unlocking the mysteries of plutonium via hpc
TRANSCRIPT
Unlocking the mysteries ofPlutonium via HPC
C.A. Marianetti LLNL-PRES-406717 HPC Conference - 10/22/2008
Collaborators and Funding
• Kristjan Haule - Rutgers University
• Gabriel Kotliar - Rutgers University
• Mike Flusss - LLNL
• Funded by LLNL
• Atlas supercomputer
Overview
• Introduction to strong correlations
• Formalism
→ Density functional theory (DFT)
→ Dynamical mean-field theory (DMFT)
→ DFT+DMFT
• Background of Plutonium
• Application to Plutonium
Electronic Many-body Hamiltonian
H =∑
σ
∫
drψ+σ (r)[−▽2 + Vext(r) − µ]ψσ(r)
+ 12
∑
σσ′
∫
drdr′ψ+σ (r)ψ+
σ′(r′) 1r−r
′ψσ′(r′)ψσ(r).
ψ+σ′(r)ψσ(r′) + ψσ(r′)ψ+
σ′(r) = δσσ′δ(r − r′)
ψσ′(r)ψσ(r′) + ψσ(r′)ψσ′(r) = 0
Local, atomic-like basis setR → Lattice site α, β, γ, δ→ orbital type (ie.s,p,d,f )
ψ(x) =∑
αR cαR(τ)χαR(r)
H =∑
αβ
∑
RR′ h(0)αRβR′
(
c+αRcβR′ + h.c.)
+
12
∑
αβγδ
∑
RR′R′′R′′′ V RR′R′′R′′′
αβγδ c+αRc+βR′cδR′′′cγR′′
where...
h(0)αRβR′ = 〈χαR| − ∇2 + Vext|χβR′〉
V RR′R′′R′′′
αβγδ = 〈χαR(r)χβR′(r′)| 1r−r′
|χγR′′(r′)χδR′′′(r)〉
Start with something simple...
• Ones orbital per lattice site...
• Only hopping between nearest neighbors...
• Only keep Coulomb repulsion ifR = R′ = R′′ = R′′′
→ Only for electrons on the same site...
• Simple model which retains relevant physics
• The Hubbard model
The Hubbard model
H =∑
ijσ
tijc+iσcjσ +
∑
i
Uni↑ni↓
Hopping Repulsion
tij = 〈χs| − ∇2 + Vext|χs〉
U = 〈χs(r)χs(r′)| 1r−r′
|χs(r′)χs(r)〉
• niσ = c+iσcjσ
• c+iσcjσ + cjσc+iσ = δij ciσcjσ = −cjσciσ
• Hopping terms yields delocalized electrons
• Repulsion term yields localized electrons
Strongly correlated electron systems
• Close to a Mott transition
• Mott insulator doped with holes
• High temperature superconductors
• Actinides
• Collosal mangeto-resistance materials
• transition-metal oxides
• many secrets locked in strongly correlated systems
Effects of electronic correlationsMagnetic Susceptibility
χ = ∂2E∂H2 = ∂M
∂H
•Metallic state - Fermi Liquid (FL)
→ interactions make electrons become HEAVY, butremain coherent
→ χ ∝ m∗ → temperature independent
• Localized state
→ interactions/temperature destroy FL!
→ χ ∝ 1T→ Curie
0 2 4 6 8 10Temperature
0
0.2
0.4
0.6
0.8
1
χ
Many-body Hamiltonian
H =∑
σ
∫
drψ+σ (r)[−▽2 + Vext(r) − µ]ψσ(r)
+ 12
∑
σσ′
∫
drdr′ψ+σ (r)ψ+
σ′(r′)vC(r − r′)ψσ′(r′)ψσ(r).
Electronic structure calculations with DMFT
Reviews of Modern Physics, 78, 865 (2006)
http://dmft.rutgers.edu
General Methods
• Density functional theory
⇒ Realistic systems
⇒ Ground state properties
• Dynamical mean-field theory
⇒ Model Hamiltonians
⇒ Captures physics of Mott transition
⇒ Ground/excited state properties
• Merge approaches→ DFT+DMFT
Density Functional Theory
• E[ρ] is minimal whenρ is ground state density
• Kohn-ShamH0 + Vks(r)
• Vks(r) is what must be added toH0 to generate correctdensity
• Many-body problem partially mapped to one-bodyproblem butVks(r) not known!
• ApproximateVks with LDA
• Vks = ǫxc + ∂ǫxc
∂ρρ
Green’s (Function) Functional Theory
• Γ[G] is stationary whenG is the interacting Greensfunction
• DysonH0 + Σ(k, ω)
• Σ(k, ω) is what must be added toH0 to generatecorrect Greens function
• Many-body problem partially mapped to timedependent one-body problem butΣ(k, ω) not known!
• ObtainΣ(k, ω) using DMFT
What is DMFT?
• DMFT maps lattice→ impurity
Hlat =∑
ijσ
tijc+iσcjσ +
∑
i
Uc+i↑ci↑c+i↓ci↓
Himp =∑
σ
ǫimpc+σ cσ + Uc+↑ c↑c
+↓ c↓ +
∑
kσ
Vk(a+kσcσ + c+σ akσ) +
∑
kσ
αka+k ak
• αk, Vk given by DMFT self-consistency
• Σlat(k, ω) ≈ Σlocal(ω)
DMFT algorithm
• guessαk, Vk → G0imp(ω)
• G0imp(ω) → impurity solver→ Gimp(ω),Σimp(ω)
• Σlat(k, ω) = Σimp(ω)
• Performself-consistency condition
Gimp′ =∑
k
Glat(k, ω) =∑
k
1ω−Hk+µ−Σimp(ω)
• G0imp(ω) = (Σimp(ω) +G−1
imp(ω))−1
→ αk, Vk
• iterate
Solving impurity problem3 main issues...
• ”Exact” but slow
• Fast but not always reliable
• real axis vs. imaginary axis
Solving impurity problem
• Hartree-Fock (DFT+DMFT→ DFT+U )
• FLEX - perturbation theory around band limit
• NCA - perturbation theory around atomic limit
• Exact Diagonalization
• Slave Boson mean-field
• Hirsch-Fye QMC
• Continuous time QMC
• NRG, DMRG, etc...
Merging DFT and DMFT
• Define local basis set
• Identify correlated subspace of Hamiltonian
⇒ Apply correlations∑
αβ
Uαβnαnβ
• Chooseρ andGloc as variables⇒ Γ[ρ,Gloc]
• H0 + VKS +∑
αβ
|α〉〈β|Σαβ(ω)
• DetermineU and ”double-counting” correction
• ObtainΣ using DMFT
• Perform both DMFT and DFT self-consistency
Applications of DFT+DMFT
• f -electron systems
• Transition-metal oxides
• Transition-metals
• Surfaces / Interfaces
• Multilayers
• Disorder
• Electron-phonon coupling
• etc, etc...
What can DFT+DMFT predict?
• Spectra
• Fermi Surface
• Density
• Total Energy
• Entropy
• Magnetic Susceptibility
• Heat capacity
• Transport
• etc, etc...
Phase diagram of Pu
0 200 400 600 800 1000Temperature (K)
0
5
10
15
20
25
30
% V
olum
e E
xpan
sion
α
βγ
δ δ’ε L
Phase diagram of Pu
200 400 600 800 1000 1200 1400 1600 1800Temperature (K)
0
5
10
15
20
25
30
% V
olum
e E
xpan
sion
Pu
Fe
Compare Pu to other metals
200 400 600 800 1000 1200 1400 1600 1800Temperature (K)
0
5
10
15
20
25
30
% V
olum
e E
xpan
sion
Pu
Fe
Mn
Experimental Magnetic Susceptibility
•
•
100 200 300 400 500 600 700Temperature (Kelvin)
0
2
4
6
8
χ (1
03 em
u/m
ol)
UBe13
(m*/m~300)
Pu
Results forδ PuPRL 101, 056403 (2008)
24 26 28 30 32
Volume (A3)
0.1
0.2
0.3
0.4
Z
J=0.5eVJ=0.0
24 25 26 27 28 29 30 31 32 33
Volume (A3)
20
25
30
35
40
45
50
55
γ (m
J/m
ol-K
2 )
200 400 600 800 1000 1200T (Kelvin)
3
4
5
6
7
8
9
10
χ loca
l
V=24.8 A3
V=28.8 A3
V=32.8 A3
0
2
4
6
8
10
eV-1
S=5/2S=7/2
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV
0
2
4
6
eV-1
S=5/2 J=0S=7/2 J=0S=5/2 J=0.5eVS=7/2 J=0.5eV
DFT
DMFT
• Performed approximation-free DMFT at ambient T
→ Including full exchange interaction
• Calculated various properties
→ Quasiparticle weight → Magnetic susceptibility
→ Heat capacity → Spectra
Photoemission spectra
0
2
4
6
8
10
eV-1
S=5/2S=7/2
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV
0
2
4
6
eV-1
S=5/2 J=0S=7/2 J=0
DFT
DMFT
• Density of electronic statesversus energy.
• DMFT renormalizes DFTspectrum
→ Transfer of weight from Fermienergy to Hubbard bands
• Exchange causes furtherrenormalization of spectrum.
Photoemission spectra
0
2
4
6
8
10
eV-1
S=5/2S=7/2
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV
0
2
4
6
eV-1
S=5/2 J=0.5eVS=7/2 J=0.5eV
DFT
DMFT
• Density of electronic statesversus energy.
• DMFT renormalizes DFTspectrum
→ Transfer of weight from Fermienergy to Hubbard bands
• Exchange causes furtherrenormalization of spectrum.
Photoemission spectra
0
2
4
6
8
10
eV-1
S=5/2S=7/2
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV
0
2
4
6
eV-1
S=5/2 J=0S=7/2 J=0S=5/2 J=0.5eVS=7/2 J=0.5eV
DFT
DMFT
• Density of electronic statesversus energy.
• DMFT renormalizes DFTspectrum
→ Transfer of weight from Fermienergy to Hubbard bands
• Exchange causes furtherrenormalization of spectrum.
Exchange has notable effect on spectra
Photoemission
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV
0
1
2
3
4
5
6
7
8eV
-1
S=5/2 J=0S=7/2 J=0S=5/2 J=0.5 eVS=7/2 J=0.5 eV
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705KT=600K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705KT=600KT=500K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400KT=352K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400KT=352KT=300K
Photoemission Temperature Dependence
-0.2 -0.1 0 0.1 0.2eV
0
0.2
0.4
0.6
0.8
1eV
-1T=1200KT=1000KT=800KT=705KT=600KT=500KT=444KT=400KT=352KT=300KT=240K
Quasiparticle WeightZ = mm∗
24 26 28 30 32
Volume (A3)
0.1
0.2
0.3
0.4
Z
J=0.5eVJ=0.0
• Z is inverse of effective mass
→ Z=1⇒ no correlations
→ Z=0⇒ electrons localize
• Pu is closer to atomic limit
• Electrons become heavier asvolume increases
• Including exchange substantiallyincreases correlations.
δ Pu is strongly correlated
Linear Coefficient of Heat Capacity
24 25 26 27 28 29 30 31 32 33
Volume (A3)
20
25
30
35
40
45
50
55
γ (m
J/m
ol-K
2 )
• Low temperature heat capacitysensitive to correlations.
• Strongly depends on volume.
• Experiments find35 − 65 mJmolK2
→ Huge expt. variation
• Cause of discrepancy:
→ Electron-phonon coupling
→ Density self-consistency
→ number of f electrons
→ Inaccurate experiments?γ =
2πk2
B
3
∑
αρα(0)Zα
Magnetic Susceptibilityχ
200 400 600 800 1000 1200T (Kelvin)
3
4
5
6
7
8
9
10
χ loca
l
V=24.8 A3
V=28.8 A3
V=32.8 A3
• First calculation ofχ in Pu
• Pauli behavior⇒ itinerantelectrons
• Curie behavior⇒ localizedelectrons
• Predict Pauli behavior forVeq
→ Agrees with experiments
→ Explains lack of magnetism
• Expanded lattice agrees withPuH2
Quantum decoherence inδ Pu
Experimental Magnetic Susceptibility
0 50 100 150 200 250 300Temperature (Kelvin)
0
0.001
0.002
0.003
0.004
0.005em
u/m
ole
δ Pu
PuH2
The effect of Structure on Correlations
• Eplore differences betweenδ andǫ
→ FCC vs. BCC
• Use the same volume to isolate effect of structure
Photoemissionǫ-Pu vs.δ-Pu
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3eV
0
1
2
3
4
5
6
7
8eV
-1
S=5/2 εS=7/2 εS=5/2 δS=7/2 δ
Quasiparticle WeightZ = mm∗
24 26 28 30 32
Volume (A3)
0.1
0.2
0.3
0.4
Z
J=0.5eVJ=0.0J=0.5eV ε
• Z is inverse of effective mass
→ Z=1⇒ no correlations
→ Z=0⇒ electrons localize
• Pu is closer to atomic limit
• Electrons become heavier asvolume increases
• Including exchange substantiallyincreases correlations.
δ Pu is strongly correlated
Magnetic Susceptibilityχ
200 400 600 800 1000 1200T (Kelvin)
3
4
5
6
7
8
9
10
χ loca
l
V=24.8 A3
V=28.8 A3
V=32.8 A3
ε-Pu V=24.8 A3
• First calculation ofχ in Pu
• Pauli behavior⇒ itinerantelectrons
• Curie behavior⇒ localizedelectrons
• Predict Pauli behavior forVeq
→ Agrees with experiments
→ Explains lack of magnetism
• Expanded lattice agrees withPuH2
Quantum decoherence inδ Pu