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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  Â 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