The group The Nuclear Problem QMC EFT Developments
Effective Field Theory and
Quantum Monte Carlo Methods
in Nuclear Physics
Paolo Armani
Università degli studi di Trento
Facoltà di scienze MM. FF. NN.
PhD WorkShop
December 5th 2008
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
Research areas People
Research areas
Strong Correlated Fermion Systems
Nuclear physics
*vacancy effects i liqiud ³He
*Gnd properties of Quantum Wires
*spin orbit effects in 2D electon systems
*vacancies effects in³He liquid
* quantum dots properties
Protein Folding
QMC
* equation of state of neutron and nuclear matter
* Dominant Pathways of Protein Folding
*nuclei binding energy and gnd state properties
* neutron drops
( * electroweak cross sections )
( *LIT transform )
* Interactive Istantos Liquid Model
( * StochasticPerturbation Theory )
( * Effective Field Theoriy )
( Chiral Perturbation Theory)
QCD
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
Research areas People
People
Strong Correlated Fermion Systems
Nuclear physics
* F. Pederiva
* P. Faccioli
* S. Gandolfi
* P. Armani
( * G. Orlandini )
( * W. Leidemann )
* F. Pederiva
* E. Lipparini
* L. Dandrea
* A. Ambrosetti
QMC
Protein Folding
* P. Faccioli
* F. Pederiva
* G. Garberolio
* M.Sega
* E. Autieri
* A. Lonardi
* S. Beccara
* P. Faccioli
* R. Millo
* M. Cristoforetti
QCD
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
Existent Methods
The Nuclear Problem
Nuclear interaction
based on:
phenomenological potentials (Argonne, Urbana).
potentials derived from Effective Field Theories (EFT).
Effective Field Theories.
Computational exact methods
Methods using potentials:
Shell–model (A ≤ 6 o A ≤ 12).
Green Function Monte Carlo (GFMC) (A ≤ 12).
Auxiliary Field Diffusion Monte Carlo (AFDMC) (A . 100).
EFT methods:
lattice simulations (A ≤ 4).
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
Existent Methods
The Nuclear Problem
Nuclear interaction
based on:
phenomenological potentials (Argonne, Urbana).
potentials derived from Effective Field Theories (EFT).
Effective Field Theories.
Computational exact methods
Methods using potentials:
Shell–model (A ≤ 6 o A ≤ 12).
Green Function Monte Carlo (GFMC) (A ≤ 12).
Auxiliary Field Diffusion Monte Carlo (AFDMC) (A . 100).
EFT methods:
lattice simulations (A ≤ 4).
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
QMC
Monte Carlo Methods
Schrödinger equation in imaginary time
X
|Ψ(0)i =
ci |φi i
H|φi i = Ei |φi i
i
−
d
|Ψ(τ )i = (H − E0 )|Ψ(τ )i
dτ
|Ψ(τ )i = e −τ (H−E0 ) |Ψ(0)i
−→
τ →∞
c0 |φ0 i
Diffusion Monte Carlo
Z
hr |Ψ(τ + dτ )i =
hr |e −dτ (H−E0 ) |r 0 ihr 0 |Ψ(τ )idr 0
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
QMC
Monte Carlo Methods
Schrödinger equation in imaginary time
X
|Ψ(0)i =
ci |φi i
H|φi i = Ei |φi i
i
−
d
|Ψ(τ )i = (H − E0 )|Ψ(τ )i
dτ
|Ψ(τ )i = e −τ (H−E0 ) |Ψ(0)i
−→
τ →∞
c0 |φ0 i
Diffusion Monte Carlo
Z
hr |Ψ(τ + dτ )i =
hr |e −dτ (H−E0 ) |r 0 ihr 0 |Ψ(τ )idr 0
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
QMC
Monte Carlo Methods
Schrödinger equation in imaginary time
X
|Ψ(0)i =
ci |φi i
H|φi i = Ei |φi i
i
−
d
|Ψ(τ )i = (H − E0 )|Ψ(τ )i
dτ
|Ψ(τ )i = e −τ (H−E0 ) |Ψ(0)i
−→
τ →∞
c0 |φ0 i
Diffusion Monte Carlo
Z
hr |Ψ(τ + dτ )i =
hr |e −dτ (H−E0 ) |r 0 ihr 0 |Ψ(τ )idr 0
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
QMC
Monte Carlo Methods
Schrödinger equation in imaginary time
X
|Ψ(0)i =
ci |φi i
H|φi i = Ei |φi i
i
−
d
|Ψ(τ )i = (H − E0 )|Ψ(τ )i
dτ
|Ψ(τ )i = e −τ (H−E0 ) |Ψ(0)i
−→
τ →∞
c0 |φ0 i
Diffusion Monte Carlo
Z
hr |Ψ(τ + dτ )i =
hr |e −dτ (H−E0 ) |r 0 ihr 0 |Ψ(τ )idr 0
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
QMC
Monte Carlo Methods
Schrödinger equation in imaginary time
X
|Ψ(0)i =
ci |φi i
H|φi i = Ei |φi i
i
−
d
|Ψ(τ )i = (H − E0 )|Ψ(τ )i
dτ
|Ψ(τ )i = e −τ (H−E0 ) |Ψ(0)i
−→
τ →∞
c0 |φ0 i
Diffusion Monte Carlo
Z
hr |Ψ(τ + dτ )i =
hr |e −dτ (H−E0 ) |r 0 ihr 0 |Ψ(τ )idr 0
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
QMC
Monte Carlo Methods
Schrödinger equation in imaginary time
X
|Ψ(0)i =
ci |φi i
H|φi i = Ei |φi i
i
−
d
|Ψ(τ )i = (H − E0 )|Ψ(τ )i
dτ
|Ψ(τ )i = e −τ (H−E0 ) |Ψ(0)i
−→
τ →∞
c0 |φ0 i
Diffusion Monte Carlo
hr , s|Ψ(τ + dτ )i =
Z X
hr , s|e −dτ (H−E0 ) |r 0 , s 0 ihr 0 , s 0 |Ψ(τ )idr 0
s0
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
Effective Field Theory
Effective Field Theory (EFT)
π fields on lattice
Nucleons
Fundamental Theory (QCD) ⇒ EFT
GFMC accurate method ⇒ determination of effective
parameters
AFDMC ⇒ many–body systems could be studied
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
4 He
State of Art Developments
Binding Energy
Nucleon Eigenenergy
0
-2.82589E+01
-5
-10
Energia di legame [MeV]
-15
-20
-25
-30
-35
-40
-45
-50
0
50000
100000
150000
200000 250000 300000
Passi monte carlo
350000
400000
450000
500000
method tested on trial 4 He system
some enhancement are necessary for an accurate fit of
effective parameters
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
State of Art Developments
Future Developments
As a first step
compute effective Hamiltonian parameters
compute binding energy of few–body nucleon systems
As next step
determine equation of state of neutron and nuclear matter
(astrophysical interest)
improve EFT Hamiltonian to next orders
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
State of Art Developments
Future Developments
As a first step
compute effective Hamiltonian parameters
compute binding energy of few–body nucleon systems
As next step
determine equation of state of neutron and nuclear matter
(astrophysical interest)
improve EFT Hamiltonian to next orders
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth
The group The Nuclear Problem QMC EFT Developments
Effective Field Theory and
Quantum Monte Carlo Methods
in Nuclear Physics
Paolo Armani
Università degli studi di Trento
Facoltà di scienze MM. FF. NN.
PhD WorkShop
December 5th 2008
Paolo Armani
Effective Field Theory andQuantum Monte Carlo Meth