Exotic Lifshitz transitions in topological materials
EExotic Lifshitz transitions in topological materials
G.E. Volovik
1, 2, 3 Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland L. D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia P.N. Lebedev Physical Institute, RAS, Moscow 119991, Russia (Dated: September 4, 2017)Topological Lifshitz transitions involve many types of topological structures in momentum andfrequency-momentum spaces: Fermi surfaces, Dirac lines, Dirac and Weyl points, etc. Each ofthese structures has their own topological invariant ( N , N , N , ˜ N , etc.), which supports thestability of a given topological structure. The topology of the shape of Fermi surfaces and Diraclines, as well as the interconnection of the objects of different dimensions, lead to numerous classesof Lifshitz transitions. The consequences of Lifshitz transitions are important in different areas ofphysics. The singularities emerging at the transition may enhance the transition temperature tosuperconductivity; the Lifshitz transition can be in the origin of the small masses of elementaryparticles in our Universe; the black hole horizon serves as the surface of Lifshitz transition betweenthe vacua with type-I and type-II Weyl points; etc. PACS numbers:
I. INTRODUCTION. FERMI SURFACE, DIRACLINE, WEYL POINT
The key word in consideration of Lifshitz transitionsis topology. Following original Lifshitz paper, Lifshitztransition has been viewed as a change of the topology ofthe Fermi surface without symmetry breaking. Later itbecame clear that topology of the shape is not the onlytopological characterization of the Fermi surface. Fermisurface itself represents the singularity in the Green’sfunction, which is topologically protected: it is the vortexline in the four-dimensional frequency-momentum spacein Fig. 1 ( top right ). The stability of the Fermi surfaceunder interaction between the fermions is in the originof the Fermi liquid theory developed by Landau. More-over, the Fermi surface appeared to be one in the series ofthe topologically stable singularities, which include inparticular the Weyl point – the hedgehog in momentumspace in Fig. 1 ( middle ) and the Dirac line – the vortexline in the three-dimensional momentum space in Fig. 1( bottom ). The stability of these objects is supported bythe corresponding topological invariants in momentumspace or in extended frequency-momentum space.The combination of topology of the shape of the Fermisurfaces, Fermi lines and Fermi points together with thetopology, which supports the stability of these objects,and also the topology of the interconnections of the ob-jects of different dimensions provide a large number ofdifferent types of Lifshitz transitions. Examples of Lif-shitz transitions coming from the interplay of differenttopological objects in momentum space are discussed inRefs. and in Secs. III C and IV. This makes the Lif-shitz transitions ubiquitous with applications to high en-ergy physics, cosmology, black hole physics and to searchfor the room- T superconductivity.In particular, the Lifshitz transition may give the so-lution of the hierarchy problem in particle physics: whythe masses of elementary particles in our Universe are so extremely small compared with the characteristic Planckenergy scale. Indeed, when we compare the mass ∼ GeV of the most heavy particle – the top quark – withthe Planck energy ∼ GeV, we can see that the vac-uum of our Universe is practically gapless. There areseveral topological scenarios, which may lead to the (al-most) gapless vacuum.In one scenario the quantum vacuum belongs to theclass of the gapless (massless) Weyl materials in Fig.1 ( middle ), where the nodes in the spectrum of ele-mentary particles – the Weyl points – are topologicallyprotected, see Sec.III A and Fig. 9. According tothis scenario the physical laws are not fundamental, butemerge in the low energy corner of the quantum vacuum,i.e. in the vicinity of the Weyl points, where the spec-trum becomes linear and all the symmetries of StandardModel including Lorentz invariance and general covari-ance emerge from nothing. In this scenaro the Lifshitztransition between type-I and type-II Weyl vacua takeplace at the black hole horizon, see Sec. III D.At even lower energy some of these symmetries expe-rience spontaneous breaking, analogous to the supercon-ducting transition. In the latter case the hierarchy prob-lem is understood: in most superconductors the tran-sition temperature T c is exponentially small, comparedto the characteristic Fermi energy scale (analog of thePlanck scale), which forces us to search for the excep-tional materials with enhanced T c . The role of the Lif-shitz transition in the enhancement of the temperature ofsuperconducting transition is in Sec. II D and Sec. III C.In the other scenario, the massless (gapless) vacuaemerge at the Lifshits transition between the fully gappedvacua with different topological charges, see Sec.VI andFig. 18. The almost perfect masslessness of elementaryparticles in our Universe suggests that the Universe isvery close to the line of the topological Lifshitz transitionbetween the fully gapped vacua, at which fermions nec-essarily become gapless, see Sec. VI. This is the topo- a r X i v : . [ c ond - m a t . o t h e r] S e p Fermi surface as vortex ring in p-space Weyl conical point: hedgehog in p-space ΔΦ = π p x p F p y ( p z ) ω Fermisurface ε < 0 occupied levels:Fermi sea ε > 0 emptylevels π i tr [ dl G l G -1 ] N = o p x E p y ( p z ) p z E p x over 2D surface Sin 3D p-space N = 1 e ijk ∫ dS k g . ( ∂ pi g x ∂ pj g ) Dirac line (Dirac point in 2D) p x p y p z p x p y p z N =1 π i tr [ K ∫ dl H -1 ∂ l H ] N = o ε = 0 FIG. 1: Topologically stable nodes in the energy spectrumof electrons in metals or fermions in general.(
Top ): Fermi surface represents the singularity in the Green’sfunction, which forms the vortex in the 3 + 1 ( p , ω )-space, seeSec. II A and Fig. 2 (in the 2 + 1 ( p x , p y , ω )-space this is thevortex line). The stability of the vortex is supported by thewinding number – the integer-valued invariant N , expressedin terms of the Green’s function. Lifshitz transitions, whichinvolve the Fermi surface, are discussed in Secs. II and IV.( Middle ): Conical point in the fermionic spectrum of the Weylmaterials (Weyl semimetals, chiral superfluid He-A, and thevacuum of Standard Model in its gapless phase, see Sec. IIIand Fig. 9). The directions of spin (or of the emergent spin,isospin, pseudo-spin, etc.) form the topological object in mo-mentum space – the hedgehog or the Berry phase monopole –described by the integer-valued topological invariant N . Lif-shitz transitions, which involve the Weyl nodes, are discussedin Secs. III, IV and VI.( Bottom ): Dirac lines – lines of zeroes in the energy spec-trum, described by the topological invariant N . The circularline is the Dirac line in the quasiparicle spectrum in the polarphase of superfluid He, which has been recently created inaerogel. The same invariant N stabilizes the point nodes in2D materials, such as graphene, see Sec.V. logical analog of the so-called Multiple Point Principle,according to which the Universe lives at the coexistencepoint (line, surface, etc.) of the first order phase transi-tion, where different vacua have the same energy. II. FERMI SURFACE AND LIFSHITZTRANSITIONSA. Fermi surface as topological object
The primary topology, which is at the origin of the Lif-shitz transitions, is the topology which is reponsible forthe stability of the Fermi surface. If the Fermi surfaceis not stable under the electron-electron interaction, theconsideration of the topology of the shape of the Fermisurface and of the corresponding Lifshitz transitions does
Fermi surface:vortex ring in p -space ΔΦ = π p x p F p y ( p z ) ω FermisurfaceEnergy spectrum ofnon-interacting gasof fermionic atoms ε < 0 occupied levels:Fermi sea p=p F phase of Green's functionGreen's function ε > 0 empty levels G ( ω , p ) = | G | e i Φ G -1 = i ω − ε ( p ) has winding number N = 1 ε ( p ) = p m – μ = p m – p F m π i tr [ dl G l G -1 ] N = o ε = 0 FIG. 2: Fermi surface is robust to interactions because itrepresents the topologically stable singularity in the Green’sfunction – the vortex the 3 + 1 ( p , ω )-space. The stability ofthe vortex is supported by the winding number of the phaseΦ of the Green’s function G = | G | e i Φ . In general the windingnumber is given by the integer-valued invariant N , expressedin terms of the Green’s function. not make much sense. To view the topological stabilityof the Fermi surface with respect to interactions let usstart with the Green’s function of an ideal Fermi gas inFig. 2 ( left ). The Fermi surface (cid:15) ( p ) = 0 of the nonin-teracting Fermi gas is the boundary in momentum space,which separates the occupied states with (cid:15) ( p ) < (cid:15) ( p ) >
0. The Green’s function G ( ω, p ) with ω on imaginary axis G − ( ω, p ) = iω − (cid:15) ( p ) , (1)has singularity at ω = 0 and (cid:15) ( p ) = 0. In Fig. 2( right ) the p z coordinate is suppressed, and the Green’sfunction singularity forms the closed line in the 2 + 1momentum-frequency space ( p x , p y , ω ). This line repre-sents the vortex line, at which the phase of the Green’sfunction Φ( p x , p y , ω ) has the 2 π winding. As in the caseof the real-space vortex in superfluids, the integer wind-ing number provides the stability of the Fermi surfacewith respect to perturbations, including the interaction(if the p z component is restored, the singularity formsthe vortex sheet in the 3 + 1 momentum-frequency ( p , ω )space).In general case, when the Green’s function has the spin,band and other indices, the winding number can be writ-ten as the following topological invariant in terms of theGreen’s function: N = tr (cid:73) C dl πi G ( ω, p ) ∂ l G − ( ω, p ) . (2)Here the integral is taken over an arbitrary contour C around the momentum-frequency vortex sheet, and tr isthe trace over all the indices.Due to topological stability one cannot make a hole inthe Fermi surface. As in the case of vortex lines, whichcannot terminate in bulk, the Fermi surface has no edges. Fermi surfacein normal 3Heflow Fermi surfacein supercritical 3He-BFermi surfacein supercritical 3He-B v > v Landau
FIG. 3: Lifshitz transition at which the closed Fermi surfacesappear in the fully gapped superfluid He-B, when the flowvelocity of the liquid with respect to the walls of containerexceeds Landau critical velocity.
B. Fermi surface Lifshitz transitions
Because of the topological stability the Fermi surfacemay be formed even in the superconducting state. Theconditions for that are multi-band structure and brokentime reversal symmetry T and parity P . These theso-called Bogolubov Fermi surfaces appear also in gaplesssuperfluids, when the Weyl points in He-A and Diracnodal line in the polar phase of He are inflated to theFermi pockets in the presence of superflow, which violatesboth T and P symmetries. The Fermi surface can be also formed in the fullygapped superfluids, if the velocity of superflow exceedsLandau critical velocity.
The crossing of Landau ve-locity with formation of closed Bogoliubov Fermi surfacesis an example of one of the two transtions suggested byLifshitz, see Fig. 3.Another original Lifshitz transition takes place whenthe Fermi surface crosses the stationary point of the elec-tronic spectrum. Near the transition the expansion of thegeneric spectrum has the form: (cid:15) p = ap x + bp y + cp z − µ . (3)For a > b > c < µ = 0 is inFig. 4 ( top ). In terms of the vortex singularities of theGreen’s function in 3+1 ( p , ω ) space, this Lifshitz tran-sition represents the interconnection of the vortex linesin Fig. 4 ( bottom ). In superfluids, the interconnectionof the real-space vortices is an important process in thevortex turbulence. C. From pole of Green’s function to zero
While in conventional Landau Fermi liquid the Green’sfunction has a pole, for Luttinger liquid the resudue of μ > 0 μ = 0 μ < 0 p x p z p y N = 1 N = − 1 FIG. 4: Since the Fermi surface represents the vortex in the3+1 ( p , ω ) space, the Lifshits transition with disruption ofthe neck of the Fermi surface ( top ) is equivalent to the inter-connection of vortices in quantum turbulence ( bottom ). From pole of Green’s function to zero
Fermi surface:vortex ring in p -space p x p F p y ( p z ) ω non-interacting fermions strongly interacting fermions ε < 0 occupied levels:Fermi sea zeroes zeroespolespoles polespole i ω − ε ( p ) G = π i tr [ dl G l G -1 ] = 1 N = o G = Z ( i ω + ε ( p ) ) zero FIG. 5: Lifshitz transition at which the whole Fermi surface ofthe poles in the Green’s function or part of the Fermi surfacetransforms to the surface of zeroes in the Green’s function.The topological charge of the surface does not change at thisquantum phase transition. As a result the Luttinger theo-rem remains valid, i.e. the particle density of interact-ing fermions is equal to the volume in the momentum spaceenclosed by the singular surface with the topological charge N = 1. the pole in the Green’s function has singularity: the pa-rameter γ in Eq.(5) is nonzero: G = Ziω − (cid:15) ( p ) , Z ∝ (cid:0) ω + (cid:15) ( p ) (cid:1) γ . (4)Nevertheless the topological invariant remains the samefor all γ : i.e. the Green’s function has the same topo-logical property as the Green’s function of conventionalmetal with Fermi surface at (cid:15) ( p ) = 0. This is the reasonwhy the Luttinger theorem is still valid. The parti-cle density of interacting fermions is equal to the volumein the momentum space enclosed by the singular surface
Lifshitz transition from Fermi surface to flat band pp p flat bandtwo solutions: δ n ( p )=0 & ε ( p ) = 0 δ n ( p )=0 δ n ( p )=0 weak interaction:Fermi surface (Landau Fermi-liquid) strong interaction:Fermi ball (flat band) p ε ( p ) ε ( p ) n ( p ) n ( p ) p F E { n ( p ) } δ n ( p )=0 δ n ( p )=0 ε ( p ) = 0 ε ( p ) = 0 δ E { n ( p ) } = ε ( p ) δ n ( p ) d d p = 0 FIG. 6: Formation of flat band in the system of stronlgly in-teracting fermions in the Landau theory approach. There areto types of the extrema of the energy functional E { n ( p ) } : (i) δn ( p ) = 0, which corresponds to the occupied n ( p ) = 1 andfree n ( p ) = 0 levels. (ii) (cid:15) ( p ) = 0; this solution takes placewhen 0 < n ( p ) <
1. On the weak interaction side, the solu-tion (i) takes place inside and the outside the Fermi surface,while the solution (ii) corresponds to the Fermi surface. Onthe strong interaction side the solution (cid:15) ( p ) = 0 is extendedto the 3D band – the flat band. Formation of the flat bandfrom the Fermi surface occurs via the new type of Lifshitztransition. with the topological charge N = 1, irrespective of therealization of the singularity.The suppression of residue Z can be so strong, that thepole in the Green’s function is transformed to the zeroof the Green’s function, which corresponds to the specialcase of γ = 1, see Fig. 5: G ∝ iω + (cid:15) ( p ) . (5)This situation in particular takes place for Mottinsulators, which means that the. topology of Fermisurface is preserved even in the insulating phase, andthus the Luttinger theorem is still valid. Thus wecan say that the transition between metals and insula-tors can be also viewed as a type of the zero-temperatureLifshitz transition, at which the property of the energyspectrum drastically changes without symmetry break-ing. However, this quantum phase transition is not topo-logical since the topological invariant does not changeacross the transition.It is not excluded that in the so-called pseudo-gapphase of cuprate superconductors and some other mate-rials, see e.g. , the part of the Fermi surface transformsto the surface of zeroes and the Fermi arcs are formed,see Fig. 5 ( bottom right ). D. From Fermi surface to flat band
The flat band – or the so-called Khodel-Shaginyanfermion condensate, where all the states have zero en-
Flat band - route to room-T superconductivity T c ∼ Δ = E c exp [-1/ gN (0) ] T c ∼ Δ = gV FB normal superconductors:exponentially suppressedtransition temperaturegap equationconventional metal with Fermi surface system with flat band g - coupling in Cooper channel f lat band superconductivity:linear dependenceof T c on coupling W couplingcoupling DOS flat band volumeflat bandvolume Δ = g h d p E ( p ) ΔΔ = g 2 h d p ε ( p ) = v F ( p − p F ) ε ( p ) = 0 E ( p ) = Δ + v F2 ( p − p F ) E ( p ) = Δ + ε ( p ) E ( p ) = Δ gN (0) E ( ε ) d ε = gN (0) ln Δ E c = g V FB in flat band FIG. 7: Flat band of electronic states with zero energy leadsto the linear dependence of transition temperature T c on thecoupling parameter, while in conventional metals T c is ex-ponentially suppressed. flat band ε ( p ) = 0 p x p y FIG. 8: Flat band emerging in the vicinity of Lifshitz tran-sition in Fig. 4 due to electron-electron interaction.
Theoriginal non-interacting spectrum is (cid:15) (0) ( p ) = p x p y /m − µ ,where the Lifshitz transition takes place at µ = 0. Due tointeraction all the states in the black region have zero energy(the flat band is shown at the point of Lifshitz transition). ergy – is caused by electron-electron interaction. This is the manifestation of the general phenomenon ofmerging of the energy level due to interaction. Such ef-fect has been observed for Landau levels in 2D quantumwells.
Since the flat band has huge density of elec-tronic states, this may considerably enhance the transi-tion temperature to superconducting state, see Fig. 7.The flat band is more easily formed in the vicinityof the conventional Lifshitz transition, see Fig. 8.Flattening of the single-particle spectrum near the Fermimomentum has been reported in 2D quantum well. Itis possible that this effect is responsible for the occur-rence of superconductivity with high T observed in thepressurized sulfur hydride: there are some theoreticalevidences that the high- T c superconductivity takes placeat such pressure, when the system is close to the Lifshitz Berry phaseDirac string(unobservable) DiracmonopoleinmomentumspaceBerry ‘magnetic’ field H ( p ) = p p ^ Φ γ ( C ) = Φ =2 π C FIG. 9: Spins of the right-handed Weyl particles (quarksand leptons) are directed along the momentum p formingthe hedgehog ( top left ). The anti-hedgehog – the hedgehogwith spines (spins) inward ( top right ) corresponds to the left-handed Weyl particles. The topological invariant N describ-ing the topologically distinct hedgehog configurations is ex-pressed either in terms of the Green’s function in Eq.(6) or interms of the unit vector field in Fig. 1 ( center ). transition. Enhanced superconductivity at Lifshitztransition has been reported in FeSe monolayer. III. LIFSHITZ TRANSITIONS GOVERNED BYWEYL POINT TOPOLOGYA. Topology of Weyl fermions
Weyl particles are the elementary particles of our Uni-verse. The Weyl spinor contains 2 complex components,and these massless particles are described by the 2 × H = c σ · p for right-handed quarksand leptons and H = − c σ · p for the left-handed parti-cles, where c is the speed of light. Their spins in mo-mentum space form correspondingly the hedgehog andanti-hedgehog in Fig. 9 ( top ). The hedgehog is the topo-logically stable object, and thus the Weyl point in thecenter of the hedgehog is topologically protected. Thecorresponding topological invariant for the hedgehogs, N , can be expressed in terms of the Green’s functionas a surface integral in the 3+1 momentum-frequencyspace p µ = ( p , ω ): N = (cid:15) µνρσ π tr (cid:73) Σ a dS σ G ∂∂p µ G − G ∂∂p ν G − G ∂∂p ρ G − . (6)Here Σ a is a three-dimensional surface around the iso-lated Weyl point in ( p , ω ) space.From the point of view of the general properties of Dirac point Weyl points p x , p y p z E E p z p z p x , p y E N = 0 N = −1 N = +1 Massive Dirac
FIG. 10: Formation of the pair of Weyl points from the vac-uum state with massive Dirac fermions. Topological charges N = +1 and N = − N = 0. the fermionic spectrum, the Weyl point represents theexceptional point of level crossing analyzed by von Neu-mann and Wigner . This analysis demonstrates thattwo branches of spectrum, which have the same sym-metry, may touch each other at the conical (or diaboli-cal) point in the three-dimensional space of parameters,which in our case are p x , p y and p z . The touching of twobranches is described in general by 2 × H = σ · g ( p ). The topological invariant N is expressedin terms of the unit vector ˆ g ( p ) = g ( p ) / | g ( p ) | in Fig.1 ( center ), which forms the hedgehog configuration inFig. 1 ( middle right ). The touching point also repre-sents the Berry phase monopole in Fig. 9 ( bottom ). It isnot excluded that the Weyl fermions of Standard Model(quark and leptons) are not the elementary particles, butemerge from the level crossing at the more fundamentallevel.
In particular, the underlying quantum vacuumcan be described by quantum field theory based on realnumbers (Majorana fermions), while the imaginary unit,which enters Schr¨odinger equation, emerges in the low en-ergy limit together with the relativistic linear spectrumof Weyl fermions. The linear (”relativistic”) spectrum emerges only forthe elementary topological charges, N = +1 or N = −
1. If the Weyl point has higher topological charge, | N | >
1, and if there is no special symmetry, which leadsto the degeneracy of the levels, the spectrum has differentdispersion relations along different axes. . For exam-ple, for | N | = 2 the spectrum is ”relativistic” in onedirection and quadratic in the other two directions. B. Lifshitz transition with splitting of Weyl points
The typical Lifshitz transition, which involves the Weylnodes in the fermionic spectrum, describes the formationof the Weyl points with opposite charges N = ± gapless spectrum in weak coupling regime of BCSeight Weyl pointsLifshits quantum phase transitionfully gapped spectrumin strong couplinggap μ gap μ = 0 N = +1 N = −1 FIG. 11: The BEC to BCS Lifshitz transition with formationof 4 right-handed and 4 left-handed Weyl points at vertices ofcube. Such arrangement of the Weyl nodes has been discussedfor the energy spectrum in superconductors, which belong tothe O ( D ) symmetry class. In relativistic theories a similaarrangement gives 8 left and 8 right Weyl fermions on thevertices of the cube in the 3+1 ( p x , p y , p z , ω ) space. topological charge N = 0. Such gapless Dirac point ismarginal, but can be protected by symmetry as this takesplace in Standard Model above the electroweak transi-tion. If the symmetry is violated or is spontaneouslybroken the Dirac spectrum either acquires mass or splitsinto the pair of Weyl points. Fig. 11 demonstrates the formation of 4 right-handedand 4 left-handed Weyl points at the Lifshitz transitionbetween the BEC strong coupling regime to the BCSweak coupling regime. Such arrangement of the Weylnodes takes place in the energy spectrum in the O ( D )symmetry class of the pair correlated systems. Inboth cases the total topological charge N (total) = 0,and thus there is an even number of Weyl fermions, whichsupports the fermion doubling principle. In relativistictheories the analogical arrangement of 8 left and 8 rightWeyl fermions on the vertices of the cube in the 3+1( p x , p y , p z , ω ) space has been discussed. It is interest-ing that each family of Standard Model fermions contains8 left and 8 right Weyl particles.
C. Lifshitz transition to type-II Weyl cone
There is the type of Lifshitz transition, which involvesboth the Fermi surface (invariant N ) and the Weyl point(invariant N ). This is the transition between the iso-lated Fermi points (type-I Weyl spectrum) to the Weylpoint connecting two Fermi surfaces (this is called thetype-II Weyl point ). In relativistic theories such tran-sitions have been discussed in .The simplest realization of the type-II Weyl pointcomes from the following Hamiltonian with two parame-ters c and v : H = c σ · p − vp z . (7)For v = 0 this is the Weyl point with the Weyl cone inFig. 12 ( top left ). For 0 < v < c the cone is tilted. At v > c the cone is overtilted, so that the cones cross the Lifshitz transition type-IIWeyl κκ = 1 type-IWeyl
Δ / 2Ω E = 0 E = 0type-I Weyl cone type-II Weyl cone FIG. 12: ( top left ): Weyl cone in the spectrum of type-I Weylfermions. ( top right ): overtilted Weyl cone in the spectrum oftype-II Weyl fermions. ( bottom ): Enhancement of supercon-ducting transition temperature T c at the Lifshitz transitionbetween the type-I Weyl and the type-II Weyl points. zero energy level forming two Fermi pockets connectedby the Weyl point – the type-II Weyl point, see Fig. 12( top right ). The Lifshitz transition between two types ofWeyl point occurs at v = c . It is demonstrated that suchLifshitz transition also leads to the enhancement of thetransition temperature to superconducting state, seeFig. 12 ( bottom ). D. Lifshitz transition at the black hole horizon
Lifshitz transition discussed in Sec. 12 takes place atthe black hole horizon. In general relativity the sta-tionary metric, which is valid both outside and insidethe black hole horizon, is provided in particular by thePainlev´e-Gullstrand spacetime. The line element of thePainlev´e-Gullstrand metric is equivalent to the so-calledacoustic metric: ds = g µν dx µ dx ν = − c dt + ( d r − v dt ) . (8)This metric is expressed in terms of the velocity field v ( r )describing the frame dragging in the gravitational field: v ( r ) = − ˆ r c (cid:114) r h r , r h = 2 M Gc . (9) event horizon v = 0 overtilted light conecorresponds to overtilted Weyl cone(Fermi pockets) v > c v = c v(r) < c black hole v(r h )=c r = r h v(r) > c g μν p μ p ν = ( E − p . v ) − c p = p r Fermi seaoccupied levels E ( p ) < 0 E ( p ) > 0 p ⊥ FIG. 13: Black hole in the Painlev´e-Gullstrand metric. Themetric g µν describes the light cone. The light cone ( top left ) isovertilted behind the horizon, where the frame drag velocity v > c , ( top right ). The metric g µν describes the Weyl cone.The Weyl cone is overtilted behind the horizon forming twoFermi pockets connected by type-II Weyl point ( bottom ). Thehorizon ar r = r h serves as the surface of the Lifshitz transi-tion between the type-I Weyl point at r > r h and the type-IIWeyl point at r < r h . This behavior of two cones allow us tosimulate the black hole horizon and Hawking radiation usingWeyl semimetals. Here M is the mass of the black hole; r h is the radiusof the horizon; G is the Newton gravitational constant.Behind the horizon the drag velocity exceeds the speedof light, | v | > c , and particles are trapped in the hole, seeFig. 13 ( top right ). The behavior of the light cone (thecone in spacetime) across the event horizon is shown inFig. 13 ( top left ). The light cone is overtilted behind thehorizon.The behavior of the Weyl cone (the cone in momentumspace) across the horizon is decsribed by Hamiltonian ofthe Weyl particles in the gravitational field of the blackhole, which for the Painlev´e-Gullstrand metric has thefollowing form: H = ± c σ · p − p r v ( r ) , v ( r ) = c (cid:114) r h r . (10)Here the plus and minus signs correspond to the righthanded and left handed Weyl fermions respectively; p r is the radial component of the linear momentum of theparticle. Behind the horizon, where v > c and the lightcone is overtilted, the Weyl cone is also overtilted, butin the way shown in Fig. 12. Two Fermi pockets areformed, which touch each other at type-II Weyl point inFig. 13 ( bottom right ). The event horizon at r = r h thusserves as the surface of the Lifshitz transition.The correspondence between Weyl semimetals andblack holes allows us to simulate the black hole hori-zon using the inhomogeneous Weyl semimetal, where thetransition between the type-I and type-II Weyl pointstakes place at some surface. This surface would playthe role of the event horizon. The formed black hole willbe fully stationary in equilibrium. However, just aftercreation of this black hole analog, the system is not in
E<0 occupied levels:Fermi sea E > 0 Fermi surface N = 1 localtopological charge E = 0 Weylpoint Fermisurface p E π i tr [ dl G l G -1 ] N = over S3 (cid:47) N = 1 e μνλ tr dV G μ G -1 G ν G -1 G λ G -1 o σ ( p ) || p hedgehog of spins N =+1 globaltopological charge FIG. 14: Fermi surface with local topological charge N andthe global topological charge N . It contains the Berry phasemonopole. the equilibrium state, and the relaxation process at theinitial stage of equilibration looks similar to the processof the Hawking radiation.In the discussed Lifshitz transition between type-I andtype-II Weyl points, the element g of the effective met-ric changes sign. Another type of Lifshitz transitionoccurs when the element g changes sign. In Weylsemimetals this corresponds to the transition to the type-III Weyl fermions, while in general relativity this is thetransition to spacetimes with closed timelike curves. IV. LIFSHITZ TRANSITIONS WITH SEVERALTOPOLOGICAL CHARGES
In sections III C and III D we considered the Lifshitztransition which involved two topological charges: thecharge N , which characterizes the Fermi surface, andthe charge N of the Berry phase monopole. There arethe other Lifshitz transitions with the interplay of thesetwo topological invariants. This happens in particular,when the closed Fermi surface is described by two in-variants: the local charge N , which provides the localstability of the Fermi surface, and the global charge N ,which describes the Weyl point inside the Fermi surfacein Fig. 14. The latter takes place for example when theWeyl point shifts from the zero energy position formingthe small Fermi sphere around the Weyl point, see Fig. 14( left ). Such Fermi sphere contains the N charge, whichcan be obtained from Eq.(6) by integration over the sur-face, which encloses the Fermi sphere.At the Lifshitz transition the Fermi surfaces can ex-change their global charges N or loose the globalcharge. Example of exchange is in Fig. 15 and theexample of the lost global charge is in Fig. 16. In bothcases the intermediate state at the point of Lifshitz tran-sition contains the type-II Weyl points.
V. LIFSHITZ TRANSITION GOVERNED BYCONSERVATION OF N CHARGE
The conical Dirac point in the 2D graphene and thenodal lines in the 3D semimetals and nodal superflu-ids and superconductors are stabilized by the topologi- N = +1 N = − FIG. 15: Lifshitz transition with exchange of Berry phasemonopoles between two Fermi surfaces. The red and the blueFermi surfaces are globally non-trivial, with N = − N = 1 respectively. In the process of the Lifshitz tramstion,two Berry phase monopoles are pushed out from the Fermisurfaces. At the Lifshitz transition the Fermi surfaces toucheach other at the Weyl points, so that the latter become thetype-II Weyl poins. Above the transition the Weyl pointsare again inside the Fermi surfaces, but now red and the blueFermi surfaces have the global charges N = +1 and N = − cal charge N in Fig. 1 ( bottom ). Dirac nodal lineswere known to exist in the polar phase of superfluid He , in cuprate superconductors, and in graphite(band crossing lines). Now they are extensively stud-ied in semimetals. The type of Lifshitz transitions governed by the con-servation of the topological charge N is shown in Fig.17 on example of bilayer graphene, when one graphenelayer is shifted with respect to the other one. Mergingof the two conical points with N = 1 leads to formationof the Dirac node with quadratic dispersion in Fig. 17( top left ), which has the toplogical charge N = 2. Thispoint in turn may split into four Dirac conical points with N = ± bottom ). This is the so-called trig-onal warping. The total topological charge is conserved, N = 1 + 1 + 1 − top right ), andthe transition occurs as a function of p z , when p z crosses the so-called nexus point. VI. LIFSHITZ TRANSITIONS BETWEENGAPPED STATES VIA GAPLESS STATE
Lifshitz transitions between gapped states includetransitions between the topological and non-topologicalinsulators; transitions between the fully gapped super-fluids/superconductors; transitions between the 2D sys-tems, which experience the intrinsic quantum Hall effect;etc. Here we consider such transition on example of the2D systems, where the Hall conductance is expressed interms of the integer-valued topological invariant ˜ N inFig. 18 ( top right ). This topological invariant has thesame structure as the invariant N in Fig. 1 ( Middle ), N = +1 FIG. 16: Lifshitz transition, at which the Fermi surfacesloose the Weyl charge N . Below the Lifshitz transitionboth surfaces contain the same Berry phase monopole with N = +1. At the transition the Weyl point connects the innerand outer Fermi surfaces. Above the transition the monopolecomes out from the Fermi surfaces, and both Fermi surfacesbecome globally trivial, with N = 0. Without global stabilitythe Fermi surface may shrink and disappear in conventionalLifshitz transition as it happens with the red Fermi surface inFig. 16 ( bottom right ). but the integration now is over the whole 2D Brillouinzone. This is an example of the dimensional reductionfrom the 3D systems with Weyl nodes to the 2D topo-logical insulators. Fig. 18 ( top left ) demonstrates the Lifshitz transi-tion between the topological insulator with ˜ N = 1 andthe trivial insulator with ˜ N = 0. Here the topolog-ical charge is not conserved across the Lifshitz transi-tion, but abrubtly changes recalling the first order phasetransition. Nevertheless the transition occurs smoothly, p y p x E N = +2 p x p y N = − N = +1 N = +1 N = +1 N = +1 + 1 + 1 −1 = +2 quadratic touchingbilayer graphenetrigonal warping Bernal graphite FIG. 17: Lifshitz transition governed by conservation ofthe topological charge N in bilayer graphene. In bilayergraphene two conical points with the same charge N = 1on two graphene layers are either merge to form the Diracpoint with topological charge N = 2 with quadratic spec-trum in Fig.17 ( top left ) or split into four Dirac conical pointsin Fig.17 ( bottom ). The latter is called the trigonal warping.The total topological charge N = 2 in both case and thusone configuration may transform to the other one by Lifshitztransition. topologicalinsulator trivialinsulator gap gap skyrmion in momentum space at μ > 0 topological invariant in p space over2D Brillouin zone N ( μ > 0 ) = 1 ~ π N = 1 ∫ d p g . ( ∂ px g x ∂ py g ) ~ g ( p x ,p y ) unit vectorsweeps unit sphere p y p x g ^ N = 1 ~ N ~ N = 0 ~ μ topological Lifshitz transitionvia gapless state μ = 0 FIG. 18: Lifshitz transition between the fully gapped topo-logically different vacua in 2D systems, which experiences theintrinsic quantum Hall effect.
The Hall conductance isexpressed in terms of the integer-valued topological invariant˜ N ( top right ). The Lifshitz transition between the topologicalinsulator with ˜ N = 1 and the trivial insulator with ˜ N = 0takes place through the state, where the gap vanishes ( topleft ).( Bottom ): The topologically nontrivial state with ˜ N = 1represents the topologically nontrivial nonsingular object –skyrmion – in the 2D momentum space. because at the point of transition the gap in the en-ergy spectrum vanishes. The nullification of the gapat the transition reflects the fact that in the 3D space( p x , p y , µ ), where µ is chemical potential or some otherparameter along which the transition occurs, the gapnode represents the Weyl point with topological charge N = ˜ N (right) − ˜ N (left). Example is the 2D p x + ip y superfluid/superconductor, where the Lifshitz tran-sition between the superfluid states with ˜ N = 1 and˜ N = 0 occurs at the same point µ = 0 as in the normalFermi liquid. The detailed consideration shows that theLifshitz transition represents the quantum transitionof third order: the third-order derivative d E/dg ofthe ground state energy E over the interaction strength g is discontinuous. Compare this with the original 2 order transition, and the 3 order transition discussedrecently. The nullification of the gap in the fermionic spectrumat the transition between the gapped vacua, suggests thescenario for the solution of the hierarchy problem: therelativistic quantum vacuum is almost massless becauseour Universe is very close to the line of the Lifshitz tran-sition. The reason why nature would prefer the criticalline may be that the gapless states on the transition lineare able to accommodate more entropy than the gappedstates. VII. CONCLUSION
Topological Lifshitz transitions are ubiquitous, sincethey involve many types of the topological structure offermionic spectrum: Fermi surfaces, Dirac lines, Diracand Weyl points, edge states, Majorana zero modes, etc.Each of these structures has their own topological in-variant, such as N , N , N , ˜ N , etc., which supportsthe stability of a given class of the topological structure.The topology of the shape of the Fermi surfaces and theDirac lines, as well as the interconnection of the objects ofdifferent dimensionalities in momentum and frequency-momentum spaces, lead to numerous classes of Lifshitztransitions.The consequences of Lifshitz transitions are importantin different areas of physics. In particular, the singulardensity of electronic states emerging at the transition isimportant for the construction of superconductors withenhanced transition temperature; the Lifshitz transitioncan be in the origin of the small masses of elementaryparticles in our Universe; the black hole horizon servesas the surface of Lifshitz transition between the vacuawith type-I and type-II Weyl points; etc. Acknowledgements
The work has been supported by the European Re-search Council (ERC) under the European Union’s Hori-0zon 2020 research and innovation programme (GrantAgreement No. 694248) and by RSCF (No. 16-42- 01100). I.M. Lifshitz, Anomalies of electron characteristics of ametal in the high pressure region, Sov. Phys. JETP ,1130 (1960). P. Hoˇrava, Stability of Fermi surfaces and K -theory, Phys.Rev. Lett. , 016405 (2005). G.E. Volovik,
The Universe in a Helium Droplet , Claren-don Press, Oxford (2003). G.E. Volovik, Quantum phase transitions from topology inmomentum space, Springer Lecture Notes in Physics ,31–73 (2007); cond-mat/0601372. G.E. Volovik, Topological Lifshitz transitions, FizikaNizkikh Temperatur , 57–67 (2017), arXiv:1606.08318. Kuang Zhang and G.E. Volovik, Lifshitz transitions viathe type-II Dirac and type-II Weyl points, Pis’ma ZhETF , 504–505 (2017), JETP Lett. , 519–525 (2017). G.E. Volovik, Zeros in the fermion spectrum in super-fluid systems as diabolical points, JETP Lett. , 98–102(1987). V.V. Dmitriev, A.A. Senin, A.A. Soldatov, and A.N.Yudin, Polar phase of superfluid He in anisotropic aerogel,Phys. Rev. Lett. , 165304 (2015). C.D. Froggatt and H.B. Nielsen,
Origin of Symmetry ,World Scientific, Singapore, 1991. G.E. Volovik, Topological invariants for Standard Model:from semi-metal to topological insulator, Pis’ma ZhETF , 61–67 (2010); JETP Lett. , 55–61 (2010);arXiv:0912.0502. B.G. Sidharth, A. Das, C.R. Das, L.V. Laperashvili andH.B. Nielsen, Topological structure of the vacuum, cosmo-logical constant and dark energy, Int. J. Mod. Phys. A ,1630051 (2016), arXiv:1605.01169. B.G. Sidharth, A. Das, C.R. Das, L.V. Laperashvili andH.B. Nielsen, Cosmological constant and the vacuum sta-bility in the Standard Model, New Advances in Physics ,1–39 (2016). L.V. Laperashvili, H.B. Nielsen and C.R. Das, New re-sults at LHC confirming the vacuum stability and MultiplePoint Principle, Int. J. Mod. Phys. A , 1650029 (2016). D.L. Bennett, H.B. Nielsen and C.D. Froggatt, Standardmodel parameters from the multiple point principle andanti-GUT, arXiv:hep-ph/9710407. G.E. Volovik, Coexistence of different vacua in the effec-tive quantum field theory and multiple point principle,JETP Lett. , 101 (2004), Pisma ZhETF , 131 (2004),arXiv:hep-ph/0309144. G.E. Volovik, Zeroes in energy gap in superconductors withhigh transition temperature, Phys. Lett. A , 282–284(1989). W.V. Liu and F. Wilczek, Interior gap superfluidity, Phys.Rev. Lett. , 047002 (2003). V. Barzykin and L.P. Gor’kov, Gapless Fermi surfaces insuperconducting CeCoIn , Phys. Rev. B , 014509 (2007) D.F. Agterberg, P.M.R. Brydon, C. Timm, BogoliubovFermi surfaces in superconductors with broken time-reversal symmetry, Phys. Rev. Lett. , 127001 (2017). C. Timm, A.P. Schnyder, D.F. Agterberg, and P.M.R. Bry-don, Inflated nodes and surface states in superconducting half-Heusler compounds, arXiv:1707.02739. G.E. Volovik, Superfluid properties of the A-phase of He,Usp. Fiz. Nauk. , 73–109; Soviet Phys. Usp. , 363–384. J.T. M¨akinen, S. Autti, V.B. Eltsov, J. Rysti andG.E. Volovik, Vortex non-dynamics and exceed-ing the Landau speed limit in the polar phaseof superfluid D. Vollhardt, K. Maki and N. Schopohl, Anisotropic gapdistortion due to superflow and the depairing critical cur-rent in superfluid He-B, J. Low Temp. Phys. , 79–92(1980). T. Zhu, M.L. Evans, R.A. Brown, P.M. Walmsley, andA.I. Golov, Interactions between unidirectional quantizedvortex rings, Phys. Rev. Fluids , 044502 (2016). J. Voit, One-dimensional Fermi liquids, Reports onProgress in Physics , 977 (1995), I. Dzyaloshinskii: Some consequences of the Luttinger the-orem: The Luttinger surfaces in non-Fermi liquids andMott insulators, Phys. Rev. Phys. Rev. B , 085113(2003). B. Farid, A.M. Tsvelik, Comment on ”Breakdown of theLuttinger sum rule within the Mott-Hubbard insulator”,by J. Kokalj and P. Prelovsek [Phys. Rev. B , 153103(2008), arXiv:0803.4468], arXiv:0909.2886. U.S. Pracht, N. Bachar, L. Benfatto, G. Deutscher, E. Far-ber, M. Dressel and M. Scheffler, Enhanced Cooper pairingversus suppressed phase coherence shaping the supercon-ducting dome in coupled aluminum nanograins, Phys. Rev.B , 100503(R) (2016). V.A. Khodel and V.R. Shaginyan, Superfluidity in systemwith fermion condensate, JETP Lett. , 553 (1990). G.E. Volovik, A new class of normal Fermi liquids,
JETPLett. , 222 (1991). P. Nozieres, Properties of Fermi liquids with a finite rangeinteraction, J. Phys. (Fr.) , 443–458 (1992). A.A. Shashkin, V.T. Dolgopolov, J.W. Clark, V.R.Shaginyan, M.V. Zverev, V.A. Khodel, Merging of Lan-dau levels in a strongly-interacting two-dimensional elec-tron system in silicon, Phys. Rev. Lett. , 186402 (2014). A.A. Shashkin, V.T. Dolgopolov, J.W. Clark, V.R.Shaginyan, M.V. Zverev, V.A. Khodel, Interaction-induced merging of Landau levels in an electron systemof double quantum wells, JETP Letters , 36 (2015) D. Yudin, D. Hirschmeier, H. Hafermann, O. Eriksson, A.I.Lichtenstein and M.I. Katsnelson, Fermi condensation nearvan Hove singularities within the Hubbard model on thetriangular lattice, Phys. Rev. Lett. , 070403 (2014). G.E. Volovik, On Fermi condensate: near the saddle pointand within the vortex core, Pis’ma ZhETF , 798–802(1994); JETP Lett. , 830–835 (1994). M.Yu. Melnikov, A.A. Shashkin, V.T. Dolgopolov, S.-H.Huang, C.W. Liu, S.V. Kravchenko, Indication of thefermion condensation in a strongly correlated electron sys-tem in SiGe/Si/SiGe quantum wells, arXiv:1604.08527. A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontov,S.I. Shylin, Conventional superconductivity at 203 K athigh pressures, Nature , 73 (2015). M.I. Eremets and A.P. Drozdov, High-temperature conven-tional superconductivity, Phys. Usp. Yundi Quan and Warren E. Pickett, Impact of van Hovesingularities in the strongly coupled high temperature su-perconductor H S, Phys. Rev. B , 104526 (2016). A. Bianconi and T. Jarlborg, Lifshitz transitions and zeropoint lattice fluctuations in sulfur hydride showing nearroom temperature superconductivity, Novel Superconduct-ing Materials , 37–49 (2015); arXiv:1507.01093. T.X.R. Souza, F. Marsiglio, The possible role of vanHove singularities in the high T c of superconducting H S,arXiv:1708.07264. X. Shi, Z.-Q. Han, X.-L. Peng, P. Richard, T. Qian, X.-X.Wu, M.-W. Qiu, S.C. Wang, J.P. Hu, Y.-J. Sun, H. Ding,Enhanced superconductivity accompanying a Lifshitz tran-sition in electron-doped FeSe monolayer, Nature Commu-nications , 14988 (2017). J. von Neumann und E.P. Wigner, ¨Uber das Verhalten vonEigenwerten bei adiabatischen Prozessen, Phys. Zeit. ,467–470 (1929). C.D. Froggatt and H.B. Nielsen,
Origin of Symmetry (World Scientific, Singapore, 1991). G.E. Volovik and M.A. Zubkov, Emergent Weyl spinorsin multi-fermion systems, Nuclear Physics B , 514–538(2014). G.E. Volovik, V.A. Konyshev, Properties of the superfluidsystems with multiple zeros in fermion spectrum, PismaZhETF , 207– 209 (1988); JETP Lett. , 250–254(1988). F.R. Klinkhamer and G.E. Volovik, Emergent CPT viola-tion from the splitting of Fermi points, Int. J. Mod. Phys.A , 2795–2812 (2005); hep-th/0403037. G.E. Volovik, L.P. Gor‘kov, Superconductivity classes inthe heavy fermion systems, JETP , 843–854 (1985). G.E. Volovik, Dirac and Weyl fermions: from Gor’kovequations to Standard Model (in memory of Lev Petro-vich Gorkov), Pis’ma ZhETF , 245–246 (2017), JETPLett. , 273–277 (2017), arXiv:1701.01075. M. Creutz, Four-dimensional graphene and chiral fermions,JHEP 0804:017 (2008). M. Creutz, Emergent spin, Annals Phys. , 21–30(2014). H.B. Nielsen, M. Ninomiya: Absence of neutrinos on alattice. I - Proof by homotopy theory, Nucl. Phys. B ,20 (1981); Absence of neutrinos on a lattice. II - Intuitivehomotopy proof, Nucl. Phys. B , 173 (1981). A.A. Soluyanov, D. Gresch, Zhijun Wang, QuanSheng Wu,M. Troyer, Xi Dai, B.A. Bernevig, Type-II Weyl semimet-als, Nature , 495–498 (2015). P. Huhtala and G.E. Volovik, Fermionic microstates withinPainlev´e-Gullstrand black hole, ZhETF , 995-1003;JETP , 853-861 (2002); gr-qc/0111055. Dingping Li, B. Rosenstein, B.Ya. Shapiro, and I. Shapiro,Effect of the type-I to type-II Weyl semimetal topologicaltransition on superconductivity, Phys. Rev. B , 094513(2017). M. Alidoust, K. Halterman, and A. A. Zyuzin, Supercon-ductivity in type-II Weyl semimetals, Phys. Rev. B ,155124 (2017). P. Painlev´e, La m´ecanique classique et la th´eorie de larelativit´e, C. R. Hebd. Acad. Sci. (Paris) , 677-680 (1921); A. Gullstrand, Allgemeine L¨osung des statischenEink¨orperproblems in der Einsteinschen Gravitations-theorie, Arkiv. Mat. Astron. Fys. , 1-15 (1922). W.G. Unruh, Experimental Black-Hole Evaporation, Phys.Rev. Lett. , 1351 (1981). W.G. Unruh, Sonic analogue of black holes and the effectsof high frequencies on black hole evaporation, Phys. Rev.D , 2827–2838 (1995). P. Kraus and F. Wilczek, Some applications of a simple sta-tionary line element for the Schwarzschild geometry, Mod.Phys. Lett. A , 3713–3719 (1994). G.E. Volovik, Black hole and Hawking radiation by type-IIWeyl fermions, Pisma ZhETF , 660–661 (2016), JETPLett. , 645–648 (2016), arXiv:1610.00521. J. Nissinen and G.E. Volovik, Type-III and IV interactingWeyl points, Pisma ZhETF , 442–443 (2017), JETPLett. , 447–452 (2017), arXiv:1702.04624. T.T. Heikkil¨a and G.E. Volovik, Nexus and Dirac linesin topological materials, New J. Phys. , 093019 (2015),arXiv:1505.03277. S. Autti, V.V. Dmitriev, J.T. M¨akinen, A.A. Soldatov,G.E. Volovik, A.N. Yudin, V.V. Zavjalov, and V.B. Eltsov,Observation of half-quantum vortices in superfluid He,Phys. Rev. Lett. , 255301 (2016). G.P. Mikitik and Yu.V. Sharlai, Dirac points of electronenergy spectrum, band-contact lines, and electron topo-logical transitions of 3 kind in three-dimensional metals,Phys. Rev. B , 155122 (2014). G.P. Mikitik and Yu.V. Sharlai, Band-contact lines in theelectron energy spectrum of graphite, Phys. Rev. B ,235112 (2006). G.P. Mikitik and Yu.V. Sharlai, The Berry phase ingraphene and graphite multilayers, Low Temp. Phys. ,794–780 (2008). D. Takane, K. Nakayama, S. Souma, T. Wada, Y.Okamoto, K. Takenaka, Y. Yamakawa, A. Yamakage, T.Mitsuhashi, K. Horiba, H. Kumigashira, T. Takahashi,and T. Sato, Observation of Dirac-like energy band andring-torus Fermi surface in topological line-node semimetalCaAgAs, arXiv:1708.06874. T. Hyart, T. T. Heikkila, Momentum-space structure ofsurface states in a topological semimetal with a nexus pointof Dirac lines, Phys. Rev. B , 235147 (2016). H. So, Induced topological invariants by lattice fermions inodd dimensions, Prog. Theor. Phys. , 585–593 (1985). K. Ishikawa and T. Matsuyama, Magnetic field inducedmulti component QED in three-dimensions and quantumHall effect, Z. Phys. C , 41–45 (1986). K. Ishikawa and T. Matsuyama, A microscopic theory ofthe quantum Hall effect, Nucl. Phys.
B 280 , 523–548(1987). F.D.M. Haldane, Model for a quantum Hall effect withoutLandau levels: Condensed-matter realization of the ”Par-ity Anomaly”, Phys. Rev. Lett. , 2015–2018 (1988). G.E. Volovik, Analog of quantum Hall effect in superfluid He film, JETP , 1804–1811 (1988). S.M.A. Rombouts, J. Dukelsky and G. Ortiz, Quantumphase diagram of the integrable p x + ip y fermionic super-fluid, Phys. Rev. B , 224510 (2010).76