Session R24: Superfluid Helium

Textural domain walls in superfluid $^3$He-B

Owing to the richness of symmetry, the superfluid $^3$He serves as a rich repository of topological quantum phenomena. This includes the emergence of surface Majorana fermions and their quantum mass acquisition at the topological critical point.\footnote{T. Mizushima {\it et al.} arXiv:1508.00787.} Furthermore, the marriage of the prototype topological superfluid with nanofabrication techniques brings about a rich variety of spontaneous symmetry breaking, such as the formation of the stripe order and nontrivial domain walls. In this work, we examine the possible formation of textural domain walls in the superfluid $^3$He-B confined to a thin slab with a sub-micron thickness. When an applied magnetic field is much higher than the dipolar field, two nearly degenerate ground states appear, which are characterized by the Ising order associated with the spontaneous breaking of a magnetic order-two symmetry, $\hat{\ell}_z=+1$ and $-1$. We here discuss the structure of the textural domain wall formed by the spatial modulation of the Ising order, such as low-lying quasiparticle excitations and spontaneous spin current. We also report bosonic modes bound to the textural domain wall.   

Chiral pair density wave phase of confined 3He-A film

The edge states of a 3He-A film are Weyl Fermions propagating in a direction determined by the chirality of the bulk phase, which leads to a non-vanishing spontaneous mass current on the edge. We report calculations of the reduction in the edge mass currents due to hybridization as a function of lateral confinement, D. Strong lateral confinement leads to a sequence of quantum phase transitions. The A phase undergoes a transition to a pair density wave (PDW) phase with broken translational symmetry at $D_{c2} \sim 13 \xi_0$, and a transition to a polar state at $D_{c1} \sim 9 \xi_0$. The order parameter for $ D_{c1} < D$ is calculated self-consistently. The resulting phase is a periodic array of chiral domains with opposite chirality separated by domain walls. The mass currents on the domain walls contradict the direction of current on the edges, which leads to separation into multiple regions of circulating current in each domain. The periodicity of PDW phase increases as confinement length $D$ increases and eventually only one domain is left in the lateral confined film when $D$ approaches $D_{c2}$. We calculated and compared the free energy of the confined single domain wall and the homogeneous A phase, and determined the phase boundary $D_{c2} - T$.   

Equilibrium helical order in radially confined superfluid $^3$He

An exciting prediction of confined superfluid $^3$He is the presence of spontaneously broken translational symmetry, resulting in a superfluid phase that has a different translational symmetry than that of the confining geometry. Such phases have been described theoretically in films, cylinders, and ribbons. We predict an inhomogeneous superfluid phase with helical order that is energetically stable within cylindrical channels of radius comparable to the Cooper pair coherence length. By incorporating extensions to standard Ginzburg-Landau (GL) strong-coupling theory that accurately reproduce the bulk phase diagram at high pressures and allow tuneable boundary conditions\footnote{J. J. Wiman \& J. A. Sauls, Phys. Rev. B 92, 144515 (2015)}, we find this new phase to be stable at both high and low pressures and favored by boundary conditions with strong pairbreaking. We present superfluid phase diagrams as functions of pressure, temperature, and channel radius showing the regions of stability for this ``spiral'' phase relative to those phases previously predicted for the channel. Transverse NMR frequency shifts are a possible experimental signature of this phase, and we present calculations of these shifts as functions of rf pulse tipping angle, field orientation, and temperature.   

Order parameter texture transition in superfluid $^3$He-B in strained aerogel

The introduction of anisotropic impurity scattering into superfluid $^3$He using high porosity silica aerogel has proven to be a fruitful method of engineering both the phase and the order parameter texture of the superfluid \footnote{J.I.A. Li {\it et al}, Phys. Rev. Lett., {\bf 114}, 105302, (2015).} \footnote{J. Pollanen {\it et al}, Phys. Rev. Lett., {\bf 107}, 235504, (2011).}. We have observed an abrupt transition between two orthogonal order parameter textures at a temperature $T_x\approx 1.9$ mK, in $^3$He-B confined in aerogel samples with anisotropy induced by mechanical compression along an axis $\vec\varepsilon$. At this transition the order parameter, characterized by the quantization axis of the orbital angular momentum $\hat{l}$, changes from a configuration with $\hat{l}\parallel\vec\varepsilon$ below $T_x$ to $\hat{l}\perp\vec\varepsilon$ above $T_x$. This transition is independent of the orientation of $\vec\varepsilon$ relative to the external magnetic field, as well as the magnitude of the applied field. This indicates that the textural transition is due to strain alone, with the anisotropic scattering from the aerogel favoring different orientations of $\hat{l}$ above and below $T_x$.   

Phase diagram of a thin film of $^3$He confined within a 1.08 $\mu$m deep cavity

We describe measurements of superfluid $^3$He confined to a high-aspect ratio cavity within the head of a high quality factor torsion pendulum. Superfluid phase diagram for the confined thin film of fluid is predicted to be radically different compared to that of the bulk. In particular, at low pressures at the onset of the A-B transition, a ``stripe phase" of alternating degenerate domains of B phase is predicted to occur [1]. By tracking the torsion pendulum frequency and quality factor, we identify a well-defined superfluid transition for the fluid within the pendulum head. At lower temperatures, sharp transitions from the A phase to the B phase on cooling and a gradual transition from the B phase to the A phase on warming are observed. The values for the ratio of the cavity depth and the coherence length $(D/\xi(T, P))$ at the transitions match well the values of the transitions seen in the NMR measurements of $^3$He confined to a 700 nm deep cavity [2]. At present, we do not see any evidence in our measurements that the ``stripe phase" is realized at the A-B phase boundary. \\[4pt] [1] A.B. Vorontsovand J.A. Sauls, Phys. Rev. Lett. 98, 045301 (2007).\\[4pt] [2] L.V. Levitin, et. al., Science 340, 841 (2013).   

Dislocation motion in solid hcp $^3$He

At temperatures above about 100 mK, dislocations reduce the shear modulus of hcp $^4$He by as much as 90$\%$. This occurs when dislocations thermally unbind from the $^3$He impurities that pin them, becoming extraordinarily mobile. The elastic softening is accompanied by a thermally activated dissipation peak due to the $^3$He impurities. At higher temperatures the dissipation has an $\omega T^4$ dependence caused by scattering of thermal phonons from moving dislocations. Previous measurements on the fermi solid, hcp $^3$He, showed a similar dislocation softening, but the corresponding dissipation was not measured. We have extended these measurements by measuring the temperature, amplitude and frequency dependence of both the shear modulus and the dissipation in hcp $^3$He. The dissipation behavior is very different from that of hcp $^4$He. Neither the impurity unbinding peak associated with the elastic softening, nor the high temperature phonon scattering dissipation, were observed. Instead, there is a large and non-thermally activated dissipation which is largest at low frequencies. We believe that this unexpected dissipation is associated with a new dislocation damping mechanism in $^3$He, perhaps associated with spin rearrangements caused by moving dislocations.   

Symmetry breaking field in UPt${_3}$ and connection to superfluid ${^3}$He

The multiple superconducting phases of UPt${_3}$ in its temperature-field phase diagram are a strong indication of its unconventional order parameter. It is generally accepted that such a complex phase diagram with 3 different vortex phases are nearly degenerate, and would be so, except for the presence of a symmetry breaking field attributed to antiferromagnetism which appears at a temperature an order of magnitude higher than the superconducting transition.[1] I propose an alternative mechanism where the symmetry breaking field can be, in large part, ascribed to anisotropic electronic scattering from stacking faults. The success of the theory[2] in accounting for stabilization of anisotropic phases of superfluid ${^3}$He in globally anisotropic aerogel[3] suggests a similar consequence from anisotropic quasiparticle scattering in UPt${_3}$. Specific heat measurements indicate that the temperature window of the more anisotropic A-phase, a direct measure of the strength of the symmetry breaking field, decreases systematically with fewer stacking faults. [1] D.W. Hess {\it et al.}, J. Phys.: Condens. Matter {\bf 1}, 8135 (1989). [2] E.V. Thuneberg {\it et al.}, Phys. Rev. Lett. {\bf 80}, 2161 (1998). [3] J. Pollanen {\it et al.}, Nat. Phys. {\bf 8}, 317 (2012).   

The Momentum Distribution of Liquid $^3$He, Revisited

Liquid $^$3He is a system of fundamental importance to condensed matter physics because it is a prototypical example of a strongly interacting fermion system whose interactions are well known. Quantum Monte Carlo calculations predict that the atomic momentum distribution of liquid $^3$He contains a Fermi surface discontinuity and an average atomic kinetic energy in the range 12-13 K at saturated vapor pressure. A number of high-resolution neutron Compton scattering studies of liquid $^3$3He have been described in the literature, with experimenters observing no Fermi surface discontinuity and obtaining kinetic energies in the range of 8-10 K. In this presentation, we reconsider measurements of the momentum distribution of liquid 3He taken at 500 mK under 0, 10, 15 bar of pressure [R.M. Dimeo et al Physica B 241-243, 952 (1998)]. We demonstrate that there is complete agreement between the experimental data and quantum Monte Carlo calculations when instrumental resolution and final state effect corrections are taken into account. We also consider the prospects for a direct observation of the Fermi surface discontinuity in liquid 3He using neutron Compton scattering.   

X-ray coherent diffractive imaging of quantum vortices in single helium droplets

Free, single, rotating superfluid $^{\mathrm{4}}$He nanodroplets (diameter D $=$ 200 - 2000 nm, temperature T $=$ 0.4 K) containing a number of quantum vortices have been studied via ultrafast X-ray coherent diffraction imaging using a free electron laser. The droplets were doped with Xe atoms, which collect on the vortex cores and serve as a contrast agent. In order to obtain the instantaneous positions and shapes of the vortices from the diffraction images, a phase retrieval technique has been developed, which utilizes the droplet boundary as a physical support. The algorithm also uses the droplet's scattering phase as an input for the iterative phase reconstruction. The obtained reconstructions reveal a plethora of transient vortex configurations within the droplet. The details of the algorithm and the possible origin of the observed vortex configuration will be discussed.   

Investigation of Grid Turbulence in Superfluid $^4$He with Improved Measurement Technique

Quantum turbulence(QT), a tangle of quantized vorticity in a macroscopically correlated quantum fluid, can have many analogous aspects to classical turbulence. Understanding QT can give us insights into classical turbulence as well as fluids in general. We generate QT by pulling a grid through a 4.6 cm x 4.6 cm cross-section channel in superfluid $^4$He. Second sound, a temperature/entropy wave, is used to monitor vorticity, $\omega$. A resonant technique with high (3000) Q increases greatly the sensitivity of the measurement, but it requires a phase and amplitude locked tracking system which adheres to the resonant peak independent of frequency shifts. According to theories, the vorticity decays as $\omega \sim$ t$^{-11/10}$ or $\omega \sim$ t$^{-17/14}$ when the energy containing eddies are growing. When they saturate at the channel size, the vorticity begins decaying as $\omega \sim$ t$^{3/2}$. These different decaying regions are examined in this large channel and compared to previous experiments that have been performed in 1 cm$^2$ square channels\footnote{M. R. Smith, R. J. Donnelly, N. Goldenfeld, and W. F. Vinen, Phys. Rev. Lett. 71, 2583 (1993).; S. R. Stalp, Ph.D. dissertation, University of Oregon 1998.; L. Munday, Ph.D. dissertation, Lancaster University 2014.}.   

Anomalous Elasticity of $^4$He Films at the Quantum Phase Transition

$^4$He films on solid substrates exhibit a quantum phase transition between localized (nonsuperfluid) and superfluid states by changing coverage $n$. We have made torsional oscillator (TO) studies for $^4$He films adsorbed on nanoporous glasses. A TO with localized films showed an apparent "supersolid" behavior, an increase in TO frequency $f$ with broad peak in $Q^{-1}$. Combining with FEM analyses for TO's with different designs, we conclude that the behavior results from the softening of adsorbed $^4$He films at high temperatures. The features in $f$ and $Q^{-1}$ are fitted well to a Debye-like activation with a distributed energy gap $\Delta$, so the elasticity is accounted by thermal excitation of localized atoms to an "extended" state. As the critical coverage $n_{c}$ approaches the gap decreases to zero with a powerlaw $\Delta \propto (n-n_{c})^{1.2}$. Assuming that the $^4$He chemical potential $\mu(n)$ is located in the middle of the gap, we can estimate the elastic constant $\kappa^{-1} = n^2 \partial \mu / \partial n$. The elasticity agrees with shear moduli of $^4$He films obtained from the FEM analysis within factor of three. The energetics proposed from the elastic behavior naturally explains other properties of He films adsorbed on disordered substrates.   

Superfluidity, Bose-Einstein condensation and dimensions of liquid $^4$He in nanopores

Path integral Monte Carlo (PIMC) calculations of the superfluid fraction, $\rho_S/\rho$, and the one-body density matrix (OBDM) (Bose-Einstein condensation (BEC)) of liquid $^4$He confined in nanopores are presented. The goal is to determine the effective dimensions of the liquid in the nanopore. We simulate a cylinder of liquid of diameter $d_L$~ surrounded by 5 \AA~ of inert solid $^4$He in a nanopore of diameter $d$; $d$ = $d_L$ + 10 \AA~ [1]. The PIMC $\rho_S(T)/\rho$ and OBDM scales as a 1D Luttinger Liquid at extremely small liquid pore diameters only, $d_L$~ = 6 \AA~ where the liquid atoms form a 1D line at the center of the pore. In the range 8 $\leq d_L \leq$ 22 \AA~ the PIMC $\rho_S(T)/\rho$ scales as a 2D liquid. In this $d_L$ range the liquid fills the pores in cylindrical layers. There is a cross over from 2D to 3D scaling at larger $d_L$ $\simeq$ 22 \AA. In the range 8 $\leq d_L \leq$ 22 \AA~, the $T_C$ predicted using the Kosterlitz-Thouless 2D scaling criterion of the OBDM agrees well with the $T_C$ obtained from $\rho_S(T)/\rho$. Superflow observed in pores of diameter (18 $< d <$ 32 \AA) is apparently standard static superflow with the low $T_C$ arising from its 2D character.\\ 1. L. Vranje\v{s} Marki\'c and H. R. Glyde, Phys. Rev. B92, 064510 (2015)   

Dissipation in Nanoscale Superfluids

Pressure driven flow of a superfluid inside a narrow channel can be maintained by the nucleation of vortices and their resulting motion across the flow lines. The maximum velocity of the superfluid is set by a nucleation rate which crucially depends on the microscopic details of the vortices and flow profile. Within the kinetic vortex theory, we have determined the critical superfluid velocity inside a nanoscale constriction and obtain agreement with experimental results for superfluid helium-4 in nanopores. In the small pore limit, when the ratio of pore radius to correlation length is of order unity, we find a drastic suppression of the superfluid velocity that can be understood within the Langer-Ambegaokar-McCumber-Halperin theory of resistive fluctuations in thin superconducting wires.   

High-Resolution Measurements of the Roton Lifetime in Nano-Confinement

At very low temperatures, the phonon-roton spectrum of bulk superfluid helium is sharp and well-defined in energy. As the temperature is increased, the roton energy gap becomes smaller and the roton peak acquires a finite linewidth. The conventional understanding of this effect is that roton-roton scattering drives the softening and broadening of the roton mode where the mean free path is governed by the thermal population of rotons. It is an open question whether the roton mode follows the same behavior when the liquid is confined within sufficiently small mesopores. It is possible that the restricted geometry introduces a new length scale which controls the roton mean free path at low temperatures. We report high-resolution (~4 $\mu$eV) measurements of the roton energy and linewidth within tubular, silica nanopores 2.8 nm in diameter. The new results provide a critical test of the idea that tight, nanoscale confinement modifies the energy and linewidth of the roton excitation.   

Pressure driven flow studies of superfluid helium-4 through single, high aspect ratio nanopipes

We have measured flow rates of helium-4 through high aspect ratio (\textgreater 10,000) single glass nanopipes and etched nanopores under the influence of a pressure drop. The initial diameter of the glass pipes is 200nm while the initial diameter of the nanopores is approximately 80nm; the diameter of both types of nanopipe were reduced using atomic layer deposition(ALD) of Al2O3. Flow rates were measured for a wide range of temperatures (0.8K to 3.0K), pressures (up to 40 atm), and pipe lengths (0.8 mm to 30 mm). We observed flow velocities in the range of 1-6 m/s which has a power law dependence on pressure. Flow appears to be governed by turbulence at low temperatures. We have found evidence for a critical pressure above which turbulent flow is eliminated. This critical pressure appears to depend on temperature.