THE UNITARY GAS: SYMMETRY PROPERTIES AND APPLICATIONS Yvan Castin, F´ elix Werner, Christophe Mora LKB and LPA, Ecole normale sup´ erieure (Paris, France) Ludovic Pricoupko LPTMC, Universit´ e Paris 6
GENERAL CONTEXT The physical system: • Fermionic atoms with two internal states ↑ , ↓ • Short-range interactions between ↑ and ↓ controlled by a magnetic Feshbach resonance • Arbitrary values for the numbers N ↑ , N ↓ • Intense experimental studies (Thomas, Salomon, Jin, Ketterle, Grimm, Hulet, Zwierlein...), e.g. BEC-BCS crossover (Leggett, Nozi` eres, Schmitt-Rink, Sa de Melo,...) What is not discussed here: • The actual many-body state of the system: superfluid or normal • The particularly intriguing strongly polarized case N ↑ ≫ N ↓ : Polaronic physics, see talk by C. Trefzger
OUTLINE OF THE TALK • What is the unitary gas ? • Simple consequences of scaling invariance • Dynamical consequences: SO (2 , 1) hidden symmetry in a trap • Separability in hyperspherical coordinates • Does the unitary gas exist ? • First deviations from unitary limit
WHAT IS THE UNITARY GAS ?
DEFINITION OF THE UNITARY GAS • Opposite spin two-body scattering amplitude f k = − 1 ∀ k ik • “Maximally” interacting: Unitarity of S matrix imposes | f k | ≤ 1 /k. • In real experiments with magnetic Feshbach resonance: − 1 = 1 a + ik − 1 2 k 2 r e + O ( k 4 b 3 ) f k unitary if “infinite” scattering length a and “zero” ranges: 1 k typ | a | > 100 , k typ | r e | and k typ b < 100 imposing | a | > 10 microns for r e ∼ b ∼ a few nm. • All these two-body conditions are only necessary.
THE ZERO-RANGE WIGNER-BETHE-PEIERLS MODEL • Interactions are replaced by contact conditions. • For r ij → 0 with fixed ij -centroid � C ij = ( � r i + � r j ) / 2 different from � r k , k � = i, j : � � 1 − 1 A ij [ � ψ ( � r 1 , . . . , � r N ) = C ij ; ( � r k ) k � = i,j ] + O ( r ij ) r ij a • Elsewhere, non interacting Schr¨ odinger equation � � − � 2 X + 1 2 mω 2 X 2 Eψ ( � ψ ( � X ) = 2 m ∆ � X ) with � X = ( � r 1 , . . . , � r N ). • Odd exchange symmetry of ψ for same-spin fermion po- sitions. • Unitary gas exists iff Hamiltonian is self-adjoint.
SIMPLE CONSEQUENCES OF SCALING INVARIANCE
SCALING INVARIANCE OF CONTACT CONDITIONS 1 ψ ( � A ij [ � X ) = C ij ; ( � r k ) k � = i,j ] + O ( r ij ) r ij r ij → 0 • Domain of Hamiltonian is scaling invariant: If ψ obeys the contact conditions, so does ψ λ with 1 ψ λ ( � λ 3 N/ 2 ψ ( � X ) ≡ X/λ ) • Consequences (also true for the ideal gas): free space box (periodic b.c.) harm. trap no bound state ( ∗ ) P V = 2 E/ 3 ( ∗∗ ) virial E = 2 E harm ( ∗∗∗ ) ( ∗ ) If ψ of eigenenergy E , ψ λ of eigenenergy E/λ 2 . Square integrable eigenfunctions (after center of mass removal) correspond to point-like spectrum, for selfadjoint H . ( ∗∗ ) E ( N, V λ 3 , S ) = E ( N, V, S ) /λ 2 , then take derivative in λ = 1. ( ∗∗∗ ) For eigenstate ψ , mean energy of ψ λ stationary in λ = 1.
TEST FOR QUANTUM MONTE CARLO For the unpolarized gas in thermodynamic limit, using Carlson’s 2009 upper bound on the ground state energy [ ξ = µ ( T = 0) /E F ≤ 0 , 41]: 0.55 thermodynamic inequalities gray area violates Burovski 0.5 µ (T)/E F Bulgac 0.45 Goulko 0.4 0.35 0.2 0.25 0.3 0.35 0.4 0.45 0.5 E(T)/NE F
DYNAMICAL CONSEQUENCES: SO (2 , 1) HIDDEN SYMMETRY IN A TRAP
IN A TIME-DEPENDENT TRAP • At t = 0 : static trap U (r) = mω 2 r 2 / 2, system in eigen- state ψ 0 ( � X ) of energy E . • For t > 0, arbitrary time dependence of trap spring constant, ω ( t ). Known solution for ideal gas: e − iθ ( t ) � im ˙ � λ 2 � λ X 2 ψ ( � ψ 0 ( � X, t ) = exp X/λ ( t )) λ 3 N/ 2 ( t ) λ = ω 2 λ − 3 − ω 2 ( t ) λ and ˙ with ¨ θ = Eλ − 2 / � . • This is a gauge plus scaling transform. • The gauge transform also preserves contact conditions: ij + 1 r 2 i + r 2 j = 2 C 2 2 r 2 ij so solution also applies to unitary gas! Y. Castin, Comptes Rendus Physique 5, 407 (2004).
IN THE MACROSCOPIC LIMIT X, t ) = e − iθ ( t ) � im ˙ � λ 2 � λ X 2 ψ ( � ψ 0 ( � λ 3 N/ 2 exp X/λ ) r/λ ) /λ 3 r ˙ density ρ ( � r, t ) = ρ 0 ( � velocity field � v ( � r, t ) = � λ/λ r, t ) = T/λ 2 r/λ ) /λ 5 local temp. T ( � pressure P ( � r, t ) = P 0 ( � local entropy per particle s ( � r, t ) = s 0 ( � r/λ ) This has to solve the hydrodynamic equations for a normal gas. Entropy production equation: v ) 2 v · � ∇ s ) = � ∇ · ( κ ∇ T ) + ζ ( � ρk B T ( ∂ t s + � ∇ · � � 2 � + ∂v j + η ∂v i − 2 � 3 δ ij � ∇ · � v 2 ∂x j ∂x i i,j so the bulk viscosity is zero: ζ ( ρ, T ) = 0 ∀ T > T c . Repro- duces the conformal invariance result of Son (2007).
LADDER STRUCTURE OF THE SPECTRUM • Infinitesimal change of ω for 0 < t < t f . For t > t f : λ ( t ) − 1 = ǫ e − 2 iωt + ǫ ∗ e 2 iωt + O ( ǫ 2 ) so an udamped mode of frequency 2 ω . • Corresponding wavefunction change: � e − iEt/ � − ǫe − i ( E +2 � ω ) t/ � L + ψ ( � X, t ) = � + ǫ ∗ e − i ( E − 2 � ω ) t/ � L − X ) + O ( ǫ 2 ) ψ 0 ( � • Raising and lowering operators: � 3 N + H � � ω − mωX 2 / � 2 i − i � L ± = ± i X · ∂ � X (in red, generator of scaling transform) • Spectrum=collection of semi-infinite ladders of step 2 � ω . SO (2 , 1) hidden symmetry (Pitaevskii, Rosch, 1997).
LADDER STRUCTURE OF THE SPECTRUM (2) E g +8 h ω / 2 h ω / E g +6 h ω / 2 h ω / E g +4 / h ω 2 h ω / E g +2 h ω / 2 / h ω E g
USEFUL MAPPING AND SEPARABILITY • Each energy ladder has a ground step of energy E g , eigenfunction ψ g . • Integration of L − ψ g = 0 gives, with � X = X� n : X ) = e − mωX 2 / 2 � × � � X E g / ( � ω ) − 3 N/ 2 f ( � ψ g ( � n ) • Limit ω → 0 : mapping to zero energy free space solu- tions. N.B.: E g / ( � ω ) is a constant. • Free space problem solved for N = 3 (Efimov, 1972)... so trapped case also solved (Werner, Castin, 2006). • Also, this is separable in hyperspherical coordinates.
SEPARABILITY IN HYPERSPHERICAL COORDINATES
SEPARABILITY IN INTERNAL COORDINATES • Use Jacobi coordinates to separate center of mass � C • Hyperspherical coordinates: r N ) ↔ ( � C, R, � ( � r 1 , . . . , � Ω ) with 3 N − 4 hyperangles � Ω and the hyperradius N R 2 = C ) 2 � r i − � ( � i =1 • Hamiltonian is clearly separable: H internal = − � 2 R + 3 N − 4 ∂ R + 1 + 1 � � ∂ 2 2 mω 2 R 2 R 2 ∆ � Ω 2 m R
Do the contact conditions preserve separability ? • For free space E = 0, yes, due to scaling invariance: ψ E =0 = R s − (3 N − 5) / 2 φ ( � Ω) E = 0 Schr¨ odinger’s equation implies � � 2 � � 3 N − 5 s 2 − Ω φ ( � φ ( � ∆ � Ω) = − Ω) 2 with contact conditions. s 2 ∈ discrete real set. • For arbitrary E , Ansatz with E = 0 hyperrangular part obeys contact conditions [ R 2 = R 2 ( r ij = 0) + O ( r 2 ij )]: ψ = F ( R ) R − (3 N − 5) / 2 φ ( � Ω) • Schr¨ odinger’s equation for a fictitious particle in 2D: EF ( R ) = − � 2 � � 2 s 2 � 2 mR 2 + 1 2 m ∆ 2 D 2 mω 2 R 2 R F ( R ) + F ( R )
SOLUTION OF HYPERRADIAL EQUATION ( N ≥ 3) � � EF ( R ) = − � 2 � 2 s 2 2 mR 2 + 1 2 m ∆ 2 D 2 mω 2 R 2 R F ( R ) + F ( R ) • Which boundary condition for F ( R ) in R = 0? Wigner- Bethe-Peierls does not say. • Key point: particular solutions F ( R ) ∼ R ± s for R → 0. • Case s 2 > 0: Defining s > 0, one discards as usual the divergent solution: R → 0 R s − F ( R ) ∼ → E q = E CoM + ( s + 1 + 2 q ) � ω, q ∈ N • Case s 2 < 0: To make the Hamiltonian self-adjoint, one is forced to introduce an extra parameter κ (inverse of a
length, calculable via microscopic model). For s = i | s | : R → 0 ( κR ) s − ( κR ) − s F ( R ) ∼ • This breaks scaling invariance of the domain. In free space, a geometric spectrum of N -mers: E n ∝ − � 2 κ 2 m e − 2 πn/ | s | , n ∈ Z For N = 3, this is the Efimov effect: • Efimov (1971): Solution for three bosons (1 /a = 0). There exists a single purely imaginary s 3 ≃ i × 1 . 00624. • Efimov (1973): Solution for three arbitrary particles (1 /a = 0). Efimov trimers for two fermions (masse m , same spin state) and one impurity (masse m ′ ) iff (Petrov, 2003) α ≡ m m ′ > α c (2; 1) ≃ 13 . 6069
DOES THE UNITARY GAS EXIST ?
MINLOS’S THEOREM (1995) Theorem: In the n + 1 fermionic problem, the Wigner- Bethe-Peierls Hamiltonian is self-adjoint and bounded from below iff � asin α ( n − 1)2 α (1 + 1 /α ) 3 1+ α π √ 1 + 2 α dt t sin t < 1 . 0 • α is mass ratio fermion/impurity • Case α = 1: No stable unitary gas for n > 9... • Proof not included in Minlos’ paper. • Proof by Teta, Finco (2010) has a hole. • A physical test: look for occurrence of s 2 < 0 for n = 3: four-body Efimov effect !?
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