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arXiv:2109.14505 · Physical review. B./Physical review. B

Optical spin transport theory of spin- 12 topological Fermi superfluids

Hiroyuki Tajima, Yuta Sekino, Shun Uchino

Published Feb 16, 2022• 7 citations
arXivHTMLPDF

Abstract

We theoretically investigate optical (frequency-dependent) bulk spin transport properties in a spin-$\frac{1}{2}$ topological Fermi superfluid. We specifically consider a one-dimensional system with an interspin $p$-wave interaction, which can be realized in ultracold-atom experiments. Developing the BCS-Leggett theory to describe the BCS to Bose-Einstein condensate (BEC) evolution and the ${\mathbb{Z}}_{2}$ topological phase transition in this system, we show how the spin transport reflects these many-body aspects. We find that the optical spin conductivity, which is a small AC response of a spin current, shows the spin-gapped spectrum in the wide parameter region and the gap closes at the ${\mathbb{Z}}_{2}$ topological phase transition point. Moreover, the validity of the low-energy effective model of the Majorana zero mode is discussed along the BCS-BEC evolution in connection with the scale invariance at $p$-wave unitarity.

DOI
10.1103/physrevb.105.064508

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We theoretically investigate optical (frequency-dependent) bulk spin transport properties in a spin-12\frac{1}{2}21​ topological Fermi superfluid. We specifically consider a one-dimensional system with an interspin ppp-wave interaction, which can be realized in ultracold-atom experiments. Developing the BCS-Leggett theory to describe the BCS to Bose-Einstein condensate (BEC) evolution and the Z2{\mathbb{Z}}_{2}Z2​ topological phase transition in this system, we show how the spin transport reflects these many-body aspects. We find that the optical spin conductivity, which is a small AC response of a spin current, shows the spin-gapped spectrum in the wide parameter region and the gap closes at the Z2{\mathbb{Z}}_{2}Z2​ topological phase transition point. Moreover, the validity of the low-energy effective model of the Majorana zero mode is discussed along the BCS-BEC evolution in connection with the scale invariance at ppp-wave unitarity.
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 show that the optical spin conductivity, which is a small ac response of a bulk spin current and elusive in condensed matter systems, can be measured in ultracold atoms. We demonstrate that this conductivity contains rich information on quantum states by analyzing experimentally achievable systems such as a spin-12\frac{1}{2}21​ superfluid Fermi gas, a spin-1 Bose-Einstein condensate, and a Tomonaga-Luttinger liquid. The obtained conductivity spectra, which are absent in the Drude conductivity, reflect quasiparticle excitations and non-Fermi-liquid properties. Accessible physical quantities include the superfluid gap and the contact for the superfluid Fermi gas, gapped and gapless spin excitations as well as quantum depletion for the Bose-Einstein condensate, and the spin part of the Tomonaga-Luttinger liquid parameter elusive in cold-atom experiments. Unlike its mass transport counterpart, the spin conductivity serves as a probe applicable to clean atomic gases without disorder or lattice potentials. Our formalism can be generalized to various systems such as spin-orbit-coupled and nonequilibrium systems.
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.
Measurement of frequency-resolved spin transport is a subject of much interest in condensed matter physics. Here we show that the optical spin conductivity being a small AC response of a spin current can be measured with existing methods in ultracold atom experiments. We point out that once interatomic interactions are turned on, the optical spin conductivity becomes nontrivial even in clean ultracold atomic gases and thereby can be a probe of generic quantum states of matter. This is a sharp contrast to the optical mass conductivity which becomes trivial in typical cold-atom systems without disorder and lattice potential. For systems with arbitrary spin degrees of freedom, we construct a general formalism of the optical spin conductivity and derive the fff-sum rule. To demonstrate the availability of the optical spin conductivity, our formalism is applied to a spin-1/2 Fermi superfluid and a spin-1 Bose-Einstein condensate. It turns out that both superfluids show nontrivial responses that cannot be captured with the Drude conductivity. The application of our proposed method to generic ultracold atomic gases with spin degrees of freedom is feasible.
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 multiorder 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 sss-wave interaction and is valid using existing experimental techniques, which unifies multiorder topological phase transitions in a simple but realistic system.
Two-dimensional spin-orbit-coupled Fermi gases subject to sss-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 ppp-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 sss-wave Cooper pairs are only weakly bound and, hence, the induced chiral ppp-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 ppp-wave superfluidity, including a well-defined underlying Fermi surface, even for large sss-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 ppp-wave order parameter is generally larger and remnants of sss-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 ppp-wave superfluid.