when θ = π/
2. Fig. 6 demonstrates the Lifshitz transition induced by the change of θ , from θ < π/ θ > π/
2, at v > c . When one continuously changes the angle θ across θ = π/
2, the Berry phasemonopoles with topological charges N = ± | p | = p F in the p y = 0 plane. At the same time, the non-vanishing local stability invariant N with v > c protects the Fermi surfacesduring this process. As a result, the Berry phase monopoles are transported between the two Fermi surfaces, withthe type-II Weyl points emerging in the intermediate state of this topological Lifshitz transition. Similar phenomenonwith the interplay between topological invariants N , N and N may take place in bbc Fe, see details in Ref. 57.2 FIG. 7:
Illustration of the process in which Fermi surfaces loose their global topological charge. ( top left ): Both blue and red Fermi surfaces enclose the Berry monopole with topological charge N = +1, when | p (0) | < p F .( top right ): The intermediate state of Lifshitz transition at | p (0) | = p F . The inner and outer Fermi surfaces touch each other,at the peculiar type-II Weyl point with topological charge N = +1.( bottom left ): On the other side of the Lifshitz transition, at at | p (0) | > p F , the Weyl point is outside of the Fermi surfaces, i.e.after the transition the Fermi surfaces lost the Berry phase flux.( bottom right ): After the second Lifshitz transition, which takes place at at | p (0) | = ( m c + p F ) / mc , the inner Fermi surfacedisappears, since it is not protected by the topological charge N . C. Fermi surface looses Berry monopole after Lifshitz transition
Let us consider another class of emergent type-II Weyl point, in which the Berry phase monopole is transportedacross the Fermi surface. It can be represented by the following Hamiltonian: H = c σ · ( p − p (0) ) + p − p F m (24)3In Fig. 7, we plot the Lifshithz transitions and the evolution of configuration of Berry monopole in momentum spacedriven by the change of the position p (0) of the Weyl point. The regime with p F > mc is considered. For | p (0) | < p F we have two Fermi surfaces, one inside the other, but both embracing the Weyl point with N = 1, the Berry phasemonopole. At the Lifshitz transition, which occurs at | p (0) | = p F , the inner and outer Fermi surfaces touch each otherat the Weyl point, which becomes the peculiar type-II point. As distinct from the conventional type-II Weyl point,which connects two Fermi pockets, this Weyl point connects the inner and outer Fermi surfaces. After the Lifshitztransition, at | p (0) | > p F , the Weyl point leaves both Fermi surfaces. The Fermi surfaces are again one inside theother, but both without the Berry flux. Finally at the second Lifshitz transition, at | p (0) | = ( m c + p F ) / mc , theinner Fermi surface collapses to the point and disappears, since the point is no more supported by the topologicalinvariant N .At the first Lifshitz transition, the cone formed at the touching point is again characterized by the topologicalinvariant N = 1, where the integral is over the infinitesimal path around the cone. V. FLAT BANDS AT LIFSHITZ TRANSITIONS
For the interacting fermions more types of Lifshitz transitions are possible – the transitions which involve the Weylpoints of type-III and type-IV. The interaction also leads to the formation of the flat band in the energy spectrum– the so-called Khodel-Shaginyan fermion condensate.
The dispersionless energy spectrum has a singular densityof states. As a results, instead of the exponential suppression of the superconducting transition temperature T c (andof the gap ∆) in the normal metal, the flat band provides T c and ∆ being proportional to the coupling constant g inthe Cooper channel: ∆ normal = E exp (cid:18) − gN F (cid:19) , ∆ flat band = gV d π ~ ) d . (25)Here N F is the density of states in normal metal; d is the dimension of the metal; and V d is the volume of theflat band. For nuclear systems, i.e. for d = 0, the linear dependence of the gap on the coupling constant has beenfound by Belyaev. The enhancement of T c in materials with the flat band opens the route to room temperaturesuperconductivity, see review Ref. .The band flattening caused by electron-electron interaction in metals is the manifestation of the general phenomenonof the energy level merging due to electron-electron interaction. This effect has been recently suggested to beresponsible for merging of the discrete energy levels in two-dimensional electron system in quantizing magnetic fields.According to Ref. the favourable condition for the formation of such flat band is when the van Hove singularitycomes close to the Fermi surface, i.e. the system is close to the Lifshitz transition (see also Ref. for the simpleLandau type model of the formation of such flat band). It is also possible that this effect is responsible for theoccurrence of superconductivity with high T observed in the pressurized sulfur hydride . There are some theoreticalevidences that the high- T c . superconductivity takes place at such pressure, when the system is near the Lifshitztransition That is why it is not excluded that the Khodel-Shaginyan flat band is formed in sulfur hydride at pressure180-200 GPa giving rise to high-T c superconductivity. The topological Lifshitz transitions with participation of theWeyl and Dirac points and Dirac lines may also lead to the formation of the flat bands in the vicinity of transitions,and thus to the enhanced T c .Here we consider the flat bands, which appear near the Lifshitz transition in Eq. (24). The arrangement of the flatbands experiences its own Lifshitz transitions in Fig. 8.The energy functional of interacting loosing Weyl model is: E [ n ( p ) , n ( p )] = X p ǫ n ( p ) + ǫ n ( p ) + 12 U ( n ( p ) −
12 ) + 12 U ( n ( p ) −
12 ) + 12 U m ( n ( p ) −
12 )( n ( p ) −
12 ) , (26)where n ( p ) and n ( p )] are distribution functions for two species of fermions (”partilces” and ”holes”). For the flatbands induced by the repulsive interaction, we have: ǫ = δEδn ( p ) = 0 , ǫ = δEδn ( p ) = 0 , (27)which give ǫ + U ( n ( p ) −
12 ) + 12 U m ( n ( p ) −
12 ) = 0 , (28) ǫ + U ( n ( p ) −
12 ) + 12 U m ( n ( p ) −
12 ) = 0 . (29)4The distribution functions of particles and hole are: n ( p ) = 4 U − U m − U ǫ + 4 U m ǫ U − U m , (30)and n ( p ) = 4 U − U m − U ǫ + 4 U m ǫ U − U m . (31)Those regions within which 0 < n ( p ) < < n ( p ) < | p | , several Lifshitz transitions occur at the critical points of Lifshitz given by: p = p F − U − U m c , (32) p = r p F − mU + mU m , (33) p T = r p F + 2 mU + mU m , (34) p = p F + 2 U − U m c , (35) p D = 2 c m (2 U + U m ) + ( U m − U ) [2 p F + m (2 U + U m )]4 mc (4 U − U m ) . (36)At these transitions the Fermi bands appear, disappear or touch each other with formation of singular configurations. VI. CONCLUSION
The interplay of different topological invariants enhances the variety of the topological Lifshitz transitons. Here wediscussed the examples of the transitions, which involve the nodes of different co-dimensions: the Fermi surfaces withtopological charge N (co-dimension 1), Weyl points with the topological charge N (co-dimension 3) and Dirac lineswith topological charge N (co-dimension 2). Depending on the type of the transition, the intermediate state has thetype-II Dirac point, the type-II Weyl point or the Dirac line. The latter is supported by combination of symmetryand topology. There are different configurations of the Fermi surfaces, involved in the Lifshitz transition with theWeyl points in the intermediate state. In Fig. 6 the type-II Weyl point connects the Fermi pockets, and the Lifshitztransition corresponds to the transfer of the Berry flux between the Fermi pockets. In Fig.7 the type-II Weyl pointconnects the outer and inner Fermi surfaces. At the Lifshitz transition the Weyl point is released from both Fermisurfaces. They loose their Berry flux and the topological charge N , which guarantees the global stability. As a resultthe inner surface disappears after shrinking to a point at the second Lifshitz transition.Many other Lifshitz transitions are expected, since we did not touch here the other possible topological features:topological invariants which describe the shape of the Fermi surface; the shape of the Dirac nodal lines; their in-terconnections; etc. The interplay of topologies can be seen in particular in the electronic spectrum of Bernal andrombohedral graphite. In particular, in the electronic spectrum of Bernal graphite the type-II Dirac line hasbeen identified, which is connected with the type-I Dirac line at some point in the 3D momentum space. If oneconsiders p z as parameter, then at some critical value of p z there is the transition from the 2D type-I Dirac point tothe 2D type-II Dirac point.However, the most important property of Lifshitz transitions is that in the vicinity of the topological transtion theelectron-electron interaction leads to the formation of zeroes in the spectrum of the co-dimension 0, i.e. to the flatbands. Because of the singular density of electronic states, materials with the flat band are the plausible candidatesfor room-temperature superconductivity.5 VII. ACKNOWLEDGEMENTS
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