Liquid Crystal-like Nature of Superfluid ³He: Textural Transitions Induced by Superfluid Flow

Introduction

Anisotropy
Liquid crystals have properties which are associated with both liquids and solids. The viscosity is one example of liquid-like property that liquid crystals have. The crystal-like behavior is seen by direction-dependent (or anisotropic) viscosity of liquid crystals.  The anisotropy may not be so surprising for liquid crystals which are made up of large non-symmetric molecules. It is very surprising that spherically symmetric atoms of  ³He, when condensed into liquid phase and cooled down to ultra-low temperatures (below ~ 3 mK) to enter its superfluid phase, also possess anisotropic properties. The ultra-low temperature liquid ³He behaves like a liquid crystal! We carried out experiments to measure how imposed superfluid flows can change the liquid property.

Order Parameter and Texture
The state of a superfluid system is specified a quantity called an order parameter. The order parameter for "conventional" low temperature superconductors such as aluminum and lead is a scalar quantity. In contrast, the order parameter for superfluid ³He is a matrix and can be specified in terms of two unit vector quantities, d vector in spin space and l vector in orbital space. The spatial variation of  l vector is called a texture. The texture is determined by the externally applied magnetic field, the boundary walls of the container and imposed superfluid flow. The ³He superfluid system adjusts the textures such that the total free energy is minimized.

Superfluid Phases of ³He

A schematic phase diagram of ³He is plotted below as function of pressure and temperature. Note that the temperature is shown with a logarithmic scale. ³He remains in liquid state to the lowest temperatures achieved up to the pressure of about 34 bar. It is a "normal" liquid down to the pressure-dependent superfluid transition temperature Tc. The normal liquid has many properties analogous to liquid water. At temperature below Tc, it is a superfluid liquid. The superfluid liquid can flow in a circular loop "for ever"(as long as it is kept cold). In the ³He superfluid, ³He atoms form many Cooper pairs and "condense" to execute coherent motion governed by a single macroscopic quantum wavefunction. If no external magnetic field is applied, there are two superfluid phases, A and B, bounded by pressure and temperature dependent phase boundary lines. When an external magnetic field is applied, the A phase region increases and the B phase region shrinks (disappears at fields greater than about 5000 gauss). In addition, a distinct phase called A₁ phases appears between the A phase and the normal phase. In our laboratory, magnetic fields as high as 150,000 gauss is applied on liquid ³He to produce A₁ phase with a temperature width of 0.75 mK. We focus on the A₁ phase in our experiment.

Properties of A₁ Phase

In A₁ superfluid phase, the nuclear magnetic moments (or the spins) of the Cooper pairs are totally aligned parallel to the externally applied magnetic field. The superfluid is a ferromagnet.  If the  A₁ superfluid is passed through a tiny porous material such that only superfluid can go through (a superleak), what comes out is totally spin-polarized!  Magnetic field gradients can be used to move the superfluid.

Free Energy of A₁ Phase
The magnetic energy favors the magnetic moment of the Cooper pairs to be aligned along the applied field direction (z direction).  The dipolar energy favors the l vector to lie in the x-y plane.  The Cooper pair depairing energy favors l vector to point perpendicularly to boundary solid walls.  The bending energy favors the spatial bending of l vector as little as possible.  The kinetic energy is minimized if the l vector is aligned along superfluid flow velocity.

Spin-Entropy Wave (Second Sound) in Superfluid ³He A₁ Phase

The spin entropy wave is analogous to the familiar second sound in superfluid ⁴He.  In the superfluid ⁴He second sound, the normal and superfluid components move in 180^o out of phase such that there is temperature oscillation but no pressure oscillation.  In superfluid ³He A₁ phase, similar out of phase motion occurs with spin density oscillation and no pressure oscillation.  The temperature (or entropy) effect is small.  Thus the mode is called spin-entropy wave in A₁ phase.  The velocity of the spin-entropy wave is given by:

Note that Planck's constant divided by 2p appears in the equation.  Here, r_s, r_n, r and are the superfluid component density, normal component density and total density, respectively.  c is the magnetic susceptibility, g the gyromagnetic ratio and m is the mass of ³He.  The important thing is that the velocity of propagation of the spin-entropy wave is anisotropic, or direction dependent, owing to the anisotropy in the superfluid component density (the first factor in the parenthesis in the above equation).  The amount of anisotropy is determined by the relative angle between the l vector and the direction of propagation vector k. The velocity is greatest when l is perpendicular to k and is smallest when l is parallel to k.

Spin-Entropy Wave Resonator Cell

The cell is rectangular in shape(9.3 mmx x 7.2 mmy x 8.3 mmz).  A second sound transducer is attached to each face of the cell.  There are six transducers in all.  The idea is to manipulate the l vector texture using one transducer to induce superflow in the cell.  The effect of l manipulation is measured by the other two pairs of transducers.  Working at the resonance frequencies of the cell makes it a sensitive probe.

The second sound transducer is an oscillating superleak variety.  The counterflow needed for second sound between the normal and superfluid components is generated by a forced oscillation of a flexible porous membrane whose motion drives the normal component but not the superfluid component.

Abrupt Change in Spin Entropy Wave Propagation

Examples of resonance spectra along x direction are shown below.  Both in-phase and quadruture signals are shown as function of drive frequency.  In (A), the drive (123 Hz) in the y direction is zero.  The resonance frequency is at 96 Hz.  Note that a  non-linearity in the resonance spectrum is already present even in y-drive is turned off.  As the y-drive level is increased up to a threshold value, the x spectra remain the same.  (B) shows the dramatic abrupt change in spectra when the threshold y-drive level is exceeded.  The "resonance" frequency shifts to 110 Hz.  The increase in frequency corresponds to a change in effective superfluid density of more than 30 %!  The large shift in resonance frequency implies a large change in l texture.  The signal levels are low owing to the small dimensions of the transducers but are reproducible.  The blue lines are the results from simulations below with no adjusted parameters.

Simulation of l vector Texture with Superflow

Minimizing Free Energy

Assuming the external magnetic field to be applied along z direction, the l vector lies in the x-y plane.  The l texture may be represented by the spatially dependent q = cos^-1(x^.l).  The sum of  flow and gradient energies is given by E below.  The spatially dependent q is numerically searched for a given sound  flow field.

for given flow field. v_x(x,y) and v_y(x,y) are the velocity amplitudes established at different frequencies, and f₀ = (3/16)(m/m^*)((^h/_2p)/2 m)²r_s||.

Textural Transition - Interpretation of Observations

The textures below show the results of simulations.  (A) is the quiescent texture when the y-drive level is relatively small.  The texture does not vary much until a threshold drive level is exceeded.  (B) below shows the sudden jump in texture when the threshold drive level is exceeded.  It is a "y-polarized" texture.  The textural transition as these simulations show are consistent with our observations.  The blue lines in the spectra data figure above are derived from these simulations.

From "Explosive Textural Polarization and Non-Equilibrium Phase Transitions in Superfluid ³He-A₁,"
P. G. N. deVegvar, K. Ichikawa and H. Kojima,
Phys. Rev. Lett., vol. 83, p. 1806 (1999).