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

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

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)

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

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

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

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,...

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

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, 095301; PRL 109, 095302)

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, 095301; PRL 109, 095302) Connection to topological materials, QHE, interplay bt. SOC & pairing

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, 095301; PRL 109, 095302) Connection to topological materials, QHE, interplay bt. SOC & pairing Clean, tunable experiments

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, 095301; PRL 109, 095302) Connection to topological materials, QHE, interplay bt. SOC & pairing Clean, tunable experiments Theoretical work mostly at mean-field level

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, 095301; PRL 109, 095302) 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.

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

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)

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( )

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

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

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

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( )

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( )

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

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

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

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

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

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

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

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

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

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

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, 250601 (2016)

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)

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)

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)

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

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

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)

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

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

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

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

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

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

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

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

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

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

Ultracold atomic Fermi gas -- 2D Exact EOS obtained, fit provided - BCS trial wf; Variance control; sampling tricks; (E-E AFQMC ) / E FG (a) 0.06 0.04 0.02 0.00 10 26 42 58 74 98 122 DMC -0.02 (b) -0.20 DMC AFQMC (E-E BCS ) / E FG -0.40-0.60-0.2-0.4 (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL 11-0.80-1.00-0.8-1.0 0.0 1.0 2.0 3.0-1 0 1 2 3 4 5 6 ln ( a k F ) Shi, Chiesa, SZ, PRA 15

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) 0.06 0.04 0.02 0.00 10 26 42 58 74 98 122 DMC -0.02 (b) -0.20 DMC AFQMC (E-E BCS ) / E FG -0.40-0.60-0.2-0.4 (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL 11-0.80-1.00-0.8-1.0 0.0 1.0 2.0 3.0-1 0 1 2 3 4 5 6 ln ( a k F ) Shi, Chiesa, SZ, PRA 15

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) 0.06 0.04 0.02 0.00 10 26 42 58 74 98 122 DMC -0.02 (b) -0.20 DMC AFQMC (E-E BCS ) / E FG -0.40-0.60-0.2-0.4 (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL 11-0.80-1.00-0.8-1.0 0.0 1.0 2.0 3.0-1 0 1 2 3 4 5 6 ln ( a k F ) Shi, Chiesa, SZ, PRA 15

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) 0.06 0.04 0.02 - careful extrap to TDL 0.00 Boettcher et al, PRL 16-0.02 10 26 42 58 74 98 122 DMC (b) -0.20 DMC AFQMC (E-E BCS ) / E FG -0.40-0.60-0.2-0.4 (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL 11-0.80-1.00-0.8-1.0 0.0 1.0 2.0 3.0-1 0 1 2 3 4 5 6 ln ( a k F ) Shi, Chiesa, SZ, PRA 15

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) 0.06 0.04 0.02 - careful extrap to TDL 0.00 Boettcher et al, PRL 16-0.02 10 26 42 58 74 98 122 DMC Precision many-body (b) -0.20 DMC AFQMC comp: benchmark thy; guide expt (E-E BCS ) / E FG -0.40-0.60-0.2-0.4 (P-P BCS )/P FG -0.6 DMC: prev. best (var) Bertaina & Giorgini, PRL 11-0.80-1.00-0.8-1.0 0.0 1.0 2.0 3.0-1 0 1 2 3 4 5 6 ln ( a k F ) Shi, Chiesa, SZ, PRA 15

Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) k k 1.00 Condensation Fraction 0.80 0.60 0.40 0.011 0.010 0.009 0.008 0.004 0.003 0.002 0.001 0.0004 0.0002 Pairing Correlation 0.20 0.0000 0 5 10 15 20 25 30 r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA 15-1 0 1 2 3 4 5 6 7 ln(a k F )

Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction 0.80 0.60 0.40 0.011 0.010 0.009 0.008 0.004 0.003 0.002 0.001 0.0004 0.0002 Pairing Correlation 0.20 0.0000 0 5 10 15 20 25 30 r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA 15-1 0 1 2 3 4 5 6 7 ln(a k F )

Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction 0.80 0.60 0.40 0.011 0.010 0.009 0.008 0.004 0.003 0.002 0.001 0.0004 0.0002 Pairing Correlation 0.20 0.0000 0 5 10 15 20 25 30 r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA 15-1 0 1 2 3 4 5 6 7 ln(a k F )

Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction 0.80 0.60 0.40 0.011 0.010 0.009 0.008 0.004 0.003 0.002 0.001 0.0004 0.0002 Pairing Correlation 0.20 0.0000 0 5 10 15 20 25 30 r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA 15-1 0 1 2 3 4 5 6 7 ln(a k F )

Ultracold atomic Fermi gas -- 2D 2D -- condensate fraction (diagonalize ) real-space pair wave function k k 1.00 Condensation Fraction 0.80 0.60 0.40 0.011 0.010 0.009 0.008 0.004 0.003 0.002 0.001 0.0004 0.0002 Pairing Correlation 0.20 0.0000 0 5 10 15 20 25 30 r QMC BCS Bogoliubov 0.00 Shi, Chiesa, SZ, PRA 15-1 0 1 2 3 4 5 6 7 ln(a k F )

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

Pairing gap Similarly for holes quasi-particle dispersion

Pairing gap Similarly for holes quasi-particle dispersion 4 2 5 ω/ε F 0 2 BEC ω/ε F 4 4 2 0 2 0 10 5 BCS 4 0 0.5 1 1.5 2 k/k F 0 0.5 1 1.5 2 k/k F 0

Pairing gap Similarly for holes quasi-particle dispersion 4 2 5 ω/ε F 0 2 BEC ω/ε F 4 4 2 0 2 0 10 5 BCS 4 0 0.5 1 1.5 2 k/k F 0 0.5 1 1.5 2 k/k F 0

Dynamical structure factors can be measured by scattering expt Response functions

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

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

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

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

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

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

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

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

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

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

Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) 1 Green function ED CPQMC <C R (τ) C + R > 0.1 0.01 0 0.5 1 1.5 2 2.5 τ Equal-time quantities have been extensively benchmarked high accuracy

Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) Green function <C R (τ) C + R > 1 0.1 ED CPQMC 4x4 5u5d U/t=4 (relatively easy, closed shell): essentially exact 0.01 0 0.5 1 1.5 2 2.5 τ Equal-time quantities have been extensively benchmarked high accuracy

Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) <Nup R (τ) Nup R > 0.35 0.3 0.25 0.2 0.15 spin-up density correlation ED CPQMC 4x4 5u5d U/t=4 (relatively easy, closed shell): essentially exact 0.1 0.05 0 0.5 1 1.5 2 2.5 τ Equal-time quantities have been extensively benchmarked high accuracy

Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) Green function <c up (R,τ) c + up (R,0)> 0.8 0.7 0.6 0.5 0.4 0.3 ED QMC 4x4 4u4d U/t=4 ( typical molecule or solid level of difficulty): good accuracy 0.2 0 0.5 1 1.5 2 τ t Equal-time quantities have been extensively benchmarked high accuracy

Imaginary-time correlation under constraint Test in repulsive Hubbard, sign problem (preliminary) density correlation function <Nup R (τ) Nup R > 0.35 0.3 0.25 0.2 0.15 ED CPQMC 4x4 4u4d U/t=4 ( typical molecule or solid level of difficulty): good accuracy 0.1 0.05 0 0.5 1 1.5 2 2.5 τ Equal-time quantities have been extensively benchmarked high accuracy

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

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

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

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

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

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

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

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 0.10 j nm = it(c m c n c n c m ) 2 4 6 8 10 12 14 16 y 18 20 22 24 26 28 30 32 0.105 0.090 0.075 0.060 0.045 0.030 0.015 0.000 0.05 0.00-0.05 5 10 15 20 25 30 Rosenberg, Shi, SZ, PRL 17

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

Example: gaps from imaginary-time GFs Example - charge gap in the Hubbard model at half-filling G(τ) 1 0.1 0.01 0.001 0.0001 1e 05 2 2.5 log(σ/g) 3 3.5 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 06 0 10 20 30 40 τ [1/t] 0 5 10 15 20 25 30 35 40 τ [1/t] G(k, )= 0 c k e Ĥ c k 0 Vitali et al, PRB, 16