Reduced Density Matrix Functional Theory for Many Electron
Transcript of Reduced Density Matrix Functional Theory for Many Electron
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Reduced Density Matrix Functional Theory forMany Electron Systems
S. Sharma, J. K. Dewhurst and E. K. U. Gross
Max-Planck-Institut fur MikrostrukturphysikWeinberg 2, D-06120 Halle
Germany
July 2011
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Transition metal oxides–prototypical Mott insulators
Solid TN Band gap
MnO 118 3.7FeO 198 2.5CoO 292 2.4NiO 523 4.1
Table: Neel temperature (in K) and band gap (in eV).
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Reduced density matrix
γ(r, r′) = N
∫Ψ(r, r2, . . . , rN )Ψ∗(r′, r2, . . . , rN ) d3r2 . . . d
3rN .
Total energy
E[γ] =
∫d3r′d3rδ(r−r′)
[−∇
2
2
]γ(r, r′)+Eext[γ]+EH[γ]+Exc[γ]
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Exchange-correlation energy functionals
“Power functional”, Sharma et al. Phys. Rev. B 78, R201103(2008)
RDM energy functional
Exc[{ni}, {φi}] = −1
2
∑ij
(ninj)α
∫φ∗i (r)φj(r)φ∗j (r
′)φi(r′)
|r− r′|d3r d3r′
φi(r) =∑l
cilΨKSl (r)
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Fixing the value of α
α is NOT system dependent
Value of α is fixed to 0.565 and all calculations for extendedsystems are performed with this FIXED value. The theory isab-initio in this sense.
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Fixing the value of α
α is NOT system dependent
Value of α is fixed to 0.565 and all calculations for extendedsystems are performed with this FIXED value. The theory isab-initio in this sense.
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Spectral information using RDMFTGreen’s function in the bais of natural orbitals:
iGµβ(t− t′) =1
〈ΨN0 |ΨN
0 〉〈ΨN
0 |T [aµ(t)a†β(t′)]|ΨN0 〉
Define a restricted but physically significant set of N+1 states
ΨN+1ζ =
1√nζa†ζ |Ψ
N0 〉, ΨN−1
ζ =1√
(1− nν)aζ |ΨN
0 〉,
spectral function
Aµβ(ω) = 2π∑ζ
1
nζ〈ΨN
0 |aµa†ζ |Ψ
N0 〉〈ΨN
0 |aζa†β|Ψ
N0 〉δ(ω − ε+ζ )
− 2π∑ν
1
nν〈ΨN
0 |a†µaν |ΨN0 〉〈ΨN
0 |a†νaβ|ΨN0 〉δ(ω − ε−ν )
DOS = 2π∑ζ
nζδ(ω − ε+ζ )− 2π∑ν
(1− nν)δ(ω − ε−ν ),
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Spectral information using RDMFT
ε±ν = Eν(N ± 1)− E(N)
with Eν(N ± 1) is energy of periodically repeated Born-vonKarman (BvK) cell.
Approximating this
The only occupation number that will change is the one thatcorresponds to the very same k
Derivative instead of energy difference
ε±ν =∂E[{φν}, {nν}]
∂nν
∣∣∣∣nν=1/2
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Spectral information using RDMFT
ε±ν = Eν(N ± 1)− E(N)
with Eν(N ± 1) is energy of periodically repeated Born-vonKarman (BvK) cell.
Approximating this
The only occupation number that will change is the one thatcorresponds to the very same k
Derivative instead of energy difference
ε±ν =∂E[{φν}, {nν}]
∂nν
∣∣∣∣nν=1/2
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Spectral information using RDMFT
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Transition metal oxides–prototypical Mott insulators
Solid TN Band gap
MnO 118 3.7FeO 198 2.5CoO 292 2.4NiO 523 4.1
Table: Neel temperature (in K) and band gap (in eV).
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Total DOS for TMOs (without long range magnetic order)
[GW: PRB 79 235114 (09), DMFT: PRB 77 195124 (08), PRB 74195115 (06), Mat. Mat. 7, 198 (06)]
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Partial DOS for TMOs (charge transfer effects)
[cond-mat/0912.1118]
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Charge density difference for NiO in [110] plane
[cond-mat/0912.1118, Phys. Rev. B 61 2506 (2000)]
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Density of states for metal and band insulator
[cond-mat/0912.1118, PRB 73, 073105 (06)]
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Summary and outlook
Summary
RDMFT has proved to be extremely successful for treatmentof finite systems.
Technique for obtaining spectral information from RDMFTgroundstate is proposed.
RDMFT captures the physics of Mott-insulators.
Outlook
Magnetic functionals for RDMFT
Temperature dependent RDMFT to study pahse diagram ofTMOs.
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Summary and outlook
Summary
RDMFT has proved to be extremely successful for treatmentof finite systems.
Technique for obtaining spectral information from RDMFTgroundstate is proposed.
RDMFT captures the physics of Mott-insulators.
Outlook
Magnetic functionals for RDMFT
Temperature dependent RDMFT to study pahse diagram ofTMOs.
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Prize questions
Green’s function in the bais of natural orbitals:
iGαβ(t− t′) =1
〈ΨN0 |ΨN
0 〉〈ΨN
0 |T [aα(t)a†β(t′)]|ΨN0 〉
Define a restricted but physically significant set of N+1 states
ΨN+1ζ =
1√nζa†ζ |Ψ
N0 〉, ΨN−1
ζ =1√
(1− nν)aζ |ΨN
0 〉,
1. Why is it important the above states are eigen-states2. When (under what conditions) are these states Eigen-states.