Solitons-solitary waves that maintain their shape as they propagate-occur as water waves in narrow canals, as light pulses in optical fibres and as quantum mechanical matter waves in superfluids and superconductors. Their highly nonlinear and localized nature makes them very sensitive probes of the medium in which they propagate. Here we create long-lived solitons in a strongly interacting superfluid of fermionic atoms and directly observe their motion. As the interactions are tuned from the regime of Bose-Einstein condensation of tightly bound molecules towards the Bardeen-Cooper-Schrieffer limit of long-range Cooper pairs, the solitons' effective mass increases markedly, to more than 200 times their bare mass, signalling strong quantum fluctuations. This mass enhancement is more than 50 times larger than the theoretically predicted value. Our work provides a benchmark for theories of non-equilibrium dynamics of strongly interacting fermions.

We observe a long-lived solitary wave in a superfluid Fermi gas of 6 Li atoms after phase imprinting. Tomographic imaging reveals the excitation to be a solitonic vortex, oriented transverse to the long axis of the cigar-shaped atom cloud. The precessional motion of the vortex is directly observed, and its period is measured as a function of the chemical potential in the BEC-BCS crossover. The long period and the correspondingly large ratio of the inertial to the bare mass of the vortex are in good agreement with estimates based on superfluid hydrodynamics that we derive here using the known equation of state in the BEC-BCS crossover. DOI: 10.1103/PhysRevLett.113.065301 PACS numbers: 67.85.-d, 03.75.Kk, 03.75.Lm, 03.75.Ss Solitary waves that do not spread as they propagate are ubiquitous in nonlinear systems, from classical fluids and fiber optics to superfluids and superconductors. These waves are localized objects with defined energy and mass, and as such they can be described as an effective single particle emerging from a many-body environment. This distinguishes them from larger-scale collective excitations such as shape oscillations of a superfluid, or from perturbative linear excitations such as phonons. Paradigmatic examples of solitary waves in superfluids are planar solitons that separate regions of differing phase, as well as vortex rings or single vortex lines [see Fig. 1(a)]. The direct creation of such localized and highly nonlinear objects "on demand" in ultracold quantum gases allows for an excellent dynamical probe of novel superfluids, such as strongly interacting Fermi gases [1] or spin-orbit coupled Bose-Einstein condensates [2,3].In a recent experiment on fermionic superfluids at MIT [1], long-lived solitary waves were produced that featured a large ratio of inertial to bare (missing) mass of over 200, evidenced by an oscillation period over 15 times longer than the period for a single atom. The observed absorption images suggested the interpretation of the waves as planar solitons, but the longevity as well as the large effective mass ratio were unexpected for this type of defect [4][5][6][7]. Indeed, the nodal plane of a soliton is energetically more costly than the nodal line of a vortex, and planar solitons can decay into lower energy excitations via the snake instability, the undulation of the soliton plane [4]. Several recent works therefore suggested that these solitary waves are vortex rings [8][9][10]. For weakly interacting BoseEinstein condensates, solitons have been created [11,12] and observed to decay into vortex rings [13,14]. The latter further decay into a vortex-antivortex pair that eventually breaks up, leaving behind a single remnant vortex [15][16][17]. The exact process was recently elucidated in a discussion of apparent soliton oscillations observed in weakly interacting BECs [18,19]. In the case of strongly interacting fermionic superfluids, the understanding of such nontrivial dynamics presents a challenging nonequilibrium many-body problem [8,20].In this Letter,...

For a zero-temperature Landau symmetry-breaking transition in n-dimensional space that completely breaks a finite symmetry G, the critical point at the transition has the symmetry G. In this paper, we show that the critical point also has a dual symmetry-a (n − 1)-symmetry described by a higher group when G is Abelian or an algebraic (n − 1)-symmetry beyond a higher group when G is non-Abelian. In fact, any G-symmetric system can be viewed as a boundary of G-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk G-gauge theory gives rise to the symmetry and the dual symmetry, respectively. So any G-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry-breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3 + 1-dimensional Z 2 gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a Z 2 0-symmetry and its dual, a Z 2 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a Z 2 1-symmetry and its dual, another Z 2 1-symmetry.