Bose-Einstein condensation (BEC) is a quantum phenomenon of formation of the collective quantum state, in which the macroscopic number of particles occupies the lowest energy state and thus is governed by a single wave function. Here we highlight the BEC in a magnetic subsystem -- the BEC of magnons, elementary magnetic excitations. Magnon BEC is manifested as the spontaneously emerging state of the precessing spins, in which all spins precess with the same frequency and phase even in the inhomogeneous magnetic field. We consider this phenomenon on example of spin precession in superfluid phases of $^3$He. The magnon BEC in these phases has all the properties of spin superfluidity. The states of the phase-coherent precession belong to the class of the coherent quantum states, which manifest themselves by superfluidity, superconductivity, quantum Hall effect, Josephson effect and many other macroscopic quantum phenomena.
Deep Dive into Magnon BEC and Spin Superfluidity: a 3He primer.
Bose-Einstein condensation (BEC) is a quantum phenomenon of formation of the collective quantum state, in which the macroscopic number of particles occupies the lowest energy state and thus is governed by a single wave function. Here we highlight the BEC in a magnetic subsystem – the BEC of magnons, elementary magnetic excitations. Magnon BEC is manifested as the spontaneously emerging state of the precessing spins, in which all spins precess with the same frequency and phase even in the inhomogeneous magnetic field. We consider this phenomenon on example of spin precession in superfluid phases of $^3$He. The magnon BEC in these phases has all the properties of spin superfluidity. The states of the phase-coherent precession belong to the class of the coherent quantum states, which manifest themselves by superfluidity, superconductivity, quantum Hall effect, Josephson effect and many other macroscopic quantum phenomena.
Last decade was marked by the fundamental studies of mesoscopic quantum states of dilute ultra cold atomic gases in the regime where the de Broglie wavelength of the atoms is comparable with their spacing, giving rise to the phenomenon of Bose-Einstein condensation (see, e.g. Ref. [1]). The formation of the Bose-Einstein condensate (BEC) -accumulation of the macroscopic number of particles in the lowest energy state -was predicted by Einstein in 1925. In ideal gas, all atoms are in the lowest energy state in the zero temperature limit. In dilute atomic gases, weak interactions between atoms produces a small fraction of the non-condensed atoms. In the only known bosonic liquid 4 He which remains liquid at zero temperature, the BEC is strongly modified by interactions. The depletion of the condensate due to interactions is very strong: in the limit of zero temperature only about 10% of particles occupy the state with zero momentum. Nevertheless, BEC still remains the key mechanism for the phenomenon of superfluidity in liquid 4 He: due to BEC the whole liquid (100% of 4 He atoms) forms a coherent quantum state at T = 0 and participates in the non-dissipative superfluid flow. Superfluidity is a very general quantum property of matter at low temperatures, with variety of mechanisms and possible nondissipative superfluid currents. These include supercurrent of electric charge in superconductors and mass supercurrent in superfluid 3 He, where the mechanism of superfluidity is the Cooper pairing; hypercharge supercurrent in the vacuum of Standard Model of elementary particle physics, which comes from the Higgs mechanism; supercurrent of color charge in a dense quark matter in quantum chromo-dynamics; etc. All these supercurrents have the same origin: the spontaneous breaking of the U (1) symmetry related to the conservation of the corresponding charge or particle number, which leads to the so called off-diagonal long-range order.
Formally, the phenomenon of superfluidity requires the conservation of charge or particle number. However, the consideration can be extended to systems with a weakly violated conservation law, including a system of sufficiently long-lived quasiparticles -discrete quanta of energy that can be treated as real particles in condensed matter. Here we shall consider the spin superfluidity -superfluidity in the magnetic subsystem of a condensed matter, which is represented by BEC of magnons -quanta of excitations of the magnetic subsystem, and is manifested as the spontaneous phase-coherent precession of spins first discovered in 1984 [2,3].
At high temperatures, spins of atoms are in a disordered paramagnetic state, which is similar to the high temperature phase of a weakly interacting gas. With cooling the magnetic subsystem typically experiences a transition into an ordered state, in which magnetic moments are correlated at long distances. In cases when the magnetic U(1) symmetry is spontaneously broken, some people describe this phenomenon in terms of BEC of magnons [4,5,6]. Let us stress from the beginning that there is the principal difference between the magnetic ordering and the BEC of quasiparticles which we are discussing in this review: i. In some magnetic systems, the symmetry breaking phase transition starts when the system becomes softly unstable towards growth of one of the magnon modes. The condensation of this mode leads finally to the formation of the true equilibrium ordered state. In the same manner, the Bose condensation of phonon modes may serve as a soft mechanism of formation of the equilibrium solid crystals [7]. But this does not mean that the final crystal state is the Bose condensate of phonons. On the contrary, BEC of quasiparticles is in principle a non-equilibrium phenomenon, since quasiparticles (magnons) have a finite life-time. In our case magnons live long enough to form a state very close to thermodynamic equilbrium BEC, but still it is not an equilibrium. In the final equilibrium state at T = 0 all the magnons will die out.
ii. The ordered magnetic states are static equilibrium states which have diagonal longrange order. The magnon BEC is a dynamic state characterized by the off-diagonal longrange order, which is the main signature of spin superfluidity.
To prove that BEC of quasiparticles does really occur in a magnetic system, one should demonstrate the spontaneous emergence of coherence, and to show the consequences of the coherence: spin superfluidity, which in particular includes the observation of interference between two condensates.
We shall demonstrate that the finite life-time of magnons, and non-conservation of spin due to the spin-orbital coupling do not prevent the coherence of the magnon BEC. The gas of magnons can live a relatively long time, particularly at very low temperatures, sufficiently enough for formation of a coherent magnon condensate. The non-conservation leads to a decrease of the number of magnons in the Bose gas until it dis
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