Bardeen Bardeen, Cooper Cooper and Schrieffer and Schrieffer 1957
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1 Unexpected aspects of large amplitude nuclear collective motion Aurel Bulgac University of Washington Collaborators: Sukjin YOON (UW) Kenneth J. ROCHE (ORNL) Yongle YU (now at Wuhan Institute of Physics and Mathematics) Yuan Lung LUO (UW) Piotr MAGIERSKI (Warsaw and UW) Ionel STETCU (UW) Funding: DOE grants No. DE FG02 97ER41014 (UW NT Group) DE FC02 07ER ER41457 (SciDAC UNEDF)
2 Bardeen, Cooper, and Schrieffer, 1957 p ( ) gs = u p + vpa a vac p p Global gauge invariance broken î gs is infinitely degenerate ˆ ( ) p p p p p ϕ = exp( iϕn ) gs = u + exp(2 iϕ) v a a vac ˆ =, ˆ ˆ pσ pσ ϕ ϕ pσ N a a H gs H gs
3 Goldstone theorem î existence of Goldstone bosons The Bogoliubov-Anderson sound modes are the Goldstone bosons Oscillations of the phase of the order parameter (pairing gap) T=0, no collisions Momentum distribution is spherical ϕ ϕ ( rt, ) = k r ωt exp ik ( r ωtn ) ˆ gs ω = ck = v F 3 k (BCS limit) Somewhat surprising, it has the same speed as regular (first/collisional collisional) sound!
4 A different type of collective excitation: Higgs mode Small amplitude oscillations of the modulus of the order parameter (pairing gap) Ω Ω = 2 Δ H 0 This mode has a bit more complex character cf. Volkov and Kogan 1972 (a bit later about it)
5 The Bogoliubov-Anderson sound modes are routinely described within Quantum Hydrodynamics approximation (Landau, here at T=0) Their properties are well confirmed experimentally n + [ vn] = 0 2 mv mv + + μ [ n] + V ext = 2 0 Landau-Ginzburg-like /effective action approaches are often advocated for the dynamics of the order parameter 2 Δ ( 2 i (,) r t (,) r t U (,) r t ) Ψ = Ψ + Ψ Ψ (,) r t + Vext (,) r t Ψ(,) r t 4m
6 Very brief/skewed summary of DFT
7 Injective map (one to one) one) Kohn Sham theorem N N N N H = T () i + U ( ij ) + U ( ijk ) V () i i i< j i< j< k i HΨ (1,2,... N) = E Ψ (1,2,... N) N nr ( ) = Ψ δ ( r r ) Ψ 0 i 0 i Ψ (1,2,... N) V ( r) n( r) 0 ext 2 3 = 0 τ + ε [ ] + ( ) * ext nr E min dr ( r ) nr ( ) V ( rnr ) ( ) 2 m ( r) N N 2 2 = = nr ( ) ϕ i ( r), τ( r) ϕ i ( r ) i Universal functional of particle density alone Independent d of external potential ti Normal Fermi systems only! i ext
8 However, not everyone is normal!
9 Superconductivity and superfluidity in Fermi systems Dilute atomic Fermi gases T c ev Liquid 3 He T c 10-7 ev Metals, composite materials T c ev Nuclei, neutron stars T c ev QCD color superconductivity T c ev units (1 ev 10 4 K)
10 Extension of Kohn Sham to superfluid fermionic systems: Superfluid Local Density Approximation (SLDA) The case of a unitary Fermi gas Why would one want to study this system? One (very good) reason: (for the nerds, I mean the hard core theorists, not the phenomenologists) Bertsch s s Many-Body X challenge, Seattle, 1999 What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, range, infinite scattering-length contact interaction. The unitary gas is really a pretty good model for dilute neutron matter
11 Dilute fermion matter The ground state energy is given by a function: E = f ( N, V,, m, a, r ) gs 0 Taking the scattering length to infinity and the range of the interaction to zero, we are left with: 3 E = F( N, V,, m) = ε N gs F 5 N V F 2, k ε F F = k = 3π 2 m Pure number ξ Bertsch s parameter
12 The renormalized SLDA energy density functional τ 2 2/3 5/3 () r c 3(3 π ) n ( r) ε() r = α Δ () r ν () r + β c * nr () = 2 v(), r τ () r = 2 v(), r ν () r = k k u()v() r r c c k k k E< E E< E c c 1/3 1 n ( r ) k () r k () r + 0 = γ πα k () r k () r c c 0 1 ln 2 g () r 2 2 k () r k () r k () r eff c c k () r k () r c 0 E + μ = α + U (), r μ = α + U () r c 2 2 π Δ 2 2 2/3 2/3 (3 ) n ( r ) () r U () r = β + V () r 2/3 ext 2 3 γn ( r) Δ () r = g () r ν () r eff c No free/fitting parameters, EDF is fully determined by ab initio calculations Bulgac, Phys. Rev. A 76, (R) (2007)
13 Time Dependent Phenomena and Formalism The time-dependent density functional theory is viewed in general as a reformulation of the exact quantum mechanical time evolution of a many-body system when only single- particle properties are considered. A.K. Rajagopal and J. Callaway, Phys. Rev. B 7, 1912 (1973) V. Peuckert, J. Phys. C 11, 4945 (1978) E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984) u i ( r, t ) [ h( r, t) + Vext ( r, t) μ ]u i( r, t) + [ Δ ( r, t) + Δ ext ( r, t)]v i( r, t) = i t * * v r t i (, ) [ Δ ( r, t) + Δ ext ( r, t )]u i ( r, t ) [ h ( r, t ) + V ext ( r, t ) μ ]v i ( r, t ) = i t
14 Full 3D implementation of TD-SLDA is a petaflop problem and is almost complete for both nuclear systems and cold dilute atomic gases Bulgac and Roche, J. Phys. Conf. Series 125, (2008) Lots of contributions ti due to Yu, Yoon, Luo, Magierski, and dst Stetcu t
15 Energy of a (unitary) Fermi system as a function of the pairing gap n + = [ vn] Δ 2 mv mv + + μ n = 2 [ ] 0 ( ) i Ψ Ψ ( r, t ) = Ψ ( r, t ) + U Ψ ( r, t ) Ψ ( r, t ) 4m
16 Response of a unitary Fermi system to changing the scattering length with time Tool: TD-SLDA All these modes have a very low frequency below the pairing gap and a very large amplitude and excitation energy as well None of these modes can be described either within Quantum Hydrodynamics or Landau-Ginzburg like approaches Bulgac and Yoon, Phys. Rev. Lett. 102, (2009)
17 3D unitary Fermi gas confined to a 1D HO potential well (pancake) New qualitative excitation mode of a superfluid Fermi system (non spherical Fermi momentum distribution) Black solid line Time dependence of the cloud radius Black dashed line Time dependence of the quadrupole moment of momentum distribution Bulgac and Yoon, Phys. Rev. Lett. 102, (2009)
18 Vortex generation and dynamics See movies at washington movies html
19 Time-Dependent Superfluid Local Density Approximation This is a general many-body problem with direct applications, which will provide the time dependent response of superfluid fermionic systems to a large variety of external probes for both cases of small and large amplitude collective motion. Nuclear physics: fission, heavy-ion collision, nuclear reactions, response electromagnetic fields, beta-decay, Neutron star crust, dynamics of vortices, vortex pinning mechanism Cold atom physics, optical lattices, Condensed d matter physics Next frontier: Stochastic TDSLDA
20 Generic adiabatic large amplitude potential energy SURFACES In LACM adiabaticity is not a guaranteed The most efficient mechanism for transitions at level crossing is due to pairing Level crossings are a great source of : entropy production (dissipation) dynamical symmetry breaking non-abelian gauge fields
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