Y TP YUKAWA INSTITUTE FOR THEORETICAL PHYSICS 1/21 Motivation - - PowerPoint PPT Presentation
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
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments
Auxiliary-Field Monte Carlo method for strongly paired fermions
Faisal Etminan
University of Birjand
String and Fields 2017
@ YITP, Kyoto, Japan
1/21
YUKAWA INSTITUTE FOR THEORETICAL PHYSICS
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments
Table of Contents
1 Motivation 2 Introduction 3 Auxiliary Field Quantum Monte Carlo Methods
Formulations
4 Results 5 Summary 6 Acknowledgments
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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.
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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 Infinite scattering length.
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 kF,↑(↓) =
L3 1/3 , EF,↑(↓) = k2
F,↑(↓)
2m .
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.
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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,
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.
ξN,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.
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
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
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.
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
Table of Contents
1 Motivation 2 Introduction 3 Auxiliary Field Quantum Monte Carlo Methods
Formulations
4 Results 5 Summary 6 Acknowledgments
8/21
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
The extended, attractive Hubbard model is
ˆ H = ˆ K + ˆ V = −t
L
c†
iσcjσ + 6t L
c†
iσciσ + U L
ni↑ni↓,
L : number of lattice sites, c†
iσ and cjσ are creation and annihilation operators of an electron of spin σ on the ith
lattice site, t = 1 : nearest-neighbor hopping energy, niσ = c†
iσciσ:density operator,
U : the on-site interaction strength.
9/21
By solving the two-body problem, the scattering length diverges at U = −7.915 t and re = −0.30572.
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
Ground-state wave function |ψ0 obtained asymptotically from any trial wave function |ψT that is not orthogonal to |ψ0 by repeated applications of the ground-state projection |ψ0 ∝ lim
β→0 exp
H − ET
Here ET is guesses of the ground-state energy.
The propagator may be evaluated using a Trotter-Suzuki approximation
K+ ˆ V)n
=
2 ∆τ ˆ
Ke−∆τ ˆ V e− 1
2 ∆τ ˆ
Kn
+ O
,
Where β = ∆τn.
Trotter error arises from the omission of the higher-order terms.
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
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;
exp
∞
dx exp
V =
p (x) e
ˆ O(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
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
The spin form of this decomposition e−∆τUn†
i↑ni↓ = e−∆τU(ni↑+ni↓−1)/2
xi ±1
p (xi) eγxi(ni↑+ni↓−1),
cosh (γ) = exp (∆τ |U| /2), p (xi ) = 1/2 as a discrete probability density function (PDF) with xi = ±1. 12/21
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
By setting ˆ B (x) = e− 1
2 ∆τ ˆ
Ke− ˆ O(x)e− 1
2 ∆τ ˆ
K,
We can rewrite the projection as |ψ0 =
P
ˆ B (xi) |ψT .
i p (xi).
The ground-state energy can be obtained by
A
A
,
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
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
1 An importance function is defined, by importance sampling OT (φk) ≡ φT |φk ,
which estimates the overlap of a Slater determinant |φ with the ground-state wave function (trial wave function).
2 We then iterate
=
B (x)
,
The walkers
are now sampled from a new distribution.
3 They schematically represent the ground-state wave function by
=
Nw
w(n)
i
i
k
, 4
i
OT
k,i
k,i−1
p (x) ,
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments Formulations
Constrained path approximation eliminates the decay of signal-to-noise ratio. Requires that each random walker at each step have a positive overlap with the trial wave function
> 0 Easily implemented by redefining the importance function OT (φk) ≡ max {φT|φk , 0} . Mixed estimator for the ground- state energy, for an ensemble {|φ} Emixed =
, local energy EL for any walker φ, EL [φT, φ] =
H
.
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments
We summarize our calculations of the energy as a function of ρ1/3 where ρ = N/L3, and the particle number is N = 6, 10, 14, 18 and 20 for Lx = Ly = Lz = 4, 5, 6, 7, 8
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.1 0.2 0.3 0.4 0.5 0.6 0.7
ξ ρ1/3 = N1/3/L N=6 N=10 N=14 N=18 N=20
Our calculations show a significant size dependence. The fits are of the form E 0
N↑,N↓
E 0,free
N↑,N↓
= ξ0 + Aρ1/3 + Bρ2/3.
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments
The behavior of energy as a function of kFre for finite effective ranges is also
ξ (re) = ξ0 + SkFre.
0.1 0.2 0.3 0.4 0.5 0.6 0.7
ξ kFre N=6 N=10 N=14 N=18 N=20
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments
ξ(1) is from extrapolation to ρ → 0, ξ(2) is from extrapolation to kF re → 0 As we see by increasing the number of particle, the decreasing of S becomes slower. Their trend are in agreement with results of S = 0.12(0.01) for N = 66 J. Carlson et al
PRA 2011. Our error is a statistical standard errors.
N N = 6 N = 10 N = 14 N = 18 N = 20 ξ(1) 0.47(4) 0.44(3) 0.47(3) 0.41(6) 0.43(5) ξ(2) 0.39(5) 0.39(4) 0.41(4) 0.37(4) 0.37(5) S 0.44(5) 0.33(5) 0.30(4) 0.22(4) 0.21(5)
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments
Summary
We have tryed a lattice technique, constrained-path Monte Carlo method using the extended, attractive Hubbard model to treat strongly paired fermion systems. We have measured the ground-state energy for N spin 1/2- fermions in the unitary limit in a cubic. Our results at N = 6, 10, 14, 18, 20 suggest that for large N the ratio of the ground-state energy to that of a free Fermi gas is 0.43(4). Results are consistent:
with the fixed node Green’s function Monte Carlo ξ = 0.44(1) J. Carlson et al PRL 2003, ξ = 0.42(1) G. E. Astrakharchik et al PRL 2004. Novel lattice approach results 0.412(4) G. Endres et al PRA 2013. more compatible with recent experimental results 0.435(15) J. Joseph, PRL 2007, 0.41(2) L. Luo et al Low Temp. Phys. 2009 and 0.41(1) N. Navon et al Sci 2010. 19/21
Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments
We also find results for
Universal dependence of the ground-state energy upon the effective range, S = 0.21(5) which is close to S = 0.12(0.03) in J. Carlson et al PRA 2011. The method we describe should be useful with modification for:
BCS-BEC transition, Probing many properties of cold Fermi gases, It can also be applied to quantum chemistry and nuclear physics.
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Motivation Introduction Auxiliary Field Quantum Monte Carlo Methods Results Summary Acknowledgments 21/21
We thank Professor Dean Lee and Dr. Hao Shi for useful discussions. And organizers of ”Strings and Fields 2017” workshop.