Fermi Flow: Ab initio study of fermionsat nite temperaturefi

Lei Wang (王磊)2105.08644

Institute of Physics, CASbuwantaiji/FermiFlow

https://wangleiphy.github.io& work in progresster is very highly compressed, such that the temperature is of the orderof (or lower than) the Fermi temperature (for the relevant parameter
range, see Fig. 1 and, for the parameter definitions, see Sec. II).
An important role in the theoretical description of quantum plasmas is being played by the quantum kinetic theory.31–38 During the last
25 years, improved and generalized quantum kinetic equations have been
derived starting from reduced density operators, e.g., Refs. 39 and 40, or
nonequilibrium Green functions (NEGFs);41–44 for text books, see Refs.
40 and 45–47 and references therein. Another direction in quantum
plasma theory is first principles computer simulations such as quantum
Monte Carlo (QMC),4,48–55 semiclassical molecular dynamics (SC-MD)
with quantum potentials, e.g., Ref. 56, electronic force fields,57,58 and various variants of quantum MD, e.g., Refs. 59–63.
A recent breakthrough occurred with the application of
Kohn–Sham density functional theory (DFT) simulations because
they, for the first time, enabled the self-consistent simulation of realistic warm dense matter that includes both plasma and condensed matter phases, e.g., Refs. 64–66. Further developments include orbital-free

Uniform electron gas

The simplest model of interacting electrons (“the zeroth element”)
N

2

2
ℏ ∇i

2

Jellium

Relevant to condensed matter, warm dense
matter, thermal density functionals …

Temperature

e
H=−
+
∑ 2m
∑ | xi − xj |i=1
i<jDensity

Bonitzplane
et al,
Phys.ofPlasmas’20
FIG. 1. Density-temperature
with examples
plasmas and characteristic
plasma parameters. ICF denotes inertial confinement fusion. Metals (semicon-

extensive investigatio
sent an overview
directions.
Motivated by t
retical description omatter is attracting
x-ray Thomson scat
powerful methods arrium Green function
All of the abovand require substant
the above-mentioned
length scales and simthat would allow todesirable. Possible cthat are obtained vi
case of classical plasm
dense plasmas have
Manfredi and Haas.8
eral assumptions, theCorrections of the cobtained in Refs. 87
equations from the tRef. 89. We also meon the computation
The goal of thi
recent ab initio simu
ter conditions, includ
rections developmen
the field of QHD forrecent developmentsof the finite-tempera
ulations of dense hy
density response funpotential of an ion in
input for the local fie
of ion-acoustic modeThis paper is oparameters of warm
density range. SectioDV=V # 0:003. Standard PIMC (black) is only useful forhigh temperatures and low densities where fermionicexchange does not play an important role and, therefore,
does not give access to the WDM regime. PB-PIMC (green)significantly extends the possible parameter combinations toLimitation of current approaches•

There is no reliable data at low
temperature and intermediate
interaction, despite of decades ofresearch•

The workhorse (Path Integral
Monte Carlo methods) su er fromthe notorious “sign problem”•

Opportunity for deep learning
Dornheim et al, Phys. Plasmas ’17

ff

FIG. 2. Density-temperature-plane around the WDM regime. Shown are the

in
a

m
U
g
a

F
fu

Warm up: Classical Coulomb gaseH=∑ | xi − xj |
i<j
T/EF

Z = dx∫?

3N −βH(x)eClassical limit

2

Quantum ground stateff

Di usion Monte Carlo, Variational Monte Carlo,…

3
rs =( 4πn )

1/3

Warm up: Classical Coulomb gas
2eH=∑ | xi − xj |i<jVariational free-energyℒ=

Z = dx∫x∼p(x)

3N −βH(x)

e

≥
−
ln
Z
ln
p(x)
+
βH(x)
[
]

variational probability distribution

Turn the sampling problem to an optimization problem.
Not necessarily easy. But may better leverage deep learning engine.

𝔼…

Warm up: Classical Coulomb gas
2eH=∑ | xi − xj |i<jVariational free-energyℒ=

Z = dx∫x∼p(x)

3N −βH(x)

e

≥
−
ln
Z
ln
p(x)
+
βH(x)
[
]

variational probability distribution

Turn the sampling problem to an optimization problem.
Not necessarily easy. But may better leverage deep learning engine.

“mean- eld” approaches
Factorized probability
Pairwise interaction
Deep generative models

fl

fi

𝔼

normalizing ow Li, LW, PRL ‘18
autoregressive model, Wu, LW, Zhang, PRL ‘19…

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“random” images“natural” images 1                      <     <  !
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C
    ,     -                 ,                                     
       HProbabilistic Generative Modeling<latexit sha1_base64="l+MU6Wop3O4gDLGg4JnxmuonPhM=">AAACEHicbVDNSsNAGNzUv1r/oj16WSxCvZREBPVW9OKxgrGFNpTNZtMu3WzC7kYaQp7CB/Cqj+BJvPoGPoGv4abNwbYOLDvMfB/fMF7MqFSW9W1U1tY3Nreq27Wd3b39A/Pw6FFGicDEwRGLRM9DkjDKiaOoYqQXC4JCj5GuN7kt/O4TEZJG/EGlMXFDNOI0oBgpLQ3NOoybAy9ivkxD/WXT/AwOzYbVsmaAq8QuSQOU6AzNn4Ef4SQkXGGGpOzbVqzcDAlFMSN5bZBIEiM8QSPS15SjkEg3m4XP4alWfBhEQj+u4Ez9u5GhUBbh9GSI1Fgue4X4n9dPVHDlZpTHiSIczw8FCYMqgkUT0KeCYMVSTRAWVGeFeIwEwkr3tXBFTYtoMq/pZuzlHlaJc966bln3F432TVlRFRyDE9AENrgEbXAHOsABGKTgBbyCN+PZeDc+jM/5aMUod+pgAcbXL/rwnUs=</latexit>p(x)How to express, learn, and sample from ahigh-dimensional probability distribution ?
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  Page 15“… the images encountered in
AI applications occupy anegligible proportion othe volume of image space.”  " 7          &    0   %  ?
    AF:)    ) ,F:)-    F         '     
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“random” images“natural” images

 

f

9

 1                      <     <  !
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C
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       HGenerative models and their physics genes
Goodfellow,
NIPS tutorial, 1701.00160<latexit sha1_base64="l+MU6Wop3O4gDLGg4JnxmuonPhM=">AAACEHicbVDNSsNAGNzUv1r/oj16WSxCvZREBPVW9OKxgrGFNpTNZtMu3WzC7kYaQp7CB/Cqj+BJvPoGPoGv4abNwbYOLDvMfB/fMF7MqFSW9W1U1tY3Nreq27Wd3b39A/Pw6FFGicDEwRGLRM9DkjDKiaOoYqQXC4JCj5GuN7kt/O4TEZJG/EGlMXFDNOI0oBgpLQ3NOoybAy9ivkxD/WXT/AwOzYbVsmaAq8QuSQOU6AzNn4Ef4SQkXGGGpOzbVqzcDAlFMSN5bZBIEiM8QSPS15SjkEg3m4XP4alWfBhEQj+u4Ez9u5GhUBbh9GSI1Fgue4X4n9dPVHDlZpTHiSIczw8FCYMqgkUT0KeCYMVSTRAWVGeFeIwEwkr3tXBFTYtoMq/pZuzlHlaJc966bln3F432TVlRFRyDE9AENrgEbXAHOsABGKTgBbyCN+PZeDc+jM/5aMUod+pgAcbXL/rwnUs=</latexit>p(x)Generative models and their physics genes
Goodfellow,
NIPS tutorial, 1701.00160<latexit sha1_base64="l+MU6Wop3O4gDLGg4JnxmuonPhM=">AAACEHicbVDNSsNAGNzUv1r/oj16WSxCvZREBPVW9OKxgrGFNpTNZtMu3WzC7kYaQp7CB/Cqj+BJvPoGPoGv4abNwbYOLDvMfB/fMF7MqFSW9W1U1tY3Nreq27Wd3b39A/Pw6FFGicDEwRGLRM9DkjDKiaOoYqQXC4JCj5GuN7kt/O4TEZJG/EGlMXFDNOI0oBgpLQ3NOoybAy9ivkxD/WXT/AwOzYbVsmaAq8QuSQOU6AzNn4Ef4SQkXGGGpOzbVqzcDAlFMSN5bZBIEiM8QSPS15SjkEg3m4XP4alWfBhEQj+u4Ez9u5GhUBbh9GSI1Fgue4X4n9dPVHDlZpTHiSIczw8FCYMqgkUT0KeCYMVSTRAWVGeFeIwEwkr3tXBFTYtoMq/pZuzlHlaJc966bln3F432TVlRFRyDE9AENrgEbXAHOsABGKTgBbyCN+PZeDc+jM/5aMUod+pgAcbXL/rwnUs=</latexit>p(x)U
Quantum
Circuits

TensorNetworks
A. Georges et al.: Dynamical mean-field theory of . . .
2
G 21
0 ~ i v n ! 5i v n 1 m 2t G ~ i v n ! .

(23)

The same density of states is also realized for a random
Hubbard model on a fully connected lattice (all N sites
pairwise connected) where the hoppings are independent random variables with variance t 2ij 5t 2 /N (seeSec. VII).
Finally, the Lorentzian density of statesD~ e !5t
p ~ e 2 1t 2 !

(24)

21

“Born Machine”

Generative modeling

Statistical Physics

Known: samples

Known: energy function

Unknown: generating distribution

Unknown: samples, partition function

Modern generative models for physics
Physics of and for generative modelingGenerative models and their physics genes
Goodfellow,
NIPS tutorial, 1701.00160<latexit sha1_base64="l+MU6Wop3O4gDLGg4JnxmuonPhM=">AAACEHicbVDNSsNAGNzUv1r/oj16WSxCvZREBPVW9OKxgrGFNpTNZtMu3WzC7kYaQp7CB/Cqj+BJvPoGPoGv4abNwbYOLDvMfB/fMF7MqFSW9W1U1tY3Nreq27Wd3b39A/Pw6FFGicDEwRGLRM9DkjDKiaOoYqQXC4JCj5GuN7kt/O4TEZJG/EGlMXFDNOI0oBgpLQ3NOoybAy9ivkxD/WXT/AwOzYbVsmaAq8QuSQOU6AzNn4Ef4SQkXGGGpOzbVqzcDAlFMSN5bZBIEiM8QSPS15SjkEg3m4XP4alWfBhEQj+u4Ez9u5GhUBbh9GSI1Fgue4X4n9dPVHDlZpTHiSIczw8FCYMqgkUT0KeCYMVSTRAWVGeFeIwEwkr3tXBFTYtoMq/pZuzlHlaJc966bln3F432TVlRFRyDE9AENrgEbXAHOsABGKTgBbyCN+PZeDc+jM/5aMUod+pgAcbXL/rwnUs=</latexit>p(x)U
Quantum
Circuits

TensorNetworks
A. Georges et al.: Dynamical mean-field theory of . . .
2
G 21
0 ~ i v n ! 5i v n 1 m 2t G ~ i v n ! .

(23)

The same density of states is also realized for a random
Hubbard model on a fully connected lattice (all N sites
pairwise connected) where the hoppings are independent random variables with variance t 2ij 5t 2 /N (seeSec. VII).
Finally, the Lorentzian density of statesD~ e !5t
p ~ e 2 1t 2 !

(24)

21

Generative models and their physics genes
Goodfellow,
NIPS tutorial, 1701.00160<latexit sha1_base64="l+MU6Wop3O4gDLGg4JnxmuonPhM=">AAACEHicbVDNSsNAGNzUv1r/oj16WSxCvZREBPVW9OKxgrGFNpTNZtMu3WzC7kYaQp7CB/Cqj+BJvPoGPoGv4abNwbYOLDvMfB/fMF7MqFSW9W1U1tY3Nreq27Wd3b39A/Pw6FFGicDEwRGLRM9DkjDKiaOoYqQXC4JCj5GuN7kt/O4TEZJG/EGlMXFDNOI0oBgpLQ3NOoybAy9ivkxD/WXT/AwOzYbVsmaAq8QuSQOU6AzNn4Ef4SQkXGGGpOzbVqzcDAlFMSN5bZBIEiM8QSPS15SjkEg3m4XP4alWfBhEQj+u4Ez9u5GhUBbh9GSI1Fgue4X4n9dPVHDlZpTHiSIczw8FCYMqgkUT0KeCYMVSTRAWVGeFeIwEwkr3tXBFTYtoMq/pZuzlHlaJc966bln3F432TVlRFRyDE9AENrgEbXAHOsABGKTgBbyCN+PZeDc+jM/5aMUod+pgAcbXL/rwnUs=</latexit>p(x)U
Quantum
Circuits

TensorNetworks
A. Georges et al.: Dynamical mean-field theory of . . .
2
G 21
0 ~ i v n ! 5i v n 1 m 2t G ~ i v n ! .

(23)

The same density of states is also realized for a random
Hubbard model on a fully connected lattice (all N sites
pairwise connected) where the hoppings are independent random variables with variance t 2ij 5t 2 /N (seeSec. VII).
Finally, the Lorentzian density of statesD~ e !5t
p ~ e 2 1t 2 !

(24)

21

Generative modeling with normalizing ows

1609.03499 1711.10433

https://deepmind.com/research/case-studies/wavenet

Glow 1807.03039
https://blog.openai.com/glow/fl

WavenetGenerative modeling with normalizing ows

1609.03499 1711.10433

https://deepmind.com/research/case-studies/wavenet

Glow 1807.03039
https://blog.openai.com/glow/fl

WavenetNormalizing ow in a nutshell
If the mapping f is 1-to-1, then the total area (or volume) should
not change after the transformation from x to z.

latent space(z)“neural net”

p(x)with 1 neuron

physical
space

fl

𝒩

12

Normalizing Flows
Change of variables x ↔ z with deep neural nets∂z(z) det( ∂x )

p(x) =

Review article
1912.02762

x

Neural Net

composable, di erentiable, and invertible mapping between manifolds

z∼

(z)

fl

ff

𝒩

𝒩

Learn probability transformations with normalizing ows

Classical Coulomb gas in a parabolic trap
N

2
xi

1
H=
+
∑ | xi − xj | ∑ 2
i<j
i
N=6

https://colab.research.google.com/drive/13wvsGtV4eTN4v6sXLf2Z52FCyCd6wQZP?usp=sharing

Other scienti c applications of ow

Noe et al, Science ‘19

Kanwar et al, PRL ‘20
fl

Lattice eld theoryfiMolecular simulation

Downloaded from http:/fi

R E S EA R C H A R T I C LE

𝒩𝔼

Two training approaches
Variational calculation“learn from data”“learn from Hamiltonian”

x∼dataset [ln p(x)]x ∼ datasetℒ = dx p(x)[ln p(x) + βH(x)]∫zSamples from the given datasetx

Neural Netℒ=−

Neural Net

Density estimationz∼(0, Σ)

Generate samples from the modelDensity estimation

Variational calculation“learn from data”“learn from Hamiltonian”ℒ=−(π | | p) =x∼dataset [ln p(x)]∑xπ ln π −∑π ln p

x

𝔼

𝕃

𝕂

ℒ
𝕃

𝕂

Two training approachesℒ = dx p(x)[ln p(x) + βH(x)]∫ℒ + ln Z =

(

p||

e

Z )

−βH

≥0

Now, march into
the quantumworldQuantum world

Probability distribution p

Density matrix ρ

Kullback-Leibler divergence

Quantum relative entropy

Variational free-energy

Variational free-energyℒ = dx p(x)[ln p(x) + βH(x)]∫ℒ = Tr(ρ ln ρ) + βTr(Hρ)(p | | q)

𝕃

𝕂Classical world

S (ρ | | σ)

Density matrix
Classical probability

Quantum state0 < μn < 1

Ψ(x) = ⟨x | Ψ⟩∑n∫μn = 1ρ=∑n

2

dx | Ψ(x) | = 1

μn | Ψn⟩⟨Ψn |Trρ = 1￼𝒫𝒫

￼How to represent variational density matrix so it is physical & optimizable ?
ρ≻0

†

ρ =ρ

⟨x | ρ | x′⟩ = (−1) ⟨ x | ρ | x′⟩𝒩

Idea: to parametrize a density,
think about the transformation

Gaussian variables(z)

Normalizing
Flow

Correlated variablesp(x)ℒ = dx p(x)[ln p(x) + βH(x)]∫Imposing constraints to the transformation for physical densities

We need unitary transformations
ρ0

Unitary
Transformation

ρ = Uρ0U
Trρ = 1 ρ ≻ 0

†

†

ρ =ρ

ℒ = Tr(ρ ln ρ) + βTr(Hρ)
How to parametrize and learn unitary transformations ?

however, simply implies
the
and
lag
independent,—
2[ax'&'/ax', p ]+PH YSICAL REVIEWVOLUM E 85, NUMBER 4
FEBRUARY
1$, 1952 by aalways be replaced'/a*' [a*"/a*",nonsingular, together wi]+
Point Transformations in Quantum Mechanics
ditions expressing the re= —,'[[ax'&/»', ax'/ax'&]+, p„]~suchofexistences
The
BRYCE SKLIGMAN DEWITT
Ecole d'Etd de Physique Theorigue de L'University
Francealter the discussion
de Grenoble, ax'&'/ax']]
res Homches, Halte Savoie, notax" /ax-&',[pI,(Received September 14, 1951)
Under general coordin=-,'[a,j', pI,'~A[ax'/ax", a'x'&/»"ax']= p, , (3.6) tities V, A;, and G;, trans
An isomorphism is shown to exist between the group of point transformations in classical mechanics anda covariant
vector, anda certain subgroup of the group of all unitary transformations in quantum mechanics. This isomorphismishas thetransformation
i:nverse signi6cantused to indicate whichthat the showsof physically
can be haveconstructed
quantum that
analogstheclassical expressions
respectively the na
coordinate
uniquely in anysame
quantumas (3.There
form system.
3). is no ambiguity in the ordering of noncommutingpotential.
likew
can
6;;
operators, and the method of constructing the quantum analogs point-transformais covariant under general coordinateoftherepresentations
The
unitaryittransformations. The method is actually only applicable to systems having Lagrangians which potential.
are at most However,
the in physics.
be allobtained
mayincludes
by aredetermining
quadratic in thetion
velocities,groupbut thissystems which
presently of interest
cannot The
be "transformed
method is applied
to two intrinsicallynonlinear examples,of whichAn
is thegravitational field, The correct
in6nitesimalof the one
infinitesimal
generatorsgroup.
whincoordinate
systemHamiltonian operator for a quantized version of Einstein's gravitational theory is constructed.
unitary
form
point transformation may be expressed in the
We shall tentatively iden
generator
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of
(3.
~~'(x)
7)
formulating the quantum analog of Eq. (3.2). For theltof&. INTRODUCTION
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5=-', [X'(x), p, .
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group of(3all.9)point transformations in cla

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=l[a

]+,

+ ,'[

x"=*+

S

'

f

5]

],

Canonical transformations
Classical world

Quantum world

Symplectic

Unitary· p)· = ∇ G(x, p)
(x,
(x,p)
(

−

)

· p)
· = i[(x, p), G(x, p)](x,point transformation“quantized” point transformation

G(x, p) = v(x) ⋅ p

1
G(x, p) = {v(x), p}
2

𝕀

𝕀

Neural canonical transformations, Li, Dong, Zhang, LW, PRX ’20

Quantum generalization to fermions, Xie, Zhang, LW, 2105.08644
see also Cranmer et al, 1904.05903

Neural canonical transformation
Normalizing

free electrons
coordinates

Flowtransformed
electron coordinates

ρ0ρ = Uρ0U†ℒ = Tr(ρ ln ρ) + βTr(Hρ)

￼

𝒫

fl

𝒫

￼

Moreover, the ow should be permutation-equavariant
to preserve fermionic statistics⟨x | ρ | x′⟩ = (−1) ⟨ x | ρ | x′⟩

Backflow corrections in QMC

Back ow transformation
Backflow transformations

2. What is backflow?

A set of collective coordinates {xi(R)} is defined so that theresulting quasi-particles “avoid” each other:3

Collective coordinates

1
2

Trial wave functions in QMC

x′i = xi +∑j≠iη( | xi − xj | )(xj − xi)

The Slater-Jastrow wave function

Dynamical view of how quasi-particles ought to behave.
As 1 moves, 2 clears the way, while 3 barely notices

Backflow transformationstransformations
Backflow
Backflow in real systems
Conclusions

Pablo López Ríos

Wigner & Seitz 1934, Feynman 1954, …

Ψ(x′): interacting particles

flNowadays, view it as a residual network, or, discretization of a ow
fl

￼

￼

Ψ(x): independent particles

Equivariant
neural network

Neural back ow transformation

xt

x) =

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Köhler et al 1910.00753
Pfau et al, 1909.02487
Hermann et al, 1909.08423 Wirnsberger et al, 2002.04913
fl

𝒫

𝒫

v(

+

Li et al 2008.02676
Biloš et al 2010.03242

Continuous neural back ow transformation
Residual network

Neural ODE

xt+1 = xt + f(xt)dx/dt = f(x)flChen et al, 1806.07366

Harbor el al 1705.03341
Lu et al 1710.10121,
E Commun. Math. Stat 17’…

Continuous unitary transformation as a ow

∂ Ψn(x, t)

2

∂t

−i
{v, ∇} Dewitt 19522+ ∇ ⋅ ( | Ψn(x, t) | v) = 0

dx
=v
dt

(c)

∂
i | Ψn⟩ = G | Ψn⟩
∂t

2

2

d ln | Ψn(x, t) |=−∇⋅v
dt
Complex density
flSimple density

FermiFlow: Equivariant ow of fermions
(a)
(c)

ρ0

(b)

free electrons
coordinates

dx
=v
dt

unitary transformation

ML technique:transformed
electron coordinatesρ = Uρ0U†

Perm-equavariant normalizing ow/Neural ODE/Invertible ResNet

Physical picture: Variational approximation of adiabatic preparation of thermal equilibrium
Optimal control a PDE with particle methodflflMathematics:

Details: Objective functionℒ=ln μn + βn∼μn [x∼pn(x)Boltzmann
distribution

Classical distribution

0 < μn < 1parametrized by μn for free electrons

Quantum distribution∑npn(x) = | Ψn(x) |μn = 1Discrete probabilistic model(Softmax, Autoregressive model,…)Continuous probabilistic model(Permutation-equivariant continuous ow)

1
1
1
2
loc
2
En (x) = − ∇ ln pn(x) − [ ∇ln pn(x)] +∑ | xi − xj |
4
8
i<j
fl

𝔼

“Local energy”
𝔼

Born
rule

2

parametrized by the equivariant drift v

loc[En (x)]]Details: Gradient estimators
Nested REINFORCEClassical
distribution

Quantum
distribution∇ϕ ℒ =n∼μn [(∇θ ℒ = βln μn + βn∼μn x∼pn(x)x∼pn(x)loc[En (x)]) ∇ϕ ln μn]

loc[En (x) ∇θ ln pn(x)]

𝔼

𝔼

𝔼

𝔼

Variance reduction in both estimators by subtracting baselinedz φi (z)φj (z) =dx ""det" φi ( f ( x ))φ ( f ( x )) = δij
j
∂x "

(8)

Related works

Thus, by using a common flow f : X → Z, the orthogonality is maintained

1.3

"
"1
Orthonormal basis of quantum flows""2∂f"det "
ψ
(
x
)
=
φ
(
f
(
x
))
i
i
"
" ψ ( x ), we can extend the preIf we wish to build a flexibly orthonormal basis of wave functionsi∂x
vious result. We begin with a fixed orthonormal basis φi (z), where the orthogonality condition
is: 1.4 Adding phase degrees of freedom

Neural nets for ground state(9)

Quantum algorithms!

∗
The orthonormal basis can be extended
multiplying
dz φby(z)
φ
ij ( z ) = δij by a common phase

(7)

"
−
1
"
If we change variables from z to x = f (z) we have:"1" 2 iη (x)∂f"det " eQuantumψi ( x ) = "φi ( f ( x ))
(10)
"
"
"
!
!
∂x
"
"
∂Ef "R AT O R ∗of quantumflows∗ U basis
F L O W1.3dz
S Orthonormal
A
S
A
N
I
TA
R
Y
O
P
"
φi (z)φj (z) = dx "det " φi ( f ( x ))φj ( f ( x )) = δij(8)Circuitfor an arbitraryIf ηweθ wish
: X to→R.a flexibly orthonormal basis
build∂x of wave functions ψi (x), we can extend the previous result. We begin with a fixed orthonormal basis φi (z), where the orthogonality condition
•Can
view
the
quantumflow
asaunitary
operator
We can also
express
this
asanintegraloperatoris:Thus, by using a common
flow f : X → Z, the orthogonality is maintained!
"1!
dz φi (z")φ∗j (z) = δij" 1"
(7)
2
"
"
∂
f
2
iη ( x )
"
"
"
"
∂
f
ψIfi (wex )change= Uvariables
[φ
(
z
)]
=
dz
δ
(
f
(
x
)
−
z
)
det
e
φ
(
z
)]
(11)
[
−1 ( z ) "
i
i
"
from
z
to
x
=
f
we
have:
"
"
ψi ( x ) = φi ( f ( x )) "det "(9)FermiNet, Pfau et al 1909.02487∂x
β-VQE,Liu,
Mao,
Zhang, LW 1912.11381
"
" ∂x
!
!
"
∂f "

dz
φi (z)φ∗j (zow…
) = dx ""det "" φi ( f ( x ))φ∗j ( f ( x )) = δij(8)PauliNet, neural back ow, Iterativeback∂xMartyn et al 1812.01015, Verdon et al 1910.02071
1.3 Orthonormal basis of quantum flows
Comparison
toanalternativeThus,
by using
a common flowapproachf : X → Z, the orthogonality is maintained
1.4 1.5
Addingphase degrees
of freedom

If we wish to build a flexibly "orthonormal
basis of wave functions ψi ( x ), we can extend the pre"
"
"
∂f "
"
An
alternative
approach
would
be
to
build
the
family
as
ψ
(
x
)
=
φ
(
f
(
x
))
det
result.byWe
begin
with
a" fixedii" orthonormal
The orthonomral basis can bevious
extended
multiplyingbya∂x
commonphasebasis φi (z(9)), where the orthogonality condition
is:
#
1
2

" 1 iη ( x)!
0ψ( x ) =" p∂X f( x" 2)e iη (x)
(12)
∗
"
"
The orthonormalcanbeextendedbymultiplyingbyacommonphaseψbasis
(
x
)
=
φ
(
f
(
x
))
det
e
(10)
dz
φ
(
z
)
φ
= δij
i
i
i
j ( z )Cranmer
"
"
et
al,
1904.05903∂xwhere the density p X ( x ) may itself be defined""" by∂ fa""" flow.
Note, in "Neural Network Quantumiη ( x )ψivariables( x ) = φi ( f ( x ))det z "toe x = f −1 ( z ) we have: (10)If
wechangefrom"for
anTomography"
arbitrary ηθ : Xp X→∂x an RBM as x was discrete. This approach is
State
( xR.
) and η ( x ) were modeled with""0
for an arbitrary ηθ : X → R.
!
!
expressive for a single
wave function, but imposing theornogonalityon∗ multiple wave"
∂conditionf"
We can also express this as an integral operator∗
1.4

Quantum ows

Adding phase degrees of freedom"

=

1
2

fl

fl

dz φi (z)φj (z) =

fl

(7)

dx ""det" φi ( f ( x ))φ ( f ( x )) = δij
j
"

(8)

dz φi (z)φj (z) =dx ""det" φi ( f ( x ))φ ( f ( x )) = δij
j
∂x "

(8)

Related works

Thus, by using a common flow f : X → Z, the orthogonality is maintained

1.3

"
"1
Orthonormal basis of quantum flows""2∂f"det "
ψ
(
x
)
=
φ
(
f
(
x
))
i
i
"
" ψ ( x ), we can extend the preIf we wish to build a flexibly orthonormal basis of wave functionsi∂x
vious result. We begin with a fixed orthonormal basis φi (z), where the orthogonality condition
is: 1.4 Adding phase degrees of freedom

Neural nets for ground state(9)

Quantum algorithms!

∗
The orthonormal basis can be extended
multiplying
dz φby(z)
φ
ij ( z ) = δij by a common phase

(7)

"
−
1
"
If we change variables from z to x = f (z) we have:"1" 2 iη (x)∂f"det " eQuantumψi ( x ) = "φi ( f ( x ))
(10)
"
"
"
!
!
∂x
"
"
∂Ef "R AT O R ∗of quantumflows∗ U basis
F L O W1.3dz
S Orthonormal
A
S
A
N
I
TA
R
Y
O
P
"
φi (z)φj (z) = dx "det " φi ( f ( x ))φj ( f ( x )) = δij(8)Circuitfor an arbitraryIf ηweθ wish
: X to→R.a flexibly orthonormal basis
build∂x of wave functions ψi (x), we can extend the previous result. We begin with a fixed orthonormal basis φi (z), where the orthogonality condition
•Can
view
the
quantumflow
asaunitary
operator
We can also
express
this
asanintegraloperatoris:Thus, by using a common
flow f : X → Z, the orthogonality is maintained!
"1!
dz φi (z")φ∗j (z) = δij" 1"
(7)
2
"
"
∂
f
2
iη ( x )
"
"
"
"
∂
f
ψIfi (wex )change= Uvariables
[φ
(
z
)]
=
dz
δ
(
f
(
x
)
−
z
)
det
e
φ
(
z
)]
(11)
[
−1 ( z ) "
i
i
"
from
z
to
x
=
f
we
have:
"
"
ψi ( x ) = φi ( f ( x )) "det "(9)FermiNet, Pfau et al 1909.02487∂x
β-VQE,Liu,
Mao,
Zhang, LW 1912.11381
"
" ∂x
!
!
"
∂f "

dz
φi (z)φ∗j (zow…
) = dx ""det "" φi ( f ( x ))φ∗j ( f ( x )) = δij(8)PauliNet, neural back ow, Iterativeback∂xMartyn et al 1812.01015, Verdon et al 1910.02071
1.3 Orthonormal basis of quantum flows
Comparison
toanalternativeThus,
by using
a common flowapproachf : X → Z, the orthogonality is maintained
1.4 1.5
Addingphase degrees
of freedom

If we wish to build a flexibly "orthonormal
basis of wave functions ψi ( x ), we can extend the pre"
"
"
∂f "
"
An
alternative
approach
would
be
to
build
the
family
as
ψ
(
x
)
=
φ
(
f
(
x
))
det
result.byWe
begin
with
a" fixedii" orthonormal
The orthonomral basis can bevious
extended
multiplyingbya∂x
commonphasebasis φi (z(9)), where the orthogonality condition
is:
#
1
2

" 1 iη ( x)!
0ψ( x ) =" p∂X f( x" 2)e iη (x)
(12)
∗
"
"
The orthonormalcanbeextendedbymultiplyingbyacommonphaseψbasis
(
x
)
=
φ
(
f
(
x
))
det
e
(10)
dz
φ
(
z
)
φ
= δij
i
i
i
j ( z )Cranmer
"
"
et
al,
1904.05903∂xwhere the density p X ( x ) may itself be defined""" by∂ fa""" flow.
Note, in "Neural Network Quantumiη ( x )ψivariables( x ) = φi ( f ( x ))det z "toe x = f −1 ( z ) we have: (10)If
wechangefrom"for
anTomography"
arbitrary ηθ : Xp X→∂x an RBM as x was discrete. This approach is
State
( xR.
) and η ( x ) were modeled with""0
for an arbitrary ηθ : X → R.
!
!
expressive for a single
wave function, but imposing theornogonalityon∗ multiple wave"
∂conditionf"
We can also express this as an integral operator∗
1.4

Quantum ows

Adding phase degrees of freedom"

=

1
2

fl

fl

dz φi (z)φj (z) =

fl

(7)

dx ""det" φi ( f ( x ))φ ( f ( x )) = δij
j
"

(8)

Demo: electrons in a 2D quantum dot
2
∇i

2
xi

λ
 ࡋF )BSNPOJDH
0TDJMMBUPS
=
−
+
+
∑( 2
∑
|
|
2
x
−
x
)
i
j
i
i<j
0

10

1

10

2

10

3

10

4

BWFSBHF TJHO hsi

0

10

10

N = 20

N =6

N =3

$1*.$
1*.$
=5
= 10
1

PhD Thesis of0

10
DPVQMJOH QBSBNFUFS

1

10Tim Schoof, ‘16

Demo: electrons in a 2D quantum dot
19.5
6 spin-polarized electrons
S. M. Reimann and M. Manninen: Electronic structure of quantum

19.0

E
.

A. Classical electron configurations

FermiFlow
PIMC2019
In 1934, Wigner predicted that electrons
crystallize
and form a lattice if the density of PIMC2020
the three- or two-

18.5

18.0

2

dimensional electron gas is lowered beyond a certain
critical value. This is due to the fact that the Coulomb
energy increases relative to the kinetic energy and correlations begin to strongly dominate the electronic structure. For a homogeneous three-dimensional electrongas, suchis
3 crystallization
4
5 expected6at very low densities,
1i.e., at large values of the average Wigner-Seitz radiusr s . For the critical value at which this transition occurs,
Ceperley and Alder (1980) reported the value Reimannrs
et al, RMP ‘02

. Ø

free
electrons

electron density ∫ dx2…dxN ⟨x | ρ | x⟩

dx
=v
dt

interacting
electrons

10 spin-polarized electrons at β=6 λ = 8

freeelectrons

electron density ∫ dx2…dxN ⟨x | ρ | x⟩

dx
=v
dt

interacting
electrons

10 spin-polarized electrons at β=6 λ = 8

Back to the uniform electron gas
Hartree-Fock

33 spin polarized electrons @rs=1.0
Reach 0.004 Hartree/electron
ground state accuracy
30 hours training on 8 GPUs

FCIQMC

Finite temperature
benchmark data

Brown et al, PRL 110, 146405 (2013)
Schoof et al, PRL 115, 130402 (2015)
Malone et al, PRL 117, 115701 (2016)

1

Application: e ective mass of quasi-particles
0

2

4

6

8

10

rs

Fig. 8.21. Effective mass enhancement in a strictly two-dimensional electron liquid calculated in theLandau’s
Fermi
liquid theory,
G0W approximation with the following effective interactions:
RPA,
KO including
only the1956
density
local field factor G + , and KO including both the density and the spin local field factors G + and G − .
The calculation is done both in the on-shell approximation (OSA), Eq. (8.249), and making use of
the formally exact Dyson equation (8.146) (D). The local field factors are given by Eq. (A11.8) inAppendix 11. The curve labeled QMC shows the results of the quantum Monte Carlo calculation by
Kwon et al. (1994). The three sets of symbols with error bars (P/S1U, P/S1L, and PS2) represent datafrom different Si inversion layers samples. (Pudalov et al., 2002). The crosses represent measured
values of the effective mass in 2-dimensional GaAs samples (Zhu et al., 2003). The inset shows anenlargement of the results for 0 ≤ rs ≤ 1. From Asgari et al., 2004.

m*=m

T.M.Rice Ann. Phys. 1965

108

RICE

density

1.2

exp.

1.15

m*/m

-High

Simion, Guiliani, PRB 2005

RPA

1.1
1.05

G + & G1

0.95

2

3

4

5

6

r
7 s
G+

.95L
FIG.

6. m*/mas a functionof the density

Inconclusive results fromdiagrammatic
approaches
Fig. 8.22. perturbative
Effective mass enhancements
for the three-dimensional electron
liquid. Effective masses
ff

forming the energy integrations, thus obtaining quadratic singularities. He thentransformed
the energy integration in the first term to the imaginary axis. However, the quadratic singularity may now lie on the contour so that the integral is
not now well-defined. We have repeated his calculation doing the energy integra-

e effective mass
12Z(k F ) andmany-body efill be concerned
ass. We will use
Sjölander-Stottby adjusting theation, so that theh is ‘‘hydrodyemploy the miequires the ‘‘corempirical input.
lectron gas willse that we use inwhich is a basis
i-surface paramt approximationr results are in aent calculations
as experimental

53

with ´ k the unperturbed energy and S s (k, v ) the irreducible self-energy. The effective mass characterizes the quasiparticle excitation spectrum,
k2
´ k5
,
2m *
and in terms of the self-energy is then given as

S

U DS

] S s ~ k, v !
m * /m5 12
]v
k5k

F

~2!

U D

m ] S s ~ k, v !
11
k
]k
k5k

F

m* = kF /(dε/dk)kF

5

21

,
~3!

where m is the unrenormalized ~bare! electron mass. The
irreducible self-energy can be approximately expressed inDyson equation form as
S s ~ k, v ! 5iEdq
~ 2p !3E

dv8
W ~ q, v 8 ! G s ~ k2q, v 2 v 8 ! ,2p~4!

where the W function incorporates the many-body effects. In
general, it can be approximated by
7352© 1996 The American Physical Society

Figure 3. Electron effective mass: The upper panel shows our calculated effective mass versus perturbation order for
Haule,
Chen, 2012.03146
Azadi,(VDMC)
Drummond,
Foulkes,rs = 1 4. The lower panel
comparesthe rs dependence of the effective mass of this workwiththe 2105.09139
prior
analytic and⇤
FIG. 4. Quasiparticle e↵ective masses m of paramagnetic
numeric work from Ref. 26.
(Para) and ferromagnetic (Ferro) 3D-HEGs at the infinitesystem-size limit as functions of density parameter rs . Padé
functions wereto the DMC
quasiparticle
energybands
Once the extremal value of l is determined, we compute the entire momentum
andfitted
frequency
dependenceof theself-energy,
to determine the e↵ective mass. The many-body GWx andwhich allows us to determine also the momentum derivative of the self-energy,
hence theMonteeffectiveof theelectronvariational anddiagrammatic
Carlomass(VDMC)
resultsareNon-perturbative methods, still inconclusive

from the ground state, the states are orthogonal
0.6 so calculation of the overlap matrix
0.6
is notneeded. Two different excitations, spin-parallel excitations and spin-antiparallel
ones, arein [1043].
Because
the
0.00
0.01
0.02electron
0.03and hole
0.04
0 possible;2
4 the 6results
8 are10reported
12
1 / Nimportant finitestates will interact in arsfinite system, the excitation energy will havesize corrections. An alternative procedure [1044, 1045] is to add or subtract a single
Figure 23.3. The left panel shows the inverse effective mass of the 2D HEG versus rs as computed
electron from the ground state. This does not have a problem with the electron–holewith two different ways of extracting the information from QMC; in [1044] (solid triangles),
PHYSICALREVIEW
045131(2013)
interaction,butthere areother
finite-size
effects
to consider,and one over
can only
calculate; in87,[1045](open
circles), averaged
excitations
computed using
k-values
very
close
to kFBparameters
do witha single-particle
suchasfactor
the effective
mass andshown
are analytic
calculationsexcitation
using a local
field[1046] (dashedline),
0 ≤ k ≤ 1.3khaving
F . Also toor using the screened RPA [1047] (solid line). The right figure shows the finite-size effects in the
bandwidth.
1.0
= 10,
from
[1044].
QMC
calculations
rsDMC
Theeffective
effectivemassmass
is definedatin
Sec.
3.4.
Two
quite different QMC results for the 2D(pres.
work)
DMC [20]
HEG
are
shown
in
Fig.
23.3
and
comparedwith screened RPA and local field method
Exp.
[1]
0.9
GW, KO, OSAwere[31] done in a similar way, but the effective
results. The two different QMC calculations
GW, KO, SC [31]606
Variational Monte Carlomass
0.8 differs because of the way it is calculated from the QMC energies. Drummond andNeeds[1045]
theexcitationenergyover afinds
wide
range
of excitations
0 ≤willk have≤ 1.3k17
Optimizing
the fitted
varianceoftheexcitationenergies[1041]
thattheoptimalguidingfunctionthe F"
2 = excluded
2valuesandspecifically
the Fermi
thecalculated.
effectiveThis
mass
was function”
determined
form
"
(R)
|#
(R)|
; i.e., it isnear
determinedby thesurface;states to be“guiding
is
G
k
k
0.7
1.2
non-negativeand zerothe
only
where allfit.
statesunder consideration
have zeros.
bydifferentiating
analytic
Holzmannetal.
[1044]
only used excitations near the
18 Due to fluctuations, H will not be symmetric. One should not symmetrize it, since that will destroy the zeroFermi
surface and applied a large finite-size correction to the effectiveQMC
mass, as shown in
0.6
variance property; see [1042].
1 panel of Fig. 23.3. Resolving these different methods of analysis
QMCcomes
corrected
the right
down to0.8
deciding
the range of validity of Fermi liquid theory. For an infinite system, Fermi liquid
0.5
0
2
4
8
10
6
theory only applies for excitations
near
the
Fermi
surface. Since excitations away from kF
rs
0.7
0.8
acquire a finite lifetime, it is not clear whether the QMC excitation energies of the finite∗
to quasi-particle
peaksmassin themspectral
function and whether using them
FIG. 9. system(Color correspond
online) Quasiparticle
effective
against0.6
can introduce
a bias in the determinationof theGW
effectivemass.
2D HEGs. The
results
density parameter0.6rs for ferromagnetic

N. D. DRUMMOND AND R. J. NEEDS

1.4

1.2

m/m*

m* (a.u.)

1.6

m* (a.u.)

DMC (pres. work)
DMC [20]
VMC [29]
VMC [23]
GW, RPA [9]GW, KO, OSA [28]GW, KO, SC [28]Exp. [27]Exp. [3]Exp. [1]

1.0

0.8

0

2

4

rs

6

8

10

FIG. 8. (Color online) Quasiparticle effective mass m∗ against
Drummond, Needs, PRB ‘13
density parameter rs for paramagnetic or partially spin-polarized
2D HEGs, as calculated or measured by different authors. The
experimental results are due to Smith and Stiles (Ref. 27), Tan et al.
(Ref. 3), and Padmanabhan et al. (Ref. 1) The GW results wereobtained using the random-phase-approximation (RPA) effective
interaction (Ref. 9) and the Kukkonen-Overhauser (KO) effective
interaction (Ref. 28) by solving the Dyson equation self-consistently(SC) or within the on-shell approximation (OSA). We show the VMCm / m*

1.8

were obtained using the Kukkonen-Overhauser (KO) effective inter0.00
0.01
0.02
0.03
action by solving the0Dyson2equation(SC)
4 self-consistently
6
8
10 or within
12
23.8 31).
Strengths
and weaknesses
the on-shell approximation (OSA) (Ref.
results of VMC
rs The experimental
1/N
are due to Padmanabhan et al. (Ref. 1). We show the DMC results
The variational method is very powerful and intuitively pleasing. By only assuming a funcreported in ourearlier
work The(Ref. left
20) inadditionto ourthecurrent
results.
Figure
23.3.
panel
shows
inversemass
2D HEG
tional form for theMartin,
wavefunction,
oneCeperley,obtains an Interacting
upper effective
boundElectrons
to theenergy.
In contrast
to versus rs as
Reining,‘16of thewithdifferent
ways ofextracting
the informationfrom QMC; in[1044]
other twomany-bodycorrelated
methods,
no further
uncontrolled approximations
need
to be (solid triangl
It should
be noted
that
the
2D on
electronsystems
in be (open; init[1045]
circles),
computedusing
k-valuesvery
to kisstudied
made.
The only
restriction
thetrialclosefunctionmust
computable.
Withaverageda Slater– over excitat
Fthat
the experimentsfromthe ideal
2D
HEG
assumedin ourwavefunction
one
can
do
calculations
withthousandsof electrons.
be sure,
.
Also
shown
are
analytic
calculations
using
a local To
field
factor [1046] (das0Jastrow≤ k differ≤trial
1.3k
Fcalculations
that thefinitewidthsthein
numerical
work
has toRPA
be systems
done
veryhave
carefully,
e.g.,
the
convergenceof therandom
walk
or
using
theexperimental
screened[1047](solidline).
The
right figure
shows
the
finite-size effec
and are subject
tobedisorder.
Itdependence
would therefore
be inappropriate
hastotestedandonsystemsize
needs
to be corrected
for. To motivate the
=
10,
from
[1044].
QMC
effective
mass
calculationsatrs QMC data
to make a methods
precise toquantitative
comparisonofourbe described in the next two chapters, we list some of the intrinsic problems
with the existing
results; nevertheless we may
with the experimental
variational method.

flMore con icting results for 2d electron gas

E ective mass from thermodynamics
Eich, Holzmann, Vignale, PRB ‘17

mkF 2
s0 = 2 kBT
3ℏ n

m*kF 2
s=
k
T
B
2
3ℏ n

m*
s
=
m
s0

Interacting/noninteracting entropy ratio

Previous calculations can not reach the low temperature region
ff

Moreover, entropy is not directly accessible to PIMC

14 electrons @ T/Ef=0.08

Outlooks
Ultracold fermi gases

Dense hydrogen

Warm dense matter

Thermal density functionals

Thank you!
2105.08644

buwantaiji/FermiFlow

Hao Xie

Linfeng Zhang