Exotic Lifshitz transitions in topological materials

TL;DR

This paper explores various topological Lifshitz transitions in materials, highlighting their complex structures, invariants, and implications across physics, including superconductivity, particle physics, and black hole horizons.

Contribution

It provides a comprehensive analysis of topological Lifshitz transitions, classifying different types and discussing their physical consequences across multiple fields.

Findings

Multiple classes of Lifshitz transitions identified.

Topological invariants support stability of structures.

Transitions influence phenomena like superconductivity and particle masses.

Abstract

Topological Lifshitz transitions involve many types of topological structures in momentum and frequency-momentum spaces: Fermi surfaces, Dirac lines, Dirac and Weyl points, etc. Each of these structures has their own topological invariant ($N_1$, $N_2$, $N_3$, $\tilde N_3$, etc.), which supports the stability of a given topological structure. The topology of the shape of Fermi surfaces and Dirac lines, as well as the interconnection of the objects of different dimensions, lead to numerous classes of Lifshitz transitions. The consequences of Lifshitz transitions are important in different areas of physics. The singularities emerging at the transition may enhance the transition temperature to superconductivity; the Lifshitz transition can be in the origin of the small masses of elementary particles in our Universe; the black hole horizon serves as the surface of Lifshitz transition between the vacua with type-I and type-II Weyl points; etc.

Equations

$$G^{-1}(\omega ,\mathbf{p})=i\omega -ϵ(\mathbf{p}),$$

$$N_1=\mathbf{tr}{\oint }_C\frac{dl}{2\pi i}G(\omega ,\mathbf{p}){\partial }_l{G}^{-1}(\omega ,\mathbf{p}).$$

$${ϵ}_{\mathbf{p}}=a{p}_x^2+b{p}_y^2+c{p}_z^2-\mu .$$

$$G=\frac{Z}{i\omega -ϵ(\mathbf{p})},Z\propto {\left({\omega }^2+{ϵ}^2(\mathbf{p})\right)}^{\gamma }.$$

$$G\propto i\omega +ϵ(\mathbf{p}).$$

$$N_3=\frac{{ϵ}_{\mu \nu \rho \sigma }}{24{\pi }^2}\mathrm{tr}{\oint }_{{\Sigma }_a}d{S}^{\sigma }G\frac{\partial }{\partial p_{\mu }}{G}^{-1}G\frac{\partial }{\partial p_{\nu }}{G}^{-1}G\frac{\partial }{\partial p_{\rho }}{G}^{-1}.$$

$$H=c{\mbox{\boldmath$\sigma$}}\cdot{\bf p}-vp_{z}\,.$$

$$d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=-c^{2}d{t}^2+(d\mathbf{r}-\mathbf{v}dt{)}^2.$$

$$\mathbf{v}(\mathbf{r})=-\hat{\mathbf{r}}c\sqrt{\frac{r_h}{r}},r_h=\frac{2MG}{c^2}.$$

$$H=\pm c{\mbox{\boldmath$\sigma$}}\cdot{\bf p}-p_{r}v(r)\,\,,\,\,v(r)=c\sqrt{\frac{r_{h}}{r}}\,.$$

Full text

I Introduction. Fermi surface, Dirac line, Weyl point

The key word in consideration of Lifshitz transitions is topology. Following original Lifshitz paper,ILifshitz1960 Lifshitz transition has been viewed as a change of the topology of the Fermi surface without symmetry breaking. Later it became clear that topology of the shape is not the only topological characterization of the Fermi surface. Fermi surface itself represents the singularity in the Green’s function, which is topologically protected: it is the vortex line in the four-dimensional frequency-momentum space in Fig. 1 (top right). The stability of the Fermi surface under interaction between the fermions is in the origin of the Fermi liquid theory developed by Landau. Moreover, the Fermi surface appeared to be one in the series of the topologically stable singularities,Horava2005 ; Volovik2003 which include in particular the Weyl point – the hedgehog in momentum space in Fig. 1 (middle) and the Dirac line – the vortex line in the three-dimensional momentum space in Fig. 1 (bottom). The stability of these objects is supported by the corresponding topological invariants in momentum space or in extended frequency-momentum space.

The combination of topology of the shape of the Fermi surfaces, Fermi lines and Fermi points together with the topology, which supports the stability of these objects, and also the topology of the interconnections of the objects of different dimensions provide a large number of different types of Lifshitz transitions. Examples of Lifshitz transitions coming from the interplay of different topological objects in momentum space are discussed in Refs. Volovik2007 ; Volovik2017 ; KuangVolovik2017 and in Secs. III.3 and IV. This makes the Lifshitz transitions ubiquitous with applications to high energy physics, cosmology, black hole physics and to search for the room-$T$ superconductivity.

In particular, the Lifshitz transition may give the solution of the hierarchy problem in particle physics: why the masses of elementary particles in our Universe are so extremely small compared with the characteristic Planck energy scale. Indeed, when we compare the mass $\sim 10^{2}$ GeV of the most heavy particle – the top quark – with the Planck energy $\sim 10^{19}$ GeV, we can see that the vacuum of our Universe is practically gapless. There are several topological scenarios, which may lead to the (almost) gapless vacuum.

In one scenario the quantum vacuum belongs to the class of the gapless (massless) Weyl materials in Fig. 1 (middle), where the nodes in the spectrum of elementary particles – the Weyl points – are topologically protected,FrogNielBook ; Horava2005 ; Volovik2003 see Sec.III.1 and Fig. 9. According to this scenario the physical laws are not fundamental, but emerge in the low energy corner of the quantum vacuum, i.e. in the vicinity of the Weyl points, where the spectrum becomes linear and all the symmetries of Standard Model including Lorentz invariance and general covariance emerge from nothing. In this scenaro the Lifshitz transition between type-I and type-II Weyl vacua take place at the black hole horizon, see Sec. III.4.

At even lower energy some of these symmetries experience spontaneous breaking, analogous to the superconducting transition. In the latter case the hierarchy problem is understood: in most superconductors the transition temperature $T_{c}$ is exponentially small, compared to the characteristic Fermi energy scale (analog of the Planck scale), which forces us to search for the exceptional materials with enhanced $T_{c}$. The role of the Lifshitz transition in the enhancement of the temperature of superconducting transition is in Sec. II.4 and Sec. III.3.

In the other scenario, the massless (gapless) vacua emerge at the Lifshits transition between the fully gapped vacua with different topological charges, see Sec.VI and Fig. 18. The almost perfect masslessness of elementary particles in our Universe suggests that the Universe is very close to the line of the topological Lifshitz transition between the fully gapped vacua, at which fermions necessarily become gapless, Volovik2010 see Sec. VI. This is the topological analog of the so-called Multiple Point Principle, according to which the Universe lives at the coexistence point (line, surface, etc.) of the first order phase transition, where different vacua have the same energy.Nielsen2016a ; Nielsen2016b ; Nielsen2016c ; Nielsen1997 ; Volovik2004

II Fermi surface and Lifshitz transitions

II.1 Fermi surface as topological object

The primary topology, which is at the origin of the Lifshitz transitions, is the topology which is reponsible for the stability of the Fermi surface. If the Fermi surface is not stable under the electron-electron interaction, the consideration of the topology of the shape of the Fermi surface and of the corresponding Lifshitz transitions does not make much sense. To view the topological stability of the Fermi surface with respect to interactions let us start with the Green’s function of an ideal Fermi gas in Fig. 2 (left). The Fermi surface $\epsilon({\bf p})=0$ of the noninteracting Fermi gas is the boundary in momentum space, which separates the occupied states with $\epsilon({\bf p})<0$ from the empty states with $\epsilon({\bf p})>0$. The Green’s function $G(\omega,{\bf p})$ with $\omega$ on imaginary axis

[TABLE]

has singularity at $\omega=0$ and $\epsilon({\bf p})=0$. In Fig. 2 (right) the $p_{z}$ coordinate is suppressed, and the Green’s function singularity forms the closed line in the $2+1$ momentum-frequency space $(p_{x},p_{y},\omega)$. This line represents the vortex line, at which the phase of the Green’s function $\Phi(p_{x},p_{y},\omega)$ has the $2\pi$ winding. As in the case of the real-space vortex in superfluids, the integer winding number provides the stability of the Fermi surface with respect to perturbations, including the interaction (if the $p_{z}$ component is restored, the singularity forms the vortex sheet in the $3+1$ momentum-frequency $({\bf p},\omega)$ space).

In general case, when the Green’s function has the spin, band and other indices, the winding number can be written as the following topological invariant in terms of the Green’s function:

[TABLE]

Here the integral is taken over an arbitrary contour $C$ around the momentum-frequency vortex sheet, and ${\bf tr}$ is the trace over all the indices.

Due to topological stability one cannot make a hole in the Fermi surface. As in the case of vortex lines, which cannot terminate in bulk, the Fermi surface has no edges.

II.2 Fermi surface Lifshitz transitions

Because of the topological stability the Fermi surface may be formed even in the superconducting state. The conditions for that are multi-band structure and broken time reversal symmetry $T$ and parity $P$. Volovik1989 ; LiuWilczek2003 ; BarzykinGorkov2007 ; Agterberg2017 ; Timm2017 These the so-called Bogolubov Fermi surfaces appear also in gapless superfluids, when the Weyl points in 3He-A and Dirac nodal line in the polar phase of 3He are inflated to the Fermi pockets in the presence of superflow, which violates both $T$ and $P$ symmetries.Volovik1984 ; Makinen2017

The Fermi surface can be also formed in the fully gapped superfluids, if the velocity of superflow exceeds Landau critical velocity.VollhardtMakiSchopohl1980 ; Volovik2003 The crossing of Landau velocity with formation of closed Bogoliubov Fermi surfaces is an example of one of the two transtions suggested by Lifshitz, see Fig. 3.

Another original Lifshitz transition takes place when the Fermi surface crosses the stationary point of the electronic spectrum. Near the transition the expansion of the generic spectrum has the form:ILifshitz1960

[TABLE]

For $a>0$, $b>0$, $c<0$ the transition with disruption of the neck of the Fermi surface at $\mu=0$ is in Fig. 4 (top). In terms of the vortex singularities of the Green’s function in 3+1 $({\bf p},\omega)$ space, this Lifshitz transition represents the interconnection of the vortex lines in Fig. 4 (bottom). In superfluids, the interconnection of the real-space vortices is an important process in the vortex turbulence.Golov2016

II.3 From pole of Green’s function to zero

While in conventional Landau Fermi liquid the Green’s function has a pole, for Luttinger liquid the resudue of the pole in the Green’s function has singularity: the parameter $\gamma$ in Eq.(5) is nonzero:Voit1995

[TABLE]

Nevertheless the topological invariant remains the same for all $\gamma$: i.e. the Green’s function has the same topological property as the Green’s function of conventional metal with Fermi surface at $\epsilon({\bf p})=0$. This is the reason why the Luttinger theorem is still valid.Dzyaloshinskii2003 ; Farid2009 The particle density of interacting fermions is equal to the volume in the momentum space enclosed by the singular surface with the topological charge $N_{1}=1$, irrespective of the realization of the singularity.

The suppression of residue $Z$ can be so strong, that the pole in the Green’s function is transformed to the zero of the Green’s function, which corresponds to the special case of $\gamma=1$, see Fig. 5:

[TABLE]

This situation in particular takes place for Mott insulators,Dzyaloshinskii2003 which means that the. topology of Fermi surface is preserved even in the insulating phase, and thus the Luttinger theorem is still valid.Dzyaloshinskii2003 ; Farid2009 Thus we can say that the transition between metals and insulators can be also viewed as a type of the zero-temperature Lifshitz transition, at which the property of the energy spectrum drastically changes without symmetry breaking. However, this quantum phase transition is not topological since the topological invariant does not change across the transition.

It is not excluded that in the so-called pseudo-gap phase of cuprate superconductors and some other materials, see e.g.PseudGap , the part of the Fermi surface transforms to the surface of zeroes and the Fermi arcs are formed, see Fig. 5 (bottom right).

II.4 From Fermi surface to flat band

The flat band – or the so-called Khodel-Shaginyan fermion condensate, where all the states have zero energy – is caused by electron-electron interaction.Khodel1990 ; Volovik1991 ; Nozieres92 This is the manifestation of the general phenomenon of merging of the energy level due to interaction. Such effect has been observed for Landau levels in 2D quantum wells. Dolgopolov2014 ; Dolgopolov2015 Since the flat band has huge density of electronic states, this may considerably enhance the transition temperature to superconducting state, see Fig. 7.

The flat band is more easily formed in the vicinity of the conventional Lifshitz transition,Yudin2014 ; Volovik1994 see Fig. 8. Flattening of the single-particle spectrum near the Fermi momentum has been reported in 2D quantum well.Dolgopolov2016 It is possible that this effect is responsible for the occurrence of superconductivity with high $T$ observed in the pressurized sulfur hydride:Drozdov2015 ; Drozdov2016 there are some theoretical evidences that the high-$T_{c}$ superconductivity takes place at such pressure, when the system is close to the Lifshitz transition.Pickett2015 ; Bianconi2015 ; Souza2017 Enhanced superconductivity at Lifshitz transition has been reported in FeSe monolayer.FeSe

III Lifshitz transitions governed by Weyl point topology

III.1 Topology of Weyl fermions

Weyl particles are the elementary particles of our Universe. The Weyl spinor contains 2 complex components, and these massless particles are described by the $2\times 2$ complex Hamiltonian: H=c{\mbox{\boldmath\sigma}}\cdot{\bf p} for right-handed quarks and leptons and H=-c{\mbox{\boldmath\sigma}}\cdot{\bf p} for the left-handed particles, where $c$ is the speed of light. Their spins in momentum space form correspondingly the hedgehog and anti-hedgehog in Fig. 9 (top). The hedgehog is the topologically stable object, and thus the Weyl point in the center of the hedgehog is topologically protected. The corresponding topological invariant for the hedgehogs, $N_{3}$, can be expressed in terms of the Green’s function as a surface integral in the 3+1 momentum-frequency space $p_{\mu}=({\bf p},\omega)$: Volovik2003

[TABLE]

Here $\Sigma_{a}$ is a three-dimensional surface around the isolated Weyl point in $({\bf p},\omega)$ space.

From the point of view of the general properties of the fermionic spectrum, the Weyl point represents the exceptional point of level crossing analyzed by von Neumann and Wigner Neumann1929 . This analysis demonstrates that two branches of spectrum, which have the same symmetry, may touch each other at the conical (or diabolical) point in the three-dimensional space of parameters, which in our case are $p_{x}$, $p_{y}$ and $p_{z}$. The touching of two branches is described in general by $2\times 2$ Hamiltonian H={\mbox{\boldmath\sigma}}\cdot{\bf g}({\bf p}). The topological invariant $N_{3}$ is expressed in terms of the unit vector $\hat{\bf g}({\bf p})={\bf g}({\bf p})/|{\bf g}({\bf p})|$ in Fig. 1 (center), which forms the hedgehog configuration in Fig. 1 (middle right). The touching point also represents the Berry phase monopole in Fig. 9 (bottom).Volovik1987 It is not excluded that the Weyl fermions of Standard Model (quark and leptons) are not the elementary particles, but emerge from the level crossing at the more fundamental level.Froggatt1991 ; Volovik2003 ; Horava2005 In particular, the underlying quantum vacuum can be described by quantum field theory based on real numbers (Majorana fermions), while the imaginary unit, which enters Schrödinger equation, emerges in the low energy limit together with the relativistic linear spectrum of Weyl fermions.VolovikZubkov2014

The linear (”relativistic”) spectrum emerges only for the elementary topological charges, $N_{3}=+1$ or $N_{3}=-1$. If the Weyl point has higher topological charge, $|N_{3}|>1$, and if there is no special symmetry, which leads to the degeneracy of the levels, the spectrum has different dispersion relations along different axes.VolovikKonyshev1988 ; Volovik2003 . For example, for $|N_{3}|=2$ the spectrum is ”relativistic” in one direction and quadratic in the other two directions.

III.2 Lifshitz transition with splitting of Weyl points

The typical Lifshitz transition, which involves the Weyl nodes in the fermionic spectrum, describes the formation of the Weyl points with opposite charges $N_{3}=\pm 1$ from the fully gapped state. Fig. 10 shows the formation of the pair of the Weyl points from the vacuum state with massive Dirac fermions. The intermediate state has the massless Dirac point in the fermionic spectrum with topological charge $N_{3}=0$. Such gapless Dirac point is marginal, but can be protected by symmetry as this takes place in Standard Model above the electroweak transition. If the symmetry is violated or is spontaneously broken the Dirac spectrum either acquires mass or splits into the pair of Weyl points.KlinkhamerVolovik2005a

Fig. 11 demonstrates the formation of 4 right-handed and 4 left-handed Weyl points at the Lifshitz transition between the BEC strong coupling regime to the BCS weak coupling regime. Such arrangement of the Weyl nodes takes place in the energy spectrum in the $O(D_{2})$ symmetry class of the pair correlated systems.VolovikGorkov1985 ; VolovikGorkov2017 In both cases the total topological charge $N_{3}({\rm total})=0$, and thus there is an even number of Weyl fermions, which supports the fermion doubling principle.NielsenNinomiya1981 In relativistic theories the analogical arrangement of 8 left and 8 right Weyl fermions on the vertices of the cube in the 3+1 $(p_{x},p_{y},p_{z},\omega)$ space has been discussed.Creutz2008 ; Creutz2014 It is interesting that each family of Standard Model fermions contains 8 left and 8 right Weyl particles.

III.3 Lifshitz transition to type-II Weyl cone

There is the type of Lifshitz transition, which involves both the Fermi surface (invariant $N_{1}$) and the Weyl point (invariant $N_{3}$). This is the transition between the isolated Fermi points (type-I Weyl spectrum) to the Weyl point connecting two Fermi surfaces (this is called the type-II Weyl pointSoluyanov2015 ). In relativistic theories such transitions have been discussed in VolovikZubkov2014 ; HuhtalaVolovik2002 .

The simplest realization of the type-II Weyl point comes from the following Hamiltonian with two parameters $c$ and $v$:

[TABLE]

For $v=0$ this is the Weyl point with the Weyl cone in Fig. 12 (top left). For $0<v<c$ the cone is tilted. At $v>c$ the cone is overtilted, so that the cones cross the zero energy level forming two Fermi pockets connected by the Weyl point – the type-II Weyl point, see Fig. 12 (top right). The Lifshitz transition between two types of Weyl point occurs at $v=c$. It is demonstrated that such Lifshitz transition also leads to the enhancement of the transition temperature to superconducting state, Shapiro2017 ; Zyuzin2017 see Fig. 12 (bottom).

III.4 Lifshitz transition at the black hole horizon

Lifshitz transition discussed in Sec. 12 takes place at the black hole horizon. In general relativity the stationary metric, which is valid both outside and inside the black hole horizon, is provided in particular by the Painlevé-Gullstrand spacetime.Painleve The line element of the Painlevé-Gullstrand metric is equivalent to the so-called acoustic metric:unruh1 ; unruh2 ; Kraus1994

[TABLE]

This metric is expressed in terms of the velocity field ${\bf v}({\bf r})$ describing the frame dragging in the gravitational field:

[TABLE]

Here $M$ is the mass of the black hole; $r_{h}$ is the radius of the horizon; $G$ is the Newton gravitational constant. Behind the horizon the drag velocity exceeds the speed of light, $|{\bf v}|>c$, and particles are trapped in the hole, see Fig. 13 (top right). The behavior of the light cone (the cone in spacetime) across the event horizon is shown in Fig. 13 (top left). The light cone is overtilted behind the horizon.

The behavior of the Weyl cone (the cone in momentum space) across the horizon is decsribed by Hamiltonian of the Weyl particles in the gravitational field of the black hole, which for the Painlevé-Gullstrand metric has the following form:HuhtalaVolovik2002

[TABLE]

Here the plus and minus signs correspond to the right handed and left handed Weyl fermions respectively; $p_{r}$ is the radial component of the linear momentum of the particle. Behind the horizon, where $v>c$ and the light cone is overtilted, the Weyl cone is also overtilted, but in the way shown in Fig. 12. Two Fermi pockets are formed, which touch each other at type-II Weyl point in Fig. 13 (bottom right). The event horizon at $r=r_{h}$ thus serves as the surface of the Lifshitz transition.

The correspondence between Weyl semimetals and black holes allows us to simulate the black hole horizon using the inhomogeneous Weyl semimetal, where the transition between the type-I and type-II Weyl points takes place at some surface.Volovik2016 This surface would play the role of the event horizon. The formed black hole will be fully stationary in equilibrium. However, just after creation of this black hole analog, the system is not in the equilibrium state, and the relaxation process at the initial stage of equilibration looks similar to the process of the Hawking radiation.

In the discussed Lifshitz transition between type-I and type-II Weyl points, the element $g_{00}$ of the effective metric changes sign. Another type of Lifshitz transition occurs when the element $g^{00}$ changes sign. In Weyl semimetals this corresponds to the transition to the type-III Weyl fermions,NissinenVolovik2017 while in general relativity this is the transition to spacetimes with closed timelike curves.

IV Lifshitz transitions with several topological charges

In sections III.3 and III.4 we considered the Lifshitz transition which involved two topological charges: the charge $N_{1}$, which characterizes the Fermi surface, and the charge $N_{3}$ of the Berry phase monopole. There are the other Lifshitz transitions with the interplay of these two topological invariants. This happens in particular, when the closed Fermi surface is described by two invariants: the local charge $N_{1}$, which provides the local stability of the Fermi surface, and the global charge $N_{3}$, which describes the Weyl point inside the Fermi surface in Fig. 14. The latter takes place for example when the Weyl point shifts from the zero energy position forming the small Fermi sphere around the Weyl point, see Fig. 14 (left). Such Fermi sphere contains the $N_{3}$ charge, which can be obtained from Eq.(6) by integration over the surface, which encloses the Fermi sphere.

At the Lifshitz transition the Fermi surfaces can exchange their global charges $N_{3}$ or loose the global charge. KlinkhamerVolovik2005a ; KuangVolovik2017 Example of exchange is in Fig. 15 and the example of the lost global charge is in Fig. 16. In both cases the intermediate state at the point of Lifshitz transition contains the type-II Weyl points.

V Lifshitz transition governed by conservation of $N_{2}$ charge

The conical Dirac point in the 2D graphene and the nodal lines in the 3D semimetals and nodal superfluids and superconductors are stabilized by the topological charge $N_{2}$ in Fig. 1 (bottom).Volovik2007 ; HeikkilaVolovik2015a Dirac nodal lines were known to exist in the polar phase of superfluid 3HeDmitriev2015 ; Autti2016 , in cuprate superconductors, and in graphite (band crossing lines).Mikitik2014 ; Mikitik2006 ; Mikitik2008 Now they are extensively studied in semimetals.Takane2017

The type of Lifshitz transitions governed by the conservation of the topological charge $N_{2}$ is shown in Fig. 17 on example of bilayer graphene, when one graphene layer is shifted with respect to the other one. Merging of the two conical points with $N_{2}=1$ leads to formation of the Dirac node with quadratic dispersion in Fig. 17 (top left), which has the toplogical charge $N_{2}=2$. This point in turn may split into four Dirac conical points with $N_{2}=\pm 1$ in Fig. 17 (bottom). This is the so-called trigonal warping. The total topological charge is conserved, $N_{2}=1+1+1-1=2$. The trigonal warping can be seen in the Bernal graphite, see Fig. 17 (top right), and the transition occurs as a function of $p_{z}$,Mikitik2006 ; Mikitik2008 ; Mikitik2014 when $p_{z}$ crosses the so-called nexus point.HeikkilaVolovik2015a ; HyartHeikkila2016

VI Lifshitz transitions between gapped states via gapless state

Lifshitz transitions between gapped states include transitions between the topological and non-topological insulators; transitions between the fully gapped superfluids/superconductors; transitions between the 2D systems, which experience the intrinsic quantum Hall effect; etc. Here we consider such transition on example of the 2D systems, where the Hall conductance is expressed in terms of the integer-valued topological invariant $\tilde{N}_{3}$ in Fig. 18 (top right).So1985 ; IshikawaMatsuyama1986 ; IshikawaMatsuyama1987 ; Haldane1988 This topological invariant has the same structure as the invariant $N_{3}$ in Fig. 1 (Middle), but the integration now is over the whole 2D Brillouin zone. This is an example of the dimensional reduction from the 3D systems with Weyl nodes to the 2D topological insulators.Volovik2003

Fig. 18 (top left) demonstrates the Lifshitz transition between the topological insulator with $\tilde{N}_{3}=1$ and the trivial insulator with $\tilde{N}_{3}=0$. Here the topological charge is not conserved across the Lifshitz transition, but abrubtly changes recalling the first order phase transition. Nevertheless the transition occurs smoothly, because at the point of transition the gap in the energy spectrum vanishes. The nullification of the gap at the transition reflects the fact that in the 3D space $(p_{x},p_{y},\mu)$, where $\mu$ is chemical potential or some other parameter along which the transition occurs, the gap node represents the Weyl point with topological charge $N_{3}=\tilde{N}_{3}({\rm right})-\tilde{N}_{3}({\rm left})$.Volovik2003 ; Kourtis2017

Example is the 2D $p_{x}+ip_{y}$ superfluid/superconductor,Volovik1988 where the Lifshitz transition between the superfluid states with $\tilde{N}_{3}=1$ and $\tilde{N}_{3}=0$ occurs at the same point $\mu=0$ as in the normal Fermi liquid. The detailed consideration shows that the Lifshitz transition represents the quantum transition of third order:Ortiz2008 the third-order derivative $d^{3}E/dg^{3}$ of the ground state energy $E$ over the interaction strength $g$ is discontinuous. Compare this with the original $2\frac{1}{2}$ order transition,ILifshitz1960 and the $3\frac{1}{2}$ order transition discussed recently.Mikitik2014

The nullification of the gap in the fermionic spectrum at the transition between the gapped vacua, suggests the scenario for the solution of the hierarchy problem: the relativistic quantum vacuum is almost massless because our Universe is very close to the line of the Lifshitz transition. The reason why nature would prefer the critical line may be that the gapless states on the transition line are able to accommodate more entropy than the gapped states.Volovik2010

VII Conclusion

Topological Lifshitz transitions are ubiquitous, since they involve many types of the topological structure of fermionic spectrum: Fermi surfaces, Dirac lines, Dirac and Weyl points, edge states, Majorana zero modes, etc. Each of these structures has their own topological invariant, such as $N_{1}$, $N_{2}$, $N_{3}$, $\tilde{N}_{3}$, etc., which supports the stability of a given class of the topological structure. The topology of the shape of the Fermi surfaces and the Dirac lines, as well as the interconnection of the objects of different dimensionalities in momentum and frequency-momentum spaces, lead to numerous classes of Lifshitz transitions.

The consequences of Lifshitz transitions are important in different areas of physics. In particular, the singular density of electronic states emerging at the transition is important for the construction of superconductors with enhanced transition temperature; the Lifshitz transition can be in the origin of the small masses of elementary particles in our Universe; the black hole horizon serves as the surface of Lifshitz transition between the vacua with type-I and type-II Weyl points; etc.

Acknowledgements

The work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694248) and by RSCF (No. 16-42-01100).