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arXiv:1005.0889 · Scientific Reports

The J-triplet Cooper pairing with magnetic dipolar interactions

Yi Li, Congjun Wu

Published May 1, 2012• 29 citations
arXivHTMLPDF

Abstract

Recently, cold atomic Fermi gases with the large magnetic dipolar interaction have been laser cooled down to quantum degeneracy. Different from electric-dipoles which are classic vectors, atomic magnetic dipoles are quantum-mechanical matrix operators proportional to the hyperfine-spin of atoms, thus provide rich opportunities to investigate exotic many-body physics. Furthermore, unlike anisotropic electric dipolar gases, unpolarized magnetic dipolar systems are isotropic under simultaneous spin-orbit rotation. These features give rise to a robust mechanism for a novel pairing symmetry: orbital p-wave (L = 1) spin triplet (S = 1) pairing with total angular momentum of the Cooper pair J = 1. This pairing is markedly different from both the (3)He-B phase in which J = 0 and the (3)He-A phase in which J is not conserved. It is also different from the p-wave pairing in the single-component electric dipolar systems in which the spin degree of freedom is frozen.

DOI
10.1038/srep00392

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Recently, cold atomic Fermi gases with the large magnetic dipolar interaction have been laser cooled down to quantum degeneracy. Different from electric-dipoles which are classic vectors, atomic magnetic dipoles are quantum-mechanical matrix operators proportional to the hyperfine-spin of atoms, thus provide rich opportunities to investigate exotic many-body physics. Furthermore, unlike anisotropic electric dipolar gases, unpolarized magnetic dipolar systems are isotropic under simultaneous spin-orbit rotation. These features give rise to a robust mechanism for a novel pairing symmetry: orbital p-wave (L = 1) spin triplet (S = 1) pairing with total angular momentum of the Cooper pair J = 1. This pairing is markedly different from both the (3)He-B phase in which J = 0 and the (3)He-A phase in which J is not conserved. It is also different from the p-wave pairing in the single-component electric dipolar systems in which the spin degree of freedom is frozen.
The rapid experimental progress of ultra-cold dipolar fermions opens up a whole new opportunity to investigate novel many-body physics of fermions. In this article, we review theoretical studies of the Fermi liquid theory and Cooper pairing instabilities of both electric and magnetic dipolar fermionic systems from the perspective of unconventional symmetries. When the electric dipole moments are aligned by the external electric field, their interactions exhibit the explicit d(r(2)-3z(2)) anisotropy. The Fermi liquid properties, including the single-particle spectra, thermodynamic susceptibilities and collective excitations, are all affected by this anisotropy. The electric dipolar interaction provides a mechanism for the unconventional spin triplet Cooper pairing, which is different from the usual spin-fluctuation mechanism in solids and the superfluid (3)He. Furthermore, the competition between pairing instabilities in the singlet and triplet channels gives rise to a novel time-reversal symmetry breaking superfluid state. Unlike electric dipole moments which are induced by electric fields and unquantized, magnetic dipole moments are intrinsic proportional to the hyperfine-spin operators with a Lande factor. Its effects even manifest in unpolarized systems exhibiting an isotropic but spin-orbit coupled nature. The resultant spin-orbit coupled Fermi liquid theory supports a collective sound mode exhibiting a topologically non-trivial spin distribution over the Fermi surface. It also leads to a novel p-wave spin triplet Cooper pairing state whose spin and orbital angular momentum are entangled to the total angular momentum J = 1 dubbed the J-triplet pairing. This J-triplet pairing phase is different from both the spin-orbit coupled (3)He-B phase with J = 0 and the spin-orbit decoupled (3)He-A phase.
Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.
We investigate Fermi liquid states of the ultracold magnetic dipolar Fermi gases in the simplest two-component case including both thermodynamic instabilities and collective excitations. The magnetic dipolar interaction is invariant under the simultaneous spin-orbit rotation but not under either the spin or the orbit one. Therefore, the corresponding Fermi liquid theory is intrinsically spin-orbit coupled. This is a fundamental feature of magnetic dipolar Fermi gases different from electric dipolar ones. The Landau interaction matrix is calculated and is diagonalized in terms of the spin-orbit coupled partial-wave channels of the total angular momentum JJJ. The leading thermodynamic instabilities lie in the channels of ferromagnetism hybridized with the ferronematic order with J=1+J={1}^{+}J=1+ and the spin-current mode with $J={1}^{\ensuremath{-}}$, where +++ and $\ensuremath{-}$ represent even and odd parities, respectively. An exotic propagating collective mode is identified as spin-orbit coupled Fermi surface oscillations in which spin distribution on the Fermi surface exhibits a topologically nontrivial hedgehog configuration.
We show that unconventional spin order arises naturally in two-component dipolar Fermi gases of atoms or molecules, which recently became accessible experimentally, in optical lattices. Using an unbiased functional renormalization-group analysis, we find that dipolar interactions lead to an instability of the gas toward an $\ensuremath{\ell}=1$ spin-density wave state. This phase is the particle-hole analog of spin-triplet, ppp-wave Cooper pairs. The order parameter for such spin-density waves of ppp-wave orbital symmetry is a vector in spin space and, moreover, is defined on lattice bonds rather than on lattice sites. We determine the rich quantum phase diagram of dipolar fermions at half filling on the square lattice as a function of the dipolar orientation and discuss how these exotic spin-density waves emerge amidst competition with superfluid and charge-density wave phases.
We show that fermionic dipoles in a two-layer geometry form Cooper pairs with both singlet and triplet components when they are tilted with respect to the normal of the planes. The mixed parity pairing arises because the interaction between dipoles in the two different layers is not inversion symmetric. We use an efficient eigenvalue approach to calculate the zero-temperature phase diagram of the system as a function of the dipole orientation and the layer distance. The phase diagram contains purely triplet as well as mixed singlet and triplet superfluid phases. We show in detail how the pair wave function for dipoles residing in different layers smoothly changes from singlet to triplet symmetry as the orientation of the dipoles is changed. Our results indicate that dipolar quantum gases can be used to unambiguously observe mixed parity pairing.
Abstract Advances in semiconductor nanostructures have enabled the creation of ultra-cold dipolar fermions, offering a new frontier for exploring many-body fermionic physics. By manipulating electric dipole interactions in these systems, we demonstrate the possibility of cooling and condensing particles. The anisotropic nature of dipole interactions leads to an anisotropic Fermi liquid state, distinct from the isotropic standard, and supports unconventional Cooper pair symmetries involving both singlet and triplet states. Long-range dipolar interactions foster strong correlations within the system. Employing methods such as Gaussian expansion and random-phase-approximation, we analyze how anisotropy influences the spectrum, collective excitations, and correlation functions. In particular, we derive the screened interaction and the polarization function, and we investigate the pairing instability via the Bethe–Salpeter equation in the ladder approximation. The critical temperature for superfluidity is obtained as function of the effective coupling strength. We discuss the the conditions for attractive pairing, the expected order parameter symmetry, and the role of confinement dimensionality. Our results provide a theoretical framework for understanding dipolar Fermi gases in nanostructures and highlight the possibility of realizing unconventional superfluid phases.