Session S53: Invited Session: Symposium on Novel Phenomena in Helium in Reduced Dimensions and Confinement

Evidence for intertwined superfluid and density wave order in two dimensional $^{4}$He

We report the identification of a new state of quantum matter with intertwined superfluid and density wave order in a system of two dimensional bosons subject to a triangular lattice potential. Using a torsional oscillator we have measured the response of the second atomic layer of $^{4}$He adsorbed on the surface of graphite over a wide temperature range down to 2 mK. Superfluidity is observed over a narrow range of film densities, emerging suddenly and collapsing towards a quantum critical point, near to layer completion where a Mott insulating phase is predicted to form. The unusual temperature dependence of the superfluid density in the $T\to $0 limit and the absence of a clear superfluid onset temperature are explained, self-consistently, by an ansatz for the excitation spectrum, reflecting density wave order, and a quasi-condensate wavefunction breaking both gauge and translational symmetry. \\[4pt] In collaboration with Jan Nyeki, Anastasia Phillis, Andrew Ho, Derek Lee, Piers Coleman, Jeevak Parpia, Brian Cowan.   

Signatures of Majorana and Weyl Fermions in confined phases of superfluid $^3$He

The B-phase of superfluid $^3$He exhibits symmetry breaking in which separate invariance under gauge-, spin- and orbital rotations is reduced to the maximal sub-group, $SO(3)_{L+S}\times{T}$. Parity is broken, but time-reversal is preserved. Broken relative spin-orbit rotational symmetry implies emergent spin-orbit coupling and non-trivial topology of the ground state, both of which are encoded in the Bogoliubov-Nambu Hamiltonian: ${\cal H} = \xi({\bf p})\tau_{3} + c\,{\bf p}\cdot\vec\sigma\,\tau_{1}$, where $c = \Delta/p_f$ is several orders of magnitude slower than the Fermi velocity. The topology of the B-phase is expressed in terms of a non-trivial winding number for the mapping between momentum space and Nambu space, $N_{\mbox{3D}} = \int\frac{d^3p}{24\pi^2}\,\epsilon_{ijk}\, \mbox{Tr}\Big\{T\,C\,({\cal H}^{-1}\partial_{p_i}{\cal H})\times ({\cal H}^{-1}\partial_{p_j}{\cal H}) ({\cal H}^{-1}\partial_{p_k}{\cal H})\Big\}\,=\, 2$, where $C$ is the particle-hole transformation. The physical consequence of $N_{\mbox{3D}}\ne 0$ is the emergence of a spectrum of Majorana fermions confined on any surface of $^3$He-B whose effective Hamiltonian is described $H = \sum_{{\bf p}_{||}}\,\psi_{-{\bf p}_{||}}^{T}{\bf p}_{||}\times\vec{\sigma}\cdot\hat{\bf s}\,\psi_{{\bf p}_{||}}$. The surface excitations are self-conjugate Majorana fermions with a gapless relativistic dispersion relation $\varepsilon({\bf p}) = c|{\bf p}_{||}|$, and their spins locked normal to the in-plane momentum and the surface normal, $\hat{\bf s}$. In this talk I describe theoretical predictions for experimental signatures based on NMR, mass flow, local ion probes and ultra-sound spectroscopy of these unique quanta that reflect the topological nature of the ground state of superfluid $^3$He.   

Observation of the Polar Phase of $^{3}$He

Exotic pairing of Fermions into a condensed bound state is now well established in many systems. In superfluid 3He it is becoming evident that the nature and stability of the emergent order parameter can be altered radically by confinement in regular geometries or by providing anisotropic disorder realized in materials whose nanoscale structure is smaller than the coherence length. The temperature dependent coherence length diverges as the superfluid transition is approached from below and sets the characteristic length scale for confinement. Thus, the degree of confinement can be varied as the temperature is varied below Tc. Additionally, unlike most superconductors, the 3He liquid's properties can be pressure-tuned over a large range. In bulk liquid, (in zero magnetic field) there are two equilibrium phases of 3He: at high pressure, as the temperature is lowered there is a transition from the chiral A phase to the isotropic B phase. In bulk liquid, the relative stability of the two phases is controlled not by confinement but by strong-coupling interactions. In our highly confined system, both factors (confinement and strong coupling) come into play and depending on pressure we see a succession of two transitions (three phases) within the superfluid as the sample is cooled from the normal state. Drawing on theoretical work we identify the phases as the Polar phase near Tc, followed by a polar distorted A or B phase (depending on pressure) with the low temperature phase being the B phase at all pressures. These observations are made with a torsional pendulum that was used to assay the superfluid fraction and the disordered medium is the so-called ``Obninsk'' alumina aerogel that has highly oriented strands aligned with the torsional axis.   

Probing the A-B interface of superfluid helium-3

At temperatures around 1 mK helium-3 forms a BCS spin triplet condensate. The order parameter is sufficiently complex that more than one superfluid phase exists, each exhibiting a different broken symmetry, and there is a model first order transition between the two most stable phases, labeled A and B. The Lancaster Ultra-Low Temperature Group has developed techniques to probe the properties of the A-B interface in the deep sub-mK regime where the superfluid is in the pure condensate limit. Shaped and controllable magnetic fields are used to induce the transition, and to stabilize and move the A-B phase boundary inside the experimental volume. The latent heat of the transition has been measured, and the nucleation behavior shown to be incompatible with conventional thermodynamic models. Since superfluid helium-3 is inherently pure, and the order parameter transforms continuously across the A-B interface, it is the most coherent two-dimensional structure to which we have experimental access. It has been proposed that this 2D surface in the surrounding 3D bulk volume is a good analog of a cosmological brane separating two distinct quantum vacuum states; experiments that simulate brane annihilation and the creation of topological defects have been carried out at Lancaster. Other investigations have included measurements of the surface tension and wetting behavior of the interface. During these studies it was discovered that a large, unpredicted frictional force was acting on the interface even though it is moving through a pure superfluid. Recent breakthrough work on the dynamics of the A-B interface has finally solved this puzzle. Current experiments include a setup where the interface region is probed directly using quartz tuning fork resonators that couple to the local density of broken Cooper pair quasiparticle excitations and thus give insight into the order parameter energy gap structure as A transforms to B.   

One-dimensional Quantum Fluids

Fifty year ago, Joachim Mazdak Luttinger generalized the Tomonaga theory of interactions in a one-dimensional metal and show that the prior restrictions imposed by Tomonaga were not necessary. This model is now known as the Tomonaga- Luttinger liquid model (TLL) and most remarkably it does have mathematically exact solutions. In the case of electrons, it predicts that the spin and charge sector should separate, with each of them propagating with their own velocities. While there has been many attempts (some with great success) to observe TLL behaviour in clean quantum wires designed on an ultra-clean semiconductor platform, overall the Luttinger physics is experimentally still in its infancy. For instance, little is known regarding the 1D physics in a strongly-interacting neutral system, whether from the point-of-view of TLL theory or even localization physics. Helium-4, the paradigm superfluid, and Helium-3, the paradigm Fermi liquid, should \textit{in principle }both become Luttinger liquids if taken to the one-dimensional limit. In the bosonic case, this is supported by large-scale Quantum Monte Carlo simulations [1] which found that a lengthscale of $\sim $ 2 nm is sufficient for the system to crossover to the 1D regime and display universal Luttinger scaling [2]. At McGill University, an experiment has been constructed to measure the liquid helium mass flow through a \textit{single nanopore}. The technique consists of drilling a single nanopore in a SiN membrane using a TEM, and then applying a pressure gradient across the membrane. Previously published data in 45nm diameter hole determined the superfluid critical velocity to be close to the limit set by the Feynman vortex rings model [3]. More recent work performed on nanopores with radii as small as 3 nm (and a length of 30nm) show the critical exponent for superfluid velocity to significantly deviate from its bulk value, 2/3. This is an important hint for the crossing over to the one-dimensional state in a strongly-correlated bosonic liquid. References\textunderscore [1] Del Maestro A, Boninsegni M, Affleck I. $^{\mathrm{4}}$He Luttinger Liquid in Nanopores. \textit{PHYSICAL REVIEW LETTERS 106}: 105303, 2011. [2] Kulchytskyy B, Gervais G, Del Maestro A. Local superfluidity at the nanoscale. \textit{PHYSICAL REVIEW B 88}: 064512, 2013. [3] Savard M, Dauphinais G, Gervais G. Hydrodynamics of Superfluid Helium in a Single Nanohole. \textit{PHYSICAL RE- VIEW LETTERS 107}: 254501, 2011.