Effective Minkowski to Euclidean signature change of the magnon BEC pseudo-Goldstone mode in polar 3He
aa r X i v : . [ c ond - m a t . o t h e r] S e p Effective Minkowski to Euclidean signature change of the magnon BECpseudo-Goldstone mode in polar He J. Nissinen and 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 28, 2017)We discuss the effective metric experienced by the Nambu-Goldstone mode propagating in thebroken symmetry spin-superfluid state of coherent precession of magnetization. This collective moderepresents the phonon in the RF driven or pulsed out-of-equilibrium Bose-Einstein condensate (BEC)of optical magnons. We derive the effective BEC free energy and consider the phonon spectrum whenthe spin superfluid BEC is formed in the anisotropic polar phase of superfluid He, experimentallyobserved in uniaxial aerogel He-samples. The coherent precession of magnetization experiencesan instability at a critical value of the tilting angle of external magnetic field with respect to theanisotropy axis. From the action of quadratic deviations around equilibrium, this instability isinterpreted as a Minkowski-to-Euclidean signature change of the effective phonon metric. We alsonote the similarity between the magnon BEC in the unstable region and an effective vacuum scalar“ghost” condensate.
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I. INTRODUCTION
There are different classes of broken symmetry statesthat experience the phenomenon of spin superfluidity .The first of them contains magnetic systems with spon-taneously broken continuous symmetry of spin rotations, SO S (3) or the planar subgroup SO S (2). The spon-taneous breaking of this symmetry leads to the asso-ciated Nambu-Goldstone (NG) modes (spin waves ormagnons), to spin supercurrents and to topological de-fects, such as spin vortices with spin supercurrent circu-lating around the cores. Examples are provided by somesolid state magnetic materials , and by spin-triplet su-perfluid phases of liquid He . In superfluid He thespin-orbit (dipole) interaction is tiny, and the spin ro-tation SO S (3) symmetry is almost exact. The broken SO S (3) × U (1) symmetry of the superfluid order parame-ter leads to the recently observed half-quantum vortices ,which have both spin and mass supercurrent circulationaround the vortex cores.Due to spin-orbit interaction, which explicitly violatesthe spin rotation symmetry, some magnons acquire smallmasses and become pseudo NG modes. In high energyphysics, the formation of such a massive boson is calledthe Little Higgs scenario , which may explain why theHiggs boson has relatively small mass of 125 GeV. In He-B the parametric decay of optical magnons to pairsof light Higgs modes has been observed . In this scenario,the spin and spin-mass vortices become the terminationlines of topological spin solitons . In the polar phase of He, the topological soliton emerges between two neigh-boring half-quantum vortices when the magnetic field istilted with respect to an anisotropy axis and is resolvedin NMR experiments .The second class of spin superfluid states encompassesstates which are periodic in time. A state with sponta-neously formed phase-coherent precession of magnetiza- tion has been first observed in He-B . The lifetime ofthis coherent precession is extremely large compared withthermalization time, and if dissipation is neglected, thisspontaneously time-periodic state represents an exampleof a time crystal . From a different point of view, thespontaneously formed coherent precession can be con-sidered in the language of an out-of-equilibrium Bose-Einstein condensate (BEC) of quasiparticles , which forthe case of He-B are optical magnons. The sponta-neous breaking of time translation symmetry leads tospin current Josephson effect, to quantized vortices inthe magnon BEC and to the new NG mode – the propa-gating oscillations of the phase of precession , whichrepresents the usual phonon mode of the magnon BECin the out-of-equilibrium BEC language .In experiments, the out-of-equilibrium magnon BEC insuperfluid He arises when the system is either continu-osly driven with an external transverse RF magnetic field H rf ⊥ H or after a short transverse RF field pulse is ap-plied. In the pulsed NMR experiment, after the RF pulseis turned off, the spin precession experiences dephasingdue to inhomogeneity of the underlying superfluid tex-ture. But then the phase coherence is rapidly restoreddue to spin supercurrents, and the spins enter a long-lived state – the magnon BEC, where the macroscopicspin S is freely precessing at an angle β with respect to H , with the off-diagonal order parameter h ˆ S + i = h ˆ S x + i ˆ S y i = S ⊥ e iωt + iα , (1) S ⊥ = S sin β, n = S − S z ~ . (2)Here n = S (1 − cos β ) / ~ is the magnon number density inthe condensate, α the condensate phase, and the preces-sion frequency ω plays the role of a chemical potential: µ ≡ ω = ω rf in the presence of continuous RF pumping(in thermodynamics this is the regime of the fixed chem-ical potential), and µ ≡ ω in pulsed RF fields, where theglobal precession frequency ω is determined by the num-ber N of magnons pumped during the pulse (the regimeof the fixed number of magnons).The nonequilibrium superfluidity of magnon BEC hasalso been observed in Yttrium Iron Garnet films In this paper we study the NG mode of the magnonBEC in the polar phase. The rest of this paper is orga-nized as follows. In Sec.II we review the magnon spec-trum in polar He and discuss the precessing magnonBEC and its phonon spectrum in Sec.III. In Sec.IV,we analyze the acoustic phonon metric and identifythe Minkowski-to-Euclidean signature change. In SecV,we compute the effective metric of quadratic deviationsaround equilibrium and conclude with an outlook inSec.VI.
II. MAGNON SPECTRUM AND EFFECTIVEMETRIC IN THE POLAR PHASE
The polar phase can be stabilized by immersing super-fluid He in an uniaxially anisotropic aerogel, where theorbital anisotropy ˆ n of the condensate aligns along theaerogel strands. The order parameter of the polar phaseis given as A αi = ∆ P ˆ d α ˆ n i e i Φ . (3)Here ∆ P is the gap amplitude with phase Φ; ˆ n the fixedorbital anisotropy along the aerogel strands; ˆ d the unitvector of the spin-anisotropy axis of the Cooper pairs.The polar phase represents the superfluid analog of spin-nematic state in antiferromagnets , since the states ˆ d and − ˆ d can be connected by the change of the phase Φ by π .The latter gives rise to the half-quantum vortices, whichhave been observed in the polar phase .Spin dynamics is governed by the Leggett equationsfor S and ˆ d , i.e. the ‘adiabatic’ Hamiltonian F , which isthe superfluid He free energy in the London limit: F = F spin + F grad + F so , (4) f spin = 12 γ m S χ − S − γ m H · S (5) f grad = 12 K ij ∇ i ˆ d α ∇ j ˆ d α (6) f so = g D (ˆ d · ˆ n ) . (7)In the polar phase, the spin susceptibility is given as χ αβ = χ k ˆ d α ˆ d β + χ ( δ αβ − ˆ d α ˆ d β ) and the spin orbitand gradient energy for the spin-vector ˆ d have K ij = K k ˆ n i ˆ n j + K ⊥ ( δ ij − ˆ n i ˆ n j ) and g D = χ Ω P γ m , where Ω P isthe Leggett frequency of the polar phase ( γ m is the Henuclei gyromagnetic ratio). It follows that S = χ H /γ m in equilibrium.When the spin-orbit interaction f so is neglected, thespectrum of longitudinal and optical magnon modes, with polarizations a = 0 , +1 respectively, can be writ-ten in relativistic form: g µνS p µ p ν + M a = 0 . (8)Here p µ is the 4-momentum of magnons, p µ = ( ω, k i )and g µνS is the effective magnon metric – the magnoniccounterpart of the acoustic metric . In the polar phase,for the homogeneous static superfluid state, it takes theform g mnS = c k S ˆ n m ˆ n n + c ⊥ S ( δ mn − ˆ n m ˆ n n ) , g S = − , (9)where the “speeds of light” for magnons propagating par-allel and transverse to ˆ n , respectively (this anisotropy hasbeen measured ) are: c k S = γ m K k /χ , c ⊥ S = γ m K ⊥ /χ . (10)In Eq.(8) the magnetic field H is chosen along ˆ z , andthe“invariant masses” are respectively: M + = ω L ≡ γ m H , M = 0 , (11)where ω L = γ m H is the Larmor frequency.With the spin-orbit interaction (7) the spectrum ofmagnons: ω = g mnS k m k n , (12) ω = Ω + ω L + g mnS k m k n . (13) III. DYNAMICS OF THE COHERENTLYPRECESSING STATEA. Magnon BEC and phonon Hamiltonian.
We are interested in the low-frequency and long wave-length dynamics of magnon BEC, which is developed inthe background of the fast precession. This dynamicsis described by the slow variables, magnon density n ofthe condensate in Eq.(2) and the phase α of precession.As in conventional BECs, these two variables are canon-ically conjugated, and the linearized equations for thesevariables describe the Goldstone mode of the coherentprecession – the phonon propagating in the magnon con-densate.The Hamiltonian H ( n, α ) = H BEC − µN = F BEC forthe slow magnon BEC modes can be obtained by averag-ing the spin-orbit and gradient terms, Eqs. (7) and (6),over the fast Larmor precession. We assume here thatΩ ≪ ω L , then to the zeroth order approximation, onehas the pure Larmor precession at ω = ω L , which can beexpressed in the general form: S ( t ) = O − ( t ) RO ( t ) S ˆ z , (14)ˆ d ( t ) = O − ( t ) RO ( t )ˆ x , S ( t ) · ˆ d ( t ) = 0 . (15)Here O ( t ) = R z ( ω L t ) is the transformation to the framerotating with the Larmor frequency ω L = γH , and R = R z ( α ) R y ( β ) R z ( γ ) is the matrix of spin rotationin that frame with Euler angles α, β, γ . Abbreviating s ( x ) ≡ sin x and c ( x ) ≡ cos x , one obtains explicit timedependence of S ( t ) and ˆ d ( t ) in the Larmor precessionˆ S ( t ) = c ( α − ω L t ) s ( β )ˆ x + s ( α − ω L t ) s ( β )ˆ y + c ( β )ˆ z , ˆ d ( t ) = [ c ( β ) c ( α − ω L t ) c ( γ + ω L t ) − s ( α − ω L t ) s ( γ + ω L t )]ˆ x + [ c ( β ) s ( α − ω L t ) s ( γ + ω L t )+ c ( α − ω L t ) s ( γ + ω L t )]ˆ y − s ( β ) c ( γ + ω L t )ˆ z . Averaging of the spin-orbit term (7) over the fast pre-cession gives h f so ( t ) i = g D (cid:18) λ + (1 − λ ) cos β − (16)12 (1 + cos β ) sin λ cos(2( α + γ )) (cid:19) , where λ is the angle of the vector of orbital anisotropywith respect to the static magnetic field, ˆ n = ˆ y sin λ +ˆ z cos λ . Minimization over α + γ gives α + γ = 0 and, as aresult, one obtains the following nonlinear contribution tothe energy density of the condensate in terms of magnondensity n = S (1 − cos β ): ǫ ( n ) = h f so ( t ) i γ = − α = g D (cid:18) λ + (1 − λ ) cos β (17) −
12 (1 + cos β ) sin λ (cid:19) . The average over the gradient term follows similarly.Taking into account that γ = − α at equilibrium, oneobtains h∇ i ˆ d ( t ) · ∇ j ˆ d ( t ) i = 12 (1 − cos β )(3 − cos β ) ∇ i α ∇ j α + 12 ∇ i β ∇ j β . (18)Finally, gathering all terms, we arrive to H ( α, n ) = Z d r S n ( n max + n ) K ij ∇ i α ∇ j α + K ij n ( n max − n ) ∇ i n ∇ j n +( ω L − µ ) n + ǫ ( n ) + γ m H rf S sin β α n max = 2 S . We also added the symmetry breakingterm for small α , which appears in case of cw-NMR andcomes from the driving RF field H rf k ˆ x , f sb ( α, β ) = − γ m H rf · S = − γ m H rf S sin β cos α . (20)It gives the mass to the phonon propagating in magnonBEC, see Eq.(27). For small n ≪ n max , the phononHamiltonian Eq. (19) transforms to the Ginzburg-Landau free energy F BEC of the magnon BEC in thepolar phase (see Eq.(51) in the Appendix), where theprecession averaged spin-orbit interaction ǫ ( n, λ ) servesas the interaction between the magnons in the BEC. B. Goldstone mode spectrum.
Introducing the dimensionless variable ˜ n = 1 − cos β ,the Poisson brackets { ˜ n ( r ) , α ( r ) } = S − δ ( r − r ) givethe following equations of motion˙˜ n = − S δHδα , ˙ α = 1 S δHδ ˜ n , (21)from which in linear order in α and δn one obtains thephonon wave equation ∂ α∂t = ǫ ′′ [ γ ij ∇ i ∇ j α − γ m H rf S sin β α ] (22) − S n − ˜ n K mn K ij ∇ i ∇ j ∇ m ∇ n α , (23) ǫ ′′ = d ǫdn = 1 S g D − λ ) , (24) γ ij = 12 (1 − cos β )(3 − cos β ) K ij (25)Let us first neglect the 4th order term in Eq. (23), thenusing Eq. (22) one obtains the “relativistic” spectrum ofthe NG mode – the phonon propagating in magnon BEC: ω ( k ) = c k k z + c ⊥ ( k x + k y ) + M . (26)Above and henceforth we set ˆ n = ˆ z and H = cos λ ˆ z − sin λ ˆ y . The small mass of the NG mode arises due to thesymmetry violating RF field H rf ≪ H : M = Ω P H rf H (1 − λ ) sin β . (27)This phonon mass has been measured in He-B andwe note that the phonon mass is absent in pulsed NMRexperiments, where the coherent precession is free.The anisotropic “speed of light” for phonons in magnonBEC in polar He is c k , ⊥ = ǫ ′′ − cos β )(3 − cos β ) K k , ⊥ . (28)In terms of spin-wave velocities c k , ⊥ S in Eq.(8): c k , ⊥ = 116 Ω P ω L (1 − λ )(1 − cos β )(3 − cos β ) c k , ⊥ S . (29)It follows that in the Ginzburg-Landau regime and weakcoupling theory, also c k = 3 c ⊥ .The important property of the spectrum is that thedispersion changes sign for ǫ ′′ ( n, λ ) at 1 − λ = 0.The same threshold has been calculated and observed inthe disordered Larkin-Imry-Ma state of He in aerogel .Clearly this implies an instability of the condensate asfunction of the parameter λ , the angle between the mag-netic field and the axis of anisotropy of the aerogel. Herewe interpret this as the transition from a Minskowskito Euclidean signature metric for the dynamical phononmodes. To see this in more detail, we compute the linearhomogenous equations of motion for the phonons in thepresence of counterflow in the next section. In Section V, we complement this with a more careful analysis of thedynamical equations of motion for small deviations of themagnon BEC around equilibrium at ǫ ′ ( n ) = µ − ω L . IV. ACOUSTIC METRIC, ERGOREGION ANDHORIZONA. Acoustic phonon metric
In the presence of a counterflow velocity w (say, in arotating cryostat, or due to a spin current), a source termfor the current is added to the free energy: F cf = Z d r n ∇ α · w . (30)The wave equation is modified and for constant w and c k , ⊥ , one can identify the following effective “acoustic”metric: 0 = ∂ α∂t − w · ∇ ∂α∂t + ( w · ∇ ) α − c k ∂ α∂z − c ⊥ ∂ α∂x − c ⊥ ∂ α∂y + M α (31) ≡ g µν ∇ µ ∇ ν α + M α . This definition of g µν works only for a homogenous equi-librium state since the full wave equation for a masslessscalar field α in the background metric g µν is g µν ˆ ∇ µ ˆ ∇ ν α = 1 √ g ∇ µ ( √ gg µν ∇ ν α ) = 0 , (32)where ˆ ∇ is the covariant derivative corresponding to g µν ,see Eq. (42) below. For constant parameters in the met-ric the contravariant metric is g = 1 , g i = − w i , g ij = w i w j − ( c k − c ⊥ )ˆ z i ˆ z j − c ⊥ δ ij . (33)This form of the metric corresponds to the Hamiltonianor ADM formalism of general relativity with the shiftvector N i = w i and the gauge fixed lapse function N = 1.This fixed-gauge metric is natural for condensed matteranalogies of general relativity, since the spectrum is ob-tained in the laboratory frame of the condensate. In thisgauge g = 1 /N = 1, and the metric determinant is: g = − c ⊥ c k . (34)This also confirms the correct choice of the gauge, since inthe laboratory frame the “speed of light” is anisotropic,and thus there is no unique definition of the propagationspeed of the Goldstone modes. However, such metric isnot suitable when the determinant changes sign. Two surfaces related to the metric (33) are of interest:the surface at which c ⊥ ( r ) = w ( r ) where g and g ww cross zero; and the surface where c ⊥ ( r ) and c k ( r ) crosszero and become negative. Let us start with the first one.The surface at which c ⊥ ( r ) − w ( r ) crosses zero is either ahorizon or an ergosurface, depending on the orientationof the interface with respect to flow velocity w . Theergosurface takes place for the circular flow w = v ( ρ ) ˆ φ .Assuming that λ is also axisymmetric, one has for thecovariant acoustic metric g µν : ds = g µν dx µ dx ν = dt − ρ c ⊥ ( ρ ) (cid:18) dφ − v ( ρ ) ρ dt (cid:19) − dρ c ⊥ ( ρ ) − dz c k ( ρ ) . (35)The ergosurface is at c ⊥ ( ρ ) = v ( ρ ), where g and g φφ cross zero.Note that for such metric there are no closed time-like curves. For the closed time-like curve to exist it isnecessary to have g φφ >
0. This occurs in the G¨odelUniverse, where the corresponding acoustic metric hasbeen discussed . At such surface g φφ and g cross zero,and thus behind such surface the closed time-like curvesappear. The vacuum in this region is unstable, as can beseen from the spectrum of photons. Similar instabilitytakes place in our case when c ⊥ ( r ) and c k ( r ) in Eqs.(28)and (29) become negative. B. Minkowski-to-Euclidean signature change of theeffective Nambu-Goldstone metric.
Let us consider the surface, where c ⊥ ( r ) = 0 and c k ( r ) = 0. At this surface g in Eq.(34) crosses infinity andchanges sign, i.e. the Minkowski signature transforms tothe Euclidean one. Such transformation via g = ∞ andalso via g = 0 has been discussed for the metric inducedon a cosmic string in the presence of black hole, where itwas mentioned that such spacelike space-time does notrepresent any solution for a physical cosmic string . Inour case, when ǫ ′′ ( n, λ ) <
0, the Euclidean space-time forthe Goldstone mode signals the instability of the coher-ent precession. Such instability has been discussed andhas been observed in the A-phase, see the review paper .In the polar phase the instability, c k , ⊥ <
0, takes placewhen tan λ < . (36)At tan λ c = 2, there is a transition from Minkowski sig-nature at tan λ > λ <
2. Such change in the signature of the phonon met-ric has been discussed for the Bose gas at the transi-tion between the repulsive and attractive interaction ofbosons . However, in our case the (small) mass be-comes simultaneously tachyonic, M <
0. Excitationswith such a spectrum can be called tachyonic ghosts .The instability is much stronger than in the ergoregion,but experimentally the lifetime of the unstable vacuumcan be made long enough near the threshold of instabil-ity.When the effective metric depends on coordinates, itsbehavior in the instability region differs from what fol-lows from the linear equations. This can be seen fromthe consideration of the full action of quadratic devia-tions from equilibrium, discussed in the next section. V. ACOUSTIC METRIC OF QUADRATICDEVIATIONS AND SIGNATURE CHANGE
We now wish to calculate the signature change of thefull acoustic metric for the phonons by considering thequadratic action of deviations around equilibrium.The phonon Hamiltonian (19) with the counterflowterm is H ( α, n ) = Z d r (cid:18) ǫ ( n ) + ( ω L − µ ) n (37)+ 12 γ ij ( n ) ∇ i α ∇ j α + n w · ∇ α + f sb ( n, α ) (cid:19) , the action follows as S = Z dt (cid:18)Z d r n ˙ α − H ( α, n ) (cid:19) (38)and defines the quantum mechanical path-integral kernel Z i → f = h f | T [ e − i R T H ( t ) ] | i i = R α T = fα = i D αe iS with saddlepoint solutions corresponding to the classical equationsof motion.To obtain the action for the phonon modes, we need thequadratic form of deviations n = n + δn and α = α + δα around some equilibrium state n ( r ), where δHδn | n = n = 0and δHδα | α = α = 0. Ignoring the small mass term f sb ( n, α )and the equilibrium spin current ∇ α , i.e. expandingaround constant α and ǫ ′ ( n ) = µ − ω L , leads to thecanonical relation δn = 1 ǫ ′′ ( n ) ( δ ˙ α − w · ∇ δα ) . (39)Then the action for the quadratic deviations δα ≡ α is S = 12 Z dt Z d r (cid:18) ǫ ′′ ( ˙ α − w · ∇ α ) − γ ij ∇ i α ∇ j α (cid:19) . (40) A. Stable Minkowski region ǫ ′′ ( n , λ ) > In the convex region, where ǫ ′′ ( n , λ ) >
0, i.e. tan λ >
2, the spin-orbit interaction reproduces the repulsive in-teraction of magnons, and the magnon BEC is stable. The quadratic action can be written in terms of effectivemetric g µν for the scalar field α : S ≡ Z dt Z d r ˜ g µν ∇ µ α ∇ ν α (41) ≡ Z dt Z d r √− gg µν ∇ µ α ∇ ν α . (42)Here the matrix ˜ g µν follows directly from (40) along withthe inverse ˜ g µν ,˜ g = 1 ǫ ′′ , ˜ g i = − w i ǫ ′′ , ˜ g ij = − γ ij + w i w j ǫ ′′ , (43)˜ g = ǫ ′′ − γ ij w i w j , ˜ g i = − γ ij w j , ˜ g ij = − γ ij . (44)where ˜ g ≡ det ˜ g µν = − ǫ ′′ γ and γ is the determinant of thematrix γ ij . Comparing Eqs.(41) and (42) one obtains theeffective metric: g µν = p − ˜ g ˜ g µν = (cid:18) ǫ ′′ γ (cid:19) / ˜ g µν , √− g = 1 √− ˜ g . (45)or g = 1 √ γǫ ′′ , g i = − w i √ γǫ ′′ , g ij = w i w j √ γǫ ′′ − γ ij s ǫ ′′ γ . (46) B. Unstable Euclidean region ǫ ′′ ( n , λ ) < In the region tan λ < ǫ ′′ < g µν Eq.(46) becomes imagi-nary, while the motion equations are still real. Moreover,the determinant of the metric changes sign and the metricsignature becomes Euclidean. From Eq. (40) the correctform of the action follows as S ≡ − Z dt Z d r ˜ g µνE ∇ µ α ∇ ν α ≡ − Z dt Z d r √ g E g µν E ∇ µ α ∇ ν α . (47)Comparing Eqs. (40) and (47) one obtains the effectivemetric g µν E = p ˜ g ˜ g µν = (cid:18) − ǫ ′′ γ (cid:19) / ˜ g µν , √ g E = 1 √ ˜ g , (48)which now has Euclidean signature, g = 1 √− γǫ ′′ , g i E = − w i √− γǫ ′′ ,g ij E = w i w j √− γǫ ′′ + γ ij s − ǫ ′′ γ . (49)The Euclidean signature corresponds to complexphonon frequencies ω ( k ) < g E ,µν doesnot correspond to the imaginary time thermal partitionfunction of the condensate but instead to the the dynami-cal phonon modes around the equilibrium state inheritedfrom the stable equilibrium state for ǫ ′′ ( n , λ ) >
0. Dy-namics is governed by the Euclidean signature metric inthe non-equilibrium region where the expansion aroundthe unstable condensate is still valid.When the instability to phonon NG modes develops attan λ <
2, it is cut-off at higher energies by the quarticterm in Eq. (23) that have the usual, Minkowski signa-ture in the dispersion. Similar role of the higher-than-quadratic terms has been discussed in Refs. . Themagnon BEC in the unstable region can be consideredas an effective non-relativistic version of a “ghost con-densate” similar to that of Ref. 33.
VI. OUTLOOK
Superfluid He has stable region of a magnon BECwhen the precession averaged spin-orbit interaction f so = ǫ ( n ) is convex, which plays the role of repulsive magnon-magnon interaction. This is the long-lived state of coher-ent precession formed after optical magnons are pumpedwith a RF pulse. The magnon number N = ( S − S z ) / ~ determines the global frequency of precession, whichplays the role of chemical potential, ω ≡ µ BEC , whilein continuous RF magnetic field the magnon BEC is sta-bilized with µ BEC = ω rf . The precession phase α corre-sponds to the conjugate degrees of freedom of the conden-sate. The most well-known example of this is the HPDin He-B. Here we have discussed the HPD magnon BECin the polar phase of He.At the critical value tan λ c = 2 of the angle λ be-tween the axis of anisotropy ˆ n of the aerogel and thestatic magnetic field H , there is a transition from repul-sive to attractive magnon spin-orbit interaction. Whentan λ <
2, we have ǫ ′′ ( n , λ ) < λ can be tuned in experiments continuously fromthe stable to the unstable region. Such transition hasbeen discussed for He-A and observed in the disor-dered Larkin-Imry-Ma state of He in aerogel . Here weconsidered this transition and condensate instability interms of the effective phonon metric.In magnon superfluids there are two effective met-rics. One is the metric for the propagaing magnons inEq.(9). The other one is the effective metric experiencedby phonons propagating in the magnon BEC in Eqs.(33)and (46). This is an analog of acoustic metric introducedby Unruh . The transition between the repulsive andattractive spin-orbit interaction corresponds to the tran-sition between Minkowski and Euclidean signature of thephonon metric. The Euclidean metric g E ,µν in Eq.(49) does not correspond to the imaginary time thermal par-tition function, but is relevant for the dynamic phononmodes of the condensate.The signature change of the metric takes place in manymodels of the early universe in cosmology, for quantumgravity and for cosmic strings . The transition of theLorentzian signature to Euclidean triggers a ghost insta-bility of the quantum vacuum . This corresponds toinstability of the magnon BEC in the Euclidean region,where the BEC decays as a false vacuum. Depending onthe experimental conditions, the decay rate may be longenough to simulate different mechanisms of the decay ofthe false magnon vacuum.We also found a difference in the phenomenology ofmagnon superfluidity for the isotropic He-B and thepolar phase with an easy axis anisotropy (see the Ap-pendix). In He-B the phenomenology corresponds toa BEC of magnons with the inertial mass M = ~ ω L / M eff = ~ ω L . However,the effective metric for these bosons coincides with themetric of the optical magnons. 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).
Appendix. Effective bosons in the precessingmagnon BEC
Here we discuss the phonon Hamiltonian H ( α, n ) = H BEC − µN = F BEC of the magnon condensate in thepolar and B-phase of He. The effective condensate freeenergy F BEC is the precession averaged free energy of thesuperfluid in the London limit. In the n ≪ n max limit, ithas the conventional form of the Ginzburg-Landau (GL)free energy of the BEC: F BEC = Z d r nS K ij ∇ i α ∇ j α + 12 S K ij n ∇ i n ∇ j n (50)+( ω L − µ ) n + ǫ ( n )= Z d r m k |∇ k Ψ | + 12 m ⊥ |∇ ⊥ Ψ | + ( ω L − µ ) | Ψ | (51)+ ǫ ( | Ψ | ) , Ψ = √ ne iα , | Ψ | = n = S (1 − cos β ) . Here m k and m ⊥ are effective inertial masses with( m − ) ij = K ij S = g ijS ~ ω L . (52)The gradient term in n corresponds to the vacuum pres-sure and gives the fourth-order correction to the spec-trum. As distinct from the conventional Bose gas, inour case the masses in the GL free energy do not neces-sary coincide with the true magnon masses. They can beconsidered as effective masses of the bosons forming themagnon BEC. The inertial mass in Eq.(52) correspondsto the effective invariant mass M eff = ~ ω L of the boson,which coincides with the invariant mass of the magnonin Eq.(11).Let us compare this with the magnon BEC of He-B inRef. . For small n ≪ n max , i.e. for β ≪
1, the gradientterms of α in the free energy correspond to the kineticenergy of the magnon BEC: h f Bgrad ( n → i = 12 ρ s ij v s i v s j , (53)where ρ s ij is the tensor of anisotropic superfluid densityand v s i is the superfluid velocity of magnon superfluid: ρ s ij = nm B ,ij , v s i = ~ (cid:0) m − (cid:1) ij ∇ j α , (54)where the matrix of masses m B ,ij for β ≪ m − ) ij = 2 g ijS ~ ω L . (55)In this case the inertial mass of effective boson corre-sponds to the invariant magnon mass M + = ~ ω L / α is J i = δF B δv s i = ~ n ∇ i α , (56) and similarly coincides with the linear momentum ofthe magnon condensate. It follows that the out-of-equilibrium magnon BEC in He-B at small n is verysimilar to the conventional BEC of bosonic particles withthe inertial mass equal to the invariant mass of the op-tical magnon M + = ~ ω L /
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