Chiral Superfluid Research Articles

Trapped Bose–Einstein condensates in the presence of a current nonlinearity

We investigate the effect of a current nonlinearity on the evolution of a trapped atomic Bose–Einstein condensate. We have implemented techniques from the field of nonlinear optics to provide new insights into the irregular dynamics associated with chiral superfluids. We have found that the current nonlinearity can be treated as a Kerr-like nonlinearity modulated by a spatiotemporal function that can lead to a number of processes such as broadening and compression of the wavefunction. In the long time scale limit, the wavefunction is drastically deformed and delocalised compared to its initial state. However, localised modes which oscillate with the period of the inverse trap frequency can still be observed.

From a semimetal to a chiral Fulde-Ferrell superfluid

The recent realization of two-dimensional (2D) synthetic spin-orbit (SO) coupling opens a broad avenue to study novel topological states for ultracold atoms. Here, we propose a new scheme to realize exotic chiral Fulde-Ferrell superfluid for ultracold fermions, with a generic theory being shown that the topology of superfluid pairing phases can be determined from the normal states. The main findings are two fold. First, a semimetal is driven by a new type of 2D SO coupling whose realization is even simpler than the recent experiment, and can be tuned into massive Dirac fermion phases with or without inversion symmetry. Without inversion symmetry the superfluid phase with nonzero pairing momentum is favored under an attractive interaction. Furthermore, we show a fundamental theorem that the topology of a 2D chiral superfluid can be uniquely determined from the unpaired normal states, with which the topological chiral Fulde-Ferrell superfluid with a broad topological region is predicted for the present system. This generic theorem is also useful for condensed matter physics and material science in search for new topological superconductors.

Critical behavior of a chiral superfluid in a bipartite square lattice

We study the critical behavior of Bose–Einstein condensation in the second band of a bipartite optical square lattice in a renormalization group framework at one-loop order. Within our field theoretical representation of the system, we approximate the system as a two-component Bose gas in three dimensions. We demonstrate that the system is in a different universality class than the previously studied condensation in a frustrated triangular lattice due to an additional Umklapp scattering term, which stabilizes the chiral superfluid order at low temperatures. We derive the renormalization group flow of the system and show that this order persists in the low energy limit. Furthermore, the renormalization flow suggests that the phase transition from the thermal phase to the chiral superfluid state is first order.

Anomalous transport and generalized axial charge

In this paper we continue studying the modification of the axial charge in chiral media by macroscopic helicities. Recently it was shown that magnetic reconnections result in a persistent current of zero mode along flux tubes. Here we argue that in general a change in the helical part of the generalized axial charge results in the same phenomenon. Thus one may say that there is a novel realization of chiral effects requiring no initial chiral asymmetry. The transfer of flow helicity to zero modes is analyzed in a toy model based on a vortex reconnection in a chiral superfluid. Then, we discuss the balance between the two competing processes effect of reconnections and the chiral instability on the example of magnetic helicity. We argue that in the general case there is a possibility for the distribution of the axial charge between the magnetic and fermionic forms at the end of the instability.

Majorana modes and topological superfluids for ultracold fermionic atoms in anisotropic square optical lattices

Motivated by the recent experimental realization of two-dimensional spin-orbit coupling through optical Raman lattice scheme, we study attractive interacting ultracold gases with spin-orbit interaction in anisotropic square optical lattices, and find that rich s-wave topological superfluids can be realized, including Z2 topological superfluids beyond the characterization of “tenfold way” in addition to chiral topological superfluids. The topological defects-superfluid vortex and edge dislocations-may host Majorana modes in some topological superfluids, which are helpful for realizing topological quantum computation and Majorana fermionic quantum computation. In addition, we also discuss the Berezinsky-Kosterlitz-Thouless phase transitions for different topological superfluids.

Electron Bubbles in Superfluid $$^3$$He-A: Exploring the Quasiparticle–Ion Interaction

When an electron is forced into liquid $^3$He it forms an "electron bubble", a heavy ion with radius, $R\simeq 1.5$ nm, and mass, $M\simeq 100\,m_3$, where $m_3$ is the mass of a $^3$He atom. These negative ions have proven to be powerful local probes of the physical properties of the host quantum fluid, especially the excitation spectra of the superfluid phases. We recently developed a theory for Bogoliubov quasiparticles scattering off electron bubbles embedded in a chiral superfluid that provides a detailed understanding of the spectrum of Weyl Fermions bound to the negative ion, as well as a theory for the forces on moving electron bubbles in superfluid $^3$He-A (Shevtsov et al. in arXiv:1606.06240). This theory is shown to provide quantitative agreement with measurements reported by the RIKEN group [Ikegami et al., Science 341:59, 2013] for the drag force and anomalous Hall effect of moving electron bubbles in superfluid $^3$He-A. In this report, we discuss the sensitivity of the forces on the moving ion to the effective interaction between normal-state quasiparticles and the ion. We consider models for the quasiparticle-ion (QP-ion) interaction, including the hard-sphere potential, constrained random-phase-shifts, and interactions with short-range repulsion and intermediate range attraction. Our results show that the transverse force responsible for the anomalous Hall effect is particularly sensitive to the structure of the QP-ion potential, and that strong short-range repulsion, captured by the hard-sphere potential, provides an accurate model for computing the forces acting on the moving electron bubble in superfluid $^3$He-A.

Phase diagram of dipolar hard-core bosons on a honeycomb lattice

In this paper, we study phase diagrams of dipolar hard-core boson gases on the honeycomb lattice. The system is described by the Haldane-Bose-Hubbard model with complex hopping amplitudes and the nearest neighbor repulsion. By using the slave-particle representation of the hard-core bosons and also the path-integral quantum Monte-Carlo simulations, we investigate the system and to show that the systems have a rich phase diagram. There are Mott, superfluid, chiral superfluid, and sublattice chiral superfluid phases as well as the density-wave phase. We also found that there exists a coexisting phase of superfluid and chiral superfluid. Critical behaviors of the phase transitions are also clarified.

Ground-state angular momentum, spectral asymmetry, and topology in chiral superfluids and superconductors

Recently, it was discovered that the ground-state orbital angular momentum in two-dimensional chiral superfluids with pairing symmetry ${({p}_{x}+i{p}_{y})}^{\ensuremath{\nu}}$ depends on the winding number $\ensuremath{\nu}$ in a striking manner. The ground-state value for the $\ensuremath{\nu}=1$ case is ${L}_{z}=\ensuremath{\hbar}N/2$ as expected by counting the Cooper pairs, while a dramatic cancellation takes place for $\ensuremath{\nu}>1$. The origin of the cancellation is associated with the topological edge states that appear in a finite geometry and give rise to a spectral asymmetry. Here, we study the reduction of orbital angular momentum for different potential profiles and pairing strengths, showing that the result ${L}_{z}=\ensuremath{\hbar}N/2$ is robust for $\ensuremath{\nu}=1$ under all studied circumstances. We study how angular momentum depends on the gap size $\mathrm{\ensuremath{\Delta}}/{E}_{F}$ and obtain the result ${L}_{z}=\frac{\ensuremath{\hbar}\ensuremath{\nu}}{2}N(1\ensuremath{-}\frac{\ensuremath{\mu}}{{E}_{F}})$ for $\ensuremath{\nu}=2,3$. Thus, the gap dependence of ${L}_{z}$ for $\ensuremath{\nu}<4$ enters at most through the chemical potential while $\ensuremath{\nu}\ensuremath{\ge}4$ is qualitatively different. In addition, we generalize the spectral asymmetry arguments to total angular momentum in the ground state of triplet superfluids where due to a spin-orbit coupling ${L}_{z}$ is not a good quantum number. We find that the ground-state total angular momentum also behaves very differently depending on total angular momentum of the Cooper pairs.

Density-induced geometric frustration of ultra-cold bosons in optical lattices

A density-dependent gauge field may induce density-induced geometric frustration, leading to a non-trivial interplay between density modulation and frustration, which we illustrate for the particular case of ultra-cold bosons in zig-zag optical lattices with a density-dependent hopping amplitude. We show that the density-induced frustration leads to a rich landscape of quantum phases, including Mott insulator, bond-order insulator, two-component superfluids, chiral superfluids, and partially paired superfluids. We show as well that the density-dependent hopping results in an effective repulsive or attractive interaction, and that for the latter case the vacuum may be destabilized leading to a strong compressibility. Finally, we discuss the characteristic momentum distribution of the predicted phases, which can be used to detect the phases in time-of-flight measurements.

Chirald-Wave Superfluid in Periodically Driven Lattices

A chiral d-wave superfluid is a preliminary example of interacting topological matter. However, unlike s-wave superfluids prevalent in nature, its existence requires a strong d-wave interaction, a criterion that is difficult to access in ordinary systems. There is no experimental observation of such unconventional superfluid at the moment. Here, we present a new principle for creating a two-dimensional (2D) chiral d-wave superfluid using periodically driven lattices. Because of an imprinted 2D pseudospin-orbit coupling, where the sublattice index serves as the pseudospin, the s-wave interaction between two hyperfine spin states naturally creates a chiral d-wave superfluid. This scheme can be directly implemented in current experiments.

Chiral topological superfluids in the attractive Haldane-Hubbard model with opposite Zeeman energy at two sublattice sites

Ultracold atoms in an optical lattice provide a platform for the realization of the topological superfluid. Motivated by the recent cold atom realization of the topological Haldane model [G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, Th. Uehlinger, D. Greif, T. Esslinger, Nature 515, 237 (2014)], in this paper we propose an alternative way to realize chiral topological superfluids with Chern number 𝒞 = 1 and 2 by considering attractive Haldane-Hubbard model with the site-dependent Zeeman field. The topological superfluids support the robust chiral edge modes, and the one-half of flux quantum-π flux in 𝒞 = 1 topological superfluid traps a pair of Majorana zero modes different from the case in the spinless p x ± i p y topological superfluid due to the extra freedom of A-B sublattices. In addition, we discuss the superfluid stability and calculate Kosterlitz-Thouless transition temperature by random-phase-approximation approach.

Density-dependent synthetic magnetism for ultracold atoms in optical lattices

Raman-assisted hopping can allow for the creation of density-dependent synthetic magnetism for cold neutral gases in optical lattices. We show that the density-dependent fields lead to a nontrivial interplay between density modulations and chirality. This interplay results in a rich physics for atoms in two-leg ladders, characterized by a density-driven Meissner-superfluid to vortex-superfluid transition, and a nontrivial dependence of the density imbalance between the legs. Density-dependent fields also lead to intriguing physics in square lattices. In particular, it leads to a density-driven transition between a nonchiral and a chiral superfluid, both characterized by nontrivial charge density-wave amplitude. We finally show how the physics due to the density-dependent fields may be easily probed in experiments by monitoring the expansion of doublons and holes in a Mott insulator, which presents a remarkable dependence on quantum fluctuations.

Relativistic gravity and parity-violating nonrelativistic effective field theories

We show that the relativistic gravity theory can offer a framework to formulate the non-relativistic effective field theory in a general coordinate invariant way. We focus on the parity violating case in 2+1 dimensions which is particularly appropriate for the study on quantum Hall effects and chiral superfluids. We discuss how the non-relativistic spacetime structure emerges from relativistic gravity. We present covariant maps and constraints that relate the field contents in the two theories, which also serve as the holographic dictionary in context of gauge/gravity duality. A low energy effective action for fractional quantum Hall states is constructed, which captures universal geometric properties and generates non-universal corrections systematically. We give another holographic example with dyonic black brane background to calculate thermodynamic and transport properties of strongly coupled non-relativistic fluids in magnetic field. In particular, by identifying the shift function in the gravity as minus of guiding center velocity, we obtain the Hall viscosity with its relation to Landau orbital angular momentum density proportional to Wen-Zee shift. Our formalism has a good projection to lowest Landau level.

Orbital angular momentum and spectral flow in two-dimensional chiral superfluids.

We study the orbital angular momentum (OAM) L_{z} in two-dimensional chiral (p_{x}+ip_{y})^{ν}-wave superfluids (SFs) of N fermions on a disk at zero temperature, in terms of spectral asymmetry and spectral flow. It is shown that L_{z}=νN/2 for any integer ν, in the Bose-Einstein condensation regime. In contrast, in the BCS limit, while the OAM is L_{z}=N/2 for the p+ip-wave SF, for chiral SFs with ν≥2, the OAM is remarkably suppressed as L_{z}=N×O(Δ_{0}/ϵ_{F})≪N, where Δ_{0} is the gap amplitude and ϵ_{F} is the Fermi energy. We demonstrate that the difference between the p+ip-wave SF and the other chiral SFs in the BCS regimes originates from the nature of edge modes and related depairing effects.

Observing Chiral Superfluid Order by Matter-Wave Interference

The breaking of time-reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool, we use a matter-wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low-energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.

Chiral bosonic phases on the Haldane honeycomb lattice

Recent experiments in ultracold atoms and photonic analogs have reported the implementation of artificial gauge fields in lattice systems, facilitating the realization of topological phases. Motivated by such advances, we investigate the Haldane honeycomb lattice tight-binding model, for bosons with local interactions at the average filling of one boson per site. We analyze the ground state phase diagram and uncover three distinct phases: a uniform superfluid (SF), a chiral superfluid (CSF) and a plaquette Mott insulator with local current loops (PMI). Nearest-neighbor and next-nearest neighbor currents distinguish CSF from SF, and the phase transition between them is first order. We apply bosonic dynamical mean field theory and exact diagonalization to obtain the phase diagram, complementing numerics with calculations of excitation spectra in strong and weak coupling perturbation theory. The characteristic density fluctuations, current correlation functions, and excitation spectra are measurable in ultracold atom experiments.

Effective theory of two-dimensional chiral superfluids: Gauge duality and Newton-Cartan formulation

We present a theory of Galilean-invariant conventional and chiral $p_x \pm ip_y$ fermionic superfluids at zero temperature in two spatial dimensions in terms of a dual gauge theory. Our formulation is general coordinate invariant. The parity-violating effects are encoded in the Wen-Zee term that gives rise to the Hall viscosity and edge current. We show that the relativistic superfluid with the Euler current reduces to the chiral superfluid in the limit $c\to\infty$. Using Newton-Cartan geometry we construct the covariant formulation of the effective theory and calculate the energy current.

Orbital momentum of chiral superfluids and the spectral asymmetry of edge states

This is comment to preprint Y. Tada, W. Nie, and M. Oshikawa, Orbital Angular Momentum and Spectral Flow in Two Dimensional Chiral Superfluids, arXiv:1409.7459, where the effect of spectral flow along the edge states on the magnitude of the orbital angular momentum is discussed. The general conclusion of the preprint on the essential reduction of the angular momentum for the higher values of chirality, |ν| > 1, is confirmed. However, we show that if parity is violated, the reduction of the angular momentum takes place also for the p-wave superfluids with |ν| = 1.

Andreev-Majorana bound states in superfluids

We consider Andreev-Majorana (AM) bound states with zero energy on surfaces, interfaces, and vortices in different phases of the p-wave superfluids. We discuss the chiral superfluid 3He-A and time reversal invariant phases: superfluid 3He-B, planar and polar phases. The AM zero modes are determined by topology in the bulk and disappear at the quantum phase transition from the topological to nontopological state of the superfluid. The topology demonstrates the interplay of dimensions. In particular, the zero-dimensional Weyl points in chiral superfluids (the Berry phase monopoles in momentum space) give rise to the one-dimensional Fermi arc of AM bound states on the surface and to the one-dimensional flat band of AM modes in the vortex core. The one-dimensional nodal line in the polar phase produces a two-dimensional flat band of AM modes on the surface. The interplay of dimensions also connects the AM states in superfluids with different dimensions. For example, the topological properties of the spectrum of bound states in three-dimensional 3He-B are connected to the properties of the spectrum in the two-dimensional planar phase (thin film).

Chiral magnetism and spontaneous spin Hall effect of interacting Bose superfluids.

Recent experiments on ultracold atoms in optical lattices have synthesized a variety of tunable bands with degenerate double-well structures in momentum space. Such degeneracies in the single-particle spectrum strongly enhance quantum fluctuations, and often lead to exotic many-body ground states. Here we consider weakly interacting spinor Bose gases in such bands, and discover a universal quantum 'order by disorder' phenomenon which selects a novel superfluid with chiral spin order displaying remarkable properties such as spontaneous spin Hall effect and momentum space antiferromagnetism. For bosons in the excited Dirac band of a hexagonal lattice, such a state supports staggered spin loop currents in real space. We show that Bloch oscillations provide a powerful dynamical route to quantum state preparation of such a chiral spin superfluid. Our predictions can be readily tested in spin-resolved time-of-flight experiments.