Density-based functional renormalizationgroup for quantum many-body problems

Takeru Yokota
RIKEN iTHEMS

Functional renormalization group at RIKEN 2023, RIKEN Wako, Jan. 21, 2023

1

Collaborators
Teiji Kunihiro

Kyoto U., YITP

Kenichi Yoshida Kyoto U. → Osaka U., RCNP
Tomoya Naito

RIKEN iTHEMS

Haruki Kasuya

Kyoto U., YITP

Osamu Sugino

U. Tokyo, ISSP

Jun Haruyama

U. Tokyo, ISSP

TY, Yoshida, Kunihiro, PRC (2019)
TY, Yoshida, Kunihiro, PTEP (2019)
TY, Naito, PRB (2019)

TY, Naito, PRResearch (2021)
TY, Kasuya, Yoshida, Kunihiro PTEP (2021)

TY, Haruyama, Sugino, PRE (2021)
TY, Naito, PRB (2022)

2

Contents• Density functional theory (DFT)• Introduction• Eﬀective-action formalism for DFT• Functional evolution equation• Demonstration: electron system

3

Contents• Density functional theory (DFT)• Introduction• Eﬀective-action formalism for DFT• Functional evolution equation• Demonstration: electron system

4

Density functional theory (DFT)Ψgsground-state
wave func.

ρgs

ground-state
densityP. Hohenberg

quantum many-body system

W. Kohn

L. J. Sham

Chem. 1998

DFT: Exact many body theory based on ρgs
instead of Ψgs.

Why possible?
5Hohenberg-Kohn theorem
Suppose a non-relativistic system
whose Hamiltonian is given byĤ = T ̂ + Û + V̂
kinetic

interacting

external
potential

Statements (non-deregerate case)
1. Equivalence of density and wave func.

Ψgs is a functional of ρgs: Ψgs = Ψgs[ρgs]̂V̂ = d xρ(x)V(x)∫{V}

proved by contradiction

There exists 1-to-1 mapping b/w ρgs, Ψgs, V

Hohenberg, Kohn, PR (1964)

{ρgs}
ρgs

V

{Ψgs}
Ψgs

2. Variational principle

There exists energy density functional (EDF), whose minimum point gives ρgs
EDF has the following form: E[ρ] =̂ Ψgs[ρ] = F[ρ] + dxρ(x)V(x)
Ψ
[ρ]
H
gs
⟨
⟩
∫

“Universal” part
(V-independent, dependent on particle mass, interaction)

6

Hohenberg-Kohn theorem
Suppose a non-relativistic system
whose Hamiltonian is given byĤ = T ̂ + Û + V̂
kinetic

interacting

Hohenberg, Kohn, PR (1964)

external
potential̂V̂ = d xρ(x)V(x)∫Degenerate case• 1-to-1 mapping b/w sets: {ρgs}degenerate, {Ψgs}degenerate, V• Well-defined EDF exists

VE[ρ] = ⟨Ψgs[ρ] Ĥ Ψgs[ρ]⟩{ρgs}degenerate

One of the degenerate
wave functions{Ψgs}degenerate• Spontaneous symmetry breaking is in principle described, but calculation of the orderparameter as a functional of ρ, O[ρ] = ⟨Ψgs[ρ] | Ô | Ψgs[ρ]⟩ is usually infeasible in practice.

• For practical analysis, additional density is introduced.
e.g.) E[ρ↑, ρ↓] to describe magnetism
7Application to numerical analysis
Kohn-Sham scheme

E[ρ] = T[ρ] + ΔE[ρ]free kinetic residual

Kohn, Sham, PR (1965)

δT[ρ] δΔE[ρ]Var. eq.
+
=μ
δρ(x)
δρ(x)

●
●

●

●

●

●

●

●

That for non-interacting system with ext. field VKS = δΔE/δρ

●

●
𝑽KS

Variational problem= finding self-consistent solution for the non-int. Schrodinger eq.
N∇2| ϕi(x) |2−+ VKS[ρgs](x) ϕi(x) = ϵiϕi(x) with ρgs(x) =
∑
( 2m
)
i=1

Application

Proteins(103∼4 atoms, electrons)• Electrons (crystal, molecule)• Nucleus• No external field. But DFT is a tool toetc..

Computationally eﬃcient!
N ≲ 104 (or 105∼6 ?)describe nucleus with various particle
number in a unified manner.

DFT

DFT
http://www.slis.tsukuba.ac.jp/
cicsj27/cicsj27/J13.pdf

Bertsch, Dean, Nazarewicz (2007), SciDAC Review, (6):42

8

Energy density functional (EDF)
How can we obtain E[ρ]?

Most important but still open problem…

Electron EDF

Nuclear EDF

Microscopic int.: Coulomb
Jacob’s ladder

Microscopic int.: Nuclear force + Coulomb
Ambiguity in nuclear force
But determination of nuclear force is in progress:
Scattering exp., lattice QCD, chiral EFT, …

Many EDFs

So far, phenomenological EDFs
obtained from fitting of experimental
data have been usually used.
Figure from Goerigk, Mehta,
Aust. J. Chem. (2019)

Figure from Burke, J. Chem. Phys. (2012)

Construction of EDF• Data-driven approach• Fitting data from Monte Carlo,•…

few-body calc., experiments,…
Many-body perturbation

Development of systematic bottom-up
approach is still requiredin order for…

• further improvement of accuracy• making the relation to microscopic int. clear
9DFT in effective action formalism
Density: composite field ρψ (x) = ψ*(x)ψ(x)Imaginary-time partition function for density correlation ρψ (τ, x) = ψ*(τ + 0,x)ψ(τ, x)

Z[J] = 𝒟ψ𝒟ψ*e∫−S[ψ,ψ*]+ ∫0β dτ ∫ dxJ(τ,x)ρψ (τ,x)

To extract the g.s. property, we replace
ρ(τ, x) → ρ(x) and take β → ∞ at the end.

Eﬀective action for ρ (two-particle-point-irreducible eﬀective action)

Γ[ρ] = supJβdτ dxJ(τ, x)ρ(τ, x) − ln Z[J]( ∫0 ∫)

EDF

Γ[ρ] ρ(τ, x) = ρ(x)E[ρ] = lim∵β→∞ β
Γ[ρ ] = βF

Fukuda, Kotani, Suzuki, Yokojima, PTP92 (1994)
Valiev, Fernando, arXiv:9702247 (1997)

for

Variational principle = quantum EOM Γ(1)[ρgs] = μ
gs

10

Helm

Approaches to DFT based on effective-action formalismc.f.) Reviews by R.J. Furnstahl+“EFT for DFT”, Furnstahl (2007)“Toward ab initio density functional theory for nuclei”, Drut, Furnstahl, Platter (2009)“Turning the nuclear energy density functional method into a proper eﬀective
field theory: reflections”, Furnstahl (2019)• Expansion w.r.t. small parameter• Power counting with fixing ρ is obtained.
• e.g., Fermi-momentum expansion(dilute system)• FRG-like evolution eq. This talkDensity profile of a dilute systemof fermions in a trap
Puglia, Bhattacharyya, Furnstahl, NPA (2003)

Polonyi, Sailer (2002), Schwenk, Polonyi (2004)

11

Contents• Density functional theory (DFT)• Introduction• Eﬀective-action formalism for DFT• Functional evolution equation• Demonstration: electron system

12Adiabatic-connection fluctuation-dissipation theorem
Levy, Perdew (1985), Levy (1991), Gorling, Levy (1992)
Switching on of two-body int.

Ĥ = T ̂ + λÛ + Vλ̂

Ext. field to fix ρ

λ:0→1

1

1
Exc[ρ] =dλ dx dx′U(x − x′)[Sλ(x, x′) − ρ(x)δ(x − x′)]2 ∫0 ∫ ∫E[ρ] = T[ρ] + EHartree[ρ] + Exc[ρ]̂ ρ(x′̂ )⟩λ − ρ(x)ρ(x′)
Density correlation Sλ(x, x′) = ⟨ρ(x)is needed as an input.
13

Evolution equation for effective action
Polonyi, Sailer (2002), Schwenk, Polonyi (2004)

β
1 β∂λΓλ[ρ] =dτ dx dτ′ dx′U(x − x′)δ(τ − τ′ + 0)∫
2 ∫0 ∫ ∫0
−1

2

δ Γλ
× ρ(τ, x)ρ(τ,′ x′) +[ρ](τ, x, τ′, x′) − ρ(τ, x)δ(x − x′)( δρδρ )• Closed equation for Γλ[ρ]• Input from other calculation is not needed.
• This in principle gives all the correlation functions.

• Calculation techniques in FRG are possibly useful.

• Vertex expansion (VE), local potential approximation (LPA)e.g.) O(4), quark-meson model• Real-time response function
Kamikado, Strodthoﬀ, von Smekal, Wambach (2020)
Strodthoﬀ, von Smekal, Wambach (2014)• Analytic continuation is easy. Tripolt,
TY, Kunihiro, Morita (2016), (2017)…
14

Studies of FRG-aided DFT• Classical & quantum anharmonic oscillators
Kemler, Braun, JPG (2013)

VE

Liang, Niu, Hatsuda, PLB (2018)

VE

Kemler, Pospiech, Braun, JPG (2017)
TY, Yoshida, Kunihiro, PRC (2019)
TY, Yoshida, Kunihiro, PTEP (2019)

VE
VE• One-dim nuclear system• Electron systems

TY, Naito, PRB (2019)
TY, Naito, PRResearch (2021)
TY, Naito, PRB (2022)

VE

Excited states(real-time correlation function)

VE
VE
VE

• Classical liquids

Lue, AIChE (2015)
TY, Haruyama, Sugino, PRE (2021)

LPA
VE• Superfluid system

TY, Kasuya, Yoshida, Kunihiro PTEP (2021)
15

Formulation

Contents• Density functional theory (DFT)• Introduction• Eﬀective-action formalism for DFT• Functional evolution equation• Demonstration: electron system

16

Contents• Density functional theory (DFT)• Introduction• Eﬀective-action formalism for DFT• Functional evolution equation• Demonstration: electron system
w/ T. Naito
17

TY, Naito, PRB (2019)
TY, Naito, PRResearch (2021)
TY, Naito, PRB (2022)

Local density approximation (LDA)
E[ρ] = F[ρ] + dxρ(x)V(x)∫Hartree1F[ρ] = T[ρ] +dx dx′U(x − x′)ρ(x)ρ(x′) + Exc[ρ]
2∫ ∫

LDA

Simple approx. for EDF
small gradient limit ∇ρ(x)/ρ(x)4/3 ≪ 1

exchange-correlation(unknown)

Proof: Lewin, Lieb, Seiringer (2020)

Exc[ρ] ≈ dxϵxc (ρ(x)) ρ(x)∫xc energy per particle of homogeneous gas

LSDA (local spin density approx.)

Exc[ρ↑, ρ↓] ≈ dxϵxc (ρ↑(x), ρ↓(x))
ρs(x)
∑
∫
Let us derive Exc with VE
18sJellium model
Electron gas in homogeneous static background positive ion

-

-

+

-

-

-

-

-

Δ
Sλ[ψ, ψ ] = dτ dxψ (τ + 0,x) ∂τ −ψ(τ, x)
(
∫ ∫
2)
λ
+
dτ dx dx′U(x − x′)(ρψ (τ, x) − n) (ρψ (τ, x′) − n)2∫ ∫ ∫††in Hartree atomic unit aBohr = me = 1

Electron field: ψ = t(ψ↑, ψ↓)−1

Coulomb force: U(x − x′) = | x − x′|Electron density: ρψ (τ, x) = ψ †(τ + 0,x)ψ(τ, x)
Background-ion density: n
19Vertex expansion
Expansion around ρ↑,↓ = ρgs↑,↓ of interestn∞1Γλ[ρ↑,↓] =⋯Γ(n)[ρgs↑,↓](X1…, Xn)
ρ(Xi) − ρgs,si)(
λ∑ n! ⨋
∏
⨋X
X
1
n
n=0
i=1

0th
EnergyX = (τ, x, s)(2)∂λΓλ[ρgs,↑↓]( = ∂λ βEgs,λ) = Flow(0)ρ,Γ[ρgs,↑↓]]λ [ gs,↑↓ λ1stChemical potential

(1)
(2)
(3)
∂λΓ(1)
[ρ
](X
)
=
∂
μ
=
Flow
X;
ρ
,
Γ
[ρ
],
Γ
[ρgs,↑↓]]
(
)
gs,↑↓
λ λ
gs,↑↓ λ
gs,↑↓
λ
λ [
λ

2nd
Density correlation(2)−1∂λΓ(2)
[ρ
](X
,
X
)
=
∂
G
(X1, X2))
gs,↑↓
1
2
λ
(
λ
λ

⋮

(2)
(3)
(4)
= Flow(2)
X
,
X
;
ρ
,
Γ
[ρ
],
Γ
[ρ
],
Γ
[ρgs,↑↓]]
gs,↑↓
gs,↑↓
λ [ 1 2 gs,↑↓ λλλApproximation for higher-order terms is needed20

Flow equations for density correlationsnδGλ(n)(X1, ⋯, Xn) =ln Zλ[J]δJ(X1)⋯δJ(Xn)

Density correlation
Momentum space (homogeneous)(n)(2π)4δ(P1 + ⋯ + Pn)G̃(n)
(P
,
⋯,
P
)
=
F
.
T
.
G
1
n−1
( λ )
λ,s ,…,s1n

P = (ω, p)
Since G̃(n) is easier to calculate than Γ(n), we rewrite the flow eq. in terms of G̃(n).
00λFlow eqs. (after spin summation)

0th
1st
2nd

Egs,λ

1
∂λ
=
Ũ(p)
G̃(2)
(P) − n
λ
N
2n ∫p
( ∫ω
)
∂λ μλ =

1∫
2G̃(2)λ (0) P

G̃(n)
=
λ

∑

s1,⋯,snG̃(n)λ,s ,…,s
1

n

1
(3)
Ũ(p)G̃λ (P, − P) − U(0)
2

1
1
(2)
(2)
(4)
(3)
2
∂λG̃λ (P) = − Ũ(p)G̃λ (P) −Ũ(p′)G̃λ (P′, − P′, P) + G̃λ (P, − P) ∂λ μλ + U(0)
(
)
∫
2 P′
2

21

Truncation
1
1
(2)
(2)
(4)
(3)
2
∂λG̃λ (P) = − Ũ(p)G̃λ (P) −Ũ(p′)G̃λ (P′, − P′, P) + G̃λ (P, − P) ∂λ μλ + U(0)
(
)
2 ∫P′
2

= Cλ(P; ζ, rs)

Spin polarization: ζ =ρgs,↑ − ρgs,↓

ρgs,↑ + ρgs,↓

Wigner-Seitz radius: rs =From the flow equation for G̃(3,4), one can showλCλ(P; ζ, rs) = C0(P; ζ, rs)(1 + 𝒪 (rs f(P, ζ)))3/ 4π (ρgs,↑ + ρgs,↓)]][ [P = (rs2ω, rs p)

We ignore this (high-density expansion)

Analytic solution can be obtained
2/3
Egs,λ=1
Egs,λ=03
9π(1 + ζ )5/3 + (1 − ζ )5/3=
Kinetic 10rs2 ( 4 )
2
N
N
1
+
Ũ(p)
G̃(2)
(P) − n Exchange
0
∫
∫
2n p
( ω
)
1
+
ln coshŨ(p)C0(P) +
(
)
∫
2n P

1/3

3
9π
−
4π ( 4rs )(1 + ζ )4/3 + (1 − ζ )4/3
2

Ũ(p) (2)G̃0 (P)sinhŨ(p)C0(P)− Ũ(p)G̃(2)
(P)
0
(
)
C0(P)

Correlation Ecorr /N
22

1/3

Behavior at high density (rs → 0)
Ecorr
1
=
ln coshŨ(p)C0(P) +
(
)
∫
N
2n P

Scaling of C0 and G̃(2) w.r.t. rs:
0

We haveŨ(p) (2)G̃0 (P)sinhŨ(p)C0(P)− Ũ(p)G̃(2)
(P)
0
(
)
C0(P)

C0(ω, p; rs, ζ) = C0(rs2ω, rsp; 1,ζ)−1 (2) 2
G̃(2)
(ω,
p;
r
,
ζ)
=
r
s
s G̃0 (rs ω, rsp; 1,ζ)
0

Ecorr
1
1
(2)
(2)
=
ln [1 + Ũ(p)G̃0 (ω, p)] − Ũ(p)G̃0 (ω, p) +Ũ(p)C0(ω, p) + 𝒪(rs)) 4n ∫PN
2n ∫P (

+

+⋯

+

RPA
2nd-order ex.
Gell-Mann-Brueckner (GB) resum.!! Gell-Mann, Brueckner, PR (1957)• Exact behavior at high density ( rs → 0 ) given by GB resum. is reproduced.
• Higher-order contribution is resumed with solving flow eq.
Our approximation seems to be good at low and moderate rs
23

Reduction of integrals (e.g. ζ = 0)Ecorr
1 dω d 3p=
ln coshŨ(p)C0(P) +()
N
2n ∫ 2π ∫ (2π)3Cλ=0 (P) = 2Ns

quadruple integral

d 3p′ d 3p′′
U p′ θ −ξ p′′θ −ξ p + p′ + p′′)) − θ (−ξ (p′ + p′′)))∫ (2π)3 ∫ (2π)3 ( ) ( ( )) ( ( (2(ξ (p′′ + p) − ξ (p′′)) − ω
2×Ũ(p) (2)G̃0 (P)sinhŨ(p)C0(P)− Ũ(p)G̃(2)
(P)
0
(
)
C0(P)

(

ω 2 + (ξ (p′′ + p) − ξ (p′′))

)

2

2

−

(ξ(p) = p2 /2 − μ0)2(ξ (p′′ + p + p′) − ξ (p′′ + p′)) (ξ (p′′ + p) − ξ (p′′)) − ω

ω 2 + (ξ (p′′ + p + p′) − ξ (p′′ + p′))
ω 2 + (ξ (p′′ + p) − ξ (p′′))
(
)(
)
2

2

sextuple integralG̃(2)(P) is obtained analytically024Reduction of integrals (e.g. ζ = 0)Ecorr
1 dω d 3p=
ln coshŨ(p)C0(P) +()
N
2n ∫ 2π ∫ (2π)3Cλ=0 (P) = 2Ns

quadruple integral

d 3p′ d 3p′′
U p′ θ −ξ p′′θ −ξ p + p′ + p′′)) − θ (−ξ (p′ + p′′)))∫ (2π)3 ∫ (2π)3 ( ) ( ( )) ( ( (2(ξ (p′′ + p) − ξ (p′′)) − ω
2×Ũ(p) (2)G̃0 (P)sinhŨ(p)C0(P)− Ũ(p)G̃(2)
(P)
0
(
)
C0(P)

(

ω 2 + (ξ (p′′ + p) − ξ (p′′))

)

2

2

−

(ξ(p) = p2 /2 − μ0)2(ξ (p′′ + p + p′) − ξ (p′′ + p′)) (ξ (p′′ + p) − ξ (p′′)) − ω

sextuple integral

ω 2 + (ξ (p′′ + p + p′) − ξ (p′′ + p′))
ω 2 + (ξ (p′′ + p) − ξ (p′′))
(
)(
)
2

2

G̃(2)
(P) is obtained analytically
0
Ecorr
1
dω ∞
=
d p p 2 ln coshŨ(p)C0(P ) +()N(2π)2 n ∫ 2π ∫0Ũ(p) (2)G̃ (P )sinhŨ(p)C0(P )− Ũ(p)G̃(2)
(P )
0
(
)
C0(P ) 0

double integral
2N1
1
Cλ=0 (P ) = − sU (x)e−i(p′−p′′)⋅x θ (−ξ (p′′ + p)) − θ (−ξ (p′′)) θ (−ξ (p + p′)) − θ (−ξ (p′))
−
[
]
[
]
∫
∫
2 p′,p′′ x( iω + ξ(p′′ + p) − ξ(p′′) iω + ξ (p′ + p) − ξ (p′) )
=−

+
=−
+

Ns

pF

(2π)6 ∫−p
Ns

F

pF

(2π)6 ∫−p

F

2Ns

pF

2Ns

pF

(2π)3 ∫−p(2π)3 ∫−p

F

F

d p′z
d p′z
d p′z
d p′z

∫0
∫0

p F2 − pz′2

p F2 − pz′2

pF

∫−p

F

pF

∫−p

F

d p′r
d p′r

2π

∫0

2π

∫0

d ϕ′ p′rd ϕ′ p′r

pF

∫−p

F

pF

∫−p

F

d p′z′∫0d p′z′∫0p F2 − pz′′2

p F2 − pz′′2

2

2π

∞
∞
2π
1
1
d p′r′d ϕ′′ p′r′Re−dz
drd θ rU ( r 2 + z 2 )e −ip′z r cos(ϕ′−θ)+ip′z′r cos(ϕ′′−θ) e −ip′z z+ip′z′ z∫0∫0( i ω + p p′z + p 2 /2 i ω + p p′z′ + p 2 /2 ) ∫−∞ ∫0
2

2π

∞
∞
2π
1
1
d p′r′d ϕ′′ p′r′Re+dz
drd θ rU ( r 2 + z 2 )e −ip′z r cos(ϕ′−θ)−ip′z′r cos(ϕ′′−θ) e −i( p′z +p′z′+p)z∫0∫0( i ω + p p′z + p 2 /2 −i ω + p p′z′ + p 2 /2 ) ∫−∞ ∫02d p′z′

p F2 − pz′2

p F2 − pz′′2 Re

∞
1
1
1
−
d r J1 rr (( i ω + p p′z + p 2 /2 i ω + p p′z′ + p 2 /2 ) ∫0

d p′z′

p F2 − pz′2

p F2 − pz′′2 Re

∞
1
1
1
+
d r J1 r
2
2∫r (( i ω + p p′z + p /2 −i ω + p p′z′ + p /2 ) 02

p F2 − pz′2 J1 r
) (p F2 − pz′2 J1 r
) (p F2 − pz′′2 K 0 (r | p′z − p′z′| ))p F2 − pz′′2 K 0 (r | p′z + p′z′ + p | ))analytic integral is possible!

[Zwillinger, “Table of integrals, series, and products”]

Drastically reduces the dimension of integrals
Fast numerical calculation!!
(only a few minutes to obtain Ecorr with laptops)
25

double integral!!

Correlation energy per particle
ϵxc = ϵx + ϵcorr

39π
ϵx = −
4π ( 4rs )1/3(1 + ζ )4/3 + (1 − ζ )4/3
2

Correlation energy in spin polarized (ζ = 0) and unpolarized (ζ = 1) cases

TY, Naito, PRResearch (2021)
TY, Naito, PRB (2022)

0.00

εcorr (a.u.)

−0.01
−0.02
−0.03
−0.04
FRG-DFT (2nd-order vertex expansion)
Gell-Mann–Brueckner resummation
DMC [Ceperley, Alder 1980]
DMC [Zong, Lin, Ceperley 2002]
DMC [Spink, Needs, Drummond 2013]

−0.05
−0.06
−0.07
−0.08

0

20

40

60

80

Gell-Mann-Bruckner
approximation(Exact at rs → 0)100

Wigner-Seitz radius rs (a.u.)
Wigner-Seitz radius rs (a . u.)
FRG-DFT results agree with Monte-Carlo (MC) results without any empirical parameter!
FRG-DFT gives many data points than MC! ⇒ LDA functional without fitting
26

Construction of EDF without empirical parameters (ζ = 0 case)
LDA
Ecorr
[ρ] = dxϵcorr(ρ(x))ρ(x)∫Correlation part of LDA EDF

e.g.) PZ81

Many of conventional EDFs (PZ81, VWN, PW92…):

The form of ϵcorr(ρ) is assumed empiricallyand free parameters are fit to few DMC data…

ϵcorr (rs) =Perdew, Zunger, PRB (1981)

A ln rs + B + Crs ln rs + Drs rs < 1 a . u .
γ / (1 + β1 rs + β2rs)

0.00

Many-data point obtained by FRG-DFT

0.00 −0.01

Non-empirical parameter in 10 ≤ rs < 10(physically relevant region)2

Details • Linear interpolation between data

εcorr (a.u.)

−6

−0.01

εcorr (a.u.)

Numerical-table method

rs ≥ 1 a . u .

−0.02

−0.02
−0.03 −0.03−0.04 −0.04
−0.05

−0.05

−0.06

resum in rs < 10−6a . u .
• Replaced by Gell-Mann-Brueckner• Extrapolation to rs ≥ 102a . u . by PZ81-type function (parameters are
determined from data in 95a . u . < rs < 100a . u . )

−0.07

−0.06

FRG-DFT (2nd-ordervertex expansion)
FRG-DFT(2nd-order vertex e
Gell-Mann–Brueckner resummation
Gell-Mann–Bruecknerre
DMC [Ceperley, Alder 1980]
DMC [Zong, Lin,DMC
Ceperley 2002][Ceperley, Alder 1980]
DMC [Spink, Needs,
Drummond 2013]
DMC[Zong, Lin, Ceperley 200−0.08 −0.07
0
20

DMC
[Spink,Drummon40
60
80 Needs,100−0.08Wigner-Seitz radius r (a.u.)
0
20
40s
60

Wigner-Seitz radius rs (a65536 points in logarithmic mesh

But the results hardly depend on these choices.

Derivative of ϵcorr are also obtained analytically.
Need for Kohn-Sham calculation
LDA
LDA
δEex
[ρ] δEcorr
[ρ]
Δ
ρ(x′)
− + dx′++ϕ (x) = ϵiϕi(x)| x − x′|δρ(x)
δρ(x) ) i
( 2 ∫LDA[ ρ]δ Ecorr= ϵcorr (ρ (x)) + ϵ′corr (ρ (x))ρ (x)δρ (x)

27

1ϵ′corr (rs) =⋯∫2nrs P

see arXiv:2010.07172

Kohn-Sham calc. for noble gas atoms
TY, Naito, PRResearch (2021)

Comparison to otherconventional functionals

Ne Ar

gs
gs
E /EPZ81 (%)

100.010100.005

VWN, PW92 (LDA obtained by fitting of MC data)
PBE (generalized gradient approximation)

Kr
Xe
VWN
PW92
Chachiyo
revChachiyo

Rn
FRG-NT
FRG-PZPBE

LDA from FRG

FRG shows comparable
results to otherconventional LDAs
despite no empiricalparameter.

100.000
100.5
100.4
100.3
100.2
100.1
100.0

99.99599.990

0

20

40

60

80 10099.985
0

20

40

60

Atomic Number Z
28

80

100

The key to improve
the accuracy is thegradient eﬀects.

Summary
Density functional theory (DFT)Eﬃcient method to analyze electrons, nuclei, …
How to derive energy density functional (EDF)?
Key quantity in DFT
Eﬀective-action formalism will provide a new way to construct EDF
Functional evolution equation in a closed form for eﬀective action
FRG techniques are useful
Application: vertex expansion (electrons, …)
Other direction: spectral function
superfluid system

TY, Yoshida, Kunihiro, PTEP (2019)
TY, Kasuya, Yoshida, Kunihiro PTEP (2021)

Outlook

Inclusion of gradient eﬀect (LPA? Neural-net ansatz?)
Application to realistic nuclear matter, nuclei
Numerical study of superfluid systems
29

Appendix
Calculation of excited states

30Analytic continuation to real time
FRG technique to obtain real-time spectral function via analytic
continuation of flow eq. for correlation function has been developed.
e.g.) meson spectral functionin O(4), quark-meson model

Kamikado, Strodthoﬀ, von Smekal, Wambach (2020)
Tripolt, Strodthoﬀ, von Smekal, Wambach (2014)
TY, Kunihiro, Morita (2016), (2017)…

This technique can be used to calculate spectral function of density
fluctuation in FRG-DFT!
Flow eq. for correlation with Matsubara freq. G̃(2)(ωi, p)λAnalytic continuation for flow eq.
Flow eq. for real-time correlation G̃(2) (ω, p)R,λSolutionSpectral function in density channel: ρd(ω, p) = − 2ImG̃(2)

R,λ=1

31

(ω, p)

One-dim. nuclear matterU(r)• Finite particle number in finite volume• Infinite matter

Kemler, Pospiech, Braun, JPG (2017)
TY, Yoshida, Kunihiro, PRC (2019)25

20

Identical fermions (e.g. spin polarized system)
interacting via very-simplified nuclear-like force

repulsive core

15

10

5

attractive force
0.2

0.4

0.6

0.8

1.0

r

U(r) =

g
σ1 π

e

r2
− 2
σ1

−

g
σ2 π

e

r2
− 2
σ2

g = 12, σ1 = 0.2, σ2 = 0.8

fermion mass=1 unit

1.2

-5

Alexandro, Myczkowski, Negele, PRC (1989)

32

Excitation in Tomonaga-Luttinger liquid
Pustilnik, Khodas, Kamenev, Glazman, PRL (2006)
Teber, PRB (2007)
Imambekov, Schmidt, Glazman, RMP (2012)

Previous results(such as bosonization)

(a)

"#

−"#

multi-pair production**+ " = "2 + 2"# " /2

"



0

(b)




(c)
-. (*, ")free particles

support for
free particles

"

0

2"#

*, " = "2 − 2"# " /2

Multi-pair production

multi-pair
production

0*, (")*+ (")*

Mahan (X-ray edge) singularity
Nozieres, De Dominicis, PR178 (1969); Mahan (1981)

33

FRG-DFT result of ρd(ω, p)

TY, Yoshida, Kunihiro, PTEP (2019)

At saturation point

No multipair
contribution∂λG̃(2)
(ω,
p)
=
R,λ
G̃(2)
R,λ

G̃(2)
R,λ

No multi-pair diagrams+34

ρd(ω, p) at fixed momentum
TY, Yoshida, Kunihiro, PTEP (2019)

103
FRG-DFTFree
RPA

Ωd(!, p = pF)

102
101
100
10°1
10°2
10°3
0.0

0.5

1.0

1.5

2.0

2.5

3.0

!/µ0
(2)
∂λG̃R,λ(ω, p) =
G̃(2)
R,λ

G̃(2)
R,λ

RPA

Reproduced in FRG-DFT!

+
35

3.5