# Twenty years of magnon Bose condensation and spin current superfluidity in He-B

###### Abstract

20 years ago a new quantum state of matter was discovered and identified 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 . The observed dynamic quantum state of spin precession in superfluid He-B bears the properties of spin current superfluidity, Bose condensation of spin waves – magnons, off-diagonal long-range order and related phenomena of quantum coherence.

###### Keywords:

superfluid He, spin-current superfluidity, magnon Bose-Einstein condensation###### pacs:

67.30.er 75.30.Ds 67.10.Ba^{†}

^{†}journal: Journal of Low Temperature Physics

## 1 Introduction

Nature knows different types of ordered states.

One major class is represented by equilibrium macroscopic ordered states exhibiting spontaneous breaking of symmetry. This class contains crystals; nematic, cholesteric and other liquid crystals; different types of ordered magnets (antiferromagnets, ferromagnets, etc.); superfluids, superconductors and Bose condensates; all types of Higgs fields in high energy physics; etc. The important subclasses of this class contain systems with macroscopic quantum coherence exhibiting off-diagonal long-range order (ODLRO), and/or nondissipative superfluid currents (mass current, spin current, electric current, hypercharge current, etc.). The class of ordered systems is characterized by rigidity, stable gradients of order parameter (non-dissipative currents in quantum coherent systems), and topologically stable defects (vortices, solitons, cosmic strings, monopoles, etc.).

A second large class is presented by dynamical systems out of equilibrium. Ordered states may emerge under external flux of energy. Examples are the coherent emission from lasers; water flow in a draining bathtub; pattern formation in dissipative systems; etc.

Some of the latter dynamic systems can be close to stationary equilibrium systems of the first class. For example, ultra-cold gases in optical traps are not fully equilibrium states since the number of atoms in the trap is not conserved, and thus the steady state requires pumping. However, if the decay is small then the system is close to an equilibrium Bose condensate, and experiences all the corresponding superfluid properties. Bose condensation of quasiparticles whose number is not conserved is a timely topic in present literature: this is Bose condensation of magnons, rotons, phonons, polaritons, excitons, etc.

There are two different schools in the study of the Bose condensation of quasiparticles. In one of them, Bose condensation (and ODLRO) of quasiparticles is used for describing an equilibrium state with diagonal long-range order, such as crystals, and magnets (see Radu ; Giamarchi and references therein). This is somewhat contradictory, since the essentially non-equilibrium phenomenon of condensation of the non-conserved quasiparticles cannot be used for the description of a true equilibrium state (see e.g. Mills and Appendix H). Actually Bose condensation here serves as the instrument for the desciption of the initial stage of the soft instability which leads to symmetry breaking and formation of the true equilibrium ordered state of the first class. For example, using the language of Bose condensation of phonons one can describe the soft mechanism of formation of solid crystals Kohn . In the same way Bose condensation of magnons can be used for the description of the soft mechanism of formation of ferromagnetic and antiferromagnetic states (see e.g. Nikuni ). The growth of a single mode in the non-linear process after a hydrodynamic instability LandauHydro can be also discussed in terms of the ‘Bose condensation’ of the classical sound or surface waves Zakharov .

A second school considers Bose condensation of quasiparticles as a phenomenon of second class, when the emerging steady state of the system is not in a full thermodynamic equilibrium, but is supported by pumping of energy, spin, atoms, etc. The distribution of quasiparticles in these dynamic states is close to the thermodynamic equilibrium with a finite chemical potential which follows from the quasi-conservation of number of quasiparticles. In this way, recent experiments in Refs. Demokritov and Kasprzak may be treated as Bose condensation of magnons and exciton polaritons, respectively (see Appendix G and also Ref. Snoke ; the possibility of the BEC of quasiequilibrium magnons has been discussed in Ref. KalafatiSafonov ). The coherence of these dynamical states is under investigation Demidov .

But not everybody knows that the coherent spin precession discovered in superfluid He more than 20 years ago, and known as Homogeneously Precessing Domain (HPD), is the true Bose-Einstein condensate of magnons (see e.g. reviews Bunkov2007 ; BunkovVolovik2008 ). This spontaneously emerging steady state preserves the phase coherence across the whole sample. Moreover, it is very close to the thermodynamic equilibrium of the magnon Bose condensate and thus exhibits all the superfluid properties which follow from the off-diagonal long-range order (ODLRO) for magnons.

In the absence of energy pumping this HPD state slowly decays, but during the decay the system remains in the state of the Bose condensate: the volume of the Bose condensate (the volume of HPD) gradually decreases with time without violation of the observed properties of the spin-superfluid phase-coherent state. A steady state of phase-coherent precession can be supported by pumping. But the pumping need not be coherent – it can be chaotic: the system chooses its own (eigen) frequency of coherent precession, which emphasizes the spontaneous emergence of coherence from chaos.

## 2 HPD – magnon Bose condensate

### 2.1 Larmor precession

The crucial property of the Homogeneously Precessing Domain is that the Larmor precession spontaneously acquires a coherent phase throughout the whole sample even in an inhomogeneous external magnetic field. This is equivalent to the appearance of a coherent superfluid Bose condensate. It appears that the analogy is exact: HPD is the Bose-condensate of magnons.

The precession of magnetization (spin) occurs after the magnetization is deflected by an angle by the rf field from its equilibium value (where is an external magnetic field and is spin susceptibility of liquid He):

(1) | |||

(2) |

The immediate analogy FominLT19 says that in precession the role of the number density of magnons is played by the deviation of the spin projection from its equilibrium value:

(3) |

The number of magnons is a conserved quantity if one neglects the spin-orbit interaction. It is more convenient to work in the frame rotating with the frequency of the rf field, where the spin is stationary if relaxation is neglected. The free energy in this frame is

(4) |

Here is the Larmor frequency; is the gyromagnetic ratio of the He atom. The precession frequency plays the role of the chemical potential for magnons:

(5) |

while the local Larmor frequency plays the role of external potential; and is the energy of spin-orbit interaction. The properties of the precession depend on : stable precession occurs when is a concave function of . This is what occurs in He-B (see Appendix A).

### 2.2 Spectrum of magnons: anisotropic mass

The important property of the Bose condensation of magnons in He-B is that the mass of magnons is anisotropic, i.e. it depends on the direction of propagation (see Appendix B). The spectrum of magnons (transverse spin waves) is

(6) |

where the longitudinal and transverse masses depend on the tilting angle. Both masses are much smaller than the mass of the He atom:

(7) |

where is the Fermi energy in He liquid (see Appendix B). In the co-rotating frame the spectrum is shifted

(8) |

which corresponds to the chemical potential .

### 2.3 Bose condensation of magnons

The value is critical: when crosses the minimum of the magnon spectrum, the Bose condensation of magnons with occurs resulting in the phase-coherent precession of spins and spin superfluidity at . In He-B, the Bose condensate of magnons is almost equilibrium. Though the number density of magnons in precessing state is much smaller than the density number of He atoms

(9) |

their mass is also by the same factor smaller (see Eq.(7)). The critical temperature of the Bose condensation of magnons, which follows from this mass is

(10) |

The more detailed calculations gives even higher transition temperature (see Appendix C). In any case the typical temperature of superfluid He of order is much smaller than the condensation temperature, and thus the Bose condensation is complete.

### 2.4 ODLRO of magnons

In terms of magnon condensation, the precession can be viewed as the off-diagonal long-range order (ODLRO) for magnons. The ODLRO is obtained using the Holstein-Primakoff transformation

(11) |

In the precessing state of Eq.(1), the operator of magnon annihilation has a non-zero vacuum expectation value – the order parameter:

(12) |

So, plays the role of the modulus of the order parameter; the phase of precession plays the role of the phase of the superfluid order parameter; and the precession frequency plays the role of chemical potential, . Note that for the equilibrium planar ferromagnet, which also can be described in terms of the ODLRO, Eq.(12) does not contain the chemical potential (see Appendix H); as a result this analogy with magnon Bose condensation Nikuni ; Giamarchi (see also Hohenberg1971 ) becomes too far distant.

The precessing angle is typically large in HPD. The profile of the spin-orbit energy is such that at (i.e. at ) the equilibrium condensate corresponds to precession at fixed tipping angle (see Appendix A):

(13) |

and thus with fixed condensate density:

(14) |

### 2.5 Spin supercurrent

The superfluid mass current carried by magnons is the linear momentum of the Bose condensate (see Appendix D):

(15) |

Here we used the fact that and are canonically conjugated variables. This superfluid mass current is accompanied by the superfluid current of spins transferred by the magnon condensate. It is determined by the spin to mass ratio for the magnon, and because the magnon mass is anisotropic, the spin current transferred by the coherent spin precession is anisotropic too:

(16) | |||

(17) |

This superfluid current of spins is one more representative of superfluid currents known or discussed in other systems, such as the superfluid current of mass and atoms in superfluid He; superfluid current of electric charge in superconductors; superfluid current of hypercharge in Standard Model; superfluid baryonic cuurent and current of chiral charge in quark matter; etc. (recent review on spin currents is in Ref. SoninReview ).

This superfluidity is very similar to superfluidity of the A phase of He where only one spin component is superfluid VollhardtWolfle : as a result the superfluid mass current is accompanied by the superfluid spin current.

### 2.6 Two-domain structure of precession

The distinguishing property of the Bose condensate of magnons in He-B is that quasi-equilibrium precession has a fixed density of Bose condensate in Eq.(14). Since the density of magnons in the condensate cannot relax continuously, the decay of the condensate can only occur via the decreasing volume of the condensate.

This results in the formation of two-domain precession: the domain with the Bose condensate (HPD) is separated by a phase boundary from the domain with static equilibrium magnetization (non-precessing domain, or NPD). The two-domain structure spontaneously emerges after the magnetization is deflected by pulsed NMR, and thus magnons are pumped into the system (Fig. 1(a)). If this happens in an applied gradient of magnetic field, the magnons are condensed and collected in the region of the sample, where (Fig. 1(b)). They form the HPD there. This process is fully analogous to the separation of gas and liquid in the gravitational field: the role of the gravitational field is played by .

In the absence of the rf field, i.e. without continuous pumping, the precessing domain (HPD) remains in the fully coherent Bose condensate state, while the phase boundary between HPD and NPD slowly moves up so that the volume of the Bose condensate gradually decreases (Fig. 1(c)). The frequency of spontaneous coherence as well as the phase of precession remain homogeneous across the whole Bose condensate domain, but the magnitude of the frequency changes with time. The latter is determined by the Larmor frequency at the position of the phase boundary between HPD and NPD, ; in other words at the phase boundary the chemical potential of magnons corresponds to the onset of condensation.

## 3 Discussion

### 3.1 Observed superfluid properties of magnon Bose condensate

As distinct from the conventional Larmor precession, the phase coherent precession of magnetizaton in He-B has all the properties of the coherent Bose condensate of magnons. The main spin-superfluid properties of HPD have been verified already in early experiments 20 years ago 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 . These include spin supercurrent which transports the magnetization (analog of the mass current in conventional superfluids); spin current Josephson effect and phase-slip processes at the critical current. Later on a spin current vortex has been observed Vortex – a topological defect which is the analog of a quantized vortex in superfluids and of an Abrikosov vortex in superconductors (see Appendix D) .

The Goldstone modes of the two-domain structure of the Larmor precession have been also observed including the “sound waves” of the magnon condensate – phonos 5 (see Appendix E) and ‘gravity’ waves - surface waves at the interface between HPD and NPD SurfaceWaves .

### 3.2 Exploiting Bose condensate of magnons

The Bose condensation of magnons in superfluid He-B has many practical applications.

In Helsinki, owing to the extreme sensitivity of the Bose condensate to textural inhomogeneity, the phenomenon of Bose condensation has been applied to studies of supercurrents and topological defects in He-B. The measurement technique was called HPD spectroscopy HPDSpectroscopy ; HPDSpectroscopy2 . In particular, HPD spectroscopy provided direct experimental evidence for broken axial symmetry in the core of a particular quantized vortex in He-B. Vortices with broken symmetry in the core are condensed matter analogs of the Witten cosmic strings, where the additional symmetry is broken inside the string core (the so-called superconducting cosmic strings ewitten ). The Goldstone mode of the vortex core resulting from the spontaneous violation of rotational symmetry in the core has been observed 26 . The so-called spin-mass vortex, which is a combined defect serving as the termination line of the topological soliton wall, has also been observed and studied using HPD spectroscopy 27 .

In MoscowDmitriev , GrenobleGrenoble1 ; Grenoble2 and TokyoTokyo1 ; Sato2008 ; Tokyo2 , HPD spectroscopy proved to be extremely useful for the investigation of the superfluid order parameter in a novel system – He confined in aerogel.

There are a lot of new physical phenomena related to the Bose condensation, which have been observed after the discovery. Other coherently precessing spin states have been observed in He-B (see review paper BunkovVolovik2008 and Ref.Grenoble2 ) and also in He-A Sato2008 . These include in particular the compact objects with finite number of the Bose condensed magnons (see Appendix I). At small number of the pumped magnons, the system is similar to the Bose condensate of the ultracold atoms in harmonic traps, while at larger the analog of -ball in particle physics develops QBall .

The important property of the condensation of quasiparticles is that the BEC is the time dependent process. That is why it may experience instabilities which do not occur in the equilibrium condensates of stable particles. In 1989 it was found that the original magnon condensate – HPD state – looses its stability below about 0.4 T CatastrophExp and experiences catastrophic relaxation. This phenomenon was left unexplained for a long time and only recently the reason was established: in the low-temperature regime, where dissipation becomes sufficiently small, a Suhl instability in the form of spin wave radiation destroys the homogeneous precession Catastroph . However, the magnon BEC in harmonic traps and -balls are not destroyed.

In conclusion, the Homogeneously Precessing Domain discovered about 20 years ago in superfluid He-B represented the first example of a Bose condensate found in nature (if one does not take into account the strongly interacting superfluid liquid He with its tiny 7% fraction of the Bose condensed atoms).

## 4 Appendix A. Ginzburg-Landau energy

In He-B the energy of spin-orbit interaction is

(18) | |||

(19) |

Here is the so-called Leggett frequency; in typical NMR experiments , i.e. the spin-orbit interaction is small compared to Zeeman energy. This leads to the following Ginzburg-Landau potential FominLT19

(20) |

where is Heaviside step function. The total Ginzburg-Landau energy functional is

(21) |

where is the anisotropic mass in the spectrum of magnons, see below Sec. 5; the local Larmor frequency plays the role of the external potential acting on magnons; and the global precession frequency plays the role of the magnon chemical potential.

The precession frequency is shifted from the Larmor value when :

(22) |

The Bose condensate starts with the tipping angle equal to the so-called Leggett angle, .

## 5 Appendix B. Magnon spectrum

The spectrum of magnons in the limit of small spin-orbit interaction is

(23) |

where the longitudinal and transverse masses depend on the tilting angle. For superfluid He-B one has the following dependence:

where the parameters and are on the order of the Fermi velocity . In the simplified cases , when one neglects the spin-wave anisotropy (or in the limit of small tilting angle), the magnon spectrum in the limit of small spin-orbit interaction is

(24) |

where the isotropic magnon mass is:

(25) |

Since , the relative magnitude of the magnon mass compared to the bare mass of the He atom is

(26) |

where the Fermi energy .

The spectrum (24) is valid when , however the typical temperature of HPD is 0.3 which is an order of magnitude larger than the magnon gap at MHz. That is why one needs the spectrum in the broader range of :

(27) |

where the sign corresponds to magnons under discussion. Since thermal magnons are spin waves with linear spectrum , with characteristic momenta . The density of thermal magnons is .

The density of the condensed of magnons is small when compared to the density of atoms in He liquid

(28) |

But it is large when compared to the density of thermal magnons:

(29) |

## 6 Appendix C. Transition temperature

At first glance, the temperature at which condensation starts can be estimated as the temperature at which and are comparable. This gives

(30) |

This temperature is much smaller than the estimate in Eq.(10) which comes from the low-frequency part of the spectrum, but still is much bigger than .

However, since the gap in magnon spectrum is small, the condensation may occur even if the number density of the condsensed magnons (see also Refs. Demokritov ; Demidov ). The number of extra magnons which can be absorbed by thermal distribution is the difference of the distribution function at and . Since , it is determined by low energy Rayleigh-Jeans part of the spectrum:

(31) |

Maximum takes place when , which gives the dependence of transition temperature on the number of pumped magnons

(32) |

This gives the transition temperature

(33) |

In any case, at the typical temperatures of superfluid He of order the Bose condensation is complete. The hierarchy of temperatures in magnon BEC is thus

(34) |

Bose condensation of magnons in He-B is similar to the Bose condensation of ultrarelativistic particles with spectrum in the regime when

(35) |

For the Bose gas in laser traps, the hierarchy of temperatures is , where is the frequency in the harmonic trap Pit1999 .

## 7 Appendix D. Superflow

In the simple case of isotropic mass the kinetic energy of superflow of magnon BEC in the London limit is

(36) |

Here and are superfluid velocity and density of magnon superfluidity. Note that the total number of magnons enters , since the temperature is low, and the number of thermal magnons is negligibly small. In the Ginzburg-Landau regime this has the form:

(37) |

and the total GL functional is

(38) |

Circulation quantum is

(40) |

This demonstrates that the observed spin vortex Vortex with nonzero winding of has also circulation of the mass current. This is similar to the A phase of He with the superfluidity of only one spin component: the superfluid mass current is accompanied by the superfluid spin current.

## 8 Appendix E. Phonons in magnon superfluid and symmetry breaking field

The speed of sound in magnon superfluid is determined by compressibility of magnons gas, which is non-zero due to dipole interaction. Applying Eq.(38) in the HPD region, i.e. in the region where , and using the isotropic spin wave approximation (for anisotropic spin wave velocity see e.g. Ref. VolovikPhonons ) one obtains the sound with velocity:

(41) |

For close to the speed of sound is

(42) |

This sound observed in Ref. 5 is the Goldstone mode of the magnon Bose condensation.

The important property of magnon BEC is that the Goldstone boson (phonon) acquires mass (gap) due to the transverse RF field . The latter plays the role of the symmetry breaking field, since it violates the symmetry of precession VolovikPhonons . As a result, the extra term in the Ginzburg-Landau energy, , induced by depends explicitly on the phase of precession with respect to the direction of the RF-field in rotating frame. For one has

(43) |

This term adds the mass (gap) to the phonon spectrum:

(44) |

The gap of the Goldstone mode induced by the symmetry breaking field has been measured in Refs. PhononMass ; Skyba .

## 9 Appendix F. Critical velocities and vortex core

The speed of sound in magnon gas determines the Landau critical velocity of the counterflow at which phonons are created:

(45) |

For conventional condensates this suggests that the coherence length and the size of the vortex core should be:

(46) |

However, for magnons BEC in He-B this gives only the lower bound on the core size. The core is larger due to specific profile of the Ginzburg-Landau (dipole) energy in Eq.(20) which is strictly zero for . This leads to the special topological properties of coherent precession (see Ref. MisirpashaevVolovik1992 ). As a result the spin vortex created and observed in Ref. Vortex has a continuous core with broken symmetry, similar to vortices in superfluid He-A SalomaaVolovik1987 . The size of the continuous core is determined by the proper coherence length Fomin1987 which can be found from the competition between the first two terms in Eq.(38):

(47) |

This coherence length determines also the critical velocity for creation of vortices:

(48) |

For large tipping angles of precession the symmetry of the vortex core is restored: the vortex becomes singular with the core radius SoninVortex .

## 10 Appendix G. BEC in YIG

Magnons in yttrium-iron garnet (YIG) films have the quasi 2D spectrum: Demokritov ; Demidov

(49) |

where magnetic field is along ; the gap in the lowest branch GHz mK at Oe Demokritov and GHz at Oe Demidov ; the position of the minimum 1/cm Demokritov ; the anisotropic magnon mass can be probably estimated as with being somewhat bigger, both are of order of electron mass.

If one neglects the contribution of the higher levels and consider the 2D gas, the Eq.(31) becomes

(50) |

In 2D, all extra magnons can be absorbed by thermal distribution at any temperature without formation of Bose condensate. The larger is the number of the pumped magnons the closer is to , but never crosses . At large the chemical potential exponentially approaches from below and the width of the distribution becomes exponentially narrow:

(51) |

If one uses the 2D number density with the film thickness m and 3D number density cm, one obtains that at room temperature the exponent is

(52) |

If this estimation is correct, the peak should be extremely narrow, so that all extra magnons are concentrated at the lowest level of the discrete spectrum. However, there are other contributions to the width of the peak due to: finite resolution of spectrometer, magnon interaction, finite life time of magnons and the influence of the higher discrete levels .

In any case, the process of the concentration of extra magnons in the states very close to the lowest energy is the signature of the BEC of magnons. The main property of the room temperature BEC in YIG is that the transition temperature is only slightly higher than temperature, ; as a result the number of condensed magnons is small compared to the number of thermal magnons: . Situation with magnon BEC in He is the opposite, one has and thus .

## 11 Appendix H. Magnon BEC vs planar ferromagnet

Coherently precessing state HPD state in He-B has

(53) |

The coherent state in YIG has

(54) |

For equilibrium planar ferromagnet one has Hohenberg1971 ; Nikuni

(55) |

This means that the equilibrium planar ferromagnet can be also described in terms of the ODLRO.

Magnons were originally determined as second quantized modes in the background of stationary state with magnetization along axis. Both the static state in Eq.(55) and precessing states (53) and (54) can be interpreted as BEC of these original magnons. On the other hand, the stationary and precessing states can be presented as new vacuum states, the time independent and the time dependent vacua respectively. The excitations – phonons – are the second quantized modes in the background of a new vacuum. What is the principle difference between the stationary vacuum of planar ferromagnet and the time dependent vacuum of coherent precession?

The major point which distinguishes the HPD state (53) in He-B and the coherent precession (54) in YIG from the equilibrium magnetic states is the conservation (or quasi-conservation) of the charge . The charge is played by the spin projection in magnetic materials (or the related number of magnons) and by number of atoms in atomic BEC. This conservation gives rise to the chemical potential , which is the precession frequency in magnetic systems. On the contrary, the static state in Eq.(55) does not contain the chemical potential , i.e. the conservation is not in the origin of formation of the static equilibrium state; the chemical potential of magnons in a fully equilibrium state is always strictly zero, .

The spin-orbit interaction violates the conservation of , as a result the life time of magnons is finite. For the precessing states (53) and (54) this leads to the finite life time of the coherent precession. To support the steady state of precession the pumping of spin and energy is required. On the contrary, the spin-orbit interaction does not destroy the long-range magnetic order in the static state (55): this is fully equilibrium state which does not decay and thus does not require pumping: the life time of static state is macroscopically large and thus by many orders of magnitude exceeds the magnon relaxation time. That is why a planar ferromagnet (55) is just one more equilibrium state of quantum vacuum, in addition to the easy axis ferromagnetic state, rather than the magnon condensate.

The property of (quasi)conservation of the charge distinguishes the coherent precession from the other coherent phenomena, such as optical lasers and standing waves. For the real BEC one needs the conservation of particle number or charge during the time of equilibration. BEC occurs due to the thermodynamics, when the number of particles (or charge ) cannot be accommodated by thermal distribution, and as a result the extra part must be accumulated in the lowest energy state. This is the essence of BEC.

Photons and phonons can also form the true BEC (in thermodynamic sense) under pumping, again if the lifetime is larger than thermalization time. These BEC states are certainly different from such coherent states as optical lasers and from the equilibrium deformations of solids.

## 12 Appendix I. Finite-size BEC & -ball

When (), the spin-orbit interaction becomes nonzero at finite polar angle of vector of He-B orbital angular momentum:

(56) |

As a result the texture of vector forms the potential well for magnons.

Four regimes of magnon condensation are possible in the potential well, which correspond to four successive ranges of the charge (magnon number ). (i) At the smallest the interaction can be neglected and the non-interacting magnons occupy the lowest energy state in the potential well. (ii) With increasing the Thomas-Fermi regime of interacting magnons is reached. (iii) When the number of magnons is sufficiently large, they start to modify the potential well; this is the regime of the so-called -ball QBall . (iv) Finally when the size of the -ball reaches the dimension of the experimental cell the homogeneous BEC is formed – the HPD.

The first two regimes are similar to what occurs in atomic BEC in laser traps, though not identical. The difference comes from the 4-th order term in the GL energy (56) which demonstrates that the attractive interaction between magnons is determined by the texture

(57) |

In the simplest case of the spherically symmetric harmonic trap one has

(58) |

where is the harmonic oscillator frequency; the magnon interaction is normalized to its magnitude in the HPD state and to the speed of sound also in the HPD state.

The first two regimes can be qualitatively described using simple dimensional analysis. Let is the dimension of magnon gas as a function of the magnon number . Taking into account that one estimates the GL energy (38) of the condensate as:

(59) |

Here is the harmonic oscillator length; the prefactor is introduced to match the oscillator frequency after minimization over in the linear regime; and . Minimization at fixed magnon number gives two regimes.

(i) In the regime linear in (the regime of spin-waves) one obtains the -independent radius ; and the Bose condensation occurs at the lowest energy level which corresponds to the precession with frequency (chemical potential)

(60) |

(ii) At larger one obtains the analog of the Thomas-Fermi droplet whose size and precession frequency depend on :

(61) |

For comparison, the atomic BEC in the nonlinear regime, which can be obtained by the same procedure, is characterized by Pit1999

(62) |

where with being the scattering length.

(iii) The regime of -ball emerges when the density of magnons in center of the droplet is such that is not small, and the spin-orbit interaction starts to modify the potential well. This regime develops when approaches the characteristic value :

(63) |

(iv) Finally, the HPD emerges when the size of the droplet reaches the size of the cell, and the magnon number reaches the maximum value , where is magnon density in HPD.

A typical texture in the vessel is determined by the vessel geometry and thus by dimension of the cell, . In such cases, the corresponding parameters are

(64) |

Here cm is the dipole length. Since , the -ball regime develops at , and then -ball transforms to HPD when approaches . In the applied external gradient of magnetic field the two-domain state in Fig. 1 emerges earlier.

One may expect a similar finite size BEC of magnons in rotating He-A, where the trap for magnons is provided by the core of individual continuous vortex SalomaaVolovik1987 . Applying gradient of magnetic field one may study individual vortices by the magnon BEC tomography.

###### Acknowledgements.

I am grateful to Yu.M. Bunkov, S. Demokritov, V.V. Dmitriev, V.B. Eltsov, I.A. Fomin, M. Krusius and V.S. L’vov for illuminating discussions. This work was supported in part by ESF COSLAB Programme and by the Russian Foundation for Basic Research (grant 06-02-16002-a).## References

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