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Lpollet@Physics.Harvard.Edu Part II : Quantum Monte Carlo Algorithms Monte Carlo and cold gases Part II : quantum Monte Carlo algorithms Lode Pollet [email protected] Friday, August 6, 2010 outline so far : sampling according to the exact energy levels for quantum systems : spectrum is not known! • mapping to classical system • diffusion Monte Carlo • Path Integral Monte Carlo • off-diagonal long-range order and superfluidity • worm algorithm (continuous space ) • stochastic series expansions & directed loop • ALPS • sign problem Friday, August 6, 2010 general idea d Consider single particle i Ψ = H Ψ dt| | Solution Ψ(t) =exp( iHt) Ψ(0) | − | Wick rotation τ = it Expand in eigenstates H ψ = E ψ ,k=0, 1,... | k k| k Ψ(τ) = exp( E τ) ψ Ψ(0) ψ | − k k| | k k •projects out ground-state for τ when not orthogonal to initial state •concept of imaginary time important→∞ for both diffusion Monte Carlo and Path Integral Monte Carlo (but they sum over all histories) Friday, August 6, 2010 Diffusion Monte Carlo δτ xk exp( H) xk 1 = | − | − δτ 1 xk xk 1 2 V (xk)+V (xk 1) exp m( − − ) + − ∼ − 2 δτ 2 idea : evolve “walkers” in time, described by its position xj,j =1,...,m at time τ = kδτ j ψ0(x) = lim δ(x x) k − →∞ - propagate walkers according to kinetic part (gaussian/diffusion; Box-Mueller) - weight of walker according to potential energy - cloning (kill/rebirth) : replace each walker j with a number of clones (weight w*) equal to wj 1 j w∗ = w int + r m w j ∗ possible to improve by good trial wave function Friday, August 6, 2010 importance sampling Suppose we know of a good trial wave function ΨT We look for the evolution of Φ = ΨT ψ Often used for fermions (electrons in crystal) construction of nodal surfaces = art Friday, August 6, 2010 Worm algorithm Prokof’ev, Svistunov and Tupitsyn, Sov. Phys. - JETP (1998) Quantum Monte Carlo algorithm to overcome critical slowing down • ergodic (i.e. no topological constraints) • continuous time (for lattice models) • no systematic errors • for continuous space : discrete time • integrated autocorrelation time of order unity idea : worldlines with two open ends, called worms, where the worms sample the space of the Green function sector Friday, August 6, 2010 How to find the properties of Helium-4 2 2 Z = Tr exp( βH) H = ∇ + UAziz − − 2m Z = dR R exp( βH) R = dR ρ(R ,R ; β) 0 0| − | 0 0 0 0 exp( β(T + U)) = exp( βT ) exp( βU) if [T,V ] =0 − − − δ = β/M exp( βH) = exp( δH)M − − exp( δ(T + U)) exp( δT ) exp( δU)+ (δ2) − ≈ − − O Friday, August 6, 2010 2 δT 3N/2 (R0 R1) R e− R = (4πλδ)− exp − 0| | 1 − 4λδ 2 2 λ = /(2m)=6.0596A K 2 3N/2 (Ri 1 Ri) ρ(Ri 1,Ri, δ) = (4πλδ)− exp − − + δU(Ri) − − 4λδ There exist higher order schemes (e.g., 4th order) bosons are indistinguishable particles; we need to sum over all possible permutations Friday, August 6, 2010 Z = dR R exp( βH) R 0 0| − | 0 δT δU M = lim dR0 R0 e− e− R0 M | | →∞ = lim dR0 ...dRM M →∞ diagonal δT δU R0 e− R1 R1 e− R1 ... | |δT | |δU <RM e− R0 R0 e− R0 | | |diagonal| Friday, August 6, 2010 Path integral and worldlines DIAGONAL periodic boundary time conditions R0 | . . large zero Rj point | motion topological constraint R0 | volume Friday, August 6, 2010 off-diagonal long-range order Consider the Green function 1 G(r, τ)= dr ˆb (τ)ˆb† (0) r+r r V V T The density matrix is then n(r)=G(r, τ = 0) − Definition off-diagonal long-range order (ODLRO): n(r)dr →∞ also in 2d Friday, August 6, 2010 superfluid density The superfluid density is the linear response coefficient of the free energy increase of a system with periodic boundary conditions when these periodic boundary conditions are twisted : iφα Ψ(...,rα + Lα,...)=e Ψ(...,rα,...) (ns)α,α V F (φ) F (0) = φαφα − 2m LαLα α,α (n ) F (φ) F (0) = s α,α ( φ) ( φ) − 2m ∇ α ∇ α α,α The system is sensitive to twisted boundary conditions only if ODLRO is present Friday, August 6, 2010 estimator for the superfluid density ω Pollock, Ceperley 1987 R annulus of thickness d much smaller than the d/R 1 radius R rotates with frequency ω equivalent of a system enclosed between two moving ωR = v planes with velocity v, periodic in one dimension d density matrix in system at rest with boundary conditions : 2 (pj mv) ρ =exp( βH ) H = − + V 2m − j unchanged when going to lab frame (distribution of states is the same) : ρv = ρ normal component is part that responds to motion of the boundaries: ρ Tr[Pρ ] N Nmv = v P total momentum ρ Tr[ρv] βFv with e− =Tr[ρv] ρ ∂ n Nmv = ln(Tr[ρ ]) + Nmv ρ β∂v v ∂F = v + Nmv − ∂v Friday, August 6, 2010 superfluid density or for the superfluid fraction : ρ = ρN + ρS ρ ∂F /N S = v 1 2 ρ ∂( 2 mv ) free energy change due to motion of the walls is : ∆F 1 ρ v = mv2 S v + (v4) N 2 ρ O How to change Path Integral formulation? Bloch equation for system with moving walls: ρv(R, R; β) 1 2 = ( i j mv) + V ρv(R, R; β) − ∂β 2m − ∇ − j with boundary conditions: ρv(r1,...,rN ,r1 ,...,rj + L, . , rN ; β)=ρv(r1,...,rN ,r1 ,...,rj ,...,rN ; β) Define: m βFv ρv(R, R; β) = exp i v (rj rj ) ρ˜(R, R; β) Tr[˜ρ]=e− · − Friday, August 6, 2010 estimator for the superfluid density ρ˜ satisfies the usual Bloch equation : ∂ρ˜(R, R; β) 1 2 = ( i j) + V ρ˜(R, R; β) − ∂β 2m − ∇ j but with boundary conditions: include exp(imvL) ρ˜(r1,...,rj,...,rN ,r1 ,...,rj + L, . , rN ; β)= m whenever a path ends exp(i v L)˜ρ(r1,...,rj,...,rN ,r1 ,...,rj ,...,rN ; β) on a periodic image · most easily done by defining the winding number N (r ri)=WL Pi − i=1 it describes how many times the paths of the particles have wound around the periodic cell The free energy is obtained from the winding number distribution: m βFv ρv(R, R; β)dR i vWL e− = = e ρv=0(R, R; β)dR For small velocities: 2 2 d m2v2 W 2 L2 ρs m W L − β∆Fv = ≈ 22 3 ρ 2 ρdβ Pollock, Ceperley 1987 Friday, August 6, 2010 worm algorithm The key is to perform local updates for the Green function: ZW = Z + Z Z = C dr dr G(r , r , δ(j j )) I M I M I − M j ,j IM easiest estimators: estimator for Green function: relative statistics given by: Boninsegni, Prokof’ev, Svistunov, PRE 2005 Friday, August 6, 2010 Path integral and worms REMOVEINSERT time volume Friday, August 6, 2010 detailed balance for insert/remove Metropolis : W (Y )P (Y X) acc = min[1,q] q = → W (X)P (X Y ) → W (X)=1 •choose arbitrary point in space and time •choose length P of new segment •generate trajectory according to the kinetic energy T (gaussian, Box-Muller) 2 δT 3N/2 (R0 R1) R e− R = (4πλδ)− exp − 0| | 1 − 4λδ 1 1 P P (X Y )= T → MV N p cut j=1 Friday, August 6, 2010 extra segment constant to give extra weight to configurations with a worm P P P W (Y )=C Tp Up exp(µP δ) j=1 j=1 For the ‘remove’ update, only do it when the worms are within a time Ncut of each other ( P <= Ncut) P (Y X)=1 → ∆U+µP δ q = CV MNcute Question : explain why the cut/glue update is different Friday, August 6, 2010 Path integral and worms GLUECUT time volume Friday, August 6, 2010 Path integral and worms mT W 2 RECONNECT n = time S dL volume Friday, August 6, 2010 pair product approximation Dilute ultracold gases are adequately described by the s-wave scattering length as a hard-sphere potential with radius R0 has a s-wave scattering length as = R0 for such potentials one uses the Pair Product Approximation: Cao-Berne (CB) have derived an analytical approximation for the pair density matrix: I refer to the literature for more details Friday, August 6, 2010 How powerful is the worm algorithm (PIMC) + - •bosonic permutations • no fermions •superfluid properties ( are ‘easier’ • periodic boundary conditions than insulating ones) required, •second order phase transition • does not help for 1st order transitions •big system sizes • mesoscopic sizes •all static thermodynamic quantities • no dynamics •numerically exact • hard to get good statistics on •first principles, no a priori energy; specific heat is impossible assumptions (for Helium, not so for cold gases) •calculating effective parameters of • QMC process is not a real-time models process • time discretization for continuous space Friday, August 6, 2010 ALPS Friday, August 6, 2010 fermions and sign problem In mapping the quantum to the classical system βH Z = Tr exp− = pi i some of the p i may pi < 0 Troyer and Wiese have shown that the sign e.g. 2 electrons might problem is NP-hard exchange places http://arxiv.org/abs/cond-mat/0408370; Phys.Rev.Lett. 94 (2005) 170201 consequence : exponential scaling Friday, August 6, 2010 Implications evaluation in case of negative weights: with s = Z/Z the average sign given by Z = pi ‘fermionic’ system i Z/Z =exp( βN∆f) − Z = p | i| ‘bosonic’ system i The variance can become exponentially large : The sign problem is basis dependent : e.g., if we know the full spectrum, all weights are positive However, still no solution The situation is reminiscent of NP hard problems (no proof of exponential scaling, but no solution that scales polynomially is known) A solution to the sign-problem is a solution that does not scale exponentially (stronger than positive weights) when the bosonic problem is easy (polynomial) Friday, August 6, 2010 Eulerian Circuit Problem • 7 bridges of Königsberg • is there a roundtrip that crosses each bridge exactly once? • Euler (1735) : it exists if and only if the graph is connected and there are no nodes of odd degree at all • can be evaluated in polynomial time; is in complexity class P Friday, August 6, 2010 Hamiltonian cycle problem • is there a path that crosses each vertex exactly once? • expensive task by evaluating all paths • no solution in polynomial time is known • is NP-complete Friday, August 6, 2010 Complexity classes Turing machine : abstract notion of a CPU Complexity class P : P is defined as the set of all languages which can be decided by a deterministic polynomial- time Turing machine Complexity class NP : • polynomial time on a non-deterministic Turing machine : it can evaluate both branches of an if-statement, but the branches cannot merge again.
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