Tackling the Sign Problem of Ultracold Fermi Gases with Mass-Imbalance

Transcription

1 Tackling the Sign Problem of Ultracold Fermi Gases with Mass-Imbalance Dietrich Roscher [D. Roscher, J. Braun, J.-W. Chen, J.E. Drut arxiv: ] Advances in quantum Monte Carlo techniques for non-relativistic many-body systems INT Program INT-13-2a Institut für Kernphysik / Technische Universität Darmstadt July 26, 2013

2 Introduction Physics and Versatility of Ultracold Fermi Gases Why we should care about physics of cold gases: Simple systems compared to e.g. QCD or nuclear physics, but similar behaviour in certain regimes (e.g. HIC) [e.g. A. Adams et al. 12]

3 Introduction Physics and Versatility of Ultracold Fermi Gases Why we should care about physics of cold gases: Simple systems compared to e.g. QCD or nuclear physics, but similar behaviour in certain regimes (e.g. HIC) [e.g. A. Adams et al. 12] Abundance of interesting phenomena: BEC/BCS crossover, (un)conventional superfluidity, topological order... [C.A.R Sa de Melo, 08]

4 Introduction Physics and Versatility of Ultracold Fermi Gases Why we should care about physics of cold gases: Simple systems compared to e.g. QCD or nuclear physics, but similar behaviour in certain regimes (e.g. HIC) [e.g. A. Adams et al. 12] Abundance of interesting phenomena: BEC/BCS crossover, (un)conventional superfluidity, topological order... Excellent experimental access: precise control in optical traps, tuning of interactions by Feshbach resonances S. Jochim, Uni Heidelberg [M. Randeria, E. Taylor 08]

5 Introduction Physics and Versatility of Ultracold Fermi Gases Why we should care about physics of cold gases: Simple systems compared to e.g. QCD or nuclear physics, but similar behaviour in certain regimes (e.g. HIC) [e.g. A. Adams et al. 12] Abundance of interesting phenomena: BEC/BCS crossover, (un)conventional superfluidity, topological order... Excellent experimental access: precise control in optical traps, tuning of interactions by Feshbach resonances Why simple does not mean easy : Lack of small parameter in strongly interacting regimes invalidates naïve perturbation theory

6 Introduction Physics and Versatility of Ultracold Fermi Gases Why we should care about physics of cold gases: Simple systems compared to e.g. QCD or nuclear physics, but similar behaviour in certain regimes (e.g. HIC) [e.g. A. Adams et al. 12] Abundance of interesting phenomena: BEC/BCS crossover, (un)conventional superfluidity, topological order... Excellent experimental access: precise control in optical traps, tuning of interactions by Feshbach resonances Why simple does not mean easy : Lack of small parameter in strongly interacting regimes invalidates naïve perturbation theory Monte Carlo calculations often severely hampered by sign problems

7 Introduction Idea of the Talk How to get rid of (some) sign problems: Identify problematic quantity x s and change into i x s Do an Auxiliary Field Quantum Monte Carlo calculation without sign problem Analytically continue results from i x s to physical value x s

8 Introduction Idea of the Talk How to get rid of (some) sign problems: Identify problematic quantity x s and change into i x s Do an Auxiliary Field Quantum Monte Carlo calculation without sign problem Analytically continue results from i x s to physical value x s This idea has been proposed for lattice QCD at finite chemical potential some time ago. [M. Alford, A.Kapustin, F.Wilczek 98; P. de Forcrand, O. Philipsen 02] Adapting this approach to imbalanced ultracold Fermi gases will be the main subject of this talk.

9 The Sign Problem Describing the Unitary Fermi Gas Action of a two component 3D Fermi gas with contact interaction: β ) S[ψ, ψ ] = dτ d 3 x ψσ ( τ 2 2m µ ψ σ + ḡψ ψ ψ ψ 0 σ=,

10 The Sign Problem Describing the Unitary Fermi Gas Action of a two component 3D Fermi gas with contact interaction: β ) S[ψ, ψ ] = dτ d 3 x ψσ ( τ 2 2m µ ψ σ + ḡψ ψ ψ ψ 0 σ=, The bare coupling ḡ is given by ḡ 1 = Λg 1 = 1 ( a 1 s c reg Λ ) 8π with UV-cutoff Λ and two-body scattering length a s. Unitary regime: a 1 S 0

11 The Sign Problem Symmetry and Spontaneous Symmetry Breaking S[ψ, ψ ] = β 0 dτ d 3 x U(1)-Symmetry of the action: σ=, ψ, e iα ψ, ; ψ σ Spontaneously broken if ψ ψ 0 ) ( τ 2 2m µ ψ, ψ, e iα ψ σ + ḡψ ψ ψ ψ

12 The Sign Problem Symmetry and Spontaneous Symmetry Breaking S[ψ, ψ ] = β 0 dτ d 3 x U(1)-Symmetry of the action: σ=, ψ, e iα ψ, ; ψ σ Spontaneously broken if ψ ψ 0 ) ( τ 2 2m µ ψ, ψ, e iα ψ σ + ḡψ ψ ψ ψ Condensation of bound states [C.A.R Sa de Melo, 08]

13 The Sign Problem Partition Function and Observables Partition function and observables from the path integral: Z = Dψ Dψ e S[ψ,ψ ] Ô = 1 Dψ Dψ Oe S[ψ,ψ ] Z

14 The Sign Problem Partition Function and Observables Partition function and observables from the path integral: Z = Dψ Dψ e S[ψ,ψ ] Ô = 1 Dψ Dψ Oe S[ψ,ψ ] Z Hubbard-Stratonovich transformation and integration of fermion fields: 1 = N DϕDϕ e dτ d 3 x m 2 ϕ ϕϕ, ϕ ϕ g ϕ mϕ 2 ψ ψ

15 The Sign Problem Partition Function and Observables Partition function and observables from the path integral: Z = Dψ Dψ e S[ψ,ψ ] Ô = 1 Dψ Dψ Oe S[ψ,ψ ] Z Hubbard-Stratonovich transformation and integration of fermion fields: 1 = N DϕDϕ e dτ d 3 x m 2 ϕ ϕϕ, ϕ ϕ g ϕ mϕ 2 ψ ψ Ĝ 1 = Z = DϕDϕ [Ĝ 1 ] det + ˆΦ e dτ d 3 x m 2 ϕ ϕϕ ( ) iω n 2 2m µ 0 ( ) 0 gϕ ϕ 0 iω n 2 2m µ, ˆΦ = g ϕ ϕ 0 Starting point for actual computations

16 The Sign Problem Positivity of the Fermion Determinant Z = DϕDϕ [Ĝ 1 det + ˆΦ ]e dτ d 3 x m 2 ϕ ϕϕ If the fermion determinant is positive for all admissible ϕ(x), a positive definite probability measure can be defined for Monte Carlo calculations - there is no sign problem. {Â } det Ĝ 1 0 det 0 Â = det [ÂÂ ] 0 Since the interaction does not break the symmetry between and fermions, the symmetry of the eigenvalue distribution is conserved for finite ϕ. Thus, the fermion determinant is positive for all ϕ(x).

17 The Sign Problem S[ψ, ψ ] = β 0 Introducing Imbalance dτ d 3 x σ=, ψ σ ) ( τ 2 µ σ 2m σ ψ σ + ḡψ ψ ψ ψ

18 The Sign Problem S[ψ, ψ ] = β Experimental relevance: 0 Introducing Imbalance dτ d 3 x σ=, ψ σ ) ( τ 2 µ σ 2m σ ψ σ + ḡψ ψ ψ ψ Population imbalance (µ σ ): Tuning of the species relative population via a magnetic field h [e.g. Zwierlein et al. 06, Partidge et al. 06] Mass imbalance (m σ ): Mixtures of different elements, e.g. 6 Li and 40 K [e.g. A. Gezerlis, S. Gandolfi, K.E. Schmidt, J. Carlson 09 ;K.B. Gubbels, J.E. Baarsma, H.T.C. Stoof 09] BUT

19 The Sign Problem S[ψ, ψ ] = β 0 Introducing Imbalance dτ d 3 x σ=, ψ σ ) ( τ 2 µ σ 2m σ ψ σ + ḡψ ψ ψ ψ Population- and/or mass-imbalance destroy the symmetry between and particles and thus also the positivity of the fermion determinant: ( ) iωn Ĝ 1 2 m = + 2 m µ h 0 0 iω n 2 m m µ+h with µ = µ + µ, h = µ µ 2 2 m + = 4m m, m = 4m m m + m m m Sign Problem!

20 The Sign Problem The Sign Problem and Imaginary Imbalance The sign problem could be resolved, if the symmetry was reinstated. Define complex-valued particle masses m C σ and chemical potentials µ C σ such that: h = ih I, h I R, m mc + m C = i m I, m I R The blocks of ĜC 1 are again complex conjugates of each other: ( ) ĜC 1 = iωn 2 m + i m I 2 µ ih I 0 0 iω n 2 m + +i m I 2 µ+ih I

21 The Sign Problem The Sign Problem and Imaginary Imbalance The sign problem could be resolved, if the symmetry was reinstated. Define complex-valued particle masses m C σ and chemical potentials µ C σ such that: h = ih I, h I R, m mc + m C = i m I, m I R The blocks of ĜC 1 are again complex conjugates of each other: ( ) ĜC 1 = iωn 2 m + i m I 2 µ ih I 0 0 iω n 2 m + +i m I 2 µ+ih I Population- and mass-imbalance behave quiet differently. For population imbalance, see [J. Braun, J.-W. Chen, J. Deng, J. Drut, B. Friman, C.-T. Ma, Y.-D. Tsai 12] For a large part of the talk: purely mass imbalanced case (h = 0)

22 The Sign Problem The Sign Problem and Imaginary Imbalance The sign could be resolved, if the symmetry was reinstated. Define complex-valued particle masses m C σ and chemical potentials µ C σ such that: h = ih I, h I R, m mc + m C = i m I, m I R The blocks of ĜC 1 are then complex conjugates of each other: ( ) ĜC 1 = iωn 2 m + i m I 2 µ ih I 0 0 iω n 2 m + +i m I 2 µ+ih I Sign Problem Circumvented!!

23 The Sign Problem The Sign Problem and Imaginary Imbalance The sign could be resolved, if the symmetry was reinstated. Define complex-valued particle masses m C σ and chemical potentials µ C σ such that: h = ih I, h I R, m mc + m C = i m I, m I R The blocks of ĜC 1 are then complex conjugates of each other: ( ) ĜC 1 = iωn 2 m + i m I 2 µ ih I 0 0 iω n 2 m + +i m I 2 µ+ih I Sign Problem Circumvented!! But... How to get back to real (physical) imbalance?

24 The Sign Problem Physical Results from Imaginary Calculations Suppose, some observable has been computed for imaginary mass-imbalance. Then fit the data points with, e.g., some polynomial Ô C N max and analytically continue m I i m. n=0 C (n) O m2n I Schematically:

25 The Sign Problem Physical Results from Imaginary Calculations Suppose, some observable has been computed for imaginary mass-imbalance. Then fit the data points with, e.g., some polynomial Ô C N max and analytically continue m I i m. n=0 C (n) O m2n I Schematically: Are there any limits to the method, once Monte Carlo data is available?

26 The Sign Problem Analytic Limits for the Method Meaningful results can only be expected, if the series representation for Ô C converges in the complex m-plane: Ô C ( i m I ) = Ô ( m) m I r m The radius of convergence is determined by the closest singularity.

27 The Sign Problem Analytic Limits for the Method Meaningful results can only be expected, if the series representation for Ô C converges in the complex m-plane: Ô C ( i m I ) = Ô ( m) m I r m The radius of convergence is determined by the closest singularity. Knowledge of r m (the singularity structure) is crucial to ascertain reliability of the results, see also examples below. Analytical pre-treatment is required

28 Mean-Field Results How can r m be obtained? Plan: Perform fully analytical calculation ϕ =const mean-field case Investigate convergence properties Get an idea of the analytic structure of the theory and see what can be used for actual MC data

29 Mean-Field Results How can r m be obtained? Plan: Perform fully analytical calculation ϕ =const mean-field case Investigate convergence properties Get an idea of the analytic structure of the theory and see what can be used for actual MC data Mean-field theory: Reduce Z = DϕDϕ e Γ[ϕ,ϕ ] to Z MF = e Γ[ϕ0,ϕ 0 ] and Γ[ϕ 0, ϕ 0] =! min

30 Mean-Field Results How can r m be obtained? Plan: Perform fully analytical calculation ϕ =const mean-field case Investigate convergence properties Get an idea of the analytic structure of the theory and see what can be used for actual MC data Mean-field theory: Reduce Z = DϕDϕ e Γ[ϕ,ϕ ] to Z MF = e Γ[ϕ0,ϕ 0 ] and Γ[ϕ 0, ϕ 0] =! min The grand canonical potential is then just given by Ω MF = 1 β ln Z MF = 1 β Γ[ϕ 0, ϕ 0]

31 Mean-Field Results Ω MF and the Gap Equation [Ĝ 1 For ϕ =const, det + ˆΦ ] can be computed analytically by performing a Bogoliubov transformation and the Matsubara sum. Result (m + = 1,regularizing terms dropped): Ω MF [ ϕ 2] { d 3 q g 2 ϕ = ϕ 2 1 [ ( (2π) 3 2q 2 β ln cosh β mq 2) ( )] } + cosh β (q 2 µ 2 ) 2 + gϕ 2 ϕ 2

32 Mean-Field Results Ω MF and the Gap Equation [Ĝ 1 For ϕ =const, det + ˆΦ ] can be computed analytically by performing a Bogoliubov transformation and the Matsubara sum. Result (m + = 1,regularizing terms dropped): Ω MF [ ϕ 2] { d 3 q g 2 ϕ = ϕ 2 1 [ ( (2π) 3 2q 2 β ln cosh β mq 2) ( )] } + cosh β (q 2 µ 2 ) 2 + gϕ 2 ϕ 2 Gap equation with g 2 ϕ ϕ0 2 µ : 2 { ( 0 =! 1 dq 2 + q 2 2 (q 2 1) e β µ ( q 2 m+ 1 ) (q 2 1) e β µ ( q 2 m 1 (q 2 1) 2 + ) )}

33 Mean-Field Results Mean-Field Phase Diagram for the Mass-Imbalanced Unitary Fermi Gas Position of the critical point: ( ) TCP µ, m CP (0.47, 0.37)

34 Mean-Field Results Comparing Phase Diagrams for m and m I Since the functional form of Ω MF is known analytically, both phase diagrams can be determined directly: No critical point/1st order transition for m I Analytic continuation of phase boundary will not reproduce result for all m radius of convergence r m?

35 Mean-Field Results Quantitative bounds on r m Functional form of Ω MF ( m) not explicitly known due to integration: { d 3 q g 2 ϕ Ω MF ( m) = ϕ 2 1 [ ( (2π) 3 2q 2 β ln cosh β mq 2) ( )] } + cosh β (q 2 µ 2 ) 2 + gϕ 2 ϕ 2 Complex analysis: r m bounded from below by singularity structure of the integrand, i.e. the log term.

36 Mean-Field Results Quantitative bounds on r m Functional form of Ω MF ( m) not explicitly known due to integration: { d 3 q g 2 ϕ Ω MF ( m) = ϕ 2 1 [ ( (2π) 3 2q 2 β ln cosh β mq 2) ( )] } + cosh β (q 2 µ 2 ) 2 + gϕ 2 ϕ 2 Complex analysis: r m bounded from below by singularity structure of the integrand, i.e. the log term. With Φ = g ϕ ϕ β r m r min = 2 Φ 2 + π 2 β 2 Φ 2 + π 2 + β 2 µ 2

37 Mean-Field Results Limiting cases: Properties and Usefulness of r min β r min = 2 Φ 2 + π 2 β 2 Φ 2 + π 2 + β 2 µ 2 r min T = 1, i.e. access to all possible m for high temperatures r min T 0 = Φ 2 Φ 2 + µ, i.e. (partial) access to symmetry broken 2 phases at T = 0

38 Mean-Field Results Limiting cases: Properties and Usefulness of r min β r min = 2 Φ 2 + π 2 β 2 Φ 2 + π 2 + β 2 µ 2 r min T = 1, i.e. access to all possible m for high temperatures r min T 0 = Φ 2 Φ 2 + µ, i.e. (partial) access to symmetry broken 2 phases at T = 0 Applicability: If Ω C itself is computed for certain ( m I, T ), T fixed, it can be continued to Ω inside the interval [0, r min (T )[ Physical observables can then be obtained from Ω. Additional expansions of Ω C around some m > 0 may vastly extend regime of applicability [F. Karbstein, M. Thies 07]

39 Mean-Field Results Limiting cases: Properties and Usefulness of r min β r min = 2 Φ 2 + π 2 β 2 Φ 2 + π 2 + β 2 µ 2 r min T = 1, i.e. access to all possible m for high temperatures r min T 0 = Φ 2 Φ 2 + µ, i.e. (partial) access to symmetry broken 2 phases at T = 0 Applicability: If Ω C itself is computed for certain ( m I, T ), T fixed, it can be continued to Ω inside the interval [0, r min (T )[ Physical observables can then be obtained from Ω. Additional expansions of Ω C around some m > 0 may vastly extend regime of applicability [F. Karbstein, M. Thies 07] If observables Ô C are computed directly, as it is most often the case with Monte Carlo, things get a little more complicated...

40 Observables & Monte Carlo Application I: Phase Boundary Information provided by Ω: Boundary T c ( m) of phase with broken symmetry is defined for every m by the lowest T with = 0 Locally smooth manifold up to critical point, global properties not accessible due to implicit form

41 Observables & Monte Carlo Application I: Phase Boundary Information provided by Ω: Boundary T c ( m) of phase with broken symmetry is defined for every m by the lowest T with = 0 Locally smooth manifold up to critical point, global properties not accessible due to implicit form Analytic continuation of boundary is meaningful if Ω[T c ( m), m, = 0] is convergent r min ( Φ 2 = 0) = T 2 π 2 T 2 π 2 + µ 2

42 Observables & Monte Carlo Application I: Phase Boundary Information provided by Ω: Boundary T c ( m) of phase with broken symmetry is defined for every m by the lowest T with = 0 Locally smooth manifold up to critical point, global properties not accessible due to implicit form Analytic continuation of boundary is meaningful if Ω[T c ( m), m, = 0] is convergent r min ( Φ 2 = 0) = T 2 π 2 T 2 π 2 + µ 2

43 Observables & Monte Carlo Phase Boundary from m I Continuation of the m I data yields: Fair reproduction of phase boundary up to the critical point Non-analytic behaviour of T c ( m) itself limits applicability

44 Observables & Monte Carlo Phase Boundary from m I Continuation of the m I data yields: Fair reproduction of phase boundary up to the critical point Non-analytic behaviour of T c ( m) itself limits applicability Way out: computation & continuation of Ω itself, critical point should then be within reach

45 Observables & Monte Carlo Application II: Bertsch Parameter ξ T =0 Definition: ξ = µ ɛ F, ξ Free T =0 = 1 m [0, 1)

46 Observables & Monte Carlo Application II: Bertsch Parameter ξ T =0 Definition: ξ = µ ɛ F, ξ Free T =0 = 1 m [0, 1) Smooth observable implicitly depends on By complex analysis: r ξ min[r min, r ] First order phase transition is expected to limit continuation of ξ C

47 Observables & Monte Carlo Bertsch Parameter from m I Good reproduction of ξ up to the discontinuity Smooth observables indeed limited by r min or r

48 Observables & Monte Carlo Detour: Population-Imbalance [J. Braun, J.-W. Chen, J. Deng, J. Drut, B. Friman, C.-T. Ma, Y.-D. Tsai 12] G 1 = iω n 2 m + i m I 2 µ ih I, ω n = (2n + 1)πT Due to interference with Matsubara frequencies: h I! < πt Continuation of ξ T =0 not possible for population imbalanced case Structurally similar to finite µ problem in lattice QCD

49 Observables & Monte Carlo Remarks on r min for Monte Carlo Schematic structure of the grand canonical potential from Monte Carlo: [Ĝ 1 Ω MC ln det + ˆΦ ] e m 2 ϕ ϕ(x) 2 Rigorous statements: {ϕ(x)} For every fixed configuration ϕ(x), the corresponding analytical calculation would correspond to the above mean-field procedure Overall radius of convergence: r MC min {ϕ(x)} [r min ] Problem: no easy way to calculate r min for general ϕ(x)

50 Observables & Monte Carlo Remarks on r min for Monte Carlo Schematic structure of the grand canonical potential from Monte Carlo: [Ĝ 1 Ω MC ln det + ˆΦ ] e m 2 ϕ ϕ(x) 2 Rigorous statements: {ϕ(x)} For every fixed configuration ϕ(x), the corresponding analytical calculation would correspond to the above mean-field procedure Overall radius of convergence: r MC min {ϕ(x)} [r min ] Problem: no easy way to calculate r min for general ϕ(x) First practical estimate: r MC r min (ϕ = 0)

51 Summary and Outlook Summary Imbalanced strongly interacting Fermi gases are tough systems to compute with Monte Carlo methods due to a sign problem Imaginary imbalance parameters remove the sign problem Physical observables may be extracted by analytic continuation Large parts of the phase diagram of the 3D unitary Fermi gas are in reach Extraction of physical results is limited by convergence issues Radius of convergence of the grand potential Ω Analyticity of the observables and dependencies Type of imbalance Analytic structure at mean-field level provides hints for treatment of genuine Monte Carlo data

52 Summary and Outlook Outlook Extend investigation of convergence properties of observables Make connection to Monte Carlo more rigorous, ideally provide rigorous quantitative bounds Apply method in an actual Monte Carlo study Work in progress by Joaquín Drut