Vielbein with mixed dimensions and gravitational global monopole in the planar phase of superfluid 3 He
aa r X i v : . [ c ond - m a t . o t h e r] S e p Vielbein with mixed dimensions and gravitational global monopole in the planarphase of superfluid He G.E. Volovik
1, 2 Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland Landau Institute for Theoretical Physics, acad. Semyonov av., 1a, 142432, Chernogolovka, Russia (Dated: September 22, 2020)The planar phase of superfluid He has Dirac points in momentum space and the analog of Diracmonopole in the real space. Here we discuss the combined effect of Dirac point and Dirac monopole.It is shown that in the presence of the monopole the effective metric acting on Dirac fermionscorresponds to the metric produced by the global monopole in general relativity: it is the conicalmetric. Another consequence is that the primary variable, which gives rise to the effective metric,is the unusual vielbein field in the form of the 4 × × PACS numbers:
I. INTRODUCTION
The planar phase is one of the possible superfluidphases of liquid He. It may exist in some region of thephase diagram of superfluid He confined in aerogels. The planar phase has two Dirac points in the quasipar-ticle spectrum, which are supported by combined actionof topology and some special symmetry, see e.g. Ref. .The quasiparticles in the planar phase with fixed spinbehave as Weyl fermions. Similar to the chiral superfluid He-A, they experience the effective gravity and gaugefield produced by the deformation of the order param-eter. But there is the following important difference.In He-A, the spin-up and spin-down fermions have thesame chirality, while in the planar phase the spin-up andspin-down fermions have the opposite chirality. As a re-sult the Weyl fermions in planar phase form the masslessDirac fermions, see Ref. .Here we study the planar phase fermions in the pres-ence of the topological defect – the hedgehog. The ef-fective gravity produced by the hedgehog appears to besimilar to the gravitational effect of the global monopolein general relativity: it gives rise to the conical space. Another consequence of the hedgehog is that the viel-bein, which describes the effective gravity, is the 4 × × II. WEYL-DIRAC POINTS AND × VIELBEIN
In the general spin triplet p -wave pairing state the sym-metric 2 × A iα σ α p i , (1)where σ α are the Pauli matrices for spin and A αi is the3 × .In the planar phase the particular representative of the order parameter is: A αi = c ⊥ e i Φ (cid:16) δ iα − ˆ l α ˆ l i (cid:17) , (2)where Φ is the phase of the order parameter and ˆ l is theunit vector. All the other degenerate states of the planarphase are obtained by spin, orbital and phase rotations ofthe group G = SO (3) S × SO (3) L × U (1) (here we ignorethe discrete symmetry, since we are only interested in theglobal monopole).The order parameter in Eq.(2) has the symmetry H = SO (2) J – the symmetry under the common spin and or-bital rotations about the axis ˆ l . As the result, the mani-fold of the degenerate states is R = ( SO (3) S × SO (3) L × U (1)) /SO (2) J , which supports the monopoles (hedge-hogs), described by the homotopy group π ( R ) = Z .The particular form of the monopole with the topologicalcharge N = 1 is: A αi ( r ) = f ( r ) (cid:0) δ iα − ˆ r α ˆ r i (cid:1) , (3)where ˆ r = r /r , and f ( r → ∞ ) = c ⊥ . We also fix thephase Φ = 0 in further considerations.The Bogoliubov-Nambu Hamiltonian for quasiparti-cles: (cid:18) ǫ ( p ) ˆ∆ˆ∆ † − ǫ ( p ) (cid:19) , (4)where ǫ ( p ) = c k ( p − p F ), c k = v F , and v F and p F arecorrespondingly the Fermi velocity and Fermi momentumof the normal Fermi liquid.The planar phase has the Weyl-Dirac points at p = ± p F ˆ l . Near the Weyl-Dirac nodes the Hamiltonian hasthe form: H = X a Γ a e ia ( p i − qA i ) . (5)Here A = p F ˆ l is the vector potential of effective gaugefield acting on the massless Dirac fermions; q = ± a with a = 1 , , , e ia are the components of thespatial vielbein with a = 1 , , , i = 1 , , × { Γ a , Γ b } = 2 δ ab , which here we choose inthe following way:Γ = τ σ x , Γ = τ σ y , Γ = τ σ z , Γ = τ , (6)and the corresponding vielbein components are e ia = c ⊥ ( δ ia − ˆ l a ˆ l i ) for a = 1 , , , e i = c k ˆ l i . (7)The important property of such vielbein is that it isthe 3 × × g ik = X a,b δ ab e ia e kb , a, b = 1 , , , , i, k = 1 , , , (8)This metric has the conventional form g ik = c k ˆ l i ˆ l k + c ⊥ ( δ ik − ˆ l i ˆ l k ) , (9)which coincides with the effective metric in the chiralsuperfluid He-A with Weyl nodes, see Ref. .The 3 + 1 effective metric is expressed in terms of the4 × g µν = X a,b η ab e µa e νb , a, b = 0 , , , , , µ, ν = 0 , , , . (10)It also has the form as in He-A. In spite of the unusualasymmetric 4 × g ik = 1 c k ˆ l i ˆ l k + 1 c ⊥ ( δ ik − ˆ l i ˆ l k ) , g = − . (11) III. GLOBAL MONOPOLE AND EFFECTIVEMETRIC
For the monopole, the asymptotic form of the metricat infinity, where f ( r ) → c ⊥ , is: g ik ( r ) = c ⊥ δ ik + ( c k − c ⊥ )ˆ r i ˆ r k , (12)and the interval is: ds = − dt + 1 c ⊥ r d Ω + 1 c k dr . (13)Eq.(13) represents conical spacetime, which in gen-eral relativity is produced by the global monopoles(monopoles without gauge fields, see Refs. ). Thisspacetime has the nonzero scalar curvature: R = 2 1 − α r , α = c k c ⊥ . (14) The analog of the global monopole was considered in su-perfluid He-A (see Refs. ). However, in He-A thishedgehog in the ˆ l field has the tail – the doubly quan-tized vortex. This is the analog of the Nambu monopole terminating cosmic string – the observable Dirac string(classification of such composite objects can be found inRef. ). In the planar phase the monopole is topologicallystable and the Nambu-Dirac string does not appear. Thisis because the Dirac string of the monopole in the orbitalvector ˆ l i ( r ) = ˆ r i in Eq.(3) is cancelled by the Dirac stringfrom the monopole in the spin vector ˆ l α ( r ) = ˆ r α .Since in superfluid He one has c k > c ⊥ , the effectivemetric corresponds to the spacetime with the solid angleexcess, α >
1, instead of the solid angle deficit dis-cussed for the global cosmic monopoles with α <
1. Forthe cosmic monopole in the scalar field of amplitude η one has the following correspondence:1 − πGη = α = c k c ⊥ . (15)The solid angle excess corresponds to the repulsive grav-ity, G <
0, and super-Planckian scalar field, η > / | G | .For c k = c ⊥ ≡ c the metric is flat, g ik = c δ ik , at leastfar from the monopole. In cosmology this corresponds tothe absence of the cosmic global monopole, or the ab-sence of the scalar field in the vacuum, η = 0. However,the planar phase monopole does not disappear: the sin-gularity remains in the vielbein field, while the metric hasonly the localized bump in the curvature and is flat (notconical) at infinity. The tetrad field monopole in the 4DEuclidean space (torsional instanton) with the localizedbump in the curvature and the flat metric at infinity wasconsidered by Eguchi and Hanson , and Hanson andRegge. IV. CONCLUSION
The planar phase provides an example, when the grav-ity for fermions and bosons can be essentially different.While the fermions are described by the 4 × e µa , the bosons are described by the conventional4D metric g µν . The vielbein with non-quadratic matrix e µa may exist in other superfluid phases, including theultracold fermionic gases. In the presence of topologicalobjects, they may give rise to exotic effective spaces andspacetimes, which are different for fermions and bosons.One may expect the similar effects in general relativitywith degenerate metric. Exotic monopole in gravity withdegenerate tetrads was discussed for example in Ref. .It would be interesting to consider the transition fromthe planar phase to the He-B, where the Dirac fermionsbecome massive.
Acknowledgements . This work has been supportedby the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovationprogramme (Grant Agreement No. 694248). D. Vollhardt and P. Woelfe,
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