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Università degli Studi di Milano Miniworkshop talk: Quantum Monte Carlo simulations of low temperature many-body systems Filippo Tramonto Physics, Astrophysics and Applied Physics Phd school Università degli Studi di Milano Supervisor: Dott. Davide E. Galli Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Outline Università degli Studi di Milano • Interests in quantum many-body systems • Three examples of topics of research: • The uniqueness of helium • Glass transition in supercooled liquids • Ultracold gases • Quantum Monte Carlo methods • Dynamic properties: the inversion problem Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems The quantum many-body problem Università degli Studi di Milano Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Università degli Studi di Milano Helium One of the most simple elements: HELIUM T=2.17 K T=4.2 K GAS LIQUID NOT SOLID at low pressure BEC in SUPERFLUID liquid phase SOLID only at high pressure 1. "Weak attractive interaction Why? 2."Large zero point motion 3. "Bose statistics • Quantum Monte Carlo simulations of 4He liquid: freezes at lower fictious system of Removing pressure then real distinguishable particles exchanges system atoms BOSE STATISTICS CONTRIBUTE TO PREVENT THE FREEZING Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Università degli Studi di Milano continue Helium Feynman`s path integrals quantum particles isomorph mapping β=1/T classical polymers Partition function Z = Tr(e− β Ĥ ) boltzmannons bosons Static density response function Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems β=1/T Glass transition in supercooled liquids Università degli Studi di Milano • Supercooled liquids = metastable liquids at T < Tm • Rapid freezing transition of supercooled liquids glass-forming • Simple and common model: binary mixtures of interacting particles • Idea: - supercooled liquid mixtures of simple bosonic elements: p-H2 and o-D2 - effects of o-D2 on crystallization of p-H2 • Large slowing down of crystallization at intermediate concentrations • Quantum Monte Carlo simulations • Collaboration with a Frankfurt experimental group Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Università degli Studi di Milano Ultracold gases • Experimental techniques, through lasers and magnetic fields: - magnetic traps - optical lattice - tuning the strength of interatomic interaction - magnetic and electric external fields tunable many-body Hamiltonian weak interaction regime • Collaboration with a Trento group: - hard sphere Bose gas - characteristic variable: gas parameter n |a|3 - elementary excitations spectrum - second sound strong interaction regime Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Quantum Monte Carlo methods 1 • How Università degli Studi di Milano can I study quantum many-body systems? • Exact computational methods? Quantum Monte Carlo methods • The key ingredient of the exact QMC methods: real time evolution operator it → τ imaginary time evolution operator Wick rotation In T=0 QMC methods projects out a trial state to the ground state In T>0 QMC methods represents the statistical operator e−τ Ĥ ψ T ⎯τ⎯⎯ →ψ0 →∞ Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Quantum Monte Carlo methods 2 Università degli Studi di Milano • In both cases we decompose the time evolution operator in many small time steps e − τ Ĥ − δτ Ĥ R0 − δτ Ĥ = e ...e Ri R1 • For T=0 we use the Path Integral ground state method (PIGS): • For T>0 the path integral Monte Carlo method (PIMC): Ô = 1 Tr(e− β Ĥ Ô) = ∫ dR R e− β Ĥ Ô R Z Ô = ψ 0 Ô ψ 0 ≅ ψ T e−τ Ĥ Ôe−τ Ĥ ψ T ψT RM ψ0 open polymers RM-1 In spatial coordinate representation M times R0 RM R2M ψ0 R1 ψT R0 ≡ RM quantum particles Ri RM-1 ring polymers Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Università degli Studi di Milano Dynamics properties 1 Neutron scattering mesurements Helium elementary excitations spectrum Partial differential cross section d 2σ | k | b2 =N S( q,ω ) dΩd ε | k0 | Dynamic structure factor S( q,ω ) = 1 2π N ∫ +∞ −∞ eiω t ρ̂q (t) ρ̂ (0) time correlation function 1 S( q,ω ) = N e β En ∑ Z ψ m ρ̂− q ψ n n,m=0 ∞ 2 δ [ω − (Em − En )] ρ̂q = ∑ e excitation energies • QMC methods N − i q⋅rˆi i=1 study dynamic properties • But we cannot obtain directly S( q,ω ) calculate imaginary time correlation functions Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Università degli Studi di Milano Dynamic properties 2 Imaginary time correlation function F(q, τ ) ≡ ρ̂q (τ ) ρ̂ (0) τ Â(τ )= e  e Ĥ Real time correlation function it → τ Wick rotation C(q,t) ≡ ρ̂q (t) ρ̂ (0) − τ Ĥ +∞ −τ ω F(q, τ ) = ∫ dω e S(q, ω ) −∞ Inverse problem Finite and discrete QMC of F(q, τ ) ? + Statistical uncertainty of data ILL POSED INVERSE PROBLEM Infinite solutions are compatible with the data Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Statistical inversion method GIFT • Scheme Università degli Studi di Milano Genetic Inversion via Falsification of Theories of the method: models space genetic algorithm selection of models compatible with QMC data Extraction of the 1 s ( ω ) = common features: N s (ω ) ∑ N i i=1 average on best models Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems Università degli Studi di Milano Thank you for the attention Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many-‐body systems
