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Quantum Monte Carlo Methods Jian-Sheng Wang Dept of Computational Science, National University of Singapore 1 Outline • • • • Introduction to Monte Carlo method Diffusion Quantum Monte Carlo Application to Quantum Dots Quantum to Classical --TrotterSuzuki formula 2 Stanislaw Ulam (19091984) S. Ulam is credited as the inventor of Monte Carlo method in 1940s, which solves mathematical problems using statistical sampling. 3 Nicholas Metropolis (1915-1999) The algorithm by Metropolis (and A Rosenbluth, M Rosenbluth, A Teller and E Teller, 1953) has been cited as among the top 10 algorithms having the "greatest influence on the development and practice of science and engineering in the 20th century." 4 Markov Chain Monte Carlo • Generate a sequence of states X0, X1, …, Xn, such that the limiting distribution is given by P(X) • Move X by the transition probability W(X -> X’) • Starting from arbitrary P0(X), we have Pn+1(X) = ∑X’ Pn(X’) W(X’ -> X) • Pn(X) approaches P(X) as n go to ∞ 5 Necessary and sufficient conditions for convergence • Ergodicity [Wn](X - > X’) > 0 For all n > nmax, all X and X’ • Detailed Balance P(X) W(X -> X’) = P(X’) W(X’ -> X) 6 Taking Statistics • After equilibration, we estimate: 1 Q (X ) Q (X ) P(X ) d X N N Q (X ) i 1 i • It is necessary that we take data for each sample or at uniform interval. It is an error to omit samples (condition on things). 7 Metropolis Algorithm (1953) • Metropolis algorithm takes W(X->X’) = T(X->X’) min(1, P(X’)/P(X)) where X ≠ X’, and T is a symmetric stochastic matrix T(X -> X’) = T(X’ -> X) 8 The Statistical Mechanics of Classical Gas/(complex) Fluids/Solids Compute multi-dimensional integral Q Q (x , y , x , y ,...) e 1 1 2 e 2 E ( x 1,y 1,...) kBT E ( x 1, y 1,...) kBT dx1dy1 ...dxN dyN dx1dy1 ...dxN dyN where potential energy N E (x1 ,...) V (dij ) i j 9 Advanced MC Techniques • • • • Cluster algorithms Histogram reweighting Transition matrix MC Extended ensemble methods (multicanonical, replica MC, Wang-Landau method, etc) 10 2. Quantum Monte Carlo Method 11 Variational Principle • For any trial wave-function Ψ, the expectation value of the Hamiltonian operator Ĥ provides an upper bound to the ground state energy E0: E0 ˆ| |H | 12 Quantum Expectation by Monte Carlo ˆ| |H | * ˆ ( X ) dX ( X )H * dX ( X ) ( X ) dX P(X )EL (X ) where 1 ˆ EL (X ) H (X ) (X ) P(X ) | (X ) |2 13 Zero-Variance Principle • The variance of EL(X) approaches zero as Ψ approaches the ground state wavefunction Ψ0. σE2 = <EL2>-<EL>2 ≈ <E02>-<E0>2 = 0 Such property can be used to construct better algorithm (see Assaraf & Caffarel, PRL 83 (1999) 4682). 14 Schrödinger Equation in Imaginary Time i Ĥ , (t ) e t i Ĥt (0) Let = it, the evolution becomes (t ) e Ĥ (0) As -> , only the ground state survive. 15 Diffusion Equation with Drift • The Schrödinger equation in imaginary time becomes a diffusion equation: 1 2 V (X ) ET 2 We have let ħ=1, mass m =1 for N identical particles, X is set of all coordinates (may including spins). We also introduce a energy shift ET. 16 Fixed Node/Fixed Phase Approximation • We introduce a non-negative function f, such that f = Ψ ΦT* ≥ 0 f is interpreted as walker density. f Ψ ΦT 17 Equation for f f 1 2 f vf E L (X ) ET f 2 where 1 1 ˆ v T and E L HT T T 18 Monte Carlo Simulation of the Diffusion Equation • If we have only the first term -½2f, it is a pure random walk. • If we have first and second term, it describes a diffusion with drift velocity v. • The last term represents birthdeath of the walkers. 19 Walker Space X The population of the walkers is proportional to the solution f(X). 20 Diffusion Quantum Monte Carlo Algorithm 1. Initialize a population of walkers {Xi} 2. X’ = X + η ½ + v(X) 3. Duplicate X’ to M copies: M = int( ξ + exp[-((EL(X)+EL(X’))/2-ET) ] ) 4. Compute statistics 5. Adjust ET to make average population constant. 21 Statistics • The diffusion Quantum Monte Carlo provides estimator for Q dX Q (X )f (X ) dX f (X ) 1 N ˆ | 0 | Q T 0 |T N Q ( X ) i 1 where i 1 ˆ Q (X ) Q T T 22 Trial Wave-Function • The common choice for interacting fermions (electrons) is the SlaterJastrow form: 1 (r1 ) 1 (r2 ) 1 (rN ) 2 (r1 ) J (X ) (X ) e N (r1 ) N (rN ) 23 Example: Quantum Dots • 2D electron gas with Coulomb interaction in magnetic field N 1 i j | ri rj | ĤN hˆk k 1 where ri (xi , yi ) and 2 2 ˆz 1 1 B B ˆ 2 2 ˆ h 0 r V (r) Lz gs 2 2 4 2 2 We have used atomic units: ħ=c=m=e=1. 24 Trial Wave-Function • A Slater determinant of Fock-Darwin solution (J(X)=0): 1 r 2 e im |m| |m| 2 n ,m ,s (r , , ) cnm r Ln (r ) e 2 s ( ) 2 where 2 B 2 02 4 • L is Laguerre polynomial • Energy level En,m,s=(n+2|m|+1)h + g B(m+s)B 25 Six-Electrons Groundstate Energy Using parameters for GaAs. The (L,S) values are the total orbital angular momentum L and total Pauli spin S. From J S Wang, A D Güçlü and H Guo, unpublished 26 Addition Spectrum EN+1-EN 27 Comparison of Electron Density N=5 L=6 S=3 Electron charge density from trial wavefunction (Slater determinant of Fock-Darwin solution), exact diagonalisation calculation, and QMC. 28 QD - Disordered Potential Random gaussian peak perturbed quantum dot. From A D Güçlü, J-S Wang, H Guo, PRB 68 (2003) 035304. 29 Quantum System at Finite Temperature • Partition function Z e E ( X ) X |e ˆ H | Tr e • Expectation value Ĥ Q Ĥ Tr Q̂ e Tre Ĥ 30 D Dimensional Quantum System to D+1 Dimensional Classical system | e Ĥ | | (e i , j , ,k |e ˆ H M Ĥ M )M | | i i | e ˆ H M | j k | e ˆ H M | Φi is a complete set of wave-functions 31 Zassenhaus formula e ˆ ˆ B A ˆ A ˆ B e e e e e  1 ˆ ˆ [A,B] 2 e 1 ˆ ˆ ˆ ˆ [A 2B,[A,B]] 6 ... ˆ B • If the operators  and Bˆ are order 1/M, the error of the approximation is of order O(1/M2). 32 Trotter-Suzuki Formula e ˆ ˆ B A lim e M ˆ M A/ e ˆ/ M B M where  and Bˆ are non-commuting operators 33 Quantum Ising Chain in Transverse Field • Hamiltonian z z x ˆ ˆ H ˆ H J ˆi ˆi 1 ˆi V 0 i i • where 0 1 0 i 1 0 y z ˆ , ˆ , ˆ 1 0 i 0 0 1 x Pauli matrices at different sites commute. 34 Complete Set of States • We choose the eigenstates of operator σz: ˆ | 1 2 z i N i | 1 2 N • Insert the complete set in the products: e ˆ H 0 M e Vˆ M e ˆ H 0 M e Vˆ M 35 A Typical Term i ,k | e a ˆix | i ,k 1 Trotter or β direction 1 sinh(2a ) e 2 1 logtanh(a ) 2 i ,k i ,k 1 (i,k) Space direction 36 Classical Partition Function Z Tr e Ĥ Z 0 { i ,k e K1 i ,k i 1,k K2 i ,k i ,k 1 i ,k i ,k } where K1 J M , K2 logcoth M Note that K1 1/M, K2 log M for large M. 37 Summary • Briefly introduced (classical) MC method • Quantum MC (variational, diffusional, and Trotter-Suzuki) • Application to quantum dot models 38