Fermions in the unitary regime at finite temperatures from path integral auxiliary field Monte Carlo simulations Aurel Bulgac,, Joaquin E. Drut and Piotr Magierski University of Washington, Seattle, WA Also in Warsaw
Expected phases of a two species dilute Fermi system BCS-BEC BEC crossover High T, normal atomic (plus a few molecules) phase T Strong interaction weak interactions weak interaction BCS Superfluid a<0 no 2-body bound state Molecular BEC and Atomic+Molecular Superfluids a>0 shallow 2-body bound state halo dimers 1/a
Early theoretical approach Eagles (1969), Leggett (1980) ( k k k k ) gs = u +v a a vacuum BCS wave function k m 1 1 = 2 4π a k 2εk 2Ek gap equation ε k µ n = 2 1 number density equation k Ek 8 π ε exp 2 F pairing gap e 2kFa 2 2 E k = ( ε k µ ) + quasi-particle energy 2 2 k ε k = 2m 2 2 2 1 ε k µ uk + vk = 1, v k= 1 2 Ek
Consequences: Usual BCS solution for small and negative scattering lengths, with exponentially small pairing gap For small and positive scattering lengths this equations describe a gas a weakly repelling (weakly bound/shallow) molecules, essentially all at rest (almost pure BEC state) Ψ ( r, r, r, r,...) A [ ϕ( r ) ϕ( r )...] 1 2 3 4 12 34 In BCS limit the particle projected many-body wave function has the same structure (BEC of spatially overlapping Cooper pairs) For both large positive and negative values of the scattering length these equations predict a smooth crossover from BCS to BEC, from a gas of spatially large Cooper pairs to a gas of small molecules
What is wrong with this approach: The BCS gap (a<0 and small) is overestimated, thus the critical temperature and the condensation energy are overestimated as well. In BEC limit (a>0 and small) the molecule repulsion is overestimated The approach neglects of the role of the meanfield (HF) interaction, which is the bulk of the interaction energy in both BCS and unitary regime All pairs have zero center of mass momentum, which is reasonable in BCS and BEC limits, but incorrect in the unitary regime, where the interaction between pairs is strong!!! (this situation is similar to superfluid 4 He) Fraction of non-condensed pairs (perturbative( result)!?! n n ex 0 8 3 n = na m mm, nm=, amm 0.6 a 3 π 2
What people use a lot? Basically this is Eagles and Leggett s model, somewhat improved. Ψ ( r, r, r, r,...) A [ ϕ( r ) ϕ( r )...] 1 2 3 4 12 34 ϕ( r ) ϕ ( X ) + ψ ( X ) 12 12 12 open channel wave function electron spins are parallel closed channel wave function (frozen) electron spins are antiparallel only one state included (effective boson) and only the amplitude is varied
Why? Meanfield calculations (even with fluctuations added on top) are easy It was unclear how to treat the unitary regime Timmermans et al. realized that a contact interaction proportional to either a very large or infinite scattering length makes no sense in meanfield approximation. 2 4π a Ur ( 1 r2) = δ ( r1 r2) m The two-channel approach, which they introduced initially for bosons, does not seem, superficially at least, to share this difficulty. However, it is known (for a long time) that corrections to such a meanfield approach will be governed by the parameter na 3 anyway, so, the problem has not been really solved.
Is there a better way? Grand Canonical Path-Integral Monte Carlo 2 2 ˆ ˆ ˆ 3 3 H = T + V = d x ψ ( x) ψ ( ) ψ ( ) ψ ( ) ˆ ( ) ˆ ( ) x + x x g 2 2 d x n x n x m m ˆ 3 N = d x nˆ ( x) + nˆ ( x), ˆ ( ) = ψ ( ) ψ ( ), =, ns x s x s x s Trotter expansion (trotterization( of the propagator) ( ) ( ) Z( β) = Tr exp β Hˆ µ Nˆ = Tr { exp τ Hˆ µ ˆ } Nτ 1 N, β = = T N τ τ ( ) 1 ET ( ) = Tr Hˆ exp β Hˆ µ Nˆ ZT ( ) ( ) 1 NT ( ) = Tr Nˆ exp β Hˆ µ Nˆ ZT ( ) No approximations so far, except for the fact that the interaction is not well defined!
How to implement the path integral? Put the system on a spatio-temporal temporal lattice!
A short detour Let us consider the following one-dimensional Hilbert subspace (the generalization to more dimensions is straightforward) P 2 = P projector in this Hilbert subspace π sin π l ( x y) dk xpy exp[ ikx ( y)] l = =, π 2 π π( x y) l ( ) ( x) = P δ x x, = ( x ) = ( x ) = K α α α β α β β α α αβ δ sin π N ( x nl) ψ( x) = c ( x) O(exp( cn)) ( nl) l α α + ψ π α = 1 n ( x nl) l 1 1 c = dx ( x) ψ( x) = ψ( x ), x = nl α K α K α α α α Littlejohn et al. J. Chem. Phys. 116,, 8691 (2002)
Schroedinger equation N ψ ( x) = d F ( x) + O(exp( cn)) α = 1 α α 1 F ( x) = ( x), x = nl, F F = δ K α α α α β αβ α β F T F + V( x ) δ d = Ed α β α αβ β α
Recast the propagator at each time slice and put the system on a 3d-spatial lattice, in a cubic box of side L=N s l,, with periodic boundary conditions ( ˆ ˆ) ( ˆ ˆ) ˆ ( ˆ ˆ) τ H µ N τ T µ N τv τ T µ N + O τ 3 exp exp / 2 exp( )exp / 2 ( ) Discrete Hubbard-Stratonovich transformation ˆ 1 exp( τv) = { 1 + σ ( ) ˆ ( ) ˆ ± x A + ( ) }, = exp( τ ) 1 2 n x n x A g x σ ( x ) =± 1 ± σ-fields fluctuate both in space and imaginary time m 1 mkc π = +, k < 2 2 2 c 4π a g 2π l Running coupling constant g defined by lattice
k y π/l Momentum space π/l k x 2π/L 2 2 π εf,, T 2 2 ml 2 2 2 π δε > 2 ml 2 2 2 π εf, 2 ml ξ L = 2π δ p > L N l s
ZT ( ) = Dσ( x, τ) Tr Uˆ ({ σ}) x, τ Uˆ({ σ}) = T exp{ τ[ hˆ({ σ }) µ ]} τ τ One-body evolution operator in imaginary time ET ( ) = x, τ Dσ(,)Tr x τ Uˆ ({ σ}) Tr HU ˆˆ({ σ}) ZT ( ) Tr Uˆ ({ σ}) Uˆ σ = + Uˆ σ = S σ > 2 Tr ({ }) {det[1 ({ })]} exp[ ({ })] 0 No sign problem! Uˆ({ σ }) n (, x y) = n (, x y) = * exp( ik x) ϕ () x ( ), ( ) k 1 + Uˆ ϕ y ϕ x = l k kl, < k ({ σ} ) V c All traces can be expressed through these single-particle density matrices k l
More details of the calculations: Lattice sizes used from 6 3 x30 (high Ts) to 6 3 x120 (low Ts) (runs with ten times smaller time steps, up to 800 time steps, also performed) 8 3 running and larger sizes to come Effective use of FFT makes all imaginary time propagators diagonal (either in real space or momentum space) and there is no need to store large e matrices Update field configurations using the Metropolis importance sampling algorithm Change randomly at a fraction of all space and time sites the signs the auxiliary fields σ(x, (x,τ)) so as to maintain a running average of the acceptance rate between 0.4 and 0.6 Thermalize for 50,000 100,000 MC steps or/and use as a start-up field configuration a σ(x, (x,τ)-field configuration from a different T At low temperatures use Singular Value Decomposition of the evolution operator U({σ}) to stabilize the numerics Use 100,000-1,000,000 1,000,000 σ(x, (x,τ)- field configurations for calculations
What we know so far from other approaches? What we know so far from other approaches? (at resonance) T BEC = 0.218 ε at a = + 0 F T Eagles = 0.277 ε at a = ± F T MF + Fluct = 0.23 ε at F a =± Cs( Tc) C ( T ) n c = 2.43 in the BCS limit
a = ± Superfluid to Normal Fermi Liquid Transition Normal Fermi Gas Bogoliubov-Anderson phonons and quasiparticle contribution (red line ) 4 3 3π T Ephonons( T) = εfn, ξ 0.44 3/2 s 5 16ξs εf 3 3 5 2π T E ( T) = ε N exp T quasi-particles F 4 5 2 ε F 7/3 2 π = ε F exp e 2 kfa 4 Bogoliubov-Anderson phonons contribution only (magenta line) People never consider this??? Quasi-particles contribution only (green line) Cs( Tc) C ( T ) n c 2 (2.43 in BCS)
Conclusions Fully non-perturbative calculations for a spin ½ many fermion system in the unitary regime at finite temperatures are feasable (One variant of the fortran 90 program, also in matlab, has about 500 lines, and it can be shortened also. This is about as long as a PRL!) Apparently the system undergoes a phase transition at T c = 0.24 (3) ε F Below the transition temperature both phonons and (fermionic( fermionic) quasiparticles contribute almost equaly to the specific heat