Superfluid He-3. Thought of the Day 130.0

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At higher temperatures 3He is a gas, while below temperature of 3K – due to van der Walls forces – 3He is a normal liquid with all symmetries which a condensed matter system can have: translation, gauge symmetry U(1) and two SO(3) symmetries for the spin (SOS(3)) and orbital (SOL(3)) rotations. At temperatures below 100 mK, 3He behaves as a strongly interacting Fermi liquid. Its physical properties are well described by Landau’s theory. Quasi-particles of the 3He (i.e. 3He atoms “dressed” into mutual interactions) have spin equal to 1/2 and similar to the electrons, they can create Cooper pairs as well. However, different from electrons in a metal, 3He is a liquid without a lattice and the electron-phonon interaction, responsible for superconductivity, can not be applied here. As the 3He quasiparticles have spin, the magnetic interaction between spins rises up when the temperature falls down until, at a certain temperature, Cooper pairs are created – the coupled pairs of 3He quasiparticles – and the normal 3He liquid becomes a superfluid. The Cooper pairs produce a superfluid component and the rest, unpaired 3He quasiparticles, generate a normal component (N -phase).

A physical picture of the superfluid 3He is more complicated than for superconducting electrons. First, the 3He quasiparticles are bare atoms and creating the Cooper pair they have to rotate around its common center of mass, generating an orbital angular momentum of the pair (L = 1). Secondly, the spin of the Cooper pair is equal to one (S = 1), thus superfluid 3He has magnetic properties. Thirdly, the orbital and spin angular momenta of the pair are coupled via a dipole-dipole interaction.

It is evident that the phase transition of 3He into the superfluid state is accompanied by spontaneously broken symmetry: orbital, spin and gauge: SOL(3)× SOS(3) × U(1), except the translational symmetry, as the superfluid 3He is still a liquid. Finally, an energy gap ∆ appears in the energy spectrum separating the Cooper pairs (ground state) from unpaired quasiparticles – Fermi excitations.

In superfluid 3He the density of Fermi excitations decreases upon further cooling. For temperatures below around 0.25Tc (where Tc is the superfluid transition temperature), the density of the Fermi excitations is so low that the excitations can be regarded as a non-interacting gas because almost all of them are paired and occupy the ground state. Therefore, at these very low temperatures, the superfluid phases of helium-3 represent well defined models of the quantum vacua, which allows us to study any influences of various external forces on the ground state and excitations from this state as well.

The ground state of superfluid 3He is formed by the Cooper pairs having both spin (S = 1) and orbital momentum (L = 1). As a consequence of this spin-triplet, orbital p-wave pairing, the order parameter (or wave function) is far more complicated than that of conventional superconductors and superfluid 4He. The order parameter of the superfluid 3He joins two spaces: the orbital (or k space) and spin and can be expressed as:

Ψ(k) = Ψ↑↑(kˆ)|↑↑⟩ + Ψ↓↓(kˆ)|↓↓⟩ + √2Ψ↑↓(kˆ)(|↑↓⟩ + |↓↑⟩) —– (1)

where kˆ is a unit vector in k space defining a position on the Fermi surface, Ψ↑↑(kˆ), Ψ↓↓(kˆ) a Ψ↑↓(kˆ) are amplitudes of the spin sub-states operators determined by its projection |↑↑⟩, |↓↓⟩ a (|↑↓⟩ + |↓↑⟩) on a quantization axis z.

The order parameter is more often written in a vector representation as a vector d(k) in spin space. For any orientation of the k on the Fermi surface, d(k) is in the direction for which the Cooper pairs have zero spin projection. Moreover, the amplitude of the superfluid condensate at the same point is defined by |d(k)|2 = 1/2tr(ΨΨH). The vector form of the order parameter d(k) for its components can be written as:

dν(k) = ∑μ Aνμkμ —– (2)

where ν (1,2,3) are orthogonal directions in spin space and μ (x,y,z) are those for orbital space. The matrix components Aνμ are complex and theoretically each of them represents possible superfluid phase of 3He. Experimentally, however, only three are stable.

phasediagramLooking at the phase diagram of 3He we can see the presence of two main superfluid phases: A – phase and B – phase. While B – phase consists of all three spin components, the A – phase does not have the component (|↑↓⟩ + |↓↑⟩). There is also a narrow region of the A1 superfluid phase which exists only at higher pressures and temperatures and in nonzero magnetic field. The A1 – phase has only one spin component |↑↑⟩. The phase transition from N – phase to the A or B – phase is a second order transition, while the phase transition between the superfluid A and B phases is of first order.

The B – phase occupies a low field region and it is stable down to the lowest temperatures. In zero field, the B – phase is a pure manifestation of p-wave superfluidity. Having equal numbers of all possible spin and angular momentum projections, the energy gap separating ground state from excitation is isotropic in k space.

The A – phase is preferable at higher pressures and temperatures in zero field. In limit T → 0K, the A – phase can exist at higher magnetic fields (above 340 mT) at zero pressure and this critical field needed for creation of the A – phase rises up as the pressure increases. In this phase, all Cooper pairs have orbital momenta orientated in a common direction defined by the vector lˆ, that is the direction in which the energy gap is reduced to zero. It results in a remarkable difference between these superfluid phases. The B – phase has an isotropic gap, while the A – phase energy spectrum consists of two Fermi points i.e. points with zero energy gap. The difference in the gap structure leads to the different thermodynamic properties of quasiparticle excitations in the limit T → 0K. The density of excitation in the B – phase falls down exponentially with temperature as exp(−∆/kBT), where kB is the Boltzman constant. At the lowest temperatures their density is so low that the excitations can be regarded as a non-interacting gas with a mean free path of the order of kilometers. On the other hand, in A – phase the Fermi points (or nodes) are far more populated with quasiparticle excitations. The nodes orientation in the lˆ direction make the A – phase excitations almost perfectly one-dimensional. The presence of the nodes in the energy spectrum leads to a T3 temperature dependence of the density of excitations and entropy. As a result, as T → 0K, the specific heat of the A – phase is far greater than that of the B – phase. In this limit, the A – phase represents a model system for a vacuum of the Standard model and B – phase is a model system for a Dirac vacuum.

In experiments with superfluid 3He phases, application of different external forces can excite the collective modes of the order parameter representing so called Bose excitations, while the Fermi excitations are responsible for the energy dissipation. Coexistence and mutual interactions of these excitations in the limit T → 0K (in limit of low energies), can be described by quantum field theory, where Bose and Fermi excitations represent Bose and Fermi quantum fields. Thus, 3He has a much broader impact by offering the possibility of experimentally investigating quantum field/cosmological theories via their analogies with the superfluid phases of 3He.

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