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Physical Review A

Topological phases and collective modes in U(1) and SU(2) sectors of spin-orbit-coupled two-dimensional superfluid Fermi gases

Jeroen P. A. Devreese, Jacques Tempere, Carlos A. R. Sá de Melo

Published Mar 7, 2022• 1 citation
Publisher

Abstract

We study the emergence of finite-temperature topological phase transitions of the Berezinskii-Kosterlitz-Thouless type and the collective mode spectrum of two-dimensional Fermi gases in the presence of attractive $s$-wave interactions, spin-orbit coupling, and Zeeman fields. We show that neglecting the spin-dependent phase shift on the fermionic wave function misses an important point. Including this spin-dependent shift, we derive the effective low-energy and long-wavelength quantum action for independent (unlocked) phase fluctuations in the U(1) (charge) and SU(2) (spin) sectors and show that at least two phase transitions occur because charge and spin degrees of freedom are coupled due to the presence of spin-orbit and Zeeman fields. Furthermore, we demonstrate that vortex and antivortex excitations are characterized by two topological quantum numbers, corresponding to the quantized circulation of charge and spin velocities. Finally, we show that there are two collective modes in the superfluid phase at low temperatures which arise due to the coupling between sound and transverse-spin waves.

DOI
10.1103/physreva.105.033304

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We study the emergence of finite-temperature topological phase transitions of the Berezinskii-Kosterlitz-Thouless type and the collective mode spectrum of two-dimensional Fermi gases in the presence of attractive sss-wave interactions, spin-orbit coupling, and Zeeman fields. We show that neglecting the spin-dependent phase shift on the fermionic wave function misses an important point. Including this spin-dependent shift, we derive the effective low-energy and long-wavelength quantum action for independent (unlocked) phase fluctuations in the U(1) (charge) and SU(2) (spin) sectors and show that at least two phase transitions occur because charge and spin degrees of freedom are coupled due to the presence of spin-orbit and Zeeman fields. Furthermore, we demonstrate that vortex and antivortex excitations are characterized by two topological quantum numbers, corresponding to the quantized circulation of charge and spin velocities. Finally, we show that there are two collective modes in the superfluid phase at low temperatures which arise due to the coupling between sound and transverse-spin waves.
We study the effect of spin-orbit coupling on both the zero-temperature and nonzero-temperature behavior of a two-dimensional Fermi gas. We include a generic combination of Rashba and Dresselhaus terms into the system Hamiltonian, which allows us to study both the experimentally relevant equal-Rashba-Dresselhaus (ERD) limit and the Rashba-only (RO) limit. At zero temperature, we derive the phase diagram as a function of the two-body binding energy and Zeeman field. In the ERD case, this phase diagram reveals several topologically distinct uniform superfluid phases, classified according to the nodal structure of the quasiparticle excitation energies. Furthermore, we use a momentum-dependent SU(2) rotation to transform the system into a generalized helicity basis, revealing that spin-orbit coupling induces a triplet pairing component of the order parameter. At nonzero temperature, we study the Berezinskii-Kosterlitz-Thouless (BKT) phase transition by including phase fluctuations of the order parameter up to second order. We show that the superfluid density becomes anisotropic due to the presence of spin-orbit coupling (except in the RO case). This leads both to elliptic vortices and antivortices, and to anisotropic sound velocities. The latter prove to be sensitive to quantum phase transitions between topologically distinct phases. We show further that at a fixed nonzero Zeeman field, the BKT critical temperature is increased by the presence of ERD spin-orbit coupling. Subsequently, we demonstrate that the Clogston limit becomes infinite: TBKT{T}_{\mathrm{BKT}}TBKT​ remains nonzero at all finite values of the Zeeman field. We conclude by extending the quantum phase transition lines to nonzero temperature, using the nodal structure of the quasiparticle spectrum, thus connecting the BKT critical temperature with the zero-temperature results.
We investigate the Berezinskii-Kosterlitz-Thouless (BKT) transition in a 2D Fermi gas with spin-orbit coupling (SOC), as a function of the two-body binding energy and a perpendicular Zeeman field. By including a generic form of the SOC, as a function of Rashba and Dresselhaus terms, we study the evolution between the experimentally relevant equal Rashba-Dresselhaus (ERD) case and the Rashba-only (RO) case. We show that in the ERD case, at a fixed nonzero Zeeman field, the BKT transition temperature T(BKT) is increased by the presence of SOC for all values of the binding energy. We also find a significant increase in the value of the Clogston limit compared to the case without SOC. Furthermore, we demonstrate that the superfluid density tensor becomes anisotropic (except in the RO case), leading to an anisotropic phase-fluctuation action that describes elliptic vortices and antivortices, which become circular in the RO limit. This deformation constitutes an important experimental signature for superfluidity in a 2D Fermi gas with ERD SOC. Finally, we show that the anisotropic sound velocities exhibit anomalies at low temperatures, in the vicinity of quantum phase transitions between topologically distinct uniform superfluid phases.
We investigate the Zeeman field effects on the bulk superfluid properties and the collective modes in two-dimensional (2D) attractive atomic Fermi gases with Rashba-type spin-orbit coupling. In the presence of a large spin-orbit coupling, the system undergoes a quantum phase transition to a topological superfluid state at a critical Zeeman field. We show that the nonanalyticities of the thermodynamic functions as well as other physical quantities at the quantum phase transition originate from the infrared singularities caused by the gapless fermionic spectrum. The same argument applies also to the BCS-BEC evolution in 2D fermionic superfluids with ppp- or ddd-wave pairing. The superfluid density ns{n}_{s}ns​ and the velocity of the Goldstone sound mode cs{c}_{s}cs​ behave oppositely in the normal and the topological superfluid phases: they are suppressed by the Zeeman field in the normal superfluid phase, but get enhanced in the topological superfluid phase. The velocity of the Goldstone sound mode also shows nonanalyticity at the quantum phase transition. For large Zeeman field, we find ${n}_{s}\ensuremath{\rightarrow}n$ and ${c}_{s}\ensuremath{\rightarrow}{\ensuremath{\upsilon}}_{\mathrm{F}}$, where nnn is the total fermion density and ${\ensuremath{\upsilon}}_{\mathrm{F}}$ is the Fermi velocity of noninteracting system. The unusual behavior of the superfluid density and the collective modes can be understood by the fact that the spin-orbit-coupled superfluid state at large Zeeman field can be mapped to the px+ipy{p}_{x}+i{p}_{y}px​+ipy​ superfluid state of spinless fermions.
We develop a theory for interacting fermions in the presence of spin-orbit coupling and Zeeman fields, and show that many new superfluids phases, which are topological in nature, emerge. Depending on values of spin-orbit coupling, Zeeman fields, and interactions, initially gapped s-wave superfluids acquire p-wave, d-wave, f-wave and higher angular momentum components, which produce zeros in the excitation spectrum, rendering the superfluid gapless. Several multi-critical points, which separate topological superfluid phases from normal or non-uniform, are accessible depending on spin-orbit coupling, Zeeman fields or interactions, setting the stage for the study of tunable topological superfluids.
Topological superfluids usually refer to a superfluid state which is gapped in the bulk but metallic at the boundary. Here we report that a gapless, topologically nontrivial superfluid with an inhomogeneous Fulde-Ferrell pairing order parameter can emerge in a two-dimensional spin-orbit coupled Fermi gas, in the presence of both in-plane and out-of-plane Zeeman fields. The Fulde-Ferrell pairing-induced by the spin-orbit coupling and in-plane Zeeman field-is responsible for this gapless feature. This exotic superfluid has a significant Berezinskii-Kosterlitz-Thouless transition temperature and has robust Majorana edge modes against disorder owing to its topological nature.
Spin-orbit coupled fermions offer a fascinating playground for realizing interesting topological quantum phases and the associated quantum critical phenomena. In two dimensions and in the presence of an attractive s-wave pairing interaction and a Zeeman field, spin-orbit coupled fermions can condense into a fully gapped non-Abelian topological superfluid phase separated from a conventional s-wave superfluid phase by a topological quantum phase transition. The critical point for this transition is marked by the proliferation of gapless nodal quasiparticles which can be tracked by studying the momentumresolved excitation spectrum from the center of the Brillouin zone. In three dimensions the same system goes through a series of topological quantum transitions with increasing Zeeman field and the low energy excitations in some of these phases realize the long-sought low-temperature analog of Weyl fermions of particle physics. In this talk I will describe in some detail the interesting topological phases and the associated quantum criticality that can be realized using spin-orbit coupled fermions in cold atom systems.