Effective Field Theory for Many-Body Systems

Transcription

1 Outline Systems EFT Dilute DFT Future Effective Field Theory for Many-Body Systems Department of Physics Ohio State University February 19, 2004

2 Outline Systems EFT Dilute DFT Future Principles of Effective Low-Energy Theories

3 Outline Systems EFT Dilute DFT Future Principles of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved

4 Outline Systems EFT Dilute DFT Future Principles of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved use low-energy dof s for low-energy processes short-distance structure can be replaced by something simpler without distorting low-energy observables

5 Outline Systems EFT Dilute DFT Future Outline Nuclear and Other Many-Body Problems Basic Effective Field Theory (EFT) Concepts EFT for Dilute Fermi Systems Density Functional Theory (DFT) Future: Challenges for

6 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Nuclear and Cold Atom Many-Body Problems Lennard-Jones and nucleon-nucleon potentials O -O potential (mev) n-n potential (MeV) n-n system O -O system n-n distance (fm) O 2-O 2 distance (nm) [figure borrowed from J. Dobaczewski] How can we deal with hard cores in many-body systems? Are there universal features of such many-body systems?

7 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Effective Range Expansion Total cross section: σ total = 4π k 2 (2l + 1) sin 2 δ l (k) l=0 What happens at low energy (k 1/R)? k cot δ 0 (k) k 0 1 a r 0k a 0 is the scattering length and r 0 is the effective range

8 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Near-Zero-Energy Bound States Bound-state or near-bound state at zero energy = large scattering lengths (a 0 ± )

9 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Near-Zero-Energy Bound States Bound-state or near-bound state at zero energy = large scattering lengths (a 0 ± ) For kr 0, the total cross section is { 4πa0 2 4πa 2 σ total = σ l=0 = 1 + (ka 0 ) 2 = 0 for ka 0 1 4π k 2 for ka 0 1 (unitarity limit)

10 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT How Are Trapped Atoms Like Neutron Stars? Regal et al., ultracold fermions Chandra X-Ray Observatory image of pulsar in 3C58

11 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Trapped Fermion Atoms Low densities and temperatures = only a 0 enters Use Feshbach resonances to tune scattering length a 0 ± Universal behavior? Superfluidity? BCS-BEC crossover?

12 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Neutron Stars n-n a 0 range of V NN (r) Dilute neutron gas at surface = neutron superfluid High density core: color superconductor? (see T. Schafer for EFT discussion)

13 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Quantum Chromodynamics and Nuclear Physics Quark Mass (m q ) Dependence of Observables S. Beane and M. Savage, Nucl. Phys. A717 (2003) 91. Crude (naive?) estimate of nonrelativistic bound state: E B p2 typical M f 2 π M 10 MeV E deuteron

14 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Is the Fine-Tuning Accidental? E. Epelbaum, U. Meißner, and W. Glöckle, Nucl. Phys. A714 (2003) B D [MeV] M π [MeV] Footnote from S. Weinberg paper [Nucl. Phys. B363 (1991) 3]: E. Witten says that large N c arguments... suggest that the very small binding energy of the deuteron is a fortuitous cancellation of potential and kinetic energies for the physical value N c = 3.

15 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Rare Isotope Accelerator (RIA) Physics Nucleosynthesis, halo nuclei, pairing,... Stable nuclei 126 Known nuclei 82 r-process Protons rp-process Terra incognita Neutron stars Neutrons Figure The nuclear landscape, defining the territory of nuclear physics research. On this chart of the nuclides, black squares represent stable nuclei and nuclei with half-lives comparable to or longer than the age of the Earth.

16 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Coulomb vs. Nuclear Many-Body Systems Common approaches to the many-body problem: Mean-field approaches (independent-particle picture) Monte Carlo methods (variational, GFMC) Hamiltonian diagonalization (CI, shell model) Density functional theory (DFT)

17 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Coulomb vs. Nuclear Many-Body Systems Common approaches to the many-body problem: Mean-field approaches (independent-particle picture) Monte Carlo methods (variational, GFMC) Hamiltonian diagonalization (CI, shell model) Density functional theory (DFT) Some important differences: Knowledge of interaction Spin-isospin dependence of interaction Importance of pairing Hartree-Fock as a starting point

18 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Coulomb vs. Nuclear Many-Body Systems Common approaches to the many-body problem: Mean-field approaches (independent-particle picture) Monte Carlo methods (variational, GFMC) Hamiltonian diagonalization (CI, shell model) Density functional theory (DFT) Some important differences: Knowledge of interaction Spin-isospin dependence of interaction Importance of pairing Hartree-Fock as a starting point Ab initio calculations?

19 Figure 2: Top: the nuclear landscape - the territory of RIA Many-Body physics. The EFT black squares represent the stable Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Different Approaches for Different Regions Limits of nuclear existence r-process protons rp-process neutrons

20 Figure 2: Top: the nuclear landscape - the territory of RIA Many-Body physics. The EFT black squares represent the stable Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Different Approaches for Different Regions Limits of nuclear existence r-process protons rp-process A=10 A=12 50 neutrons A~60 82 Density Functional Theory Selfconsistent Mean Field Ab initio few-body calculations 0Ñω Shell Model No-Core Shell Model G-matrix Many-body approaches for ordinary nuclei

21 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Density Functional Theory (DFT) Hohenberg-Kohn: There exists an energy functional E v [n]... E v [n] = F HK [n] + d 3 x v(x)n(x) F HK is universal (same for any external v) = H 2 to DNA!

22 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Density Functional Theory (DFT) Hohenberg-Kohn: There exists an energy functional E v [n]... E v [n] = F HK [n] + d 3 x v(x)n(x) F HK is universal (same for any external v) = H 2 to DNA! Useful if you can approximate the energy functional

23 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Density Functional Theory (DFT) Hohenberg-Kohn: There exists an energy functional E v [n]... E v [n] = F HK [n] + d 3 x v(x)n(x) F HK is universal (same for any external v) = H 2 to DNA! Useful if you can approximate the energy functional Kohn-Sham procedure similar to nuclear mean field calculations

24 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Problems with Extrapolations Stable nuclei 126 Known nuclei 82 r-process Protons rp-process Terra incognita Neutron stars Neutrons Figure The nuclear landscape, defining the territory of nuclear physics research. On this chart of the nuclides, black squares represent stable nuclei and nuclei with half-lives comparable to or longer than the age of the Earth. These nuclei define the valley of stability. By adding either protons or neutrons, one moves away from the valley of stability, finally reaching the drip lines where nuclear binding forces Many-Body are no longer EFT strong enough to hold these nuclei

25 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Problems with Extrapolations

26 Outline Systems EFT Dilute DFT Future a 0 Atoms RIA DFT Problems with Extrapolations Mass formulas and energy functionals do well where there is data, but elsewhere... two-neutron separation energies S 2n (MeV) Sn data exist Experiment HFB-SLy4 HFB-SkP HFB-D1S SkX RHB-NL3 LEDF Mass Formulae Neutron Number data do not exist Neutron Number Neutron Number exp FRDM CKZ JM MJ T+ Figure 6: Predicted two-neutron separation energies for the even-even Sn isotopes using several microscopic models based on effective nucleon-nucleon interactions and obtained with phenomenological

27 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Why Use EFT For Many-Body Physics? Connections between different processes and to QCD Guide to making systematic calculations with error estimates For RIA physics = reliable, model independent extrapolation

28 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Why Use EFT For Many-Body Physics? Connections between different processes and to QCD Guide to making systematic calculations with error estimates For RIA physics = reliable, model independent extrapolation Analogy between EFT and basic numerical analysis naive error analysis: pick a method and reduce the mesh size (e.g., increase grid points) until the error is acceptable sophisticated error analysis: understand scaling and sources of error (e.g., algorithm vs. round-off errors) = Does it work as well as it should? representation dependence = not all are equally effective! extrapolation: completeness of an expansion basis

29 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Numerical Analysis Numerical Derivatives f (x) = [f(x+h)-f(x)]/h + O(h) relative error mesh size h

30 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Numerical Analysis relative error Numerical Derivatives f (x) = [f(x+h)-f(x)]/h + O(h) f (x) = [f(x+h/2)-f(x-h/2)]/h + O(h 2 ) mesh size h

31 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Numerical Analysis relative error Numerical Derivatives f (x) = [f(x+h)-f(x)]/h + O(h) f (x) = [f(x+h/2)-f(x-h/2)]/h + O(h 2 ) Richardson extrapolation O(h 4 ) mesh size h

32 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Numerical Analysis Numerical Integration relative error h n to h n trapezoid rule O(h 2 ) mesh size h

33 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Numerical Analysis Numerical Integration relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) mesh size h

34 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Numerical Analysis Numerical Integration relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) mesh size h

35 Outline Systems EFT Dilute DFT Future Analogs RG Chiral The Representation Can Make A Difference! E.g., elliptic integral: 1 0 (1 x 2 )(2 x) dx

36 Outline Systems EFT Dilute DFT Future Analogs RG Chiral The Representation Can Make A Difference! E.g., elliptic integral: 1 0 (1 x 2 )(2 x) dx How do the numerical errors behave? relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration mesh size h

37 Outline Systems EFT Dilute DFT Future Analogs RG Chiral The Representation Can Make A Difference! E.g., elliptic integral: 1 0 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 0 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

38 Outline Systems EFT Dilute DFT Future Analogs RG Chiral The Representation Can Make A Difference! E.g., elliptic integral: 1 0 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 0 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

39 Outline Systems EFT Dilute DFT Future Analogs RG Chiral The Representation Can Make A Difference! E.g., elliptic integral: 1 0 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 0 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

40 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Completeness, Fitting, and Extrapolation What if I fit my model in one region, then predict in another? Take our function: (1 x 2 )(2 x) and fit it for 0.3 x 0.6 f(x) exact function a 0 + a 1 x + a 2 x 2 b 0 + b 1 x + b 3 x 3 c 0 + c 1 x + c 3 x 3 + c 4 x x

41 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Completeness, Fitting, and Extrapolation What if I fit my model in one region, then predict in another? Take our function: (1 x 2 )(2 x) and fit it for 0.3 x 0.6 f(x) exact function a 0 + a 1 x + a 2 x 2 b 0 + b 1 x + b 3 x 3 c 0 + c 1 x + c 3 x 3 + c 4 x x

42 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Completeness, Fitting, and Extrapolation What if I fit my model in one region, then predict in another? Take our function: (1 x 2 )(2 x) and fit it for 0.3 x 0.6 Compare complete and incomplete models and weighting relative error a 0 + a 1 x + a 2 x 2 b 0 + b 1 x + b 3 x x

43 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Completeness, Fitting, and Extrapolation What if I fit my model in one region, then predict in another? Take our function: (1 x 2 )(2 x) and fit it for 0.3 x 0.6 Compare complete and incomplete models and weighting relative error a 0 + a 1 x + a 2 x 2 b 0 + b 1 x + b 3 x 3 c 0 + c 1 x + c 3 x 3 + c 4 x x

44 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Completeness, Fitting, and Extrapolation What if I fit my model in one region, then predict in another? Take our function: (1 x 2 )(2 x) and fit it for 0.3 x 0.6 Compare complete and incomplete models and weighting relative error a 0 + a 1 x + a 2 x 2 b 0 + b 1 x + b 3 x 3 c 0 + c 1 x + c 3 x 3 + c 4 x 4 weighted x

45 Outline Systems EFT Dilute DFT Future Analogs RG Chiral NN Potential and Scattering Phase Shifts n-n potential (MeV) n-n system O -O system n-n distance (fm) low 0 energies 0.4 = 0.8 large a O 2-O 2 distance (nm) zero crossing = hard core phase shift (degrees) np 1 S 0 (PWA93) E lab (MeV)

46 V NN (k,k) [fm] Outline Systems EFT Dilute DFT Future Analogs RG Chiral NN Potential and Scattering Phase Shifts Now look at NN potentials in momentum space: s-wave k V k Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ] phase shift (degrees) np 1 S 0 (PWA93) E lab (MeV) Many different potentials with same phase shifts (χ 2 /dof 1)

47 Outline Systems EFT Dilute DFT Future Analogs RG Chiral NN Scattering in the COM Frame k +k k +k T 3 2 T k +k = V B k +k + q Λ B χ 2 /dof 1 bare potential V B Probes intermediate states to q Λ B = 25 fm 1 =. 5 GeV V B V NN (k,k) [fm] Model dependent: q 3 fm k [fm -1 ] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I

48 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 V Λ k +k k +k Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1 E lab. = 350 MeV V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

49 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 V Λ k +k k +k Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1 E lab. = 350 MeV V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

50 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 V Λ k +k k +k Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1 E lab. = 350 MeV V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

51 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Momentum NN Potential Bogner, Kuo, Schwenk k +k k +k T 3 T = V Λ + q Λ 2 V Λ k +k k +k Require dt dλ = 0 = renormalization group equation for V Λ Run from Λ B = 25 fm 1 to Λ = 2 fm 1. E lab = 350 MeV Same long distance physics = collapse! V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

52 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Energy Observables Unchanged Phase shift δ vs. E Lab [MeV] 0 1 S P S1 P0 5 P D1 1 ε P1 PSA93

53 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Energy Observables Unchanged Phase shift δ vs. E Lab [MeV] 0 1 S P S1 P0 5 P D1 1 ε P1 PSA93 CD Bonn

54 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Energy Observables Unchanged Phase shift δ vs. E Lab [MeV] 0 1 S P S1 P0 5 P D1 1 ε P1 PSA93 CD Bonn V Λ=2 fm -1

55 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Low-Momentum Pot l in the Many-Body Problem Removing hard core = simpler many-body starting point for neutron matter [Schwenk, Friman, Brown] Simple Hartree-Fock (triangles) instead of BHF (circles)

56 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Three-Body Forces are Inevitable! What if we have three nucleons interacting? Successive two-body scatterings with short-lived high-energy intermediate states unresolved = must be absorbed into three-body force Binding Energy (MeV) Triton Binding Energy AV18 CD Bonn Experiment Λ (fm -1 ) [Bogner, Nogga, Schwenk]

57 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Three-Body Forces are Inevitable! What if we have three nucleons interacting? Successive two-body scatterings with short-lived high-energy intermediate states unresolved = must be absorbed into three-body force Binding Energy (MeV) Triton Binding Energy AV18 CD Bonn Experiment How do we organize three-body forces? Λ (fm -1 ) [Bogner, Nogga, Schwenk]

58 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Renormalization and Short-Distance Physics 3 V B = C0 2 q > Λ V B q > Λ intermediate states = replace with contact term: C 0 δ 3 (x x ) L eft = (ψ ψ) 2 + V NN (k,k) [fm] Paris Bonn A Argonne v18 Idaho A CD Bonn Nijmegen94 II Nijmegen94 I k [fm -1 ]

59 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Effective Field Theory Ingredients From Crossing the Border [nucl-th/ ] 1. Use the most general L with low-energy dof s consistent with the global and local symmetries of the underlying theory 2. Declaration of regularization and renormalization scheme 3. Well-defined power counting = small expansion parameters

60 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Effective Field Theory Ingredients: NN From Crossing the Border [nucl-th/ ] 1. Use the most general L with low-energy dof s consistent with the global and local symmetries of the underlying theory L eft = L ππ + L πn + L NN chiral symmetry = systematic long-distance pion physics 2. Declaration of regularization and renormalization scheme 3. Well-defined power counting = small expansion parameters

61 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Effective Field Theory Ingredients: NN From Crossing the Border [nucl-th/ ] 1. Use the most general L with low-energy dof s consistent with the global and local symmetries of the underlying theory L eft = L ππ + L πn + L NN chiral symmetry = systematic long-distance pion physics 2. Declaration of regularization and renormalization scheme momentum cutoff and Weinberg counting use cutoff sensitivity as measure of uncertainties! 3. Well-defined power counting = small expansion parameters

62 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Effective Field Theory Ingredients: NN From Crossing the Border [nucl-th/ ] 1. Use the most general L with low-energy dof s consistent with the global and local symmetries of the underlying theory L eft = L ππ + L πn + L NN chiral symmetry = systematic long-distance pion physics 2. Declaration of regularization and renormalization scheme momentum cutoff and Weinberg counting use cutoff sensitivity as measure of uncertainties! 3. Well-defined power counting = small expansion parameters use the separation of scales = {p, m π} with Λ Λ χ 1 GeV χ Weinberg: power-count V NN and solve S-equation chiral symmetry = V NN = ν=ν min c ν Q ν with ν 0

63 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N 1S0 3S P0 1D D3 3G

64 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q 0 1S0 3S P0 1D D3 3G

65 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q 0 Q 1 1S0 3S P0 1D D3 3G

66 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q 0 Q 1 Q 2 1S0 3S P0 1D D3 3G

67 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q 0 Q 1 Q 2 Q 3 1S0 3S P0 1D D3 3G

68 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Effective Field Theory G. P. Lepage, How to Renormalize the Schrödinger Equation! "$# % % & Errors in the 1 S 0 phase shifts versus energy through orders Λ 2 and Λ 4. Results from the theory with just pion exchange (V 1π ) are also shown. The cutoff was Λ=330 MeV in each case.

69 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Error Plots in Effective Field Theory G. P. Lepage, How to Renormalize the Schrödinger Equation!#"%$ & & '(' & ) & Errors in the 1 S 0 phase shifts versus energy for different values of the cutoff Λ.

70 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Segue: From EFT Potentials to EFT Many-Body So far: Systematic few-body potentials for input to exact methods for solving many-body Schrödinger equations The effort to develop many-body potentials for nuclei continues No-core shell model calculations are applying these potentials Steady progress expected but limited range in A = N + Z

71 Outline Systems EFT Dilute DFT Future Analogs RG Chiral Segue: From EFT Potentials to EFT Many-Body So far: Systematic few-body potentials for input to exact methods for solving many-body Schrödinger equations The effort to develop many-body potentials for nuclei continues No-core shell model calculations are applying these potentials Steady progress expected but limited range in A = N + Z Turn now to more basic interactions but direct many-body EFT Start with simplest systems; protoype is cold Fermi atoms in a trap Technical advantages from physics simplifications Large scattering length problem Finite systems

72 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Simple Many-Body Problem: Hard Spheres Infinite potential at radius R sin(kr+δ) R 0 R r ~ 1/ k F Scattering length a 0 = R Dilute nr 3 1 = k F a 0 1 What is the energy / particle?

73 Outline Systems EFT Dilute DFT Future Spheres Large a 0 In Search of a Perturbative Expansion For free-space scattering at momentum k 1/R, we should recover a perturbative expansion in kr for scattering amplitude: f 0 (k) 1 k cot δ(k) ik a 0 ia 2 0k (a 3 0 a2 0r 0 /2)k 2 + O(k 3 a 3 0 ) with a 0 = R and r 0 = 2R/3 for hard-core spheres

74 Outline Systems EFT Dilute DFT Future Spheres Large a 0 In Search of a Perturbative Expansion For free-space scattering at momentum k 1/R, we should recover a perturbative expansion in kr for scattering amplitude: f 0 (k) 1 k cot δ(k) ik a 0 ia 2 0k (a 3 0 a2 0r 0 /2)k 2 + O(k 3 a 3 0 ) with a 0 = R and r 0 = 2R/3 for hard-core spheres Perturbation theory in the hard-core potential won t work: = k V k dx e ik x V (x) e ik x 0 R

75 Outline Systems EFT Dilute DFT Future Spheres Large a 0 In Search of a Perturbative Expansion For free-space scattering at momentum k 1/R, we should recover a perturbative expansion in kr for scattering amplitude: f 0 (k) 1 k cot δ(k) ik a 0 ia 2 0k (a 3 0 a2 0r 0 /2)k 2 + O(k 3 a 3 0 ) with a 0 = R and r 0 = 2R/3 for hard-core spheres Perturbation theory in the hard-core potential won t work: = k V k dx e ik x V (x) e ik x 0 R Standard solution: Solve nonperturbatively, then expand

76 Outline Systems EFT Dilute DFT Future Spheres Large a 0 In Search of a Perturbative Expansion For free-space scattering at momentum k 1/R, we should recover a perturbative expansion in kr for scattering amplitude: f 0 (k) 1 k cot δ(k) ik a 0 ia 2 0k (a 3 0 a2 0r 0 /2)k 2 + O(k 3 a 3 0 ) with a 0 = R and r 0 = 2R/3 for hard-core spheres Perturbation theory in the hard-core potential won t work: = k V k dx e ik x V (x) e ik x 0 R Standard solution: Solve nonperturbatively, then expand EFT approach: k 1/R means we probe at low resolution = replace potential with a simpler but general interaction

77 Outline Systems EFT Dilute DFT Future Spheres Large a 0 EFT for Natural Dilute Fermi Gas A simple, general interaction is a sum of delta functions and derivatives of delta functions. In momentum space, k V eft k = C C 2(k 2 + k 2 ) + C 2 k k + Or, L eft has most general local (contact) interactions: L eft = ψ [ i t + 2 2M ] ψ C 0 2 (ψ ψ) 2 + C 2 [ (ψψ) (ψ 2 ψ) + h.c. ] 16 + C 2 8 (ψ ψ) (ψ ψ) D 0 6 (ψ ψ) Dimensional analysis = C 2i 4π M R2i+1, D 2i 4π M R2i+4

78 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Renormalization Consider the leading potential V (0) EFT (x) = C 0δ(x) or k V (0) eft k = = C 0 Choosing C 0 a 0 gets the first term. Now k VG 0 V k : = d 3 q 1 (2π) 3 k 2 q 2! + iɛ = Linear divergence!

79 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Renormalization Consider the leading potential V (0) EFT (x) = C 0δ(x) or k V (0) eft k = = C 0 Choosing C 0 a 0 gets the first term. Now k VG 0 V k : = Λc d 3 q 1 (2π) 3 k 2 q 2 + iɛ Λ c 2π 2 ik 4π + O(k 2 /Λ c ) = If cutoff at Λ c, then can absorb into V (0), but all powers of k 2

80 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Renormalization Consider the leading potential V (0) EFT (x) = C 0δ(x) or k V (0) eft k = = C 0 Choosing C 0 a 0 gets the first term. Now k VG 0 V k : = d D q 1 D 3 (2π) 3 k 2 q 2 ik + iɛ 4π Dimensional regularization with minimal subtraction = only one power of k!

81 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Dim. reg. + minimal subtraction = simple power counting: P/2 + k P/2 + k = + P/2 k P/2 k it (k, cos θ) ic 0 M 4π (C0)2 k O(k 3 ) ( M )2 +i (C 0) 3 k 2 ic 2k 2 ic 4π 2k 2 cos θ Matching: C 0 = 4π M a 0 = 4π M R, C 2 = 4π M a 2 0 r 0 2 = 4π M R3 3,

82 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Dim. reg. + minimal subtraction = simple power counting: P/2 + k P/2 + k = + P/2 k P/2 k it (k, cos θ) ic 0 M 4π (C0)2 k O(k 3 ) ( M )2 +i (C 0) 3 k 2 ic 2k 2 ic 4π 2k 2 cos θ Matching: C 0 = 4π M a 0 = 4π M R, C 2 = 4π M a 2 0 r 0 2 = 4π M R3 3, Recovers effective range expansion order-by-order with perturbative diagrammatic expansion one power of k per diagram estimate truncation error from dimensional analysis

83 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Now Sum Over Fermions in the Fermi Sea Leading order V (0) EFT (x) = C 0δ(x) = a 0 k 6 F

84 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Now Sum Over Fermions in the Fermi Sea Leading order V (0) EFT (x) = C 0δ(x) = a 0 k 6 F At the next order, we get a linear divergence again: = k F d 3 q 1 (2π) 3 k 2 q 2

85 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Now Sum Over Fermions in the Fermi Sea Leading order V (0) EFT (x) = C 0δ(x) = a 0 k 6 F At the next order, we get a linear divergence again: = k F d 3 q 1 (2π) 3 k 2 q 2 Same renormalization fixes it! Particles holes k F 1 k 2 q 2 = 0 1 k 2 q 2 kf 0 1 D 3 k 2 q 2 kf 0 1 k 2 q 2 a2 0k 7 F

86 Outline Systems EFT Dilute DFT Future Spheres Large a 0 T = 0 Energy Density from Hugenholtz Diagrams E V = n k F 2 [ 3 2M 5 ]

87 Outline Systems EFT Dilute DFT Future Spheres Large a 0 T = 0 Energy Density from Hugenholtz Diagrams O ( kf 6 ) E : V = n k F 2 2M [ (ν 1) 5 3π (k Fa 0 ) ]

88 Outline Systems EFT Dilute DFT Future Spheres Large a 0 T = 0 Energy Density from Hugenholtz Diagrams O ( k 6 F) : O ( k 7 F) : + E V = n k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 ]

89 Outline Systems EFT Dilute DFT Future Spheres Large a 0 T = 0 Energy Density from Hugenholtz Diagrams O ( k 6 F) : O ( k 7 F) : + E V = n k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 O ( k 8 F) : + ]

90 Outline Systems EFT Dilute DFT Future Spheres Large a 0 T = 0 Energy Density from Hugenholtz Diagrams O ( k 6 F) : O ( k 7 F) : + O ( k 8 F) : + E V = n k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 + (ν 1) ( (ν 3) ) (k F a 0 ) ]

91 Outline Systems EFT Dilute DFT Future Spheres Large a 0 T = 0 Energy Density from Hugenholtz Diagrams O ( kf) 6 : O ( kf) 7 : + O ( kf) 8 : E V = n k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 + (ν 1) ( (ν 3) ) (k F a 0 ) 3 + (ν 1) 1 10π (k Fr 0 )(k F a 0 ) 2 + (ν + 1) 1 5π (k Fa p ) 3 + ]

92 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Looks Like a Power Series in k F! Is it?

93 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Looks Like a Power Series in k F! Is it? New logarithmic divergences in 3 3 scattering + (C 0 ) 4 ln(k F /Λ c )

94 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Looks Like a Power Series in k F! Is it? New logarithmic divergences in 3 3 scattering + (C 0 ) 4 ln(k F /Λ c ) Changes in Λ c must be absorbed by 3-body coupling D 0 (Λ c ) = D 0 (Λ c ) (C 0 ) 4 ln(a 0 Λ c ) + const. [Braaten & Nieto] [ ] d = 0 = same coefficient! dλ c

95 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Looks Like a Power Series in k F! Is it? New logarithmic divergences in 3 3 scattering + (C 0 ) 4 ln(k F /Λ c ) Changes in Λ c must be absorbed by 3-body coupling D 0 (Λ c ) = D 0 (Λ c ) (C 0 ) 4 ln(a 0 Λ c ) + const. [Braaten & Nieto] [ ] d = 0 = same coefficient! dλ c What does this imply for the energy density? O ( k 9 F ln(k F ) ) : + + (ν 2)(ν 1) (k F a 0 ) 4 ln(k F a 0 )

96 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Large Scattering Length Problem Attractive with a 0 If R 1/k F a 0, then expect scale invariance R Energy and gap are pure numbers times E FG = 3 5 kf 2 2M ~ 1/ k F a 0

97 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Large Scattering Length Problem Attractive with a 0 If R 1/k F a 0, then expect scale invariance R Energy and gap are pure numbers times E FG = 3 5 kf 2 2M EFT power counting says: sum everything with leading vertices Easy in free space = geometric sum of bubbles Many more diagrams in the medium ~ 1/ a 0 k F

98 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Calculations Summing All Leading Diagrams Green s Function Monte Carlo (GMFC) [J. Carlson et al.] Solve many-body S-eqn with variational wf improved by GFMC GFMC applied to real nuclei up to A = 12 with energies to 1% Large a 0 : Convenient potential tuned to a 0

99 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Calculations Summing All Leading Diagrams Green s Function Monte Carlo (GMFC) [J. Carlson et al.] Solve many-body S-eqn with variational wf improved by GFMC GFMC applied to real nuclei up to A = 12 with energies to 1% Large a 0 : Convenient potential tuned to a 0 Lattice calculation for two spins = no fermion sign problem! Path integral evaluated on discretized space-time lattice QCD lattice calculations = best nonperturbative results Large a 0 : Proposed by J.-W. Chen and D. Kaplan In progress by QCD lattice gauge theorist [M. Wingate]

100 Outline Systems EFT Dilute DFT Future Spheres Large a 0 GFMC Results [J. Carlson et al.] Extrapolate to large numbers of fermions 20 Pairing gap ( ) = 0.9 E FG E/E FG 10 odd N E = 0.44 N E FG even N N Energy per particle: E/N = 0.44(1)E FG

101 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Trapped Atoms Low densities and temperatures = only a 0 enters Use Feshbach resonances to tune scattering length a 0 ± Universal behavior? Superfluidity? BCS-BEC crossover?

102 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Is There An Additional Expansion?

103 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Is There An Additional Expansion? No parameters with dimensions. What about geometry?

104 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Is There An Additional Expansion? No parameters with dimensions. What about geometry? J. Steele: Organize according to space-time dimension P/2 q k F 1 P/2 q k F 1 2 D/2

105 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Is There An Additional Expansion? No parameters with dimensions. What about geometry? J. Steele: Organize according to space-time dimension P/2 q k F 1 P/2 q k F 1 2 D/2 Leading 1/D analytic expansion yields: ( E A = E FG k ) Fa 0 /9 a + O(1/D) 0 4 π 2k F a 0 9 E FG [cf. 0.44(1) E FG ]

106 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Is There An Additional Expansion? No parameters with dimensions. What about geometry? J. Steele: Organize according to space-time dimension P/2 q k F 1 P/2 q k F 1 2 D/2 Leading 1/D analytic expansion yields: ( E A = E FG k ) Fa 0 /9 a + O(1/D) 0 4 π 2k F a 0 9 E FG [cf. 0.44(1) E FG ] Subleading??? Pairing gap???

107 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Only an EFT Sampler: Many Omitted Topics Bose many-body systems (BEC s and all that) Three-body physics with large scattering lengths Halo nuclei, pairing, relativistic systems,...

108 Outline Systems EFT Dilute DFT Future Spheres Large a 0 Only an EFT Sampler: Many Omitted Topics Bose many-body systems (BEC s and all that) Three-body physics with large scattering lengths Halo nuclei, pairing, relativistic systems,... Turn now to Density Functional Theory (DFT)

109 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Density Functional Theory Hohenberg-Kohn: There exists an energy functional E v [n]... E v [n] = F HK [n] + d 3 x v(x)n(x) E v [n] minimized for ground state n(x) F HK is universal (same for any external v) = H 2 to DNA!

110 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham DFT V HO = V KS Interacting density with v HO Non-interacting density with v KS

111 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham DFT V HO = V KS Interacting density with v HO Orbitals {φ i (x)} in local potential v KS ([n], x) Non-interacting density with v KS [ 2 /2m + v KS (x)]φ i = ɛ i φ i = n(x) = N φ i (x) 2 i=1 find Kohn-Sham potential v KS (x) from δe v [n]/δn(x) Solve self-consistently

112 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Density Functional Theory Dominant application: inhomogeneous electron gas Interacting point electrons in static potential of atomic nuclei Ab initio calculations of atoms, molecules, crystals, surfaces number of retrieved records per year Hartree Fock Density Functional Theory year

113 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Density Functional Theory Dominant application: inhomogeneous electron gas Interacting point electrons in static potential of atomic nuclei Ab initio calculations of atoms, molecules, crystals, surfaces % deviation from experiment Atomization Energies of Hydrocarbon Molecules Hartree-Fock DFT Local Spin Density Approximation DFT Generalized Gradient Approximation H 2 C 2 C 2 H 2 CH 4 C 2 H 4 C 2 H 6 C 6 H 6 molecule

114 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Quotes From the DFT Literature

115 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Quotes From the DFT Literature A Chemist s Guide to DFT (Koch & Holthausen, 2000) To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations.

116 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Quotes From the DFT Literature A Chemist s Guide to DFT (Koch & Holthausen, 2000) To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. Density Functional Theory (AJP, Argaman & Makov, 2000) It is important to stress that all practical applications of DFT rest on essentially uncontrolled approximations, such as the LDA...

117 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Quotes From the DFT Literature A Chemist s Guide to DFT (Koch & Holthausen, 2000) To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. Density Functional Theory (AJP, Argaman & Makov, 2000) It is important to stress that all practical applications of DFT rest on essentially uncontrolled approximations, such as the LDA... Meta-Generalized Gradient Approximation (Perdew et al., 1999) Some say that there is no systematic way to construct density functional approximations. But there are more or less systematic ways, and the approach taken... here is one of the former.

118 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Thermodynamic Interpretation of DFT Consider a system of spins S i on a lattice with interaction g The partition function has the information about the energy, magnetization of the system: Z = Tr e βg P {i,j} S i S j

119 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Thermodynamic Interpretation of DFT Consider a system of spins S i on a lattice with interaction g The partition function has the information about the energy, magnetization of the system: Z = Tr e βg P {i,j} S i S j The magnetization M is M = S i i = 1 Z Tr [ ( i S i ) e βg P {i,j} S i S j ]

120 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Add A Magnetic Probe Source H The source probes configurations near the ground state Z[H] = e βf [H] = Tr e β(g P {i,j} S i S j +H P i S i )

121 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Add A Magnetic Probe Source H The source probes configurations near the ground state Z[H] = e βf [H] = Tr e β(g P {i,j} S i S j +H P i S i ) Variations of the source yield the magnetization M = S i i H F [H] = H

122 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Add A Magnetic Probe Source H The source probes configurations near the ground state Z[H] = e βf [H] = Tr e β(g P {i,j} S i S j +H P i S i ) Variations of the source yield the magnetization M = S i i H F [H] = H F[H] is the Helmholtz free energy. Set H = 0 (or equal to a real external source) at the end

123 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Legendre Transformation to Effective Action Find H[M] by inverting M = S i = F[H] H H i

124 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Legendre Transformation to Effective Action Find H[M] by inverting M = S i = F[H] H H i Legendre transform to the Gibbs free energy Γ[M] = F [H] + H M

125 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Legendre Transformation to Effective Action Find H[M] by inverting M = S i = F[H] H H i Legendre transform to the Gibbs free energy Γ[M] = F [H] + H M The ground-state magnetization M gs follows by minimizing Γ[M]: H = Γ[M] Γ[M] M M = 0 Mgs

126 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results DFT as Analogous Legendre Transformation In analogy to the spin system, add source J(x) coupled to density operator n(x) ψ (x)ψ(x) to the partition function: Z[J] = e W [J] Tr e β(b H+J bn) D[ψ ]D[ψ] e R [L+J ψ ψ]

127 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results DFT as Analogous Legendre Transformation In analogy to the spin system, add source J(x) coupled to density operator n(x) ψ (x)ψ(x) to the partition function: Z[J] = e W [J] Tr e β(b H+J bn) D[ψ ]D[ψ] e R [L+J ψ ψ] The density n(x) in the presence of J(x) is n(x) n(x) J = δw [J] δj(x)

128 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results DFT as Analogous Legendre Transformation In analogy to the spin system, add source J(x) coupled to density operator n(x) ψ (x)ψ(x) to the partition function: Z[J] = e W [J] Tr e β(b H+J bn) D[ψ ]D[ψ] e R [L+J ψ ψ] The density n(x) in the presence of J(x) is n(x) n(x) J = δw [J] δj(x) Invert to find J[n] and Legendre transform from J to n: Γ[n] = W [J] + J n with J(x) = δγ[n] δγ[n] δn(x) δn(x) = 0 ngs(x) = For static n(x), Γ[n] the DFT energy functional F HK!

129 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results What can EFT do for DFT? Effective action as a path integral = construct W [J] order-by-order in EFT expansion For dilute system, same diagrams as before

130 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results What can EFT do for DFT? Effective action as a path integral = construct W [J] order-by-order in EFT expansion For dilute system, same diagrams as before Inversion method: order-by-order inversion from W [J] to Γ[n] E.g., J(x) = J 0 (x) + J LO (x) + J NLO (x) +... Two conditions on J 0 : n(x) = δw 0[J 0 ] J 0 (x) and J 0 (x) n=ngs = δγ interacting[n] δn(x) n=ngs

131 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results What can EFT do for DFT? Effective action as a path integral = construct W [J] order-by-order in EFT expansion For dilute system, same diagrams as before Inversion method: order-by-order inversion from W [J] to Γ[n] E.g., J(x) = J 0 (x) + J LO (x) + J NLO (x) +... Two conditions on J 0 : n(x) = δw 0[J 0 ] J 0 (x) and J 0 (x) n=ngs = δγ interacting[n] δn(x) Interpretation: J 0 is the external potential that yields for a noninteracting system the exact density This is the Kohn-Sham potential! Two conditions on J 0 = Self-consistency n=ngs

132 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham J 0 According to the EFT Expansion Simplifying with the local density approximation (LDA) J 0 (x) = [

133 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham J 0 According to the EFT Expansion Simplifying with the local density approximation (LDA) LO : J 0 (x) = [ (ν 1) ν 4πa 0 M ρ(x)

134 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham J 0 According to the EFT Expansion Simplifying with the local density approximation (LDA) LO : NLO : + J 0 (x) = [ (ν 1) ν 4πa 0 M ρ(x) c 1 a 2 0 2M [ρ(x)]4/3

135 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham J 0 According to the EFT Expansion Simplifying with the local density approximation (LDA) LO : NLO : + J 0 (x) = [ (ν 1) ν 4πa 0 M ρ(x) c 1 a 2 0 2M [ρ(x)]4/3 NNLO : +

136 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham J 0 According to the EFT Expansion Simplifying with the local density approximation (LDA) LO : NLO : + J 0 (x) = [ (ν 1) ν 4πa 0 M ρ(x) c 1 a 2 0 2M [ρ(x)]4/3 NNLO : + c 2 a 3 0 [ρ(x)]5/3 + +

137 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Kohn-Sham J 0 According to the EFT Expansion Simplifying with the local density approximation (LDA) LO : NLO : + J 0 (x) = [ (ν 1) ν 4πa 0 M ρ(x) c 1 a 2 0 2M [ρ(x)]4/3 NNLO : c 2 a0 3 [ρ(x)]5/3 c 3 a0 2 r 0 [ρ(x)] 5/3 ] c 4 ap[ρ(x)] 3 5/3 +

138 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Dilute Fermi Gas in a Harmonic Trap Iteration procedure:

139 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Dilute Fermi Gas in a Harmonic Trap Iteration procedure: 1. Guess an initial density profile n(r) (e.g., Thomas-Fermi)

140 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Dilute Fermi Gas in a Harmonic Trap Iteration procedure: 1. Guess an initial density profile n(r) (e.g., Thomas-Fermi) 2. Evaluate local single-particle potential v KS (r) v(r) J 0 (r)

141 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Dilute Fermi Gas in a Harmonic Trap Iteration procedure: 1. Guess an initial density profile n(r) (e.g., Thomas-Fermi) 2. Evaluate local single-particle potential v KS (r) v(r) J 0 (r) 3. Solve for lowest N states (including degeneracies): {ψ α, ɛ α } [ 2 2M + v KS(r) ] ψ α (x) = ɛ α ψ α (x)

142 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Dilute Fermi Gas in a Harmonic Trap Iteration procedure: 1. Guess an initial density profile n(r) (e.g., Thomas-Fermi) 2. Evaluate local single-particle potential v KS (r) v(r) J 0 (r) 3. Solve for lowest N states (including degeneracies): {ψ α, ɛ α } [ 2 2M + v KS(r) ] ψ α (x) = ɛ α ψ α (x) 4. Compute a new density n(r) = N α=1 ψ α(x) 2 other observables are functionals of {ψ α, ɛ α }

143 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Dilute Fermi Gas in a Harmonic Trap Iteration procedure: 1. Guess an initial density profile n(r) (e.g., Thomas-Fermi) 2. Evaluate local single-particle potential v KS (r) v(r) J 0 (r) 3. Solve for lowest N states (including degeneracies): {ψ α, ɛ α } [ 2 2M + v KS(r) ] ψ α (x) = ɛ α ψ α (x) 4. Compute a new density n(r) = N α=1 ψ α(x) 2 other observables are functionals of {ψ α, ɛ α } 5. Repeat until changes are small ( self-consistent )

144 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Dilute Fermi Gas in a Harmonic Trap Iteration procedure: 1. Guess an initial density profile n(r) (e.g., Thomas-Fermi) 2. Evaluate local single-particle potential v KS (r) v(r) J 0 (r) 3. Solve for lowest N states (including degeneracies): {ψ α, ɛ α } [ 2 2M + v KS(r) ] ψ α (x) = ɛ α ψ α (x) 4. Compute a new density n(r) = N α=1 ψ α(x) 2 other observables are functionals of {ψ α, ɛ α } 5. Repeat until changes are small ( self-consistent ) Looks like a simple Hartree calculation!

145 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Check Out An Example 4 Dilute Fermi Gas in Harmonic Trap N F =7, A=240, ν=2, a 0 = C 0 = 0 (noninteracting) 3 ρ(r/b) 2 E/A <k F a s > r/b

146 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Check Out An Example 4 3 Dilute Fermi Gas in Harmonic Trap N F =7, A=240, ν=2, a 0 = C 0 = 0 (noninteracting) Kohn-Sham LO ρ(r/b) 2 E/A <k F a s > r/b

147 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Check Out An Example 4 Dilute Fermi Gas in Harmonic Trap N F =7, A=240, ν=2, a 0 = C 0 = 0 (noninteracting) Kohn-Sham LO Kohn-Sham NLO (LDA) ρ(r/b) 2 1 E/A <k F a s > r/b

148 Outline Systems EFT Dilute DFT Future Intro Thermo DFT/EFT Results Check Out An Example 4 Dilute Fermi Gas in Harmonic Trap N F =7, A=240, ν=2, a 0 = C 0 = 0 (noninteracting) Kohn-Sham LO Kohn-Sham NLO (LDA) Kohn-Sham NNLO (LDA) ρ(r/b) 2 1 E/A <k F a s > r/b