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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