Unconventional Symmetries Of Fermi Liquid And Cooper .

J. Phys.: Condens. Matter 26 (2014) 493203Topical Review[60, 66]. Even in the simplest case of F 21 , the magneticdipolar interaction provides a novel and robust mechanismfor the p-wave (L 1) spin triplet (S 1) Cooper pairingwhich arises from the attractive channel of the magnetic dipolarinteraction. It turns out that its pairing symmetry structure ismarkedly different from that in the celebrated p-wave pairingsystem of 3 He: the orbital angular momenta L and spin Sof Cooper pairs are entangled into the channel of the totalangular momentum J 1, dubbed as the J -triplet pairing. Incomparison, the 3 He-B phase is isotropic in which J 0,while the A-phase is anisotropic in which J is not welldefined [56].In this article, we review the recent progress of the novelmany-body physics with dipolar fermions, such as the Fermiliquid properties and Cooper pairing structures, focusing onunconventional symmetries. In section 2, we review theanisotropy of the electric dipolar interaction and the SOstructure of the magnetic dipolar interactions, respectively,from the viewpoint of their Fourier components. In section 3,the anisotropic Fermi liquid theory of the electric dipolarfermions is reviewed. And the SO coupled Fermi liquidtheory of the magnetic dipolar fermion systems is reviewedin section 4. The pz -wave Cooper pairing in the single andtwo-component electric dipolar systems and the TR reversalsymmetry breaking effect are reviewed in section 5. TheSO coupled Cooper pairing with the J -triplet structure in themagnetic dipolar fermion systems is reviewed in section 6.Conclusions and outlooks are presented in section 7.Due to limit of space and also the view point from theunconventional symmetry, we are not able to cover manyimportant research directions of dipolar atoms and moleculesin this review. For example, the progress on topics ofstrong correlation physics with dipolar fermions [67–69],the Feshbach resonance with dipolar fermions [52, 53], thesynthetic gauge field with dipolar fermions [70, 71] andthe engineering of exotic and topological many-body states[72–74]. Some of these progresses have been excellentlyreviewed in [17, 30]. The properties of dipolar bosoncondensations are not covered here either and there are alreadymany important reviews on this topic [1, 3, 3, 30, 75, 76].where d is the magnitude of the electric dipole moment;r 12 r 1 r 2 is the displacement vector between two dipoles;θ12 is the polar angle of r 12 ; P2 (cos θ12 ) is the standard secondLegendre polynomial asP2 (cos θ12 ) The zeros of the second Legendre polynomial lie around thelatitudes of θ0 and π θ0 with1θ0 cos 1 55 .3A lot of information can be obtained solely based on symmetryand dimensional analysis. First, ei q · r is invariant under spatialrotations, thus Vd ( q ) transforms the same as Vd ( r ) underspatial rotations. It should exhibit the same symmetry factorof the spherical harmonics. Second, since Vd ( r ) decays witha cubic law, Vd ( q ) should be dimensionless.If Vd ( r12 ) were isotropic, Vd ( q ) would logarithmicallydepends on q. However, a more detailed calculation showsthat actually it does not depend on the magnitude of q. Let usintroduce a short distance cutoff that the dipolar interactionequation (1) is only valid for r and a long distance cutoffR as the radius of the system. A detailed calculation showsthat [34] j1 (qR)2 j1 (q )Vd ( P2 (cos θq )q ) 8π d(4)q qRwhere j1 (x) is the first order spherical Bessel function with theasymptotic behavior as x ,as x 0;3 j1 (x) 1πsin x , as x .x2In this section, we review the Fourier transformations of boththe electric and magnetic dipolar interactions in section 2.1 andsection 2.2, respectively. The anisotropy of the electric dipolarinteraction and the SO coupled feature of the magnetic dipolarinteraction also manifest in their momentum space structure.These Fourier transforms are important for later analysis ofmany-body physics.(5)After taking the limits of q 0 and qR , we arrive atq) Vd ( 8π d 2P2 (cos θq ).3(6)q 0) is defined as 0 based on the fact that theAt q 0, Vd ( angular average of the 3D dipolar interaction vanishes, thusVd is singular as q 0. Even in the case that R is large butfinite, the smallest nonzero value of qR is at the order of O(1).Thus, Vd ( q ) remains non-analytic as q 0.An interesting feature of the above Fourier transformequation (6) is that the anisotropy in momentum space isopposite to that in real space: it is most negative when q liesin the equatorial plane and most positive when q points to the2.1. Electric dipolar interactionWithout loss of generality, we assume that all the electricdipoles are aligned by the external electric field E along thez-direction, then the dipolar interaction between two dipolemoments is [28, 29]2d 2P2 (cos θ12 ), r12 3(2)Within θ0 θ12 π θ0 , the dipolar interaction is repulsiveand otherwise, it is attractive. The spatial average of the dipolarinteraction in 3D is zero.For later convenience, we introduce the Fourier transformof the dipolar interaction equation (1), Vd ( q ) d3 r e i q · r Vd ( r ).(3)2. Fourier transform of dipolar interactionsVd ( r12 ) 1(3 cos2 θ12 1).2(1)3

J. Phys.: Condens. Matter 26 (2014) 493203Topical ReviewThe magnetic dipolar interaction between two spin-F atomslocated at r 1 and r 2 isVαβ;β α ( r ) gF2 µ2B Fαα · F ββ 3(F αα · r̂)(F ββ · r̂) ,r3(10) Figure 1. The Fourier components of the dipolar interaction Vd (k).where r r 1 r 2 and r̂ r /r is the unit vector alongr . Similarly to the case of the electric dipolar interaction,the Fourier transform of equation (10) possesses the samesymmetry structure as that in real space [43, 59]The left-hand-side is for k ẑ and the right-hand-side is for k ẑ.north and south poles. An intuitive picture is explained infigure 1. Consider a spatial distribution of the dipole densityρ(r), then the classic interaction energy is 1 dr1 dr2 ρ( r1 )ρ( r2 )Vd ( r1 r 2 ) ρ( q ) 2V0q) Vαβ;β α ( q Vd ( q ),Again, it only depends on the direction of the momentumtransfer but not on its magnitude and it is also singular asq 0) 0.q 0. If q is exactly zero, Vαβ;β α ( In the current experiment systems of magnetic dipolaratoms, the atomic spin is very large. For example, for161Dy, its atomic spin reaches F 21and thus an accurate2theoretical description of many-body physics of the magneticdipolar interactions of such a large spin system would be quitechallenging [21, 26]. Nevertheless, as a theoretical startingpoint, we can use the case of F 21 as a prototype model whichexhibits nearly all the qualitative features of the magneticdipolar interactions [43, 60].(7)where V0 is the system volume. If the wave vector q is alongthe z-axis, then the dipole density oscillates along the dipoleorientation, thus the interaction energy is repulsive. On theother hand, if q lies in the equatorial plane, the dipole densityoscillates perpendicular to the dipole orientation and thus theinteraction energy is attractive.2.2. Magnetic dipolar interactionNow let us consider the magnetic dipolar interaction[18–21, 26, 27]. Different from the electric dipole moment,the magnetic one originates from contributions of severaldifferent angular momentum operators. The total magneticmoment is not conserved and thus its component perpendicularto the total spin averages to zero. For the low energyphysics below the coupling energy among different angularmomenta, the magnetic moment can be approximated as justthe component parallel to the spin direction and thus theeffective magnetic moment is proportional to the hyperfine spinoperator up to a Lande factor and thus is a quantum mechanicaloperator. Due to the large difference of energy scales betweenthe fine and hyperfine structure couplings, the effective atomicmagnetic moment below the hyperfine energy scale can becalculated through the following two steps. The first step is theLande factor for the electron magnetic moment respect to totalangular momentum of electron defined as µ e gJ µB J , where µB is the Bohr magneton; J L S is the sum of electron and spin S; and the value of gJorbital angular momentum Lis determined asgL gs gL gs L(L 1) S(S 1)gJ .22J (J 1)3. Anisotropic Fermi liquid theory of electric dipolarfermionsIn this section, we will review the new ingredients of theFermi liquid theory brought by the anisotropic electric dipolarinteraction [31–39, 41, 42], including the single-particleproperties such as Fermi surface distortions and two-bodyproperties including thermodynamic properties and collectivemodes.A general overview of the Landau–Fermi liquid theoryis presented in section 3.1. In section 3.2, we review thedipolar interaction induced Fermi surface distortions. TheLandau interaction matrix is presented in section 3.3 andits renormalization on thermodynamic properties includingPomeranchuk instabilities are review in section 3.4. Theanisotropic collective excitations are reviewed in section 3.5.3.1. A quick overview of the Fermi liquid theory(8)One of the most important paradigms of the interacting fermionsystems is the Landau Fermi liquid theory [77–79]. The Fermiliquid ground state can be viewed as an adiabatic evolutionfrom the non-interacting Fermi gas by gradually turning oninteractions. Although the ideal Fermi distribution functioncould be significantly distorted, its discontinuity remainswhich still defines a Fermi surface enclosing a volume inmomentum space proportional to the total fermion number, asstated by the Luttinger theorem. Nevertheless the shape of theFermi surface can be modified by interactions. The low energyexcitations become the long-lived quasi-particles around theFermi surface, whose life-time is inversely proportional toFurther considering the hyperfine coupling, the total magneticmomentum is defined µ µB (gJ

External electric fields are needed to polarize electric dipole moments, which mixes rotational eigenstates with opposite parities. However, the dipole moment of these mixed states is unquantized and, thus the electric dipole moment is a classic vector. When two dipole moments are aligned, say, along the