Pis'ma v ZhETF, vol. 98, iss. 8, pp. 539-542
© 2013 October 25
Spin-motive force and orbital-motive force: from magnon Bose—Einstein
condensation to chiral Weyl superfluids
G. E. Volovik^
Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 AALTO, Finland Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia Submitted 4 September 2013
Spin-motive geometric force acting on electrons in metallic ferromagnets is extended to spin-motive force in magnon BEC, which is represented by phase-coherent precession of magnetization, and to the orbital-motive force in superfluid 3He-A. In 3He-A there are two contributions to the orbital-motive force. One of them comes from the chiral nature of this liquid. Another one originates from chiral Weyl fermions living in the vicinity of the topologically protected Weyl points, and is related to the phenomenon of chiral anomaly.
DOI: 10.7868/S0370274X13200095
1. Introduction. A spin-motive force is a force acting on electron from the magnetization. It was introduced for metallic ferromagnets [1-7], where it reflects the conversion of the magnetic energy of a fer-romagnet into the electrical energy of the conduction electrons. The spin-motive force includes the geometric force which originates from the Berry phase in spin dynamics [2]. In metals, the spin-motive force reflects the intricate coupling between the electric current and the collective magnetic degrees of freedom. The inverse phenomenon is the effect of electric currents on the dynamics of magnetization in ferromagnets. It is now a concept relevant to electronic devices.
The collective magnetic degrees of freedom exist not only in ferromagnets, but also in other systems, such as Bose-Einstein condensation of magnons manifested in the phase-coherent precession (the so-called magnon BEC, see review [8]) and the chiral Weyl superfluid 3He-A, which is the orbital ferromagnet. We extend here the concept of the spin-motive geometric force to spin-motive force in magnon BEC and to the geometric orbital-motive force in 3He-A. In 3He-A there are two contributions to the orbital geometric force: one is related to the chiral nature of this liquid and another one is the property of the topological Weyl superfluid, which experiences the chiral anomaly.
2. Spin-motive force in magnon BEC. The geometric force acting on atoms of 3He in the precessing state can be obtained from the geometric spin-motive force discussed in Ref. [2] for ferromagnets:
F!pi
in-motive
h
= 2 • (dtm x Vjih). (1)
Here Fspin-motiVe is the geometric spin-motive force; m is the unit vector parallel to the local spin magnetization; n+ — n- is spin polarization. The origin of this force is the Berry phase. The collective spin dynamics is governed by SU(2) symmetry group. The conduction electrons feel the U (1) subgroup of spin rotations about the local magnetization as the gauge field. This is one of the numerous examples of emergent gauge fields in many-body systems (other types of the gauge field originating from spin degrees of freedom can be found in [9, 10]). The "electric" field component of the effective U (1) gauge field is
Efeom = ■
(dtm x Vjih).
(2)
e-mail: volovik@boojum.hut.fi
This electric field drives spin-up and spin-down conduction electrons in opposite directions, inducing a spin current. In the presence of spin polarization due spontaneous magnetization emerging in metallic ferromagnets, this field produces the spin-motive force in Eq. (1) giving rise to electric current.
Eq. (1) can be also applied to magnon BEC - the spontaneously precessing magnetization, which emerges in supefluid 3He-B in pulsed NMR experiments. Shortly after the pulse the state of the coherent spin precession is formed:
S(r,t) = S m(r,t), (3)
m(r,t) = zcos ¡3 (r) + sin¡(r)(Xcos wt + ysin wt). (4)
Here w is the global frequency of free precession: it is the same in the whole sample even if the tipping angle of the deflected magnetization ¡3 (r) is coordinate dependent. The phase coherent precession is the signature
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G. E. Volovik
of magnon BEC. The direction of magnetic field - the axis of precession - is assumed along the axis I. The modulus of the deflected spin is the same as in equilibrium non-precessing states: S = xH/7, where x is magnetic susceptibility, and 7 is the gyromagnetic ratio of 3He atom. Since S is the difference between the density of spin-up and spin-down atoms in applied magnetic field S = | (n+ — n_), one obtains from Eq. (1) the following spin-motive force - the transfer of linear momentum density from the collective magnetic subsystem (magnon BEC) to the mass current:
Fspi
n-motive
Sm • (dtrii x V^m),
(5)
which for the space coherent precession in Eq. (4) reads:
Fspin_motive=Sm • (dtm x Viih) = SuVi cos 3= — ¡ViU.
(6)
Here in the last equation we used the language of magnon BEC, in which n = (S — Sz)/fr is the magnon density and j = fru plays the role of their chemical potential. In a full equilibrium, i.e. without precession, the chemical potential of magnons is zero, since their number is not conserved. In 3He-B the life time of coherent precession is large compared to the time of the formation of the coherent state. That is why for such quasiequilib-rium state the magnon number is quasi-conserved.
In the limit of vanishing dissipation the magnon chemical potential j is nonzero and is well determined, being equal to the global frequency of precession, which in turn depends on the number of the pumped magnons. In this limit, when losses of magnetization can be neglected and the spin projection Sz on magnetic field can be considered as conserved quantity, the precess-ing state becomes an example of the system with spontaneously broken time translation symmetry and offdiagonal long range order (ODLRO). In a given case the ODLRO is based on the operator of spin creation (S+) = Ssin/ ei^t+ia [11]. In principle, all the systems with off-diagonal long range order demonstrate the broken time translation symmetry in the ground state, assuming that it is the ground state under condition of fixed value of some conserved (or quasi-conserved) quantity, or the fixed value of the corresponding chemical potential.
Example of ODLRO is also provided by the precession of partially trapped vortex line in 3He-B, where hours long oscillations have been experimentally observed [12-14]. In the limit of vanishing dissipation the projection Lz of the orbital angular momentum of the liquid can be considered as conserved quantity. Then the precessing state of a vortex can be represented in terms of the ODLRO based on the operator of creation
of the orbital angular momentum (L+) « eiUt+ia. This coherent precession can be also considered as BEC of excitations propagating along the vortex - Kelvin waves or kelvons.
3. Orbital-motive force and chiral anomaly.
The Berry-phase geometric force emerges not only from spin. It may also come from other internal degrees of freedom, such as pseudospin - valley spin, isotopic band spin, Bogoliubov-Nambu spin, etc. The low-frequency dynamics of the order parameter in the broken symmetry systems often induces the transport of mass, electric charge and spin, see e.g. [15]. We consider here the geometric force generated by dynamics of the orbital angular momentum. Example is provided by chiral superfluid 3He-A, which is polarized in terms of the orbital angular momentum (the chirality of 3He-A is observed in recent experiments, see [16-18]). The corresponding geometric effect of transformation of orbital degees of freedom to the mass current is known under the name of Mermin-Ho relation [19], which leads to the following dynamic equation for superfluid velocity [20]:
dt{rrivs) — V/x = 1 • (dtl x VI). (7)
Here vs is superfluid velocity and l is the unit vector parallel to the local orbital momentum density L. The inverse effect - oscillations of the orbital vector l generated by applied heat current - have been observed in [21] (see also [22]).
At T = 0 the orbital momentum of superfluid 3He-A is fully polarized with momentum fr/2 per each atom (or fr per each Cooper pair). This means that n+ — n- = n3, where n3 is the particle density of 3He atoms, and L = (fr/2)n3l . When multiplied by n3, the right-hand side of Eq. (7) gives the orbital-motive geometric force acting from the collective orbital momentum degrees of freedom to the mass current:
^orbital-motive = . x (g)
However, this is not the whole story. Chiral super-fluid 3He-A belongs to the class of topological Weyl su-perfluids. These substances have topologically protected gap nodes in the fermionic spectrum - Weyl points, which represent the Berry phase magnetic monopoles in momentum space [23]. Close to any of two Weyl points, fermionic quasiparticles behave as chiral Weyl fermions. As a consequence, there is another geometric orbitalmotive force, which comes from the Adler-Bell-Jackiw chiral anomaly [24] (on anomalies in dynamics of chiral liquids including the dense quark-gluon matter in QCD and hypothetical Weyl materials see Refs. [25-29] and references therein):
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Spin-motive force and orbital-motive force.
541
7-ichiral-anomaly 1 Z r> ^
i'i = 2^rPFkii ■ E =
Here B = (pF/h)V x 1 and E = — (pF/h)dt1 are effective "magnetic" and "electric" fields acting on chiral Weyl quasiparticles in 3He-A. This is another example of emergent U(1) gauge field in condensed matter. In a given case the gauge field emerges as collective mode of the fermionic vacuum, which contains the Berry magnetic monopoles in p-space.
Though the two geometric forces have different origin, they have common property, which is manifested in their effect on dynamics of a continuous vortex - doubly quantized vortex, which represents the skyrmion in the 1-field. When integrated over the smooth core of the vortex-skyrmion movi
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