Ground-state properties, excitations, and response of the 2D Fermi gas

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1 Ground-state properties, excitations, and response of the 2D Fermi gas Introduction: 2D FG and a condensed matter perspective Auxiliary-field quantum Monte Carlo calculations - exact* here Results on spin-balanced system: ground-state properties pairing gaps, spectral information, response Comment on (e.g. spin-imbalance) cases with sign problem Results on optical lattices with SOC Summary Shiwei Zhang Flatiron Institute and College of William & Mary Outline

2 Collaborators: Mingpu Qin -> Hao Shi -> Flatiron Peter Rosenberg -> FSU MagLab Ettore Vitali -> Cal State Fresno Shanghai Jiaotong U Support: NSF Simons Foundation DOE -- SciDAC; ThChem Simone Chiesa (Citi group)

3 Ultracold atomic Fermi gas H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V In 3-dimensitons, can tune V to modify 2-body s-wave scattering length: V depth large unitarity small 2-body scattering length >0 infinity <0 physics molecule unbound Image from D. Jin group

4 Ultracold atomic Fermi gas - 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V

5 Ultracold atomic Fermi gas - 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V In 2-dimensions, always bound state -- no unitarity Pair size vs. d: Image from D. Jin group

6 Ultracold atomic Fermi gas - 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V In 2-dimensions, always bound state -- no unitarity Pair size vs. d: Image from D. Jin group Expt realized (recall tremendous precision in 3D) -- 2D important in condensed matter: cuprates,...

7 Ultracold atomic Fermi gas - 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V In 2-dimensions, always bound state -- no unitarity Pair size vs. d: Image from D. Jin group Metric : x ln(k F a) basically, scattering length/d

8 Ultracold atomic Fermi gas -- 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V Spin-orbit coupling has been realized (PRL 109, ; PRL 109, )

9 Ultracold atomic Fermi gas -- 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V Spin-orbit coupling has been realized (PRL 109, ; PRL 109, ) Connection to topological materials, QHE, interplay bt. SOC & pairing

10 Ultracold atomic Fermi gas -- 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V Spin-orbit coupling has been realized (PRL 109, ; PRL 109, ) Connection to topological materials, QHE, interplay bt. SOC & pairing Clean, tunable experiments

11 Ultracold atomic Fermi gas -- 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V Spin-orbit coupling has been realized (PRL 109, ; PRL 109, ) Connection to topological materials, QHE, interplay bt. SOC & pairing Clean, tunable experiments Theoretical work mostly at mean-field level

12 Ultracold atomic Fermi gas -- 2D H = 2 2m N/2 i N/2 2 i + j 2 j + i,j V (r ij ) inter-particle spacing d >> range of V Spin-orbit coupling has been realized (PRL 109, ; PRL 109, ) Hamiltonian incl. synthetic Rashba SOC in dilute gas (and optic. latt.!) t H = X k k 2 c k c k + U X i n i" n i# + X k (k y ik x )c k# c k" + h.c.

13 Ground-state properties, excitations, and response of the 2D Fermi gas Auxiliary-field quantum Monte Carlo calculations - exact* here Results on spin-balanced system: ground-state properties pairing gaps, spectral information, response Comment on (e.g. spin-imbalance) cases with sign problem Results on optical lattices with SOC Summary Shiwei Zhang Flatiron Institute and College of William & Mary Outline Introduction: 2D FG and a condensed matter perspective

14 An auxiliary-field perspective To obtain ground state, use projection in imaginary-time: T He H e H e H T e H e H e H E.g. Ĥ = ˆp 2 i 2m + ˆV R = r 1, r 2,, r M e Ĥ = p( )B( )d 0 = R 0(R)R B( )R R ==> diffusion Monte Carlo (GFMC) (and path-integral MC) - initialize { } from R (R) - random walks with { R } - distribution -> 0(R)

15 An auxiliary-field perspective To obtain ground state, use projection in imaginary-time: T He Auxiliary-field QMC: H e H e H T e H e H e H e Ĥ = p( )B( )d B( )

16 An auxiliary-field perspective To obtain ground state, use projection in imaginary-time: T He Auxiliary-field QMC: H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) Slater determinant

17 An auxiliary-field perspective To obtain ground state, use projection in imaginary-time: T He H e H e H T e H e H e H Auxiliary-field QMC: e Ĥ = p( )B( )d B( ) use basis 0 = 0( ) Slater determinant

18 An auxiliary-field perspective To obtain ground state, use projection in imaginary-time: T He H e H e H T e H e H e H Auxiliary-field QMC: e Ĥ = p( )B( )d B( ) use basis 0 = 0( ) Slater determinant Like a TDDFT propagator

19 Many-body propagator --> many 1-body prop s Consider the propagator Ĥ = N i,j T ij c j c j + N i,j,k,l e Ĥ V ijkl c i c j c kc l. = e Ĥ 1 e Ĥ 2 + O( 2 ) e.g. Ĥ 2 = ˆv 2 V ijkl. = J max =1 L ij L kl ˆv = i,j L ij c i c j Hubbard-Stratonovich transform.: e v2 = 1 e 2 e 2 v d e Ĥ = p( )B( )d B( )

20 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( )

21 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations

22 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1. ψ Ν MnO

23 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1. ψ Ν MnO

24 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1 1-body op. ψ Ν MnO

25 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1 1-body op. ψ Ν MnO AF variable -- sample

26 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1. ψ Ν MnO

27 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1. ψ Ν MnO N is size of basis

28 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv NxN matrix ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1. ψ Ν N is size of basis MnO

29 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv ψ 1.. ψ Ν ψ 2. ψ 2 ψ 1. ψ Ν MnO

30 Path integral over AF s by MC Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) A step advances the SD by rotations e ˆv ψ 1 ψ 2 ψ 1.. ψ Ν ψ 1 ψ 1 ψ 2 ψ 2 ψ 2. ->..... ψ Ν ψ Ν ψ Ν MnO

31 The sign problem Imaginary-time projection --> random walk: T He The sign problem H e H e H T e H e H e H e Ĥ = p( )B( )d B( ) * happens whenever B...B exists * symmetry can prevent this - sign-problem-free cases: - attractive interaction, spin-balanced ( det[ ] )^2 - repulsive half-filling bipartite (particle-hole) - attractive, spin-balanced, w/ spin-orbit coupling - a more general formulation w/ Majorana fermions PRL 116, (2016)

32 Equal-time correlations and observables Imaginary-time projection --> random walk: T He H e H e H T e H e H e H e Ĥ = p(x)b(x)dx B(X)

33 Equal-time correlations and observables Imaginary-time projection --> random walk: T He Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx B(X)

34 Equal-time correlations and observables Imaginary-time projection --> random walk: T He Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx B(X)

35 Equal-time correlations and observables Imaginary-time projection --> random walk: T He Ô H e H e H T e H e H e H Ô h L Ri h T i T ˆB(X 2M ) ˆB(X 1 ) e Ĥ = p(x)b(x)dx B(X) m walk in the manifold of N particles Slater Determinants, that can

36 Equal-time correlations and observables Imaginary-time projection --> random walk: T He Ô H e H e H T e H e H e H Ô h L Ri h T i T ˆB(X 2M ) ˆB(X 1 ) e Ĥ = p(x)b(x)dx B(X) Note m walk in the manifold of N particles Slater Determinants, that can Apply force bias - importance sampling - much more efficient than standard algorithm

37 Equal-time correlations and observables Imaginary-time projection --> random walk: T He Ô H e H e H T e H e H e H Ô h L Ri h T i T ˆB(X 2M ) ˆB(X 1 ) e Ĥ = p(x)b(x)dx B(X) Note m walk in the manifold of N particles Slater Determinants, that can Apply force bias - importance sampling - much more efficient than standard algorithm Infinite variance for sign-problem-free cases (Hao Shi talk)

38 Equal-time correlations and observables Imaginary-time projection --> random walk: T He Ô H e H e H T e H e H e H Ô h L Ri h T i T ˆB(X 2M ) ˆB(X 1 ) e Ĥ = p(x)b(x)dx B(X) Note m walk in the manifold of N particles Slater Determinants, that can Apply force bias - importance sampling - much more efficient than standard algorithm Infinite variance for sign-problem-free cases (Hao Shi talk) If sign problem, apply constraint in forward direction. In that case back-propagation is required

39 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Vitali et al, PRB 16

40 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Imaginary-time GFs: G(k, )= 0 c k e Ĥ c k 0 Vitali et al, PRB 16

41 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Imaginary-time GFs: The random walk is no longer confined in the N particles Slater determinants manifold G(k, )= T i 0 c k e Ĥ c k 0 h T h L ĉ G N+1 particles ĉ G Ri N particles Standard algorithm: commutators, (basis size)^3 Vitali et al, PRB 16

42 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Imaginary-time GFs: The random walk is no longer confined in the N particles Slater determinants manifold G(k, )= T i 0 c k e Ĥ c k 0 h T h L ĉ G N+1 particles ĉ G Ri N particles Standard algorithm: commutators, (basis size)^3 New: linear*n^2 (important in FG dilute) Vitali et al, PRB 16

43 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Imaginary-time density/spin corr: 0ˆn i, e Ĥ ˆn j, 0 Vitali et al, PRB 16

44 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Imaginary-time density/spin corr: 0ˆn i, e Ĥ ˆn j, 0 ˆn i, = eˆn i, 1 e 1 Vitali et al, PRB 16

45 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Imaginary-time density/spin corr: h T The random walk is no longer confined in the N particles Slater determinants manifold h L ĉ G N+1 particles ˆn j, ĉ G T i Ri N particles 0ˆn i, e Ĥ ˆn j, 0 ˆn i, = eˆn i, 1 e 1 Vitali et al, PRB 16

46 Imaginary-time correlations Imaginary-time projection --> random walk: T He Equal-time GFs: Ô H e H e H T e H e H e H e Ĥ = p(x)b(x)dx h T ˆB(X 2M ) h L Ô Ri ˆB(X 1 ) T i B(X) m walk in the manifold of N particles Slater Determinants, that can Imaginary-time density/spin corr: h T The random walk is no longer confined in the N particles Slater determinants manifold h L ĉ G N+1 particles ˆn j, ĉ G T i Ri N particles 0ˆn i, e Ĥ ˆn j, 0 ˆn i, = eˆn i, 1 e 1 Standard algorithm: commutators, (basis size)^3 New: linear*n^2 (important in FG dilute) Vitali et al, PRB 16

47 Ground-state properties, excitations, and response of the 2D Fermi gas Results on spin-balanced system: ground-state properties pairing gaps, spectral information, response Comment on (e.g. spin-imbalance) cases with sign problem Results on optical lattices with SOC Summary Shiwei Zhang Flatiron Institute and College of William & Mary Outline Introduction: 2D FG and a condensed matter perspective Auxiliary-field quantum Monte Carlo calculations - exact* here

48 Ultracold atomic Fermi gas -- 2D Exact EOS obtained, fit provided - BCS trial wf; Variance control; sampling tricks; (E-E AFQMC ) / E FG (a) DMC (b) DMC AFQMC (E-E BCS ) / E FG (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL ln ( a k F ) Shi, Chiesa, SZ, PRA 15

49 Ultracold atomic Fermi gas -- 2D Exact EOS obtained, fit provided - BCS trial wf; Variance control; sampling tricks; - careful extrap to TDL (E-E AFQMC ) / E FG (a) DMC (b) DMC AFQMC (E-E BCS ) / E FG (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL ln ( a k F ) Shi, Chiesa, SZ, PRA 15

50 Ultracold atomic Fermi gas -- 2D Exact EOS obtained, fit provided - BCS trial wf; Variance control; sampling tricks; - careful extrap to TDL (E-E AFQMC ) / E FG (a) DMC (b) DMC AFQMC (E-E BCS ) / E FG (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL ln ( a k F ) Shi, Chiesa, SZ, PRA 15

51 Ultracold atomic Fermi gas -- 2D Exact EOS obtained, fit provided New expt and comparison - BCS trial wf; Variance control; sampling tricks; (E-E AFQMC ) / E FG (a) careful extrap to TDL 0.00 Boettcher et al, PRL DMC (b) DMC AFQMC (E-E BCS ) / E FG (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL ln ( a k F ) Shi, Chiesa, SZ, PRA 15

52 Ultracold atomic Fermi gas -- 2D Exact EOS obtained, fit provided New expt and comparison - BCS trial wf; Variance control; sampling tricks; (E-E AFQMC ) / E FG (a) careful extrap to TDL 0.00 Boettcher et al, PRL DMC Precision many-body (b) DMC AFQMC comp: benchmark thy; guide expt (E-E BCS ) / E FG (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL ln ( a k F ) Shi, Chiesa, SZ, PRA 15

53 Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) k k 1.00 Condensation Fraction Pairing Correlation r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA ln(a k F )

54 Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction Pairing Correlation r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA ln(a k F )

55 Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction Pairing Correlation r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA ln(a k F )

56 Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction Pairing Correlation r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA ln(a k F )

$Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space$ 008 0.004 0.003 0.002 0.001 0.0004 0.0002 Pairing Correlation 0.20 0.

57 Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction Pairing Correlation r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA ln(a k F )

58 Ground-state properties, excitations, and response of the 2D Fermi gas pairing gaps, spectral information, response Comment on (e.g. spin-imbalance) cases with sign problem Results on optical lattices with SOC Summary Shiwei Zhang Flatiron Institute and College of William & Mary Outline Introduction: 2D FG and a condensed matter perspective Auxiliary-field quantum Monte Carlo calculations - exact* here Results on spin-balanced system: ground-state properties

59 Pairing gap Similarly for holes quasi-particle dispersion

60 Pairing gap Similarly for holes quasi-particle dispersion ω/ε F 0 2 BEC ω/ε F BCS k/k F k/k F 0

Pairing gap Similarly for holes quasi-particle dispersion 4 2 5 ω/ε F 0 2

61 Pairing gap Similarly for holes quasi-particle dispersion ω/ε F 0 2 BEC ω/ε F BCS k/k F k/k F 0

62 Dynamical structure factors can be measured by scattering expt Response functions

63 Response functions Dynamical structure factors can be measured by scattering expt 0ˆn i, e Ĥ ˆn j, 0

64 Response functions Dynamical structure factors can be measured by scattering expt 0ˆn i, e Ĥ ˆn j, 0

65 Response functions Dynamical structure factors can be measured by scattering expt Analytic cont. * GIFT (Vitali 10) 0ˆn i, e Ĥ ˆn j, 0

66 Response functions Dynamical structure factors can be measured by scattering expt Analytic cont. * GIFT (Vitali 10) 0ˆn i, e Ĥ ˆn j, 0 at 4*k_F main: density inset: spin

67 Response functions Dynamical structure factors can be measured by scattering expt Analytic cont. * GIFT (Vitali 10) 0ˆn i, e Ĥ ˆn j, 0 at 4*k_F main: density inset: spin 2 k 2 R = : atom recoil 2m

68 Response functions Dynamical structure factors can be measured by scattering expt Analytic cont. * GIFT (Vitali 10) 0ˆn i, e Ĥ ˆn j, 0 at 4*k_F main: density inset: spin 2 k 2 R = : atom recoil 2m BCS

69 Response functions Dynamical structure factors can be measured by scattering expt Analytic cont. * GIFT (Vitali 10) 0ˆn i, e Ĥ ˆn j, 0 at 4*k_F main: density inset: spin 2 k 2 R = : atom recoil 2m

70 Response functions Dynamical structure factors can be measured by scattering expt Analytic cont. * GIFT (Vitali 10) 0ˆn i, e Ĥ ˆn j, 0 at 4*k_F main: density inset: spin 2 k 2 R = : atom recoil 2m BEC

71 Ground-state properties, excitations, and response of the 2D Fermi gas Comment on (e.g. spin-imbalance) cases with sign problem Results on optical lattices with SOC Summary Shiwei Zhang Flatiron Institute and College of William & Mary Outline Introduction: 2D FG and a condensed matter perspective Auxiliary-field quantum Monte Carlo calculations - exact* here Results on spin-balanced system: ground-state properties pairing gaps, spectral information, response

72 Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) Equal-time quantities have been extensively benchmarked high accuracy

73 Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) 1 Green function ED CPQMC <C R (τ) C + R > τ Equal-time quantities have been extensively benchmarked high accuracy

74 Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) Green function <C R (τ) C + R > ED CPQMC 4x4 5u5d U/t=4 (relatively easy, closed shell): essentially exact τ Equal-time quantities have been extensively benchmarked high accuracy

75 Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) <Nup R (τ) Nup R > spin-up density correlation ED CPQMC 4x4 5u5d U/t=4 (relatively easy, closed shell): essentially exact τ Equal-time quantities have been extensively benchmarked high accuracy

76 Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) Green function <c up (R,τ) c + up (R,0)> ED QMC 4x4 4u4d U/t=4 ( typical molecule or solid level of difficulty): good accuracy τ t Equal-time quantities have been extensively benchmarked high accuracy

77 Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) density correlation function <Nup R (τ) Nup R > ED CPQMC 4x4 4u4d U/t=4 ( typical molecule or solid level of difficulty): good accuracy τ Equal-time quantities have been extensively benchmarked high accuracy

78 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y k x Rosenberg Shi, SZ, PRL 17

79 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y k x Attractive interaction, U<0 Rosenberg Shi, SZ, PRL 17

80 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y k x Attractive interaction, U<0 Supersolid phase: Rosenberg Shi, SZ, PRL 17

81 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y k x Attractive interaction, U<0 Supersolid phase: - charge density wave Rosenberg Shi, SZ, PRL 17

82 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y k x Attractive interaction, U<0 Supersolid phase: - charge density wave - superfluid order Rosenberg Shi, SZ, PRL 17

83 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y k x Attractive interaction, U<0 t s Rosenberg, Shi, SZ, PRL 17

84 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y - n(k) - spin k x Attractive interaction, U<0 t s Rosenberg, Shi, SZ, PRL 17

85 Rashbba SOC in 2D optical lattice Hubbard dispersion, half-filling k y - n(k) - spin - pairing wfs k x Attractive interaction, U<0 t s Rosenberg, Shi, SZ, PRL 17

Rashbba SOC in 2D optical lattice Edge currents: - Open BC here - equal magnitude in opposite dir. for spins x 8 7 6 5 4 3 2 1 0.

86 Rashbba SOC in 2D optical lattice Edge currents: - Open BC here - equal magnitude in opposite dir. for spins x j nm = it(c m c n c n c m ) y Rosenberg, Shi, SZ, PRL 17

87 Summary 2D Fermi gas Clean & tunable; exciting new possibilities, especially useful to CM We use auxiliary-field QMC to carry out exact simulations in large systems (>120 particles, > 3000 sites, large beta) Metropolis with force bias to accelerate sampling and improve acceptance ratio (Note standard deteminantal MC has infinite variance) Method to compute gaps and imaginary-time correlations 2D: equation of state; n(k); pairing wf; cond frac... Pairing gaps, spectral info, and response (analytic cont) Rashba spin-orbit coupling in 2D optical lattice: super solid phase, singlet vs triplet pairing, topological signatures

88 Example: gaps from imaginary-time GFs Example - charge gap in the Hubbard model at half-filling G(τ) e log(σ/g) U/t=0.5 momentum space real lattice exponential fit exponential fit Gap is slope at large tau Can work with realspace or k-space GF k-space (k near FS) works better at low U 1e τ [1/t] τ [1/t] G(k, )= 0 c k e Ĥ c k 0 Vitali et al, PRB, 16

Auxiliary-field quantum Monte Carlo calculations of excited states and strongly correlated systems

Auxiliary-field quantum Monte Carlo calculations of excited states and strongly correlated systems Formally simple -- a framework for going beyond DFT? Random walks in non-orthogonal Slater determinant