Filippo Tramonto. Miniworkshop talk: Quantum Monte Carlo simula9ons of low temperature many- body systems

Transcription

1 Miniworkshop talk: Quantum Monte Carlo simulations of low temperature many-body systems Physics, Astrophysics and Applied Physics Phd school Supervisor: Dott. Davide E. Galli

2 Outline 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

3 The quantum many-body problem Università degli Studi di Milano Filippo Tramonto Quantum Monte Carlo simula9ons of low temperature many- body systems

4 Helium One of the most simple elements: HELIUM T=4.2 K T=2.17 K GAS LIQUID SUPERFLUID BEC in liquid phase NOT SOLID at low pressure SOLID only at high pressure 1."Weak attractive interaction Why? 2."Large zero point motion 3. "Bose statistics Quantum Monte Carlo simulations of 4 He liquid: Removing fictious system of freezes at lower particles exchanges distinguishable pressure then real atoms system BOSE STATISTICS CONTRIBUTE TO PREVENT THE FREEZING

5 continue Helium Feynman`s path integrals quantum particles isomorph mapping Partition function Z = Tr(e βĥ ) classical polymers boltzmannons β=1/t bosons β=1/t Static density response function

6 Glass transition in supercooled liquids Supercooled liquids = metastable liquids at T < T m 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-h 2 and o-d 2 - effects of o-d 2 on crystallization of p-h 2 Large slowing down of crystallization at intermediate concentrations Quantum Monte Carlo simulations Collaboration with a Frankfurt experimental group

7 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

8 Quantum Monte Carlo methods 1 How 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 e τ Ĥ ψ T τ ψ 0 In T>0 QMC methods represents the statistical operator

9 Quantum Monte Carlo methods 2 In both cases we decompose the time evolution operator in many small time steps e τ Ĥ = e δτ Ĥ...e δτ Ĥ M times R 0 R i R M-1 R 1 In spatial coordinate representation R M For T=0 we use the Path Integral ground state method (PIGS): ψ T Ô = ψ 0 Ô ψ 0 ψ T e τ Ĥ Ôe τ Ĥ ψ T R 0 R M R 2M ψ T For T>0 the path integral Monte Carlo method (PIMC): Ô = 1 Z Tr(e β Ĥ Ô) = dr R e β Ĥ Ô R R 0 R M R 1 R i ψ 0 ψ 0 quantum particles R M-1 open polymers ring polymers

10 Dynamics properties 1 Neutron scattering mesurements Partial differential cross section d 2 σ dωdε = N k k b 2 0 S( q,ω ) Dynamic structure factor S( q,ω ) = 1 + e iωt ˆρ 2π N q (t) ˆρ(0) time correlation function Helium elementary excitations spectrum S( q,ω ) = 1 N e β E n 2 ψ Z m ˆρ q ψ n δ[ ω (Em E n )] n,m=0 excitation energies ˆρ q = N i=1 e i q ˆ r i QMC methods study dynamic properties But we cannot obtain directly S( q,ω ) calculate imaginary time correlation functions

11 Dynamic properties 2 Imaginary time correlation function Real time correlation function Â(τ )= e τ Ĥ Â e τ Ĥ F( q,τ ) ˆρ q (τ ) ˆρ(0) it τ Wick rotation C( q,t) ˆρ q (t) ˆρ(0) F( + q,τ ) = dω e τ ω 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

12 Statistical inversion method GIFT Genetic Inversion via Falsification of Theories Scheme of the method: models space genetic algorithm Extraction of the common features: s (ω ) = 1 N N selection of models compatible with QMC data (ω ) i=1 s i average on best models

13 Thank you for the attention