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.
The two-dimensional spin-imbalanced Fermi gas subject to
s-wave pairing and spin-orbit coupling is considered a promising platform for realizing a topological chiral-
p-wave superfluid. In the BCS limit of
s-wave pairing, i.e., when Cooper pairs are only weakly bound, the system enters the topological phase via a second-order transition driven by increasing the Zeeman spin-splitting energy. Stronger attractive two-particle interactions cause the system to undergo the BCS-BEC crossover, in the course of which the topological transition becomes first order. As a result, topological and nontopological superfluids coexist in spatially separated domains in an extended region of phase space spanned by the strength of
s-wave interactions and the Zeeman energy. Here we investigate this phase-coexistence region theoretically using a zero-temperature mean-field approach. Exact numerical results are presented to illustrate basic physical characteristics of the coexisting phases and to validate an approximate analytical description derived for weak spin-orbit coupling. Besides extending our current understanding of spin-imbalanced superfluid Fermi systems, the present approach also provides a platform for future studies of unconventional Majorana excitations that, according to topology, should be present at the internal interface between coexisting topological and nontopological superfluid parts of the system.
The twodimensional chiral p-wave superfluid is known to be topological, with gapless Majorana states expected to exist at its edge. The possibility to utilise such Majorana excitations as fault-tolerant quantum bits has spurred interest in robust experimental realisation of such systems. One proposed pathway is to use spin-orbit coupled and spin-imbalanced ultracold gases of fermionic atoms subject to conventional s-wave pairing, as inducing a large-enough Zeeman energy splitting between pseudo-spin-1/2 states drives the system into a topological superfluid (TSF) phase with chiral-p-wave characteristics. In this thesis, we apply and extend advanced theoretical methods to study this topological phase transition in detail. Specifically, our work elucidates what happens when the s-wave interaction strength between atoms is increased so that the system undergoes the BCS-to-BEC crossover. While previous studies have largely focused on the weakly interacting BCS regime, studying the full crossover reveals new properties of the TSF phase that may aid its experimental realisation in the future. In particular, we obtain phase diagrams in the parameter space of the s-wave interaction strength and Zeeman energy. As long as the system is in the nontopological superfluid (NSF) phase, it shows the ordinary features expected during the BCS-to-BEC crossover, such as the shrinking and disappearance of the Fermi surfaces as the s-wave interaction strength is increased. The TSF phase, however, exhibits strikingly different behavior: it retains the BCS-limit characteristics of a TSF even as the s-wave interaction strength reaches extremely large values. The phase diagrams we obtain also show that increasing the s-wave interaction strength changes the nature of the topological transition from second-order continuous to first-order. During the first-order topological transition, the system splits into coexisting regions of NSF and TSF phases. We study this phase coexistence theoretically and obtain analytical formulae that capture the behaviour of all relevant thermodynamic quantities during the first-order topological transition. In that, our work extends current fundamental knowledge about spin-imbalanced Fermi gases to situations with finite spin-orbit coupling. We also investigate the gapless edge states emerging at the system’s boundaries, as these are crucial for experiments that aim to observe topological phases and realise novel quantum-information-processing architectures. The principle of bulk-boundary correspondence guarantees that gapless Majorana excitations should be localised to the interface between coexisting NSF and TSF regions. We develop a detailed theoretical description of this novel topological interface state and compare its physical features with those exhibited by the Majorana states expected to exist at the system’s boundary with vacuum.
Two-dimensional spin-orbit-coupled Fermi gases subject to
s-wave pairing can be driven into a topological phase by increasing the Zeeman spin splitting beyond a critical value. In the topological phase, the system exhibits the hallmarks of chiral
p-wave superfluidity, including exotic Majorana excitations. Previous theoretical studies of this realization of a two-dimensional topological Fermi superfluid have focused on the BCS regime where the
s-wave Cooper pairs are only weakly bound and, hence, the induced chiral
p-wave order parameter has a small magnitude. Motivated by the goal to identify potential new ways for the experimental realization of robust topological superfluids in ultracold-atom gases, we study the BCS-to-BEC crossover driven by increasing the Cooper-pair binding energy for this system. In particular, we obtain phase diagrams in the parameter space of two-particle bound-state energy and Zeeman spin-splitting energy. Ordinary characteristics of the BCS-to-BEC crossover, in particular the shrinking and eventual disappearance of the Fermi surface, are observed in the nontopological phase. In contrast, the topological phase retains all features of chiral
p-wave superfluidity, including a well-defined underlying Fermi surface, even for large
s-wave pair-binding energies. Compared to the BCS limit, the topological superfluid in the BEC regime turns out to be better realizable even for only moderate magnitude of spin-orbit coupling because the chiral
p-wave order parameter is generally larger and remnants of
s-wave pairing are suppressed. We identify optimal parameter ranges that can aid further experimental investigations and elucidate the underlying physical reason for the persistence of the chiral
p-wave superfluid.
We theoretically predict an exotic topological superfluid state with a spatially modulated pairing gap in one-dimensional spin-orbit-coupled Fermi gases. This inhomogeneous topological superfluidity is induced by applying simultaneously a perpendicular Zeeman magnetic field and an equally weighted Rashba and Dresselhaus spin-orbit coupling in one-dimensional optical lattices. Based on the self-consistent Bogoliubov-de Gennes theory, we confirm that this novel topological phase is a unique condensation of Cooper pairs, which manifests the interplay between the inhomogeneity of a superfluid and its nontrivial topological structure. The properties of the emergent Majorana bound states are investigated in detail by examining the associated Z2 topological number, the eigenenergy and density of states spectra, as well as the wave functions of the localized Majorana end modes. The experimental feasibility of observing this new topological state of matter is also discussed.
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.
Abstract By solving the Bogoliubov–de Gennes equations analytically, we derive the fermionic zero-modes satisfying the Majorana property that exist in vortices of a two-dimensional s-wave Fermi superfluid with spin–orbit coupling and Zeeman spin-splitting. The Majorana zero-mode becomes normalisable and exponentially localised to the vicinity of the vortex core when the superfluid is topologically non-trivial. We calculate the energy splitting due to Majorana hybridisation and identify that the s-wave Majorana vortices obey non-Abelian statistics.