Density Waves and Supersolidity in Rapidly Rotating Atomic Fermi Gases

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1 Density Waves and Supersolidity in Rapidly Rotating Atomic Fermi Gases Nigel Cooper T.C.M. Group, Cavendish Laboratory, University of Cambridge Quantum Gases Conference, Paris, 30 June Gunnar Möller and NRC, arxiv/

2 BEC-BCS crossover Two hyperfine states of the same fermionic atom:, BEC of tightly-bound bosonic molecules smooth crossover BCS state fermionic atoms. What is the effect of rapid rotation on the balanced Fermi gas?

3 av Vortex density 2MΩ 1 a 2 V [Zwierlein, Abo-Shaeer, Schirotzek, Schunck & Ketterle, Nature 435, (2005)] What sets the upper critical rotation rate on the BCS side?

4 Semiclassical approximation (small λ F ): Ω > 2 ǫ [Gorkov, JETP 9, 1364 (1959)] F [BCS gap ǫ F e2k π F a ] av Vortex density, 2MΩ 1 a 2 V superfluid velocity, K.E. per pair, 1 2 M ( Ma V Ma V ) 2 Ω High-field superconductivity [e.g. Rasolt & Tešanović, RMP 64, 709 (1992)] Beyond mean-field theory.

5 Overview What is the fate of the BCS superfluid at rapid rotation? BCS Mean Field Theory: high field superconductivity Beyond MFT: Density waves and supersolidity ( Strong-coupling: Evolution across the Feshbach resonance)

6 Uniform 3D regime: Rapidly Rotating Fermi Gas Ω ω ω k ǫ nk = (n + 1/2)2 Ω + 2 k 2 2m quasi-1d Fermi surface for each LL. k F1 k F2 k F2 Density of States E 1/2 k F1 hω 3hΩ 5hΩ 7hΩ 9hΩ E What is the groundstate for (weak) attractive s-wave interactions?

7 Related studies: BCS Mean Field Theory for a Rotating Fermi Gas [Veillette, Sheehy, Radzihovsky & Gurarie, PRL 97, (2006); Zhai & Ho, PRL 97, (2006)] Linearized Gap equation 1 a s = Ω n,n =0 ( ) n + n 1 n 2 n+n dk tanh ǫ nk µ 2k B T + tanh ǫ n k µ 2k B T 2π ǫ nk + ǫ n k 2µ 2 ǫ nk +ǫ n k k B T c Ω T c η Ω { } 2π exp G(η) 1 k B a s ; G(η) 1 η [Gunnar Möller & NRC, arxiv/ ] n max n=0 (2n)! (2 n n!) 2 ( 1 n η) 1 2. η (µ Ω)/(2 Ω)

8 Numerical Solution 10 [Gunnar Möller & NRC, arxiv/ ] increasing rotation frequency 1 k B T c / [µ hω] e-3 =1.5 =1.0 =0.9 =0.8 =0.7 =0.6 1e T c never vanishes. µ / hω T c /µ is an increasing function of Ω/µ for atoms in the lowest Landau level. Ω c2 Ω c2 Ω c2

9 Lowest Landau Level limit, n = 0 [Gunnar Möller & NRC, arxiv/ ] k ǫ 0k = Ω + 2 k 2 2m quasi-1d Fermi surface 2mΩ 1 2πl 2 states per unit area. Attractive interactions Fermi-surface instabilities at low T. Within mean field theory BCS and CDW instabilities both occur at the same T c.

10 Parquet diagrams [Brazovskii, Zh. Eksp. Teor. Phys. 61, 2401 (1972); Yakovenko, PRB 47, 8851 (1993)] ( ) g ξ = ln ǫf (2π) 3 v F l 2 k B T. g = 4π 2 a s /M γ = + g 2 g ξ g ξ 3 2 g ξ g ξ g ξ /2 3 2 g ξ /2 RG equations for spin-degenerate case [Gunnar Möller & NRC, arxiv/ ] dγ 1 dξ dγ 2 dξ = 2γ 1 γ 1 + 2γ 1 γ 2 2 γ 1 γ 2 = 2γ 2 γ 2 γ 1 γ 1 γ 2 γ 2

11 Numerical solution [Gunnar Möller & NRC, arxiv/ ] Attractive contact interactions, n = 0 CDW at ξ c = 0.726/(2π). T CDW c ) ǫ F kb exp ( (2π)3 v F l 2 g ξ c Density per layer, n 2d n π k = 4mΩ F0 π/ i.e. filling factor, ν a n 2d (2πl 2 ) = 2 A CDW of ν a = 2 states, with uniform 2D density in each layer

12 Comparison with BCS theory 10 [Gunnar Möller & NRC, arxiv/ ] increasing rotation frequency 1 k B T c / [µ hω] e-3 1e-4 =1.5 =1.0 =0.9 =0.8 =0.7 = µ / hω Ω c2 Ω c2 Ω c2 The upper critical rotation rate is set by the transition into a CDW.

13 k SC k F1 CDW k F2 k F2 k F1 Coexistence of CDW in the upper LL and SC in lower LLs: supersolidity.

14 Implications for Experiment 3D regime: ω < ω a > a. Breakdown of semiclassical BCS theory: Ω > 2 ǫ F Lowest Landau level: µ, k B T < 2 Ω c.f. bosons [Schweikhard, Coddington, Engels, Mogendorff & Cornell, PRL 92, (2004)] Low filling factors more easily accessible than for (weakly-interacting) bosons. [Antezza, Cozzini & Stringari PRA 75, (2007)] Spontaneous formation of density wave order, with period λ n = π k Fn π 2 a.

15 2D Fermi Gas with strong interactions Strong attractive interactions atoms form small bosonic molecules. [Duncan Haldane, Ed Rezayi] n m = n a 2, l m = 2(2m)Ω = l a ν m = ν a molecules atoms ν m = 1/2 bosonic Laughlin liquid ν a = 2 fermion QH liquid These two states are separated by a phase transition

16 Summary Rapidly rotating cold atomic Fermi gases offer the opportunity to explore superconductivity beyond the conventional semiclassical regime. BCS theory applied to a rapidly rotating gas shows a non-zero T c for any rotation rate: high-field superconductivity. The upper critical rotation rate is determined by physics that goes beyond mean-field theory. Superfluidity is destroyed by the appearance of CDW order. At intermediate rotation rates CDW and SC coexist (supersolidity); In the lowest Landau level the groundstate is a CDW of layers of ν a = 2. In 2D, crossing the Feshbach resonance drives a phase transition between a ν a = 2 state of atoms and a ν m = 1/2 Laughlin state of molecules.