reduceddensity-matrixfunctionaltheory: … · 2018-01-28 · Overview Goals •fast ground-state...
Transcript of reduceddensity-matrixfunctionaltheory: … · 2018-01-28 · Overview Goals •fast ground-state...
reduced density-matrix functional theory:
Energies and forces for materials with strong correlations
Robert Schade1, Ebad Kamil2, Peter Blochl1, Thomas Pruschke2
1Institute for Theoretical Physics, Clausthal University of Technology, Germany2Institute for Theoretical Physics, Georg-August-University Gottingen, Germany
OverviewGoals
• fast ground-state calculations for correlated electron systems
• integration into the infrastructure of existing density-functionaltheory calculations for structure relaxations and ab-initiomolecular dynamics
Framework
• variational formulation based on reduced density-matrix func-tional theory (rDMFT)
• direct evaluation of the density-matrix functional with a DMFT-likelocal approximation
Ensemble rDMFTReduced ensemble density-matrix functional theory
Gilbert [1975], Levy [1979], Lieb [1983]
• the main quantity is the one-particle reduced density-
matrix ρ(1) = c†αcβ
• grand canonical potential Ωβ,µ(h)
ΩWβ,µ
(
h)
= −1
βln
(
Tr e−β(
h+W−µN))
= minρ(1)
(∑
α,β
hα,βρ(1)β,α
︸ ︷︷ ︸
one−particle terms
+ F Wβ
(
ρ(1))
︸ ︷︷ ︸
density−matrix functional︸ ︷︷ ︸
interaction energy−T·entropy
)
• the density-matrix functional F Wβ (ρ(1)) can be calcu-
lated/approximated in several ways:
1. via Legendre-Fenchel (Lieb [1983]) transform of the grand
canonical potential with respect to the one-particle operator h
F Wβ
(
ρ(1))
= maxhα,β
(
ΩWβ
(
h)
− Tr(
hρ(1)))
2. via constrained search (Levy [1979]) over an ensemble ofmany-particle wave functions |Ψi〉:
F Wβ
(
ρ(1))
=
minPi,|Ψi〉
stathα,β,Λi,j,λ
[∑
i
Pi〈Ψi|W |Ψi〉
︸ ︷︷ ︸interaction energy
+ β−1∑
i
Pi ln (Pi)
︸ ︷︷ ︸
−T ·entropy
+∑
α,β
hα,β
∑
i
Pi〈Ψi|c†αcβ|Ψi〉 − ρ
(1)β,α
︸ ︷︷ ︸
density−matrix constraints
−∑
i,j
Λj,i(〈Ψi|Ψj〉 − δi,j
)
︸ ︷︷ ︸
orthogonality constraints
−λ
∑
i
Pi − 1
︸ ︷︷ ︸
probability constraint
]
3. via a relation to the Luttinger-Ward functional andGreens functions [Blochl, Pruschke, Potthoff (2013)]
F Wβ (ρ(1)) =
1
βTr[
ρ(1) ln(ρ(1)) + (1− ρ(1)) ln(1− ρ(1))]
︸ ︷︷ ︸entropy of a non−interacting electron gas
+ stath′,G,Σ
(
ΦLWβ (G, W )−
1
β
∑
ν
Tr
ln[1−G(iων)i
(h′ − h + Σ(iων)
)]
+(h′ − h + Σ(iων)
)G(iων)−
[G(iων)−G(iων)
](h′ − h)
)
h = µ1 +1
βln
(1− ρ(1)
ρ(1)
)
G(iων) =((iων + µ)1− h
)−1
4. via parametrized approximations like Mueller-functional(Muller [1984]) resp. Power-functional (Sharma et al. [2008])
P. E. Blochl, C. F. J. Walther, and T. Pruschke. Method to include explicit correlations into density-
functional calculations based on density-matrix functional theory. Phys. Rev. B, 84:205101, Nov
2011.
P. E. Blochl, T. Pruschke, and M. Potthoff. Density-matrix functionals from green’s functions.
Phys. Rev. B, 88:205139, Nov 2013. doi: 10.1103/PhysRevB.88.205139.
T. L. Gilbert. Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B, 12:2111–
2120, Sep 1975.
E. Kamil, R. Schade, T. Pruschke, and P. E. Blochl. Reduced density-matrix functionals applied
to the Hubbard dimer. ArXiv e-prints, accepted for pub. in PRB, 1509.01985, Sept. 2015.
M. Levy. Universal Variational Functionals of Electron Densities, First-Order Density Matrices,
and Natural Spin-Orbitals and Solution of the v-Representability Problem. Proceedings of the
National Academy of Science, 76:6062–6065, Dec. 1979.
A. M. K. Muller. Explicit approximate relation between reduced two- and one-particle density
matrices. Phys. Lett., 105A:446, Aug 1984. doi: 10.1016/0375-9601(84)91034-X.
S. Sharma, J. K. Dewhurst, N. N. Lathiotakis, and E. K. U. Gross. Reduced density matrix
functional for many-electron systems. Phys. Rev. B, 78:201103, Nov 2008. doi: 10.1103/Phys-
RevB.78.201103.
W. Tows and G. M. Pastor. Lattice density functional theory of the single-impurity anderson
model: Development and applications. Phys. Rev. B, 83:235101, Jun 2011. doi: 10.1103/Phys-
RevB.83.235101.
DFT+rDMFT[Blochl,Walther,Pruschke 2011]
E0(N) ≈ min|φn〉,fn∈[0,1]
statµ,Λnm
(
EDFT [|φn〉, fn]
+(
F WHF [ρ
(1)]− F WDFT,DC [ρ
(1)])
︸ ︷︷ ︸hybrid functional
+(
FˆW [ρ(1)]− F
ˆWHF [ρ
(1)])
︸ ︷︷ ︸high−level correction
− µ
(∑
n
fn −N
)
︸ ︷︷ ︸particle−number constraint
−∑
i,j
Λi,j (〈φi|φj〉 − δi,j)︸ ︷︷ ︸
orthogonality constraints
)
with ρ(1)α,β =
∑
n
〈χα|φn〉fn〈φn|χβ〉
Local approximation [Blochl,Walther,Pruschke 2011]
Define cluster of interacting orbitals CR and restrict interaction tothese clusters
W ≈∑
R
WR WR =1
2
∑
a,b,c,d∈CR
Ua,b,d,cc†ac†bcccd
F Wβ [ρ(1)] ≈ F
∑
R WR
β [ρ(1)] ≈∑
R
F WR
β [ρ(1)].
Adaptive cluster approximation[Schade, Blochl 2016]
Main idea: transform density-matrix with a unitary transformationbefore doing a cluster approximation
neglect coupling between interacting orbitals (impu-rity)+effective bath and remaining system in the transformed
density matrix ρ(1)T (N = NA · (M + 1), NA interacting orbitals):
F WR[ρ(1)] ≈ F WR
ACA(M)[ρ(1)] = F WR[T
†Nρ
(1)T TN ]
⇒ reduced sensitivity to truncation of one-particle basis
example: histogram of D0/T,α,β =∂FWcenter [ρ0/T ]
∂ρ0/T,αβof half-filled 3-by-3
Hubbard cluster with U/t = 5
−10
0
10
20
30
40
0.001 0.01 0.1 1 2
|D0,αβ |/t,−|DT,αβ|/t
⇒ rapid convergence to the exact result with increasingtruncation level MThe ACA can be seen as a generalization of the two-level-approximation by Tows and Pastor [2011] to
• multi-band and multi-site interactions (TLA only one-band).
• arbitrary truncation parameters M (TLA only M = 1).
Corrected adaptive cluster approximation:Additional correction based on parametrized approximate functional
F W≈ [ρ] to mediate effect of truncation for small M :
F WR
cACA(M)[ρ(1)] = F WR
ACA(M)[ρ(1)]− F WR
≈ [T†Nρ
(1)T TN ] + F WR
≈ [ρ(1)T ]
Application to a finite single-orbital SIAM:
H =∑
σ
ǫf f†σfσ + Unf,↑nf,↓ +
∑
σ,i
ǫini,σ +V√Nb
∑
i,σ
(
f†σci,σ + cc.
)
(Nb = 11, Ne = 12, ǫi = −2t cos(2πn/Nb))
Interaction strength dependence (V/t = 0.4, ǫf = 0, t > 0):
−14.3
−14.2
−14.1
−14
−13.9
0
0.05
0.1
0.15
0.75
1
1.25
1.5
0 2 4 6 8
0
0.5
1
E0/t
W0/t
nf
U/t
mf
−20
−10
0
−10
−5
0
5
0
0.01
0.02
0.03
0 2 4 6 8
103∆E/t
103∆W/t
∆nf
U/t
Impurity onsite-energy dependence (V/t = 0.4, U/t = 8,t > 0):
−27
−20
−13
0
4
8
0
1
2
−10 −5 0 5
0
0.5
1
0
0.025
0.05E0/t
W0/t
nf
ǫf/t
mf
∆E
0/t
−30
−20
−10
0
−300
−200
−100
0
−0.025
0
0.025
−10 −5 0 5
103∆E/t
103∆W/t
∆nf
ǫf/t
Bandwidth dependence (ǫf/V = −1, U/V = 5, V > 0):
−300
−200
−100
0
0
1
2
3
4
0.7
1
1.3
1.6
1.9
0 10 20
0
0.5
1
00.10.20.3
E0/V
W0/V
nf
t/V
mf
∆E
0/V
−80
−60
−40
−20
0
−40
−20
0
0
0.01
0.02
0.03
0.04
0 10 20
0
0.02
0.04
0 1 2
103∆E/V
103∆W/V
∆nf
t/V
Application to a half-filled 24-site Hubbard ring:
• impurity size in the local approximation NA
• truncation parameter in the ACA M (N = NA · (M + 1))
Convergence with truncation parameter for exact ground state densitymatrix (single site local approx.):
0.28
0.29
0.3
0.31
0.32
1 2 3 4 5 6
FW
1
(c)A
CA(M
)[ρ
0]/t
M
Exact results and local approximation with ACA for total energy E0,interaction energyW0 and next-neighbor density matrix elements ρ12:
−1
−0.5
0NA = 1
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0 10 20
E0/(N
t) exact
M=1
M=2
M=3
W0/(N
t)ρ12
U/t
−1
−0.5
0NA = 2
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0 10 20
E0/(N
t)
exact
M=1
M=2
W0/(N
t)ρ12
U/t
HFexact
HF
↓ ACA M=3
↓ cACA M=2
↓ cACA M=1
↓ ACA M=2
ACA M=1
HF
exact/HF
HF
HF
exact
ACA M=1
ACA M=2
cACA M=2
↑ cACA M=1
HF
exact/HF
HF
exact
HF
↓ cACA M=2
← ACA M=1
← ACA M=2
← ← cACA M=1
↑ ACA
↓ cACA