Topology in superfluid3He: Recent results

We discuss: the topological phase transition in nonsingular vortices in³He-A; vortices in³He-B and solitons terminating on strings; topological defects of the A-B interface: the interaction of continuous A-phase vortices with singular B-phase vortices across the interface; extended degeneracy and topology of Larmor precession; and internal topology in thin³He-A film, responsible for chiral edge states of fermions and QHE.     

Transition to Superfluid Turbulence

Turbulence in superfluids depends crucially on the dissipative damping in vortex motion. This is observed in the B phase of superfluid ³He where the dynamics of quantized vortices changes radically in character as a function of temperature. An abrupt transition to turbulence is the most peculiar consequence. As distinct from viscous hydrodynamics, this transition to turbulence is not governed by the velocity-dependent Reynolds number, but by a velocity-independent dimensionless parameter 1/q which depends only on the temperature-dependent mutual friction—the dissipation which sets in when vortices move with respect to the normal excitations of the liquid. At large friction and small values of the dynamics is vortex number conserving, while at low friction and large vortices are easily destabilized and proliferate in number. A new measuring technique was employed to identify this hydrodynamic transition: the injection of a tight bundle of many small vortex loops in applied vortex-free flow at relatively high velocities. These vortices are ejected from a vortex sheet covering the AB interface when a two-phase sample of ³He-A and ³He-B is set in rotation and the interface becomes unstable at a critical rotation velocity, triggered by the superfluid Kelvin–Helmholtz instability.      

The superflow state of 3He-B at a diffusive wall. Quasiclassical calculations

We report on first computations considering effects of a rough wall on the counterflow state in superfluid ³He-B for high flow velocities. Using the quasiclassical Green's-function formalism supplemented by the boundary conditions for a diffusive wall, we calculate the order-parameter field and the supercurrent near a container wall for various pressures and temperatures. One of our results is that the current density at the wall as a function of the flow has a maximum at the velocity which is about half of the pair breaking velocity.     

A–BTransition of Superfluid 3He with a Film Geometry

We have performedcwNMR experiments on superfluid ³ He confined to a parallel-plate geometry with a m scale spacing for a wide pressure range. A static field was applied parallel or perpendicular to the plate surface. The spectra of two absorption signals, a main and a satellite, have been observed below the superfluid transition temperature in a parallel field. As the temperature decreased, the main signal decreased with shifts to higher frequencies, and the satellite grew with shifts to much higher frequencies. From the temperature dependence of these signals and the result in the perpendicular field, it is confirmed that the main signal and the satellite correspond to the A phase signal (ABM state) and the B phase signal (BW state), respectively. The temperature dependence of the two signals indicates that a phase transition from the A phase to the B phase occurs with decreasing temperature. By analyzing these signals, we determine A–B transition temperatures experimentaly. TheA–Btransition temperature normalized by the superfluid transition temperature is 0.95 at 20 bar, and decreased further to 0.70 at 0 bar for a thickness of 0.88 m for pure ³ He. The values of T_AB/T_C were slightly elevated when covering the surface with 4.5 layers of ⁴ He film, which suggests that this transition is also influenced by the surface condition.     

Precessing Vortex Motion and Instability in a Rotating Column of Superfluid 3He-B

In a rotating circular cylinder of superfluid ³He-B, an evolving vortex expands longitudinally such that its end point describes a helically spiralling trajectory along the cylinder wall. The spiral motion is found to give rise to a periodically oscillating NMR signal, which is brought about by the modulation in the superfluid counterflow and its influence on the “flare-out” order parameter texture. The new NMR signal becomes observable within a narrow temperature interval close to the onset temperature of turbulence, when new vortices are continuously generated by the single-vortex instability at the cylindrical wall at a slow rate, ～1 vortex/s. We use numerical vortex filament calculations to examine the precessing motion of the evolving vortices, while they expand towards their stable state as rectilinear line vortices.     

Measurements on a Dynamic A-B Phase Boundary in Superfluid 3He at Very Low Temperatures

We present preliminary measurements of the dynamics of a moving A-B phase interface in superfluid ³ He at temperatures below 0.2T _c We initially stabilise the interface at low temperatures with a shaped magnetic field. We can then move the interface in a controlled manner by applying small additional time-dependent fields. The interface is created inside a quasiparticle radiator consisting of a cylindrical chamber in weak thermal contact with the refrigerant. Vibrating wire resonators inside the radiator allow us to monitor the temperature of the superfluid and to infer the heat generated by the interface motion. When we oscillate the interface at low frequencies, we measure spectacular oscillatory swings of the liquid temperature arising from the enormous change in the low lying density of states as the volume of the A-phase superfluid is alternately compressed and expanded. We have also observed hysteresis in the transition as a function of magnetic field. In particular, we observe a small history-dependent super-magnetisation of the B-phase prior to A-phase nucleation in the experimental chamber. When the system is in the metastable super-magnetised state we are able to observe a higher nucleation probability of the A-phase when the cryostat is exposed to neutrons.     

Boojums Connecting Singular and Continuous Vortices in Two-Component Bose-Einstein Condensates

A boojum is a topological defect that can form only on the surface of an ordered medium such as superfluid ³He and liquid crystals. By analogy with superfluid ³He, we numerically create boojums between two phases with different vortex structures in two-component BECs where the intercomponent interaction is spatially dependent. The detailed structure of the boojums is revealed by investigating its density distribution, effective superflow vorticity and pseudospin texture.     

Surface Effects of 4He Coating on the Viscosity and the Slip Length of Normal and Superfluid 3He

We have measured the viscosity, , and the slip length, , of normal and superfluid ³ He using a torsional oscillator with a thick sample space. We coated the oscillating surface with 2.5 layers of ⁴ He film to study how the ⁴ He thin film changes the scattering mechanism of ³ He quasiparticles at the cell wall at 5 bar and 21 bar. In the normal phase, the temperature dependence of the viscosity was changed a little by the ⁴ He film at 21 bar but no change was observed at 5 bar. The slip length was enhanced by ⁴ He coating at 5 bar. This enhancement indicates the increase of specularity of ³ He quasiparticles scattering at the oscillating surface. On the other hand, a reduction of the slip length was observed at 21 bar. In the superfluid phase, the temperature dependence of supports the existence of Andreev reflection even with ⁴ He film on the surface at 5 bar and 21 bar.     

Textural Effects and Spin-Wave Radiation in Superfluid 3He Weak Links

The Josephson coupling between two volumes of superfluid ³He-B also couples the phase difference to the spin-orbit texture of the order parameter. As a result, the equilibrium configuration of the macroscopic texture depends on the phase difference. If the junction is biased by a pressure head, the Josephson oscillations of the supercurrent are accompanied by an oscillation of the texture close to the junction. This leads to a radiation of spin waves from the weak link, and thus contributes to the dissipative dc currents flowing through it. PACS numbers: 67.57.De, 67.57.Fg, 67.57.Np.     

Measuring the Decay of Vorticity in Superfluid 3He-B - A Progress Report

Recent measurements in superfluid ³ He-B have been interpreted as evidence in support of the Kibble-Zurek hypothesis for the formation of defects in the early Universe. The superfluid ³ He analogue to the cosmic string is the quantum vortex. We report here the preliminary attempts to detect the heat evolution from the decay of this vorticity in superfluid ³ He-B at temperatures down to 110 K. We produce vorticity by neutron irradiation of a small superfluid ³ He-B sample and attempt to measure the heat released as the vorticity decays. We detect a long time scale heat release and find that the lifetime of this decay varies inversely proportional to the quasiparticle excitation number density in the superfluid. However, we have yet to be convinced that this heat is from the decay of vorticity.     

Thermal Detection of Turbulent and Laminar Dissipation in Vortex Front Motion

We report on direct measurements of the energy dissipated in the spin-up of the superfluid component of ³He-B. A vortex-free sample is prepared in a cylindrical container, where the normal component rotates at constant angular velocity. At a temperature of 0.20T _c, seed vortices are injected into the system using the shear-flow instability at the interface between ³He-B and ³He-A. These vortices interact and create a turbulent burst, which sets a propagating vortex front into motion. In the following process, the free energy stored in the initial vortex-free state is dissipated leading to the emission of thermal excitations, which we observe with a bolometric measurement. We find that the turbulent front contains less than the equilibrium number of vortices and that the superfluid behind the front is partially decoupled from the reference frame of the container. The final equilibrium state is approached in the form of a slow laminar spin-up as demonstrated by the slowly decaying tail of the thermal signal.     

Self-Organized Superfluid States in Gravity and Heat Flow

When ⁴ He is heated from above near the superfluid transition, there can appear a self-organized region, either in normal fluid or superfluid, with the temperature gradient equal to the transition temperature gradient. When it is in a superfluid state, there can be two regimes. Regime M is realized relatively far from the superfluid transition, where thermal resistance due to vortices is described in terms of the conventional Gorter–Mellink mutual friction. In regime G vortices are densely generated, where the line density in units of ^–2 is much larger than in any other previous experiments. Such a self-organized superfluid can coexist with a normal fluid or a superfluid containing only a small number of vortices in a dynamical steady state.     

Textures of Rotating Superfluid 3He–A in a Single Narrow Cylinder

We have studied textures of the rotating superfluid ³He–A in a single narrow cylinder by NMR measurement. In a narrow cylinder, the characteristic textures such as Mermin-Ho texture (Mermin and Ho in Phys. Rev. Lett. 36:594, 1976) can be formed in order to minimize the free energy of the system determined by the effect of the wall, the magnetic field, the dipole interaction, the flow of the superfluid and so on. We observed three types of NMR absorption spectra dependent on the processes used to form A-phase in a narrow cylinder. A particular texture shows a characteristic spectrum and we can determine the texture from the observed spectrum by comparing the resonance frequency of the peak in NMR spectrum with the calculated one of the spin wave mode. We present the identification and the phase diagram of textures of the rotating superfluid ³He–A in a single narrow cylinder.     

NMR experiments on rotating superfluid3He-A and3He-B and their theoretical interpretation

We have constructed a rotating nuclear demagnetization cryostat and used it for continuous-wave NMR experiments on superfluid³He-A and³He-B. The measurements were performed in a long cylindrical geometry of 5 mm diameter, with the cylinder axis parallel to the axis of rotation and with the external magnetic field H₀=284 or 142 Oe in the same direction. The angular velocity of rotation was varied between 0.2 and 1.5 rad/sec, and the experiments were done under 29.3 bar pressure at temperatures between T_c=2.72 and about 1.4 mK. As a guide to the new and esoteric field of superfluid³He in rotation, we first review the general theory at some length in relatively simple terms. Pictorial explanations are often given.In³He-A, a rotation-dependent NMR satellite was found; its intensity a rotation-dependent NMR satellite peak was discovered; its relative intensity increases linearly with . The position of the satellite is independent of and H, and does not depend on whether the sample was cooled from the Fermi-liquid region to the A phase while rotating or at rest. At temperatures 0.1<1–T/T_c<0.3, the frequency shift of the satellite can be described by the parameter R_t=0.86–1.1(1–T/T_c). Cooldown under rotation produced systematically larger satellite intensities than cooldown at rest. A second, metastable satellite, best seen at rest and disappearing in less than 30 min, was also discovered. Furthermore, the main NMR peak broadens during rotation, while the total NMR absorption remains the same. The behavior of the rotation-dependent satellite strongly supports the existence of vortices in³He-A, their number being proportional to ; the satellite is caused by localized spin wave modes trapped by vortex cores. Theoretical calculations agree quite well with our experimental data if continuous vortices, without a singularity in the order parameter, are assumed. Their presence is also responsible for the additional broadening of the main peak, due either to increased spin diffusion or to scattering of spin waves. The metastable satellite is caused by textural boundaries, probably by twist solitons in the superfluid, created by the rapid cooldown of the sample.In³He-B, a series of nearly equally spaced NMR satellites was found on the high-frequency side of the main peak with the cryostat at rest. Under rotation the separation between the satellites increases linearly with . The spacing displays a jump, proportional to , at 1–T/T_c=0.40. The discontinuity occurred only during start/stop experiments, not if the cryostat was continuously rotated while warming over the transition region. Immediately after rotation had been started the whole NMR spectrum shifted toward higher frequencies for about 30 sec; these transients were seen only at >0.25 rad/sec. In³He-B, the order parameter is strongly influenced by the wall of the container, producing the so-called flareout texture, with the angle between the vector andH equal to 63° at the walls. The satellites can be explained as spin wave modes arising from an almost harmonic potential well formed by the texture. The creation of vortices changes the texture and increases the steepness of the potential and therefore increases the satellite spacing during rotation. The vortices themselves perturb the texture due to the long-range orientating effect of their cores on the order parameter. The discontinuity in the satellite splitting at 1–T/T_c=0.40 is explained as being due to a first-order phase change in the vortex core at this temperature. The transient shift in the NMR spectrum, immediately after the start of rotation when vortices are not yet present, is caused by the large superfluid vs. normal liquid counterflow; this phenomenon thus gives an estimate for the time needed to create vortices in³He-B.     

Stability,orientational effects,and temperature dependence of nonunitary vortex structures in superfluid3He

First we set up the nine Ginzburg-Landau (GL) equations for the ninecomplex radial functions of an axisymmetric vortex line in superfluid³He. These GL equations necessarily yield a nine-componentnonunitary state for the A-phase vortex that corresponds to the Mermin-Ho trial state. We then solve the GL equations and calculate the energies for the five B-phase vortices of different symmetries that have been discussed by Salomaa and Volovik. We find that the only stable vortex is theirv-vortex, which has a superfluid and ferromagnetic core. Theo-vortex, which has a normal core, turns out to bemetastable (it is a saddle point on the energy surface describing theuvw-vortices). Theu-,w-, anduvw-vortices are found to beunstable. In the second part of the paper we calculate, with the help of the generalized GL theory, the temperature corrections to the GL structures of theo- andv-vortices. It turns out that the corresponding temperature variations of the two orientational energies for the -vector that are linear and quadratic in the field disagree with the measured temperature variations of the parameters and (see Hakonenet al.). However, a novel new orientational energy due to the interaction of the field and superflow around the vortex line increases linearly asT is decreased, in agreement with the measured behavior of .     

Experiments on the Twisted Vortex State in Superfluid 3He-B

We have performed measurements and numerical simulations on a bundle of vortex lines which is expanding along a rotating column of initially vortex-free ³He-B. Expanding vortices form a propagating front: Within the front the superfluid is involved in rotation and behind the front the twisted vortex state forms, which eventually relaxes to the equilibrium vortex state. We have measured the magnitude of the twist and its relaxation rate as function of temperature above 0.3T _c. We also demonstrate that the integrity of the propagating vortex front results from axial superfluid flow, induced by the twist.      

Dissipation and Phase Slip in Confined Superfluid Helium: Evidence for an Equilibrium Distribution of Vortices

The effects of phase slip due to the thermal creation of vortices in confined liquid ⁴He below the point is discussed. In a narrow tube there is a finite probability of thermal activation of a vortex in the superfluid helium, with the vortex not parallel to the axis of the tube. In the presence of a heat flow along the tube, the vortices experience the Magnus force, which prevents exact cancellation of the motion of thermally activated vortices traversing the cross section of the tube in opposite directions. Each crossing of the tube by a vortex causes a 2 phase slip of the superfluid order parameter along the tube. A temperature gradient results, which is proportional to the rate of phase slip, thus yielding a nonvanishing thermal resistivity below the bulk point. The calculated temperature dependence compares well with experimental data, thereby providing indirect evidence of the presence of vortices in thermal equilibrium.     

Phase transitions in the growth of4He films

Using a microscopic, variational approach we examine the growth of⁴He absorbed to graphite and alkali substrates. We find that superfluid layers are formed and their behavior as a function of coverage is closely related to the one of a purely two-dimensional superfluid. The growth of a new layer undergoes a phase transition from a cluster formation into the connected superfluid when the coverage is increased. Based on the important connection to the two-dimensional fluid we propose a microscopic theory of quantum vortices in⁴He films at zero temperature, in which single vortices are treated as quasiparticles. We calculate the energy needed to create the single vortex, vortex inertial mass, microscopic interaction between vortices and binding energy of the vortex-antivortex pair as a function of density. We predict that at the⁴He superfluid density less than about 0.037 Å^–2 the binding energy of the pair becomes negative, indicating a phase transition into a new state where vortex-antivortex pairs are spontaneously created.     

The Thermal Conductivity of Superfluid 3He in Aerogel: A Measurement of the Energy Gap

Transport properties of superfluid ³ He in aerogel are governed by a fixed mean free path set by the typical dimensions of the strand separation. We describe preliminary measurements of the thermal conductivity of superfluid ³ He confined in 98% silica aerogel. The majority of the measurements are made in relatively high magnetic fields where the superfluid within the aerogel is in the A-phase like state. Since the quasiparticle mean free path is fixed, the temperature dependence of the thermal conductivity depends only on the superfluid energy gap and on the texture in the case of the A-phase. The results so far obtained are broadly consistent with the aerogel-confined superfluid having a temperature dependent energy gap close to the BCS prediction although there are significant deviations at the lowest temperatures.     

Conductivity of the Wigner Solid on the Free Surface of Superfluid 3He-B

We have measured the conductivity of the 2D electron crystal (the Wigner solid) trapped on the free surface of superfluid ³ He-B down to 230 K. The resistance R(T) greatly decreases as temperature T decreases, obeying Arrhenius' law, R(T) exp (–(T)/k _B T), with (T) which we assign to the superfluid energy gap. This reveals that the scattering of ³ He quasiparticles by the WS determines the transport. As the input voltage increases, an anomalous behavior dominates the resistance just below the superfluid transition. The WS provides a new experimental method for probing both the bulk and free surface of superfluid ³ He.