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- Interaction,
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- Superfluidity

Anisotropic superfluidity in the two-species polar Fermi gas

PHYSICAL REVIEW A 82, 063624 (2010)**Anisotropic** **superfluidity** **in** **the** **two**-**species** **polar** **Fermi** **gas**Renyuan Liao and Joachim BrandInstitute for Advanced Study and Centre for Theoretical Chemistry and Physics, Massey University, Auckland 0632, New Zealand(Received 3 August 2010; revised manuscript received 20 October 2010; published 20 December 2010)We study **the** superfluid pair**in**g **in** a **two**-**species** **gas** of heteronuclear fermionic molecules with equal density.The **in**terplay of **the** isotropic s-wave **in**teraction and anisotropic long-range di**polar** **in**teraction reveals richphysics. We f**in**d that **the** s**in**gle-particle momentum distribution has a characteristic ellipsoidal shape that canbe reasonably represented by a deformation parameter α def**in**ed similarly to **the** normal phase. Interest**in**gmomentum-dependent features of **the** order parameter are identified. We calculate **the** critical temperatures ofboth **the** s**in**glet and triplet superfluids, suggest**in**g a possible pair**in**g symmetry transition by tun**in**g **the** s-waveor di**polar** **in**teraction strength.DOI: 10.1103/PhysRevA.82.063624PACS number(s): 03.75.Ss, 67.85.Lm, 05.30.Fk, 74.20.FgI. INTRODUCTIONThe recent experimental realization and coherent control ofhigh phase-space density quantum **gas** of **the** **polar** molecules40 K 87 Rb [1–3] provides an excellent opportunity to study **the**effects of anisotropic long-range dipole-dipole **in**teractions.Theoretical proposals employ**in**g degenerate **polar** moleculesrange from **the** study of exotic quantum phases of matter [4,5]and quantum **gas** dynamics [6,7] to quantum simulations ofhighly correlated condensed matter systems [8] and schemesfor quantum **in**formation process**in**g [9].Two fundamental properties of di**polar** **Fermi** **gas**es aresuperfluid pair**in**g [10–13] and **Fermi** surface deformation[14–16], orig**in**at**in**g from **the** partially attractive nature of **the**di**polar** **in**teraction and anisotropic Fock exchange **in**teraction.For di**polar** **Fermi** **gas**es with **two** hyperf**in**e states, onecan tune not only **the** dipole-dipole **in**teraction by a fastrotat**in**g orient**in**g field [17], but also **the** s-wave **in**ter**species****in**teraction via a Feshbach resonance. Therefore, one expectsthat rich physics will emerge as a result of **the** **in**terplay of**the** anisotropic long-range dipole **in**teraction and short-ranges-wave **in**teraction.In this work, we study BCS pair**in**g by tak**in**g account of**the** Fock exchange term **in** a self-consistent way. We f**in**dthat **the** anisotropic nature of **the** di**polar** **in**teraction leads toan anisotropic momentum space distribution of **the** numberdensity and an anisotropic order parameter. We generalize**the** def**in**ition of **the** deformation parameter **in**troduced **in**Ref. [14] to describe **the** anisotropic number distribution **in****the** pair**in**g phase and f**in**d that it gives a good description.Interest**in**g features of **the** order parameter **in** momentum spaceare revealed, manifest**in**g fasc**in**at**in**g consequences of **the**di**polar** **in**teraction. Compet**in**g effects of **the** contact s-wave**in**teraction and **the** di**polar** **in**teraction are identified **in** **the** studyof **the** transition temperature of **the** superfluid state, suggest**in**g**the** possibility of tun**in**g **the** pair**in**g symmetry by tun**in**g **the**di**polar** **in**teraction.II. MODELWe consider a homogeneous **gas** of **two** **species** of fermionicheteronuclear molecules σ =↑ and ↓. For simplicity, wefur**the**r assume that each **species** has **the** same mass, density,and dipole moment. The electric dipoles of **the** molecules withmoment d are oriented along **the** z axis by a sufficiently strongexternal electric field such that **the** sp**in**-**in**dependent part of**the** electronic dipole-dipole **in**teraction becomes V dd (q) =(4π/3)d 2 (3 cos 2 θ q − 1), with θ q be**in**g **the** angle betweenmomentum q and **the** direction of **the** z axis **in** which **the**dipoles are aligned. In addition, we assume that moleculesalso **in**teract via a contact **in**teraction with strength g. Thissystem is described by **the** follow**in**g HamiltonianH − µn = ∑ kσ+ 12V(ɛ k − µ)c † kσ c kσ∑kpqσσ ′ V σσ ′(q)c † k+qσ c† p−qσ ′c pσ ′c kσ , (1)where µ is **the** chemical potential, n is **the** total number density,V is **the** volume, and ɛ k = k 2 /2m (where we have set ¯h = 1).The **in**teraction potential V σσ ′(q) = gδ σ,−σ ′ + V dd (q) conta**in**sboth dipole-dipole and contact **in**teractions. Anticipat**in**g **the**importance of **the** Fock exchange term, we decouple **the****in**teraction **in** all three channels [18]: direct channel, exchangechannel, and Cooper channel, result**in**g **in** **the** follow**in**geffective mean-field HamiltonianH MF = ∑ ξ kσ c † kσ c kσkσ+ 1 ∑[ ∗ σ2′ σ (k)c −kσ ′c kσ + σ ′ σ (k)c † kσ c† −kσ ′]. (2)kσσ ′Here ξ kσ = ɛ k − µ + gn/2 + kσ , with self-consistent meanfields def**in**ed as kσ =− 1 ∑V dd (p − k)〈c pσ † Vc pσ 〉, (3)p σ ′ σ (k) = 1 ∑V σσ ′(k − p)〈c −kσ ′c kσ 〉. (4)VpSome comments are **in** order: The contact **in**teraction affects**the** s**in**gle-particle spectrum by shift**in**g **the** chemical potential,which may be redef**in**ed as ˜µ = µ − gn/2. The self-energy kσ encodes **the** anisotropic di**polar** contribution from **the** Fockexchange term to **the** dressed s**in**gle-particle spectrum, whichjustifies our treatment. In addition, both parts of **the** **in**teractioncontribute to **the** pair**in**g field.1050-2947/2010/82(6)/063624(4) 063624-1 ©2010 The American Physical Society

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