Topological Superfluid Phase Research Articles

Andreev bound states at boundaries of polarized 2D Fermi superfluids with s-wave pairing and spin-orbit coupling

A topological superfluid phase characterized by an emergent chiral-p-wave pair potential is expected to form in a two-dimensional Fermi superfluid subject to s-wave pairing, spin-orbit coupling and a large-enough Zeeman splitting. Andreev bound states appear at phase boundaries, including Majorana zero modes whose existence is assured by the bulk-boundary correspondence principle. Here we study the physical properties of these subgap-energy bound states at step-like interfaces using the spin-resolved Bogoliubov-de Gennes mean-field formalism and assuming small spin-orbit coupling. Extending a recently developed spin-projection technique based on Feshbach partitioning [SciPost Phys. 5, 016 (2018)] combined with the Andreev approximation allows us to obtain remarkably simple analytical expressions for the bound-state energies as well as the majority and minority spin components of their wave functions. Besides the vacuum boundary, where a majority-spin Majorana excitation is encountered, we also consider the boundary between the topological and a nontopological superfluid phase that can appear in a coexistence scenario due to the first-order topological phase transition predicted for this system. At this superfluid-superfluid interface, we find a localized chiral Majorana mode hosted by the minority-spin sector. Our theory further predicts majority-spin subgap-energy bound states similar to those found at a Josephson junction between same-chirality p-wave superfluids. Their presence affects the Majorana mode due to a coupling of minority and majority spin sectors only in the small energy range where their spectra overlap. Our results may inform experimental efforts aimed at realizing and characterizing unconventional Majorana quasiparticles.

Spin–orbit coupling adjusting topological superfluid of mass-imbalanced Fermi gas

Topological superfluid state is different from the normal superfluid one due to the excitation energy gap on the boundary. How to obtain the topological superfluid state by using spin–orbit coupling to control the s-waves paired mass-imbalanced Fermi gas is a recent novel topic. In this paper, we study the topological superfluid phase diagram of two-dimensional mass-imbalanced Fermi gas with Rashba spin–orbit coupling at zero temperature. We find that due to the competition among mass imbalance, pairing interaction and spin–orbit coupling, there is a double-well structure in the thermodynamic potential, which affects the properties of the ground state of the system. We comprehensively give the phase diagrams of the system on the plane of spin–orbit coupling and chemical potential, and the phase diagrams on the plane of the reduced mass ratio and two-body binding energy. This study not only points out the stable region of topological superfluid state of mass-imbalanced Fermi gas, but also provides a detailed theoretical basis for better observation of topological superfluid state in experiments.

Stabilizing topological superfluidity of lattice fermions

Attractive interaction between spinless fermions in a two-dimensional lattice drives the formation of a topological superfluid. But the topological phase is dynamically unstable towards phase separation when the system has a high density of states and large interaction strength. This limits the critical temperature to an experimentally challenging regime where, for example, even ultracold atoms and molecules in optical lattices would struggle to realize the topological superfluid. We propose that the introduction of a weaker longer-range repulsion, in addition to the short-range attraction between lattice fermions, will suppress the phase separation instability. Taking the honeycomb lattice as an example, we show that our proposal significantly enlarges the stable portion of the topological superfluid phase and increases the critical temperature by an order of magnitude. Our work opens a route to enhance the stability of topological superfluids by engineering inter-particle interactions.

Multiorder topological superfluid phase transitions in a two-dimensional optical superlattice

Higher-order topological superfluids have gapped bulk and symmetry-protected Majorana zero modes with various localizations. Motivated by recent advances, we present a proposal for synthesizing multi-order topological superfluids that support various Majorana zero modes in ultracold atomic gases. For this purpose, we use the two-dimensional optical superlattice that introduces a spatial modulation to the spin-orbit coupling in one direction, providing an extra degree of freedom for the emergent higher-order topological state. We find the topologically trivial superfluids, first-order and second-order topological superfluids, as well as different topological phase transitions among them with respect to the experimentally tunable parameters. Besides the conventional transition characterized by the Chern number associated with the bulk gap closing and reopening, we find the system can support the topological superfluids with Majorana corner modes, but the topological phase transition undergoes no gap-closing of bulk bands. Instead, the transition is refined by the quadrupole moment and signaled out by the gap-closing of edge states. The proposal is based on the $s$-wave interaction and is valid using existing experimental techniques, which unifies multi-order topological phase transitions in a simple but realistic system.

Thermal transport of helium-3 in a strongly confining channel

The investigation of transport properties in normal liquid helium-3 and its topological superfluid phases provides insights into related phenomena in electron fluids, topological materials, and putative topological superconductors. It relies on the measurement of mass, heat, and spin currents, due to system neutrality. Of particular interest is transport in strongly confining channels of height approaching the superfluid coherence length, to enhance the relative contribution of surface excitations, and suppress hydrodynamic counterflow. Here we report on the thermal conduction of helium-3 in a 1.1 μm high channel. In the normal state we observe a diffusive thermal conductivity that is approximately temperature independent, consistent with interference of bulk and boundary scattering. In the superfluid, the thermal conductivity is only weakly temperature dependent, requiring detailed theoretical analysis. An anomalous thermal response is detected in the superfluid which we propose arises from the emission of a flux of surface excitations from the channel.

Topological superfluids for spin-orbit coupled ultracold Fermi gases

We examine the topological superfluid phases for ultracold fermions in the presence of two-dimensional (2D) synthetic spin-orbit (SO) couplings which have been well realized in recent experiments. Specifically, we focus on the 2D Dirac semimetal and the SO coupled quantum anomalous Hall model, together with an attractive Hubbard interaction which can induce a superfluid phase in SO coupled systems. The superfluid phases are studied systematically based on self-consistent theories. In particular, an interesting transition between $(\ensuremath{\pi},\ensuremath{\pi})$-momentum superfluid phase and zero center-of-mass momentum superfluid phase is predicted as the strength of SO coupling is increased. The full phase diagrams of the superfluid phases are obtained, with the nontrivial topological superfluids being determined by precisely calculating the topological invariants. We also determine the optimal parameter conditions of the topological superfluid phases, which could be helpful for future real experimental studies.

Accurate projective two-band description of topological superfluidity in spin-orbit-coupled Fermi gases

The interplay of spin-orbit coupling and Zeeman splitting in ultracold Fermi gases gives rise to a topological superfluid phase in two spatial dimensions that can host exotic Majorana excitations. Theoretical models have so far been based on a four-band Bogoliubov-de Gennes formalism for the combined spin-\mathbf{1/2}1/2 and particle-hole degrees of freedom. Here we present a simpler, yet accurate, two-band description based on a well-controlled projection technique that provides a new platform for exploring analogies with chiral p-wave superfluidity and detailed future studies of spatially non-uniform situations.

From nodal-ring topological superfluids to spiral Majorana modes in cold atomic systems

In this work, we consider a 3D cubic optical lattice composed of coupled 1D wires with 1D spin-orbit coupling. When the s-wave pairing is induced through Feshbach resonance, the system becomes a topological superfluid with ring nodes, which are the ring nodal degeneracies in the bulk, and supports a large number of surface Majorana zero energy modes. The large number of surface Majorana modes remain at zero energy even in the presence of disorder due to the protection from a chiral symmetry. When the chiral symmetry is broken, the system becomes a Weyl topological superfluid with Majorana arcs. With 3D spin-orbit coupling, the Weyl superfluid becomes a novel gapless phase with spiral Majorana modes on the surface. The spatial resolved radio frequency spectroscopy is suggested to detect this novel nodal ring topological superfluid phase.

Higher first Chern numbers in one-dimensional Bose–Fermi mixtures

We propose to use a one-dimensional system consisting of identical fermions in a periodically driven lattice immersed in a Bose gas, to realise topological superfluid phases with Chern numbers larger than 1. The bosons mediate an attractive induced interaction between the fermions, and we derive a simple formula to analyse the topological properties of the resulting pairing. When the coherence length of the bosons is large compared to the lattice spacing and there is a significant next-nearest neighbour hopping for the fermions, the system can realise a superfluid with Chern number ±2. We show that this phase is stable in a large region of the phase diagram as a function of the filling fraction of the fermions and the coherence length of the bosons. Cold atomic gases offer the possibility to realise the proposed system using well-known experimental techniques.

Odd-parity topological superfluidity for fermions in a bond-centered square optical lattice

We propose a physical scheme for the realization of two-dimensional topological odd-parity superfluidity in a spin-independent bond-centered square optical lattice based upon interband fermion pairing. The D4 point-group symmetry of the lattice protects a quadratic band crossing, which allows one to prepare a Fermi surface of spin-up fermions with odd parity close to the degeneracy point. In the presence of spin-down fermions with even parity populating a different energetically well separated band, odd-parity pairing is favored. Strikingly, as a necessary prerequisite for pairing both Fermi surfaces can be tuned to match well. As a result, topological superfluid phases emerge in the presence of merely s-wave interaction. Due to the Z2 symmetry of these odd-parity superfluids, we infer their topological features simply from the symmetry and the Fermi-surface topology as confirmed numerically.

Topological superfluid in a two-dimensional polarized Fermi gas with spin–orbit coupling and adiabatic rotation

We study the properties of superfluid in a two-dimensional (2D) polarized Fermi gas with spin–orbit coupling and adiabatic rotation which are trapped in a harmonic potential. Due to the competition between polarization, spin–orbit coupling, and adiabatic rotation, the Fermi gas exhibits many intriguing phenomena. By using the Bardeen–Cooper– Schrieffer (BCS) mean-field method with local density approximation, we investigate the dependence of order parameter solution on the spin–orbit coupling strength and the rotation velocity. The energy spectra with different rotation velocities are studied in detail. Besides, the conditions for the zero-energy Majorana fermions in topological superfluid phase to be observed are obtained. By investigating distributions of number density, we find that the rotation has opposite effect on the distribution of number density with different spins, which leads to the enhancement of the polarization of Fermi gas. Here, we focus on the region of BCS pairing and ignore the Fulde–Ferrell–Larkin–Ovchinnikov state.

Harmonically trapped quasi-two-dimensional Fermi gases with synthetic spin-orbit coupling

We study the properties of spin-orbit coupled and harmonically trapped quasi-two-dimensional Fermi gas with tunable s-wave interaction between the two spin species. We adapt an effective two-channel model which takes the excited states occupation in the strongly confined axial direction into consideration by introducing dressed molecules in the closed channel, and use a Bogoliubovde Gennes (BdG) formalism to go beyond local density approximation. We find that both the in-trap phase structure and density distribution can be significantly modified near a wide Feshbach resonance compared with the single-channel model without the dressed molecules. Our findings will be helpful for the experimental search for the topological superfluid phase in ultracold Fermi gases.

Soliton-induced Majorana fermions in a one-dimensional atomic topological superfluid

We theoretically investigate the behavior of dark solitons in a one-dimensional spin-orbit coupled atomic Fermi gas in harmonic traps by solving self-consistently the Bogoliubov--de Gennes equations. The dark soliton---to be created by phase imprinting in future experiments---is characterized by a real order parameter, which changes sign at a point node and hosts localized Andreev bound states near the node. By considering both cases of a single soliton and multiple solitons, we find that the energy of these bound states decreases to zero when the system is tuned to enter the topological superfluid phase by increasing an external Zeeman field. As a result, two Majorana fermions emerge in the vicinity of each soliton, in addition to the well-known Majorana fermions at the trap edges associated with the nontrivial topology of the superfluid. We propose that the soliton-induced Majorana fermions can be directly observed by using spatially resolved radio-frequency spectroscopy or indirectly probed by measuring the density profile at the point node. For the latter, the deep minimum in the density profile will disappear due to the occupation of the soliton-induced zero-energy Majorana fermion modes. Our prediction could be tested in a resonantly interacting spin-orbit coupled ${}^{40}\text{K}$ Fermi gas confined in a two-dimensional optical lattice.

Gapless topological Fulde-Ferrell superfluidity induced by an in-plane Zeeman field

Topological superfluids are recently discovered quantum matters that host topologically protected gapless edge states known as Majorana fermions - exotic quantum particles that act as their own anti-particles and obey non-Abelian statistics. Their realizations are believed to lie at the heart of future technologies such as fault-tolerant quantum computation. To date, the most efficient scheme to create topological superfluids and Majorana fermions is based on the Sau-Lutchyn-Tewari-Das Sarma model with a Rashba-type spin-orbit coupling on the }\textbf{\textit{x-y}}\textbf{ plane and a large out-of-plane (perpendicular) Zeeman field along the }\textbf{\textit{z}}\textbf{-direction. Here we propose an alternative setup, where the topological superfluid phase is driven by applying an in-plane Zeeman field. This scheme offers a number of new features, notably Cooper pairings at finite centre-of-mass momentum (i.e., Fulde-Ferrell pairing) and gapless excitations in the bulk. As a result, a novel gapless topological quantum matter with inhomogeneous pairing order parameter appears. It features unidirected Majorana surface states at boundaries, which propagate in the same direction and connect two Weyl nodes in the bulk. We demonstrate the emergence of such an exotic topological matter and the associated Majorana fermions in spin-orbit coupled atomic Fermi gases and determine its parameter space. The implementation of our scheme in semiconductor/superconductor heterostructures is briefly discussed.

Realization of 2D Spin-Orbit Interaction and Exotic Topological Orders in Cold Atoms

Majorana zero bound mode exists in the vortex core of a chiral $p+ip$ superconductor or superfluid, which can be driven from an s-wave pairing state by two-dimensional (2D) spin-orbit (SO) coupling. We propose here a novel scheme based on realistic cold atom platforms to generate 2D SO interactions in a blue-detuned square optical lattice, and predict both the quantum anomalous Hall effect and chiral topological superfluid phase in the experimentally accessible parameter regimes. This work opens a new direction with experimental feasibility to observe non-Abelian topological orders in cold atom systems.

Chiral ladders and the edges of quantum Hall insulators

The realization and detection of topological phases with ultracold atomic gases is at the frontier of current theoretical and experimental research. Here, we identify cold atoms in optical ladders subjected to synthetic magnetic fields as readily realizable bridges between one-dimensional spin-orbit (time reversal) topological insulators and two-dimensional Chern insulators. We reveal three instances of their promising potential: i) they realize spin-orbit coupling, with the left-right leg degree of freedom playing the role of an effective spin, ii) their energy bands and eigenstates exactly reproduce the topological chiral edge modes of two-dimensional Chern insulators, and iii) they can be tailored to realize a topological phase transition from a trivial to a topological insulating phase. We propose realistic schemes to observe the chiral and topological properties of ladder systems with current optical lattice-based experiments. Our findings open a door to the exploration of the physics of the edges of Chern insulators and to the realization of spin-orbit coupling and topological superfluid phases with ultracold atomic gases.

Topological uniform superfluid and Fulde-Ferrell-Larkin-Ovchinnikov phases in three-dimensional to one-dimensional crossover of spin-orbit-coupled Fermi gases

We consider the quasi-one-dimensional (quasi-1D) system realized by an array of weakly coupled parallel one-dimensional ``tubes'' in a two-dimensional lattice which permits free motion of atoms in an axial direction in the presence of a Zeeman field, Rashba type spin-orbit coupling (SOC), and an $s$-wave attractive interaction, while the radial motion is tightly confined. We solve the zero-temperature ($T=0$) Bogoliubov--de Gennes (BdG) equations for the quasi-1D Fermi gas with the dispersion modified by tunneling between the tubes and show that the $T=0$ phase diagram hosts the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase with nonzero center-of-mass momentum Cooper pairs for small values of the SOC while for larger values of the SOC and high Zeeman fields the uniform superfluid phase with zero center-of-mass momentum Cooper pairs has an instability towards the topological uniform superfluid phase with Majorana fermions at the tube ends. Also, we show that tuning the two-dimensional optical lattice strength in this model allows one to explore the crossover behaviors of the phases during the transition between the three-dimensional and 1D systems and in general the FFLO (for small SOC) and the topological uniform superfluid phase (for large SOC) are favored as the system becomes more one dimensional. We also find evidence of the existence of a Zeeman-tuned topological quantum phase transition (TQPT) within the FFLO phase itself and, for large values of the Zeeman field and small SOC, the TQPT gives rise to a topologically distinct FFLO phase.

Phase diagram of a spin–orbit coupled Fermi gas in a bilayer optical lattice

We investigate the stability of helical superfluid phase in a spin–orbit coupled Fermi gas loaded in a bilayer optical lattice. The phase diagram of the system is constructed in the mean-field framework. We investigate the topological properties of the superfluid phases by a nontrivial application of the Fermi surface topological invariant to our time-reversal invariant system with degeneracies on the Fermi surface. We find that there is a first-order phase boundary in the phase diagram of half-filling case and the superfluid phases are all topological trivial. The superfluid phase is topological nontrivial when the filling fraction deviates from the half-filling. In the topological nontrivial superfluid phase, a full pairing gap exists in the bulk and gapless helical Majorana edge states exist at the boundary.

Topological Fulde-Ferrell superfluid in spin-orbit-coupled atomic Fermi gases

We theoretically predict a new topological matter - topological inhomogeneous Fulde-Ferrell superfluid - in one-dimensional atomic Fermi gases with equal Rashba and Dresselhaus spin-orbit coupling near s-wave Feshbach resonances. The realization of such a spin-orbit coupled Fermi system has already been demonstrated recently by using a two-photon Raman process and the extra one-dimensional confinement is easy to achieve using a tight two-dimensional optical lattice. The topological Fulde-Ferrell superfluid phase is characterized by a nonzero center-of-mass momentum and a non-trivial Berry phase. By tuning the Rabi frequency and the detuning of Raman laser beams, we show that such an exotic topological phase occupies a significant part of parameter space and therefore it could be easily observed experimentally, by using, for example, momentum-resolved and spatially resolved radio-frequency spectroscopy.

Superfluidity and collective modes in Rashba spin–orbit coupled Fermi gases

We present a theoretical study of the superfluidity and the corresponding collective modes in two-component atomic Fermi gases with s-wave attraction and synthetic Rashba spin–orbit coupling. The general effective action for the collective modes is derived from the functional path integral formalism. By tuning the spin–orbit coupling from weak to strong, the system undergoes a crossover from an ordinary BCS/BEC superfluid to a Bose–Einstein condensate of rashbons. We show that the properties of the superfluid density and the Anderson–Bogoliubov mode manifest this crossover. At large spin–orbit coupling, the superfluid density and the sound velocity become independent of the strength of the s-wave attraction. The two-body interaction among the rashbons is also determined. When a Zeeman field is turned on, the system undergoes quantum phase transitions to some exotic superfluid phases which are topologically nontrivial. For the two-dimensional system, the nonanalyticities of the thermodynamic functions and the sound velocity across the phase transition are related to the bulk gapless fermionic excitation which causes infrared singularities. The superfluid density and the sound velocity behave nonmonotonically: they are suppressed by the Zeeman field in the normal superfluid phase, but get enhanced in the topological superfluid phase. The three-dimensional system is also studied.