NNoname manuscript No. (will be inserted by the editor) He Universe 2020
G.E. Volovik
Received: October 1, 2020/ Accepted: date
Abstract
The latest news from He Universe are presented together with theextended map of the Universe.
Contents
The old information on the He Universe can be found in Refs. [1,2,3], Themore recent information is in Refs.[4,5,6,7].Here are the latest news from He Universe 2020.
Superfluid phases of He opened the new area of the application of topologicalmethods to condensed matter systems, see recent reviews in Refs.[5,7].
G.E. VolovikDepartment of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 Aalto, FinlandLandau Institute for Theoretical Physics, acad. Semyonov av., 1a, 142432, Chernogolovka,Russia a r X i v : . [ c ond - m a t . o t h e r] S e p G.E. Volovik
Condensed Matter top. insulatorssemimetalsBEC high-T & chiral super-conductivity blackholesvacuumgravity cosmic strings physicalvacuumneutronstars nuclearphysics hydrodynamicsdisorderphasetransitions High EnergyPhysicsPlasmaPhysics PhenomenologyQCDGravity cosmology intrinsic QHE, spin & FQHE, Chern-Simons, 3D QHEskyrmion & vortex statistics, Hall viscosity, edge statessuperfluidity of neutron star vortices, glitches shear flow instability
Nuclei vs 3He droplet shell model pair-correlations collective modes time crystals & quasicrystals time crystal Josephson effect supersolid, topological Floquet phases quarkmatter quark condensate Nambu--Jona-Lasinio Vaks--Larkin color superfluidity MIT bag model Savvidi vacuum quark confinement, QCD cosmology intrinsic orbital momentum, chiral magnetic & chiral vortical effects velocity independent Reynolds numbergeneral, relativistic, multi-fluid turbulence of vortex lines propagating vortex frontmagnetohydrodynamicsshear flow instabilityrotating superfluidspin superfluidity relativistic plasmaphoton massvortex Coulomb plasma mixture of condensates, vector & spinor BEC of quasipartcles, photon BEC skyrmion, 1/2 vortex, monopole, nexuscosmological constant, q-theory Kibble-Zurek mechanism primordial magnetic field anisotropic dark matterbaryogenesis by textures & stringsinflation, matter creation branes gravitational Aharonov-Bohm effect (Iordanskii force)Kibble-Lazarides-Shafi walls bounded by stringstorsion & spinning strings, torsion instantonfermion zero modes on strings & wallsantigravitating (negative-mass) stringstring terminating on domain wallelectroweak monopole, Q-balls monopoles on string & boojumsWitten superconducting stringnexus, soft core stringZ &W stringsskyrmionsAlice stringPion stringNewton constantdark energy, dark mattereffective gravity, tetradsbi-metric & conformal gravityvacuum dynamics, 4-form fieldconformal anomaly, graviton, dilaton, spin connectionrotating vacuum, Mach principleNieh-Yan anomaly, thermal anomalyantispacetime, Euclidean signature emergence & effective theories, Weyl, Majorana & Dirac fermionsvacuum polarization, screening - antiscreening, running couplingmomentum-space topology, vacuum as topological insulatorsymmetry breaking (anisotropy of vacuum), gauge bosonsfermionic charge of vacuum, 4-form vacuum variablevacuum instability in strong fields, pair productionhierarchy problem, supersymmetry, hidden symmetryHiggs fields, light Higgs, Little Higgs, Nambu sum ruleparity violation, chiral fermions, Planck physicsCasimir forces, quantum friction, neutrino oscillationschiral anomaly, Schwinger pair production, axions spin, isospin, string theory, CPT-violation, SM, GUT 2+1 electrodynamics gap nodeslow -T scalingbroken time reversalmixed state, stripe phaseHiggs amplitude modeschiral anomaly, CME, CVEBogoliubov Fermi surface Khodel-Shaginyan flat band flat band from Dirac nodes flat band room-T superconductivity 1/2-vortex, fractional flux, vortex sheet Andreev-Majorana states on vortex vortex dynamics, spectral flow, Kopnin force ergoregion, event horizonHawking & Unruh effectsblack hole entropy, vacuum instabilitytype-II Weyl fermions behind horizon quantum Lifshitz phase transitions p-space topology random anisotropy Larkin- Imry-Ma states classes of random matrices skyrmion glass, Weyl glass spin & Hopf glass, torsion glass Anderson-Fomin theorem
Abrikosov Universe: Weyl semimetalsChern number, nexus, type II Weyl,higher order topology, Berry monopole, Dirac line, Fermi arc, Lifshitz transition synthetic gauge field & gravity spintronics, magnonics, spinmotive forcespin currents, spin nematic, spin superfluidity skyrmionics, magnon BEC, coherent precession
Fig. 1 He Universe-2020. On the Far East is a connection with the neighboring AbrikosovUniverse, which was also born in 1971 [8,9]. The He Universe-1997 is in Fig. 1 of Ref. [10],and the comparison demonstrates inflating expansion of the He Universe. He are the best representatives of different families oftopological materials. In bulk liquid He there are two topologically differentsuperfluid phases [1]. One is the chiral superfluid He-A with topologicallyprotected Weyl points in the quasiparticle spectrum. In the vicinity of theWeyl points, quasiparticles behave as Weyl fermions moving in the effectivegauge and gravitational fields. Another phase is the fully gapped time reversalinvariant superfluid He-B. It has topologically protected gapless Majoranafermions living on the surface. In He confined in the nematically orderedaerogel, the polar phase of He has been stabilized [11,12,13]. It is the timereversal invariant superfluid, which contains Dirac nodal ring in the fermionicspectrum and flat band on the surface.
He Universe 2020 3 He-A, Weyl fermions, flat band of Majorana fermions onvortices
Chirality of He-A has been probed in the torsional oscillator measurementsthat distinguished between states of opposite chiralities [14,15]. The topologi-cal manifestation of chirality is the separation of left-handed and right-handedWeyl points in momentum space. Due to bulk-vortex correspondence, the sep-aration of the Weyl points leads to the flat band of Majorana fermions livingin the vortex core [16,17]. In topological Weyl semimetals the similar bulk-surface correspondence produces the so-called Fermi arc on the surface of thematerial [18]. He-B, higher order topology
The topological superfluid He-B is the prototype of topological insulators andprovides example of the higher order topology. The boundary of the B-phasecontains 2D gapless Majorana fermions, which are supported by topology andsymmetry of the B-phase. When magnetic field is applied, the time reversalsymmetry of the B-phase is violated. The gapless fermions are not protectedany more by topology: they become massive 2D Dirac fermions. However,the topology is not destroyed completely: the boundary states acquire theirown topology, where the topological charge is determined by the sign of theDirac mass. The line on the boundary, which separates the surface domainswith opposite signs of mass term, contains its own topologically protected 1Dgapless Majorana fermions [19]. The presence of such lines on the boundaryof He-B is seen in NMR experiments [20].Such composite 3D-2D-1D correspondence characterizes the so-called sec-ond order topology [21]. In topological insulators and superconductors thethird order topology is also possible, with 3D-2D-1D-0D correspondence [22].This may lead to the zero dimensionless Majorana modes in the corners ofsuperconductor.
Polar phase is the nodal line superfluid, which is similar to cuprate d -wavesuperconductors and nodal line semimetals. In both systems the Dirac nodalline in the spectrum is supported by topology: the π change of the Berry phasealong the loop around the nodal line. There are several important consequencesof the Dirac line.One of them is the existence of the flat band (or approximate flat band –drumhead states) on the surface due to the bulk-surface correspondence. Thephenomenon of flat band is important for search of room- T superconductivitybecause of the singular density of states [23,24,25]. The transition temperature T c is not exponentially suppressed as in conventional metals, but is the linearfunction of the coupling in the Cooper channel. T c is proportional to the volumeof the flat band, if the flat band is formed in the bulk [23], or to the area of G.E. Volovik the flat band if it is formed on the surface of the sample [24,25]. In nodal linesemimetals the area of the flat band is determined by the projection of thenodal line to the surface of the sample. The largest area is obtained when thenodal lines move to the boundaries of the Brillouin zone, where they canceleach other, and the nodal line semimetal is transformed to the topologicalinsulator with the surface flat band [26].
The other important consequence of the nodal line takes place in superfluidsand superconductors in the presence of supercurrent. Supercurrent violatesparity and time reversal symmetries, as a result the Dirac line in the spectrumof Bogoliubov quasiparticles transforms to the Fermi surface of quasiparticleswithin the superconducting state – the so-called Bogoliubov Fermi surface [27,28,29].In the moving polar phase of He, the Bogoliubov Fermi surface has anexotic shape: it consists of two Fermi pockets which touch each other at twopseudo-Weyl points [30]. In cuprate superconductors, the local BogoliubovFermi surfaces caused by supercurrents around Abrikosov vortices lead to the √ H field dependence of the electronic density of states in the vortex state ofsuperconductor [31].2.2 Quantum anomaliesThe nontrivial topology of the superfluid phases of He lead to different typesof quantum anomalies; chiral anomaly, gravitational anomaly, and mixed anoma-lies.
Chiral anomaly is the anomalous production of fermions from the vacuum,which is described by the Adler-Bell-Jackiw equation [32,33,34]. The anoma-lous non-conservation of the chiral current has been verified in experimentswith vortex-skyrmions in rotating He-A [35]. The anomalous production ofWeyl fermions by moving skyrmions leads to the anomalous production ofthe linear momentum, and thus to the extra force, which acts on the vortex.This spectral-flow or Kopnin force is measured, which allows to extract thefundamental prefactor in the Adler-Bell-Jackiw equation, which depends onlyon symmetry and fermionic content of the quantum vacuum. The measuredinteger number is in the full agreement with the Adler-Bell-Jackiw equationapplied to Weyl fermions in He-A (two left-handed fermions at one Weylpoint and two right-handed fermions at another Weyl point).In Weyl semimetals the manifestation of the chiral anomaly in experimentsis not so spectacular, it leads to negative magnetoresistance, when the mag-netic field is parallel to the current [36].
He Universe 2020 5
Chiral anomaly solves the paradox of the orbital angular momentum inchiral superfluids. The deviation of the orbital momentum from its naturalvalue, L z = (cid:126) νN/ N is the number of particles, and ν is the angularmomentum of Cooper pair) is fully determined by the spectral flow either inbulk or on the surface of the superfluid, see recent papers [37,38]. The similarspectral asymmetry in the vortex core leads to the modification of the angularmomentum of quantized vortices [39].The paradox of the orbital angular momentum in chiral superfluids mayhave something common with the proton spin puzzle, however the presentunderstanding suggests that the chiral anomaly effects are too small to explainthe spin crisis [40].Another phenomenon related to the orbital angular momentum is the Hallviscosity – the non-dissipative response of stress tensor to the velocity gradi-ents. In the Eq.(9.81) of the book [1] six Hall viscosity terms in He-A withcoefficients γ ⊥ , γ (cid:107) and β R a with a = 1 , , , The interplay of quantum anomalies with magnetic field and vorticity re-sults in a variety of non-dissipative transport phenomena in systems withchiral fermions [42]. This is popular for consideration of different effects inthe quarkgluon plasma created in relativistic heavy ion collisions. The ChiralMagnetic Effect (CME) and the Chiral Vortical Effect (CVE) describe thegeneration of non-dissipative electric current along an external magnetic fieldor along the vortex. They are described by the topological quantum numbers,similar to that, which operate in the intrinsic quantum and spin Hall effects.The experimental signature of the CME in He-A is the helical instabilityof the superflow generated by the ChernSimons term, expressed via effectivegauge fields acting on Weyl fermions [43,44], S ∼ (cid:82) µ A · B . This CME islinear in the chiral chemical potential µ . In He-A the effective chiral chemicalpotential is formed by superflow, see Sec. 2.2.3. The corresponding term in thefree energy is linear in the superluid velocity, which leads to instability of flowtowards creation of vortex-skyrmions (continuous doubly quantized vortices).CVE is manifested by the current along the vortex, which is concentrated inthe core of vortex-skyrmion [45]. It is important that the total current is zero:the current along a given vortex in the vortex lattice is compensated either bythe countercurrent in the core of another vortex, or by the countercurrent inthe bulk. This supports the Bloch theorem (see e.g. Ref. [46]), which prohibitsthe total current in the equilibrium state.
G.E. Volovik
Experimental observation of CVE in the A-phase is still waiting in thewings. The same is with the Chiral Separation Effect (CSE), which is dual tothe CME with S ∼ (cid:82) µ A · B , see e.g. [47,48]. The relativistic nature of the Weyl fermions in the A-phase is explicitly mani-fested in the thermal contributions of Weyl fermions to the free energy at lowtemperature, T (cid:28) T c . In particular, three different T terms in the gradientenergy of He at low T can be rewritten in the fully relativistic form: F = T √− g R − T √− gµ − µ T (cid:15) ijk e ia T ajk . (1)Here µ is the chiral chemical potential of Weyl fermions; e ia are tetrads, whichdescribe the effective spacetime in which Weyl fermions are moving, R is theeffective scalar curvature of this spacetime, and T ajk is the analog of torsionfield [49].The first term in Eq.(1) describes the temperature correction to the inverseNewton ”constant” 1 /G in the effective gravity. The last term in Eq.(1) is themanifestation of the Nieh-Yan gravitational anomaly, which is expressed interms of torsion field [50]: ∂ µ j µ = − T (cid:15) µναβ T aµν T aαβ . (2)As distinct from the unknown ultraviolet parameter Λ in the conventionaltorsional Nieh-Yan anomaly [51,52,53] (the ultraviolet parameter for the Nieh-Yan anomaly in He-A see in Refs. [54,55]), the thermal Nieh-Yan term con-tains T and thus is well defined. The prefactor in this term is funamental,being determined by the geometry, topology and number of chiral quantumfields in the system. For the effective quantum relativistic fields in He-A thisparameter is 1 /
48, while the other two parameters in Eq.(1) are 1 / / He language, the chiral chemical potential of Weyl fermions isrepresented by the Doppler shift µ = p F ˆ l · v s , where p F is Fermi momentum,ˆ l is the unit vector along the angular momentum of Cooper pairs, ± p F ˆ l arepositions of two Weyl points and v s is superfluid velocity. The Ricci scalar R and torsion T a in the effective gravity are expressed in terms of the gradientsof the order parameter ( v s and ∇ × ˆ l ) [49].2.3 Composite topological objectsDue to the multi-component order parameter which characterizes the bro-ken SO (3) × SO (3) × U (1) symmetry in superfluid phases of He, there aremany inhomogeneous objects – textures and defects in the order parameter
He Universe 2020 7 field – which are protected by topology and are characterized by topologi-cal quantum numbers. Among them there are quantized vortices, skyrmionsand merons, solitons and vortex sheets, monopoles and boojums, etc. Thereare also composite topological objects, which combine defects of different di-mension. Among them there are Alice strings with soliton tail and analog ofKibble-Lazarides-Shafi cosmic walls terminated by Alice strings [57], see recentreview in Ref. [58].
Half-quantum vortices (analogs of Alice string in cosmology) have been sug-gested more than 40 years ago, but have been observed in superfluid Heonly recently, first in the polar phase [59] and then in the A-phase [60]. Half-quantum vortex (HQV) itself represents the combination of the linear objects:it is partly a vortex (the vortex with half of circulation quanta) and partly aspin vortex (the vortex with π change of spin vector) [61]. As a spin vortex, it isinfluenced by the spin-orbit interaction. As a results in the A-phase the HQVis always accompanied by the solitonic tail, i.e. it becomes the termination lineof the topological spin soliton which makes it energetically unfauvorable com-pared to other vortices (singly quantized vortex and vortex skyrmion). Thiswas the reason, why it was so difficult to stabilize half-quantum vortices in theA-phase. In the polar phase the solitonic tails are absent if the magnetic fieldis along the nafen strands, and the half-quantum vortices become energeticallyfavourable. They can be created either by cooling through T c under rotationor by fast cooling through T c without rotation, when the topological defectsare formed by Kibble-Zurek mechanism [62].When half-quantum vortices are created, they are pinned by nafen strands,and we can do with them whatever we want. If we tilt the magnetic field withrespect to strands, the solitons appear, which are terminated by half-quantumvortices. Then by measuring the intensity of the satellite peak in the NMRspectrum, which comes from soliton, we can find the total length of solitons,and thus the total number of half-quantum vortices in the cell.We can make the phase transition from the polar phase to the He-A, andthe half quantum vortices are still there. Moreover, we can make the phasetransition from the polar phase to the He-B, where half-quantum vorticescannot exist as topological objects. But again they remained pinned [60].
The object, which is formed in He-B, after the transition from the polar orA-phase with the pinned half-quantum vortices, is the domain wall terminatedby pinned vortices [60,63]. This composite object is the exact analog of theKibble-Lazarides-Shafi wall bounded by cosmic strings in cosmology [57].
G.E. Volovik
The other composite objects can be constructed and pinned by nafen strands,including nexus (monopole or hedgehog, which connects two or more strings[63]), necklaces [64] and lattices of composite objects [58]. But at the momentthese analogs of Nambu monopoles [65] and their further extensions are stillnot resolved in NMR experiments.Randomly pinned topological objects can provide different types of topo-logical glasses.2.4 From topological classes to topological glassesThe quenched random anisotropy provided by the confining material strandsproduces several different glass states resolved in NMR experiments in thechiral superfluid He-A and in the time-reversal-invariant polar phase. Thesmooth textures of spin and orbital order parameters in these glasses can becharacterized in terms of the randomly distributed topological charges, whichdescribe skyrmions, spin vortices and hopfions. In addition, in these skyrmionglasses the momentum-space topological invariants are randomly distributedin space. The Chern mosaic [66], Weyl glass, torsion glass and other exotictopological sates are examples of close connections between the real-space andmomentum-space topologies in superfluid He phases in aerogel, see review inRef.[67].
One of the most spectacular discoveries made in superfluid He confined in ananostructured material like aerogel or nafen was the observation of the de-struction of the long-range orientational order by a weak random anisotropy[68,69], the so-called Larkin-Imry-Ma state [70,71,72,73]. In the chiral A-phaseit is the orbital vector ˆ l , which looses the long range orientational order. Ac-cording to Mermin-Ho relation the disordered texture of the ˆ l -vector generatesthe random distribution of superfluid velocity. This state has random distribu-tion of π and π topological charges, which describe skyrmions and hopfionscorrespondingly. Thus the Larkin-Imry-Ma states can be realized in the formof the skyrmion glass [74] or/and hopfion glass.On the other hand, the orbital vector ˆ l determines the position of two Weylpoints in momentum space. That is why the skyrmion glass also represents thetopological Weyl glass, which is different from the conventional Weyl disorderwith random shifts in the position of Weyl nodes [75]. According to the Anderson theorem [76], the s -wave superconductors with non-magnetic impurities are robust to weak disorder, i.e. the critical temperature T c is the same as in the clean limit. He Universe 2020 9
For a long time it was presumed that the Anderson theorem is not appli-cable to spin triplet superconductors, or to superconductors with nodes in thegap. However, it was found by Fomin [77], that if impurities in the polar phasehave the form of infinitely long non-magnetic strands, which are straight andparallel to each other, the transition temperature also coincides with that inthe clean limit. The reason is that in the presence of columnar defects the polarphase can be considered as a set of independent 2D superfluids. The behaviorof each 2D superfluid in the presence of the corresponding 2D defects is simi-lar to that of the s -wave superconductors. This robustness of the polar phaseto the columnar disorder is the main reason why the polar phase survives innafen even for strong disorder, when all the other phases are removed fromthe phase diagram.Unusual properties of the polar phase in nafen have been discussed in Refs.[30,78]. He Universe, gravity emerges together with the gauge fields in the vicinityof the topologically stable Weyl point. In He-A, gravity emerges in the formof tetrads, which are obtained as the spacetime dependent parameters of ex-pansion of the Green’s function of Bogoliubov quasiparticles in the vicinity ofWeyl points. This suggests that in our Universe, gravity and gauge fields arealso the emergent phenomena, which come from the topology of the quantumvacuum.The He Universe suggests also the alternative scenario of the origin ofgravity: in He-B the tetrad field emerges as a composite field. Such origin ofgeometry of spacetime (the pregeometry), has been first discussed by Akama[79] and more recently by Diakonov [80,81,82], where the tetrad field emergesas bilinear combination of the fermionic fields: e aµ = i (cid:10) ψ † γ a ∇ µ ψ + ∇ µ ψ † γ a ψ (cid:11) , (3)where γ a are Dirac matrices.This mechanism was discussed in Ref.[83] in terms of the symmetry break-ing scenario, when two separate Lorentz groups of coordinate and spin ro-tations are spontaneously broken to the combined Lorentz symmetry group, L L × L S → L J , and the tetrad field e aµ serves as the order parameter of thetransition. This is the analog of the broken spin-orbit symmetry SO (3) L × SO (3) S → SO (3) J introduced by Leggett for He-B [84]. The order param-eter in the B-phase, A αi = ∆ B R αi e iΦ , contains the matrix of rotation R αi ,which connects spin and orbital degrees of freedom of the liquid. The orderparameter e aµ connects the spin and orbital degrees of the quantum vacuum,thus realizing the the extension of the B-phase condensed matter vacuum tothe 3 + 1 vacuum of our Universe. According to Eq.(3), the tetrad field e aµ transforms as a derivative and thushas the dimension of inverse length. This gives the unexpected consequencefor gravity and actually for any other fields living in such geometry. All thephysical quantities, which obey diffeomorphism invariance, such as the New-ton constant, the scalar curvature, the cosmological constant, particle masses,fermionic and scalar bosonic fields, etc., are dimensionless (see details in Refs.[85,86]).3.2 Type-II Weyl, black and white hole horizonsIn the moving He-A the Weyl cone is tilted. When the flow velocity exceedsthe effective speed of light, the Weyl cone is overtilted and the Fermi surfaceis formed [87]. In the modern language this corresponds to the Lifshitz topo-logical transition between the type-I Weyl system with isolated Weyl pointsto the type-II Weyl system, where Weyl points connect the Fermi pockets [88,89].The type-II Weyl fermions also emerge behind the event horizon of blackholes. That is why the black hole horizon serves as the boundary betweentype-I and type-II quantum vacua. In Weyl semimetals and Weyl superfluidsthe analog of event horizon is formed at the interface, which separates theregions of type-I and type-II Weyl points. This allows us on one hand to probethe Hawking radiation using Weyl superfluids and semimetals [88,89]. On theother hand, one can study the interior of the black hole using the experience ofcondensed matter, where the ultraviolet physics is known [90,91]. The effective(acoustic) metric which describes the black hole analog in condensed matter,is known in general relativity as the Painleve-Gullstrand metric [92,93].Using the junction of type-I and type II Weyl semimetals one may probethe more exotic horizon of the white hole [94]. On the other hand using theacoustic (Painleve-Gullstrand) metric in general relativity one can study thetransformation of the black hole to white hole in the process of quantumtunneling [95].3.3 Routes to antispacetime and to Euclidean spacetimeUsing different types of the Weyl points one can model exotic spacetimes andthe routes between them. For example two roads to antispacetime exist inthe presence of the the Kibble-Lazarides-Shafi wall bounded by strings in theB-phase [96]: the safe route is around the Alice string (half-quantum vortex)).The dangerous route is across the domain wall. This dangerous route throughthe Alice looking glass is similar to the route of our Universe from spacetimeto antispacetime via Big Bang. In the A-phase the route to antispacetimetakes place across the polar phase, where the metric of general relativity isdegenerate [97].The transition from Minkowski spacetime to Euclidean space time, in whichthe signature of the metric changes, is also possible to probe in superfluids,
He Universe 2020 11 using the collective modes of Bose-Einstein condensation (BEC) of magnonsin the polar phase [98]. Let us also mention that the magnon BEC – thespontaneously formed coherent precession of spins – provides the experimentalrealization of time crystal and time quasicrystal [99,100] following the Wilczekidea [101].3.4 Dark matterMagnon BEC also provides the realization of the condensed matter analog of Q -balls [102]. In relativistic quantum field theories, Q -ball is the nontopologicalsoliton, which is stabilized by conservation of an additive quantum number.In our case it is the number of self-trapped magnons. In cosmology, Q -ballscould have participated in baryogenesis, formation of bosonic stars, and thedark matter.3.5 Dark energy, cosmological constantAccording to standard physics, the vacuum has an enormous energy density ρ vac . A positive contribution comes from the zero-point energy of bosonicfields, such as electromagnetic field, and a negative contribution comes fromthe fermionic fields – from the so called Dirac vacuum, and there is no reasonwhy they should cancel [103]. Again according to standard physics, ρ vac shouldact as a gravitational source – effectively an enormous cosmological constant Λ vac . With the Planck scale providing a natural cutoff it is roughly 120 or-ders of magnitude larger than is compatible with observations. According toBjorken, this is the oft-repeated mantra that ”no one has any idea as to whythe cosmological constant is so small” [104].However, anyone who is familiar with superfluid He at zero temperatureand at zero pressure can immediately find a simple solution of the problem.The ground state of this superfluid is described by the thermodynamic po-tential (cid:15) ( n ) − µn , where (cid:15) ( n ) is the energy density, n is particle density and µ the chemical potential. According to the Gibbs-Duhem identity, at zerotemperature one has (cid:15) − µn = − P , where P is pressure. That is why thisthermodynamic potential is equivalent to ρ vac , which obeys the equation ofstates characterizing the dark energy, ρ vac = − P vac .The low energy modes of superfluid He are described in terms of bosonicand fermionic quantum fields (Higgs fields, Nambu-Goldstone fields and Bo-goliubov fermions). One may think that zero-point energies of these modescontribute to (cid:15) ( n ). However this is not so, since the energy (cid:15) ( n ) fully comesfrom the ultraviolet physics, in our case from physics of the He atoms. Thecollective excitations of the liquid He contribute only to such effects, whichdepend on the low-energy (infrared) physics, such as the Casimir effect. Inthe absence of environment, i.e. at zero pressure, the thermodynamic poten-tial of the system is exactly zero, (cid:15) ( n ) − µn = − P = 0. That is why in the full equilibrium, the microscopic (Planck scale) energy (cid:15) ( n ) is cancelled bythe counter-term µn without any fine-tuning. This comes purely from ther-modynamics, and does not depend on the phases of superfluid He (A-phase,B-phase, polar phase, etc.) and on the quantum fields living in these He vacua.The same situation takes place for the relativistic quantum vacuum. Thiscan be seen using the nonlinear extension of the Hawking description of thequantum vacuum phenomenology in terms of the 4-form field. In this de-scription, the cosmological constant in Einstein equations is equivalent to thethermodynamic potential, Λ vac = (cid:15) ( q ) − µq = − P vac [105,106], where q isthe 4-form field. While (cid:15) ( q ) is determined by the microscopic trans-Planckianphysics of the ”atoms of the vacuum” and is very large, the pressure belongsto the infrared physics. That is why Λ vac = 0 if our Universe is isolated fromthe environment, as it happens for the He Universe isolated from the environ-ment. The Gibbs-Duhem thermodynamic identity ensures the cancellation oflarge vacuum energy in an equilibrium vacuum, regardless of the microscopicstructure of the vacuum. p -wave superfluid He with 3 × He-B they are separated into 4 Nambu-Goldstone modes and 14Higgs amplitude modes.
Nambu [107] noticed that in He-B these 18 modes, which are distributed intofamilies with different angular momentum J = 0 , ,
2, obey the rule valid foreach family. In the relativistic form this Nambu sum rule reads: M J + + M J − = 4 M F . (4)Here M + and M − are the gaps in the spectrum of modes, in which correspond-ingly the real and imaginary parts of the order parameter are oscillating; M F is the fermionic mass, which in He-B is the gap in fermionic spectrum of Bo-goliubov quasiparticles. This rule has been further extended to the thin film ofthe A-phase [108,109]. The connection between the fermion and boson massescan be attributed to the manifestation of hidden supersymmetry in superfluid He [107].The Nambu sum rule is valid for the BCS weak coupling regime [110,111].However, this rule can be applied to the extensions of the Standard Model,assuming that they are also in the weak coupling regime. In the simplest casesthis assumption suggests the existence of the second Higgs particle, with themass 325 GeV in the He-B scenario or with the mass 245 GeV in the He-Ascenario, in which M + = M − [108,109]. He Universe 2020 13 In He-B, due the tiny spin-orbit interaction one of the four Nambu-Goldstonemodes becomes the Higgs boson with small mass, which is determined by spin-orbit coupling. This is the mode measured in experiments with the longitudinalNMR. The formation of the Higgs mass is the result of the violation of the He-B symmetry by spin-orbit interaction. In the modern language the enhancedsymmetry which takes place when some terms in energy are small and areneglected, is called the hidden symmetry, and the corresponding Higgs withsmall mass is called pseudo Nambu-Goldstone boson.The presently known Higgs boson has mass 125 GeV, which looks rathersmall compared to the typical electroweak energy scale of 1 TeV. That iswhy the natural guess from the He physics is that this boson is actually thepseudo Nambu-Goldstone mode [112,113,114], and one should search for thereal Higgs, which could be as heavy as 1 TeV.It is interesting that the hints of the Higgs bosons with 245 GeV, 325 GeVand with the TeV scale, have been reported (see e.g. [115,116,117]), but notconfirmed.4.2 Nuclear PhysicsConnection between the chiral phenomena in the quarkgluon plasma and inthe chiral superfluid He-A in relation to chiral magnetic and chiral vorticaleffects has been discussed in Sec. 2.2.2. Here we consider another connectionwhich is related to the models of hadrons and quark confinement. In He-B we can imitate the MIT bag model for hadrons [118,119]. In thismodel [120,121] the hadron is considered as macroscopic box (bag) with thedeconfinement phase inside, where the massless quarks are free, and with thevacuum in the confinement phase outside, where free quarks cannot exist. Thismodel is similar to the electron bubble in a helium liquid, where the zero pointenergy of electron in the ground state in the box potential compensates theexternal pressure and surface tension. In the MIT bag the compensation alsocomes from the ground state energy of free quarks in the box.In He-B, we constructed the bosonic analog of the bag model: instead ofthe quarks in the ground state in a box potential, there are magnons whichalso fill the ground state forming the magnon Bose condensate [118,119].
At the moment, He cannot say anything reasonable on the origin of quantummechanics and quantum field theory. However, the possible origin of √− in quantum mechanics is suggested [87,122]. The microscopic physics of thequantum vacuum is fully described in terms of the real numbers, while theimaginary unit emerges in the low energy corner together with Weyl fermions.The reason for that is the same topology, which protects Weyl points in thechiral superfluid He-A.
The Helium can answer most of the questions presented in the paper by Allenand Lidstr¨om ”Life, the Universe, and everything-42 fundamental questions”[123]. At least Helium has opinion on these problems. However, at the momentHelium cannot say anything physical on such questions as: What is quantummechanics? What is life? What is consciousness? These topics still remainsupernatural for Helium, and this is the reason why they are not on the physicalmap in Fig. 1. The main task of Helium is to push forward the border linebetween the natural and supernatural parts of the Universe.
Acknowledgements
This work has been supported by the European Re-search Council (ERC) under the European Union’s Horizon 2020 research andinnovation programme (Grant Agreement No. 694248).
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