Path Integral (Auxiliary Field) Monte Carlo approach to ultracold - PowerPoint PPT Presentation

Path Integral (Auxiliary Field) Monte Carlo approach to ultracold atomic gases Piotr Magierski Warsaw University of Technology

Collaborators: A. Bulgac - University of Washington J.E. Drut - University of North Carolina K.J. Roche – PNNL G. Wlazłowski - Univ. of Washington/Warsaw Univ. of Techn.

Outline  BCS-BEC crossover. Unitary regime.  Theoretical approach: Path Integral Monte Carlo (QMC)  Equation of state for the Fermi gas in the unitary regime. Critical temperature.  Correlation functions from QMC: analytic continuation and inverse problem.  Pairing gap and pseudogap.  Viscosity, spin conductivity and diffusion.

What is a unitary gas? A gas of interacting fermions is in the unitary regime if the average separation between particles is large compared to their size (range of interaction), but small compared to their scattering length. n - particle density 3 << 1 n |a| 3 >> 1 n r 0 a - scattering length r 0 - effective range NONPERTURBATIVE REGIME System is dilute but strongly interacting!       T Universality: ; x= E x x E  FG F   - Exp. estimate (0) 0.376(5) - Energy of noninteracting Fermi gas E FG

One fermionic atom in magnetic field F m F     ; F I J J L S Nuclear spin Electronic spin Two hypefine states are populated in the trap Collision of two atoms: At low energies (low density of atoms) only L=0 (s-wave) scattering is effective. • Due to the high diluteness atoms in the same hyperfine state do not interact with one another. • Atoms in different hyperfine states experience interactions only in s-wave.

Fes eshb hbac ach h res eson onance ance 2 2 p        hf Z d ( ) ( ) ( ) H V V V r P V r P V 0 0 1 1  i i Tiesinga, Verhaar, 2  1 i Stoof, Phys. Rev. A47 47, 4114 (1993) a       hf hf Z , ( ) V I J V J I B e z n z 2 Channel coupling  E resonance: a  Interatomic distance Regal and Jin, PRL 90 90, , 230404 (2003)

6 system of fermionic atoms Li Feshbach resonance: B=834G BEC side: a>0 UNITARY REGIME BCS side: M.W. Zwierlein et al ., a<0 Nature, 435, 1047 (2005)

3 T     ENERGY: ( ) ( ) ; E x x N x F  5 F  2 E N      PRESSURE: ( ) P x  F 5 V V 2 Note the similarity to  PV E 3 the ideal Fermi gas C  lim ( ) , n k - contact C 4  k k   3 2 2 d k k C Total energy     ( )   E n k    3  4   2 of the system m k 2   1   2 dE     C Adiabatic relation   (1/ )  4 d a m s S C and 1/a are conjugate thermodynamic variables 1/a – „ generalized force ” C - „ generalize displacement ” – capture physics at short length scales. Shina Tan, Ann.Phys.323,2971(2008), Ann.Phys.323,2952(2008) Other theory papers: Tan, Leggett, Braaten, Combescot, Baym, Blume, Werner, Castin, Randeria,Strinati ,…

Hamiltonian    2  ˆ ˆ ˆ   3 † 3        ˆ ˆ ˆ ˆ   ( ) ( ) ( ) ( ) H T V d r r r g d r n r n r   s s 2  m   s   ˆ  3 †      ˆ ˆ ˆ ˆ ˆ ( ) ( ) ; ( ) ( ) ( ) N d r n r n r n r r r   s s s Path Integral Monte Carlo for fermions on 3D lattice  3 Volume L Coordinate space   lattice spacing x correlations in the system L – limit for the spatial - Spin up fermion: - Spin down fermion:    ; k x External conditions:  cut x - temperature T  - chemical potential Periodic boundary conditions imposed

Path Integral Monte Carlo for Fermions on 3D lattice  Momentum space k y  x   x k x 2 π /L 2 π /L  n(k)    UV momentum cutoff UV x  2   IR momentum cutoff IR L k cut = π / D x   2 2 2 2     IR , UV F 2 m 2 m k FFT REAL SPACE MOMENTUM SPACE

Hamiltonian    2  ˆ ˆ ˆ   3 † 3        ˆ ˆ ˆ ˆ   ( ) ( ) ( ) ( ) H T V d r r r g d r n r n r   s s 2   m  s   ˆ  3 †      ˆ ˆ ˆ ˆ ˆ ( ) ( ) ; ( ) ( ) ( ) N d r n r n r n r r r   s s s 1 mk m    cut Running coupling constant g defined by lattice   2 2 2 g 4 2 a 1 m  - UNITARY LIMIT   2 2 g x

            ˆ ˆ ˆ ˆ ˆ ˆ ˆ                exp exp / 2 exp( )exp / 2 H N T N V T N         3 ( ) O Discrete Hubbard-Stratonovich transformation 1   ˆ               ˆ ˆ exp( ) 1 ( ) ( ) 1 ( ) ( ) , exp( ) 1 V r An r r An r A g       2   ( ) 1 r r σ - fields fluctuate both in space and imaginary time N   ˆ ˆ    ( ) ( ); U W j  1 j          ˆ ( ˆ ˆ ˆ ˆ                   ˆ ˆ ) exp / 2 1 ( ) ( ) 1 ( ) ( ) exp / 2 W T N  r An r   r An r  T N       j r

 ˆ     Z T ( ) D ( , ) Tr r U ({ }); 1          D ( , ) r ... ; N  T       { ( ,1) r 1} { ( ,2) r 1} { ( , r N ) 1}   One-body evolution ˆ  ˆ        U ({ }) T exp{ d h [ ({ }) ]}  operator in imaginary time 0 ˆ ˆ    Tr HU ({ }) ˆ    D ( , )Tr r U ({ })     E T ( ) ˆ  Z T ( ) Tr U ({ }) ˆ ˆ         2 Tr U ({ }) {det[1 U ( )]} exp[ S ({ })] 0 No sign problem!      U ({ }) exp( ik x )        * n x y ( , ) n x y ( , ) ( ) x ( ), y ( ) x       k l k 1 U ({ }) 3   L  k l k , k l c All traces can be expressed through these single-particle density matrices

 ˆ  ˆ ˆ          U ({ }) T exp{ d h [ ({ }) ]}; ({ }) h one-body operator  0 ˆ       U ({ }) U ({ }) ; - single-particle wave function kl k l l     S [ ]   D [ ( , )] r e ˆ   E T ( ) H [ ({ })] E U 1 kT Z T ( )  E U [ ({ })]- energy associated with a given sigma field Quantum Monte-Carlo: Sigma space sampling N 1       ( ) ({ }) E T E U k N   1 k ( ) - stochastic variable E T    ( ) ( ) E T E T 2   1 2 ( ) ( ) E T E T N  - number of N  uncorrelated samples     [ ] S ( ) P e

Details of calculations, improvements and problems • Currently we can reach 14 3 lattice and perform calcs. down to x = 0.06 (x – temperature in Fermi energy units) at the densities of the order of 0.03. • Effective use of FFT(W) makes all imaginary time propagators diagonal (either in real space or momentum space) and there is no need to store large matrices. • Update field configurations using the Metropolis importance sampling algorithm. QMC calculations can be split into two independent processes: 1) sample generation (generation of sigma fields), 2) calculations of observables. • Change randomly at a fraction of all space and time sites the signs the auxiliary fields σ (r,  ) so as to maintain a running average of the acceptance rate between 0.4 and 0.6 . • At low temperatures use Singular Value Decomposition of the evolution operator U({ σ }) to stabilize the numerics. • MC correlation “time” ≈ 200 time steps at T ≈ T c for lattices 10 3 . Unfortunately when increasing the lattice size the correlation time also increases. One needs few thousands uncorrelated samples (we usually take about 10 000) to decrease the statistical error to the level of 1%.

Finite size scaling:   0.671   0.8   0.4 0.5 k r Effective range corrections: F eff CM momentum of a fermion pair CM corrections: Total systematic error does not exceed 10-15% Bulgac, Drut, Magierski, PRA78, 023625(2008), Wlazłowski , Magierski, Bulgac, Drut, Roche, arXive:1212.1503 (to appear in Phys. Rev. Lett.)

Diagram. + analytic Diagram. MC QMC Bulgac, Drut, Magierski, Haussmann et al. Burovski et al. PRL99, 120401(2006) PRA75, 023610(2007) PRL96, 160402(2006) Experiment S. Nascimbene et al. Nature 463, 1057 (2010)   exp( ) Courtesy of C. Salomon

Equation of state of the unitary Fermi gas - current status Experiment: M.J.H. Ku, A.T. Sommer, L.W. Cheuk, M.W. Zwierlein , Science 335, 563 (2012) QMC (PIMC + Hybrid Monte Carlo): J.E.Drut, T.Lähde , G.Wlazłowski , P.Magierski, Phys. Rev. A 85, 051601 (2012)