Topological phase transitions in ultracold Fermi superfluids: The evolution from Bardeen-Cooper-Schrieffer to Bose-Einstein-condensate superfluids under artificial spin-orbit fields
We discuss topological phase transitions in ultracold Fermi superfluids induced by interactions and artificial spin-orbit fields. We construct the phase diagram for population-imbalanced systems at zero and finite temperatures, and analyze spectroscopic and thermodynamic properties to characterize various phase transitions. For balanced systems, the evolution from Bardeen-Cooper-Schrieffer to Bose-Einstein-condensate superfluids in the presence of spin-orbit effects is only a crossover as the system remains fully gapped, even though a triplet component of the order parameter emerges in the helicity basis. However, for imbalanced populations, spin-orbit fields induce a triplet component for the order parameter in the helicity basis, and may produce nodes in the quasiparticle excitation spectrum leading to bulk topological phase transitions of the Lifshitz type. Additionally a fully gapped phase exists, where a crossover from indirect to direct gap occurs, but a topological transition to a gapped phase possessing Majorana fermion edge states does not occur.
We discuss topological phase transitions in ultracold Fermi superfluids induced by interactions and artificial spin-orbit fields. We construct the phase diagram for population-imbalanced systems at zero and finite temperatures, and analyze spectroscopic and thermodynamic properties to characterize various phase transitions. For balanced systems, the evolution from Bardeen-Cooper-Schrieffer to Bose-Einstein-condensate superfluids in the presence of spin-orbit effects is only a crossover as the system remains fully gapped, even though a triplet component of the order parameter emerges in the helicity basis. However, for imbalanced populations, spin-orbit fields induce a triplet component for the order parameter in the helicity basis, and may produce nodes in the quasiparticle excitation spectrum leading to bulk topological phase transitions of the Lifshitz type. Additionally a fully gapped phase exists, where a crossover from indirect to direct gap occurs, but a topological transition to a gapped phase possessing Majorana fermion edge states does not occur.
We discuss the emergence of rings of zero-energy excitations in momentum space for superfluid phases of ultracold fermions when spin-orbit effects, Zeeman fields, and interactions are varied. We show that phases containing rings of nodes possess nontrivial topological invariants, and that phase transitions between distinct topological phases belong to the Lifshitz class. Upon crossing phase boundaries, existing massless Dirac fermions in the gapless phase annihilate to produce bulk zero-mode Majorana fermions at phase boundaries, and then become massive Dirac fermions in the gapped phase. We characterize these tunable topological phase transitions via several spectroscopic properties, including excitation spectrum, spectral function, and momentum distribution.
We analyze the evolution of superfluidity for nonzero orbital angular momentum channels from the Bardeen-Cooper-Schrieffer (BCS) to the Bose-Einstein condensation (BEC) limit in three dimensions. First, we analyze the low-energy scattering properties of finite range interactions for all possible angular momentum channels. Second, we discuss ground-state (T=0) superfluid properties including the order parameter, chemical potential, quasiparticle excitation spectrum, momentum distribution, atomic compressibility, ground-state energy, and low-energy collective excitations. We show that a quantum phase transition occurs for nonzero angular momentum pairing, unlike the s-wave case where the BCS to BEC evolution is just a crossover. Third, we present a Gaussian fluctuation theory near the critical temperature (T=Tc), and we analyze the number of bound, scattering, and unbound fermions as well as the chemical potential. Finally, we derive the time-dependent Ginzburg-Landau functional near Tc, and compare the Ginzburg-Landau coherence length with the zero-temperature average Cooper pair size.
This thesis focuses on the analysis of Bardeen-Cooper-Schrieffer (BCS) \nto Bose-Einstein condensation (BEC) evolution in ultracold superfluid \nFermi gases when the interaction between atoms is varied. The tuning of \nattractive interactions permits the ground state of the system to evolve from a weak \nfermion attraction BCS limit of loosely bound and largely overlapping Cooper pairs \nto a strong fermion attraction limit of tightly bound small bosonic molecules \nwhich undergo BEC. \nThis evolution is accompanied by anomalous behavior of many superfluid \nproperties, and reveals several quantum phase transitions. \nThis thesis has two parts: \nIn the first part, I analyze zero and nonzero orbital angular momentum \npairing effects, and show that a quantum phase transition occurs for \nnonzero angular momentum pairing, unlike the s-wave case where the BCS \nto BEC evolution is just a crossover. \nIn the second part, I analyze two-species fermion mixtures with mass and \npopulation imbalance in continuum, trap and lattice models. In contrast with \nthe crossover physics found in the mass and population balanced mixtures, \nI demonstrate the existence of phase transitions between normal and \nsuperfluid phases, as well as phase separation between superfluid (paired) \nand normal (excess) fermions in imbalanced mixtures as a function of scattering \nparameter and mass and population imbalance.
We describe how color superfluidity is modified in the presence of color-flip and color-orbit fields in the context of ultracold atoms and discuss connections between this problem and that of color superconductivity in quantum chromodynamics. We study the case of s-wave contact interactions between different colors and we identify several superfluid phases, with five being nodal and one being fully gapped. When our system is described in a mixed-color basis, the superfluid order parameter tensor is characterized by six independent components with explicit momentum dependence induced by color-orbit coupling. The nodal superfluid phases are topological in nature and the low-temperature phase diagram of the color-flip field versus the interaction parameter exhibits a pentacritical point, where all five nodal color superfluid phases converge. These results are in sharp contrast to the case of zero color-flip and color-orbit fields, where the system has perfect U(3) symmetry and possesses a superfluid phase that is characterized by fully gapped quasiparticle excitations with a single complex order parameter with no momentum dependence and by inert unpaired fermions representing a nonsuperfluid component. In the latter case, just a crossover between a Bardeen-Cooper-Schrieffer and a Bose-Einstein-condensation superfluid occurs. Furthermore, we analyze the order parameter tensor in a total pseudospin basis, investigate its momentum dependence in the singlet, triplet, and quintet sectors, and compare the results with the simpler case of spin-1/2 fermions in the presence of spin-flip and spin-orbit fields, where only singlet and triplet channels arise. Finally, we analyze in detail spectroscopic properties of color superfluids in the presence of color-flip and color-orbit fields, such as the quasiparticle excitation spectrum, momentum distribution, and density of states to help characterize all the encountered topological quantum phases, which can be realized in fermionic isotopes of lithium, potassium, and ytterbium atoms with three internal states trapped.
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.
We theoretically investigate one-dimensional three-component spin-orbit-coupled Fermi gases in the presence of the Zeeman field. By solving the Bogoliubov-de Gennes equations, we obtain the phase diagram at a given chemical potential and order parameter. We show that, with increasing the intensity of the Zeeman field, in addition to undergoing a phase transition from Bardeen-Cooper-Schrieffer (BCS) superfluid to topological superfluid, similar to the two-component system, the three-component system may exhibit some other interesting topological phase transitions. For example, by appropriately adjusting the chemical potential $\ensuremath{\mu}$, the system can be in a nontrivial topological superfluid in the whole region of the Zeeman field h. It also may initially be a topological superfluid and then translate to a topologically trivial BCS superfluid with increasing the field h. Even more exotically, the system may exhibit a re-entrance behavior, being a topological superfluid at small and large fields but a topologically trivial BCS superfluid in between at a mediate Zeeman field. It can therefore have two regions with zero-energy Majorana fermions. As a consequence of these interesting topological phase transitions, the system of the three-component spin-orbit-coupled Fermi gases in a certain parameter range is more optimizing for the experimental realization of the topological phase due to the smaller magnetic field needed. Thus, a promising candidate for the realization of the topological phase is proposed.