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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Auxiliary-Field Monte Carlo method for strongly paired fermions Faisal Etminan M. M. Firoozabadi University of Birjand String and Fields

  1. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Auxiliary-Field Monte Carlo method for strongly paired fermions Faisal Etminan M. M. Firoozabadi University of Birjand String and Fields 2017 Aug. 10, 2017 @ YITP, Kyoto, Japan Y TP YUKAWA INSTITUTE FOR THEORETICAL PHYSICS 1/21

  2. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Table of Contents Summary Acknowledgments 1 Motivation 2 Introduction 3 Auxiliary Field Quantum Monte Carlo Methods Formulations 4 Results 5 Summary 6 Acknowledgments 2/21

  3. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments 1 Understanding of strongly correlated quantum matter, 2 Experimental ability to use a Feshbach resonance to to explore physics over many length scales, 3 Prototypical example: Superfluidity in unpolarized cold atomic Fermi gases, 4 The model is of interest in both condensed-matter and nuclear physics. 3/21

  4. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Fermi gas at unitary limit for strongly interacting fermions: Zero-range attractive interaction and � � 1 / 3 k 2 Infinite scattering length. 6 π 2 N ↑ ( ↓ ) F , ↑ ( ↓ ) k F , ↑ ( ↓ ) = , E F , ↑ ( ↓ ) = . L 3 2 m In nuclear physics is when the interparticle spacing is about 5 − 10 fm, relevant to the physics of the inner crust of neutron stars. 4/21

  5. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments ξ Values from experiments: 0 . 51(4) J. Kinast et al Sci. 2005, 0 . 46(5) G. B. Partridge et al Sci. 2006, 0 . 46 +05 − 12 J. Stewart PRL 2006, 0 . 435(15) J. Joseph, PRL 2007, 0 . 41(1) N. Navon et al Sci. 2010, 0 . 41(2) L. Luo et al Low Temp. Phys. 2009. 5/21

  6. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments ξ Values from different methods: 0 . 383(1) Diffusion Monte Carlo (DMC) calculations by Bardeen- Cooper-Schrieffer (BCS) trial wave function, M. M. Forbes et al, PRL 2011 and S. Gandolfi et al PRA 2011 . Between 0 . 07 and 0 . 42 Lattice simulations of two-component fermions, D. Lee and T. Schfer, PRC 2006 , PRB 2007; T. Abe PRC 2009 . ξ = 0 . 31(1) Non ab initio method, symmetric heavy-light ansatz, D. Lee, PRC 2008 . ξ = 0 . 322(2) Density-functional theory method include shell effects, M. M. Forbes et al PRL 2011. ξ N , N = 0 . 412(4) Novel lattice approach for studying large numbers of fermions, M. G. Endres et al PRA 2013 . The most predicted values for ξ , range from 0 . 3 to 0 . 4. 6/21

  7. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments Quantum Monte Carlo (QMC) methods only suitable tool for microscopic calculations of strongly interacting many-body systems. Sign problem arises from the combination of Pauli principle and the use of random sampling. Constrained-path Monte Carlo (CPMC) : For systems by sign problem, constraining the random walks in sampling the space of auxiliary fields, will led to considerable progress. The idea is to constrain the sign or phase of the overlap of the sampled Slater determinants with a trial wave function, S. Zhang PRL 2003 . Applications to a variety of systems have shown that the methods are very accurate, W. Purwanto et al PRB 2009 ; C. C. Chang and S. Zhang, PRL 2010. 7/21

  8. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Table of Contents Summary Acknowledgments 1 Motivation 2 Introduction 3 Auxiliary Field Quantum Monte Carlo Methods Formulations 4 Results 5 Summary 6 Acknowledgments 8/21

  9. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments The extended, attractive Hubbard model is L L L H = ˆ ˆ K + ˆ c † c † V = − t � i σ c j σ + 6 t � i σ c i σ + U � n i ↑ n i ↓ , i σ i � i , j � σ L : number of lattice sites, c † i σ and c j σ are creation and annihilation operators of an electron of spin σ on the ith lattice site, t = 1 : nearest-neighbor hopping energy, n i σ = c † i σ c i σ :density operator, U : the on-site interaction strength. By solving the two-body problem, the scattering length diverges at U = − 7 . 915 t and r e = − 0 . 30572. 9/21

  10. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments Ground-state wave function | ψ 0 � obtained asymptotically from any trial wave function | ψ T � that is not orthogonal to | ψ 0 � by repeated applications of the ground-state projection � � �� ˆ | ψ 0 � ∝ lim β → 0 exp − β H − E T | ψ T � , Here E T is guesses of the ground-state energy. The propagator may be evaluated using a Trotter-Suzuki approximation V ) � n K � n � e − ∆ τ ( ˆ K + ˆ � e − 1 2 ∆ τ ˆ K e − ∆ τ ˆ V e − 1 2 ∆ τ ˆ � ∆ τ 2 � = + O , Where β = ∆ τ n . Trotter error arises from the omission of the higher-order terms. 10/21

  11. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments Contact interactions can be represented by auxiliary fields. The Hubbard-Stratonovich (HS) transformation decoupled two-body propagator into one-body propagators by auxiliary fields J. Hubbard, PRL 1959; J. E. Hirsch, PRB 1983. ∞ � ρ 2 / 2 − x 2 / 2 − x ρ � � � � exp dx exp ∼ −∞ V = e − ∆ τ ˆ ˆ � O ( x ) , p ( x ) e x ˆ O ( x ) is a one-body operator that depends on the auxiliary field x and p ( x ) is a probability density function with the normalization � p ( x ) = 1. 11/21

  12. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments The spin form of this decomposition e − ∆ τ Un † i ↑ n i ↓ = e − ∆ τ U ( n i ↑ + n i ↓ − 1 ) / 2 � p ( x i ) e γ x i ( n i ↑ + n i ↓ − 1 ) , x i ± 1 cosh ( γ ) = exp (∆ τ | U | / 2), p ( x i ) = 1 / 2 as a discrete probability density function (PDF) with x i = ± 1. 12/21

  13. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments By setting B ( x ) = e − 1 2 ∆ τ ˆ K e − ˆ O ( x ) e − 1 2 ∆ τ ˆ ˆ K , We can rewrite the projection as n � � � � � ˆ | ψ 0 � = P X B ( x i ) | ψ T � . � i =1 X � � � � X is ( x 1 , x 2 , ..., x n ), and P = � i p ( x i ). X The ground-state energy can be obtained by � � � � � ˆ ψ 0 � ψ 0 A � � � � ˆ A 0 = , � ψ 0 | ψ 0 � MC methods calculate this many-dimensional integrals (3nL dimensions in the Hubbard model) by sampling the probability density function using the Metropolis algorithm. 13/21

  14. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments 1 An importance function is defined, by importance sampling O T ( φ k ) ≡ � φ T | φ k � , which estimates the overlap of a Slater determinant | φ � with the ground-state wave function (trial wave function). 2 We then iterate � � φ ( n +1) � φ ( n ) � � � � �� P ( x ) ˆ � �� = B ( x ) , x � φ ( n ) � � �� The walkers are now sampled from a new distribution. 3 They schematically represent the ground-state wave function by � � � � φ ( n ) � � ψ ( n ) � N w � � i w ( n ) = � � , i φ ( n ) O T i k P ( x ) = � M 4 � p ( x i ), Probability for sampling auxiliary-field at each lattice site is � i � � φ ( n ) O T k , i � p ( x ) = � � p ( x ) , φ ( n ) O T k , i − 1 � P ( x ) is function of current and future positions in Slater-determinant space. 14/21

  15. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Formulations Results Summary Acknowledgments Constrained path approximation eliminates the decay of signal-to-noise ratio. Requires that each random walker at each step have a positive overlap � φ T | φ ( n ) � with the trial wave function > 0 Easily implemented by redefining the importance function O T ( φ k ) ≡ max {� φ T | φ k � , 0 } . Mixed estimator for the ground- state energy, for an ensemble {| φ �} � k w k E L [ φ T , φ k ] E mixed = , � k w k local energy E L for any walker φ , � � � � � ˆ φ T H � φ � � E L [ φ T , φ ] = . � φ T | φ � 15/21

  16. Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments We summarize our calculations of the energy as a function of ρ 1 / 3 where ρ = N / L 3 , and the particle number is N = 6 , 10 , 14 , 18 and 20 for L x = L y = L z = 4 , 5 , 6 , 7 , 8 0.55 0.5 N=6 N=10 N=14 0.45 N=18 N=20 0.4 0.35 ξ 0.3 0.25 0.2 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ρ 1/3 = N 1/3 /L Our calculations show a significant size dependence. The fits are of the form E 0 = ξ 0 + A ρ 1 / 3 + B ρ 2 / 3 . N ↑ , N ↓ E 0 , free N ↑ , N ↓ 16/21

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