submitted manuscriptOn the chemical potential of many-body perturbation theory in extended systems Felix Hummel

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submitted manuscript

On the chemical potential of many-body

perturbation theory in extended systems

Felix Hummel∗

Institute for Theoretical Physics, TU Wien,

Wiedner Hauptstraße 8-10/136, 1040 Vienna, Austria

E-mail: felix.hummel@tuwien.ac.at

Abstract

Many methods for computing electronic correlation eﬀects at ﬁnite temperature

are related to many-body perturbation theory in the grand-canonical ensemble. In

most applications, however, the average number of electrons is known rather than the

chemical potential, requiring that expensive correlation calculations must be repeated

iteratively in search for the chemical potential that yields the desired average number

of electrons. In extended systems with mobile charges, however, the long-ranged elec-

trostatic interaction should guarantee that the average ratio of negative and positive

charges is one for any ﬁnite chemical potential. All properties per electron are virtually

independent of the chemical potential, as for instance in an electric wire at diﬀerent

voltage potentials.

This work shows that the inﬁnite-size limit of the exchange-correlation free energy

agrees with the inﬁnite-size limit of the exchange-correlation grand potential at a non-

interacting chemical potential. The latter requires only one expensive correlation calcu-

lation for each system size. Analogous to classical simulations of long-range-interacting

particles, this work uses a regularization of the Coulomb interaction such that each

electron on average interacts only with as many electrons as there are electrons in the

simulation, avoiding interactions with periodic images.

Numerical calculations of the warm uniform electron gas have been conducted with

the Spencer–Alavi regularization employing the ﬁnite-temperature Hartree approxima-

tion for the self-consistent ﬁeld and linearized ﬁnite-temperature direct-ring coupled

cluster doubles for treating correlation.

1 Background

In the warm-dense matter (WDM) regime the

relevant many-body states exceed the ground

state and the density is suﬃciently large to re-

quire a quantum mechanical treatment of the

electrons interacting with each other. WDM

conditions are found, for instance, during in-

arXiv:2210.05548v2 [cond-mat.mtrl-sci] 20 Oct 2022

submitted manuscript

ertial conﬁnement fusion (ICF), in the core

region of gas giants, or in matter interacting

with high intensity laser ﬁelds.1Even at room

temperature the thermal energy must be con-

sidered to be large compared to the vanishing

band gap of bulk metals.

The mobility of electrons at warm-dense

conditions poses challenges for ab-initio simu-

lations of extended systems that are absent in

zero-temperature calculations. Unlike at zero

temperature, the number of electrons in a vol-

ume of ﬁxed shape ﬂuctuates rendering such

a volume not necessarily charge neutral at all

times. Thus, the long-ranged Coulomb inter-

action cannot be used under periodic bound-

ary conditions due to the diverging electro-

static energy per volume for net-charged con-

ﬁgurations. There are mainly two methods in

current state of the art ab-initio simulations

at warm-dense conditions to circumvent this

divergence: (i) The simulation is done in the

canonical ensemble where electrons are not

permitted to enter or leave the simulated vol-

ume. While this ensures charge neutrality it

also reduces the number of possible conﬁgura-

tions, aﬀecting the system’s entropy.2Path-

integral quantum Monte Carlo (PIQMC) cal-

culations are usually conducted in the canon-

ical ensemble.3,4 (ii) Another possibility is to

disregard the parts of the electrostatic in-

teraction stemming from the average elec-

tron and background densities, thus remov-

ing the divergence. This allows for grand-

canonical simulations with a ﬂuctuating num-

ber of electrons including its eﬀect on the en-

tropy. Many-body perturbation theory calcu-

lations usually apply this method5,6 following

the work of Kohn and Luttinger, in particular

the assumption for arriving at Eq. (20) in Ref.

7. A physical justiﬁcation for this procedure

would be if the ﬂuctuations of the positive

background were fully correlated with the

ﬂuctuations of the electrons. Diﬀerent mobil-

ities of electrons and ions, however, question

this assumption.

In this work a third alternative is studied

to treat long-range electrostatic interactions

with thermal many-body perturbation the-

ory. Liang and coworkers8have studied clas-

sical simulations of mobile electrostatically

interacting particles under periodic boundary

conditions. They look at the pair correla-

tion function and observe the theoretically

expected Debey–Hueckel screening at long

distances only under two conditions: (i) when

simulating in the grand-canonical ensemble,

and (ii) when limiting the range of the elec-

trostatic interaction, such that the particles

do not interact with all of their own periodic

images. Periodic boundary conditions cannot

model charge ﬂuctuations at length scales be-

yond the size of the simulation cell. In reality,

the charges would move from one cell to the

neighboring cell, keeping the average charge

constant. Under periodic boundary condi-

tions, however, charges can only appear or

disappear simultaneously in all periodic im-

ages of the simulation cell. Still, the range of

the electrostatic interaction can be limited to

allow for charge ﬂuctuations.

A spherical truncation scheme has already

been developed by Spencer and Alavi9to pre-

vent spurious Fock-exchange interactions of

the electrons with their periodic images for

zero-temperature calculations as an alterna-

tive to other methods treating the occurring

integrable singularity.10,11 Here, the trunca-

tion scheme is applied to all parts of the

electrostatic interaction in the self-consistent

ﬁeld calculations, as well as in the subsequent

perturbation calculation. Other regulariza-

tion schemes that limit the interaction range

are also possible, such as the Minimal Im-

age Convention for atom centered orbitals,

or the Wigner–Seitz truncation scheme.12,13

For point-like charges the spherical trunca-

tion is not continuous which may pose diﬃ-

culties when considering diﬀerent atomic con-

ﬁgurations.

submitted manuscript

Related work

Finite-temperature many-body perturba-

tion theory (FT-MBPT) oﬀers an ele-

mentary framework for ab-initio calcula-

tions of WDM.5,6,14–16 Numerous approx-

imation schemes employ thermal MBPT,

such as thermal second-order MBPT,17–20

ﬁnite-temperature random phase approxi-

mation,21–24 Green’s function based meth-

ods,25,26 as well as some ﬁnite-temperature

generalizations of coupled-cluster meth-

ods.27–30 An alternative formulation of the

coupled-cluster methods has been brought

forward recently in the framework of thermo-

ﬁeld dynamics.31–33 Finite-temperature per-

turbation theory is originally formulated

in the grand-canonical ensemble, however

formulations in the canonical ensemble ex-

ist.34,35 Equally, thermo ﬁeld dynamics can

be employed in the canonical ensemble.36

Analogous to ab-initio calculations at

zero-temperature, thermal Hartree–Fock and

density functional theory (DFT) calculations

are among the most widely used meth-

ods.37–39 In general, it is not suﬃcient to use a

zero-temperature exchange-correlation func-

tional and introduce temperature merely by

smearing. Temperature must be a parameter

of the exchange-correlation functional.40 At

higher temperatures, a large number of one-

body states is occupied with non-negligible

probabilities. Orbital-free density functional

theories (ofDFT) aim at mitigating this with

functionals that do not depend on the usual

Kohn–Sham orbital description of DFT.41,42

Canonical or grand-canonical full conﬁgura-

tion interaction methods can be used for

benchmarking more approximate theories.43

Finally, path-integral quantum Monte Carlo

(PIQMC) methods are available and often

complement other calculations, as they have

entirely diﬀerent error sources in the approx-

imation of the many-body problem. PIQMC

calculations are usually conducted in the

canonical ensemble.3,4 High accuracy calcula-

tions of the warm uniform electron gas are of

particular interest since they can serve for ac-

curate temperature dependent parametriza-

tions of DFT exchange-correlation poten-

tials.44–46

The Kohn–Luttinger conundrum is also

closely related to this work. It states that the

inﬁnite-size zero-temperature limit of ﬁnite-

temperature many-body perturbation theory

not necessarily agrees with the inﬁnite-size

limit of zero-temperature many-body pertur-

bation theory. In the common approach

where the zero-momentum part of the elec-

trostatic interaction is disregarded, certain

terms called anomalous diagrams aﬀect both,

the chemical potential and the grand poten-

tial in a way such that their contributions

cancel in the zero-temperature limit of the

free energy under certain, but not all condi-

tions.7Discussions on this conundrum can be

found in Refs. 18–20,47,48.

With the method of this work the situ-

ation is diﬀerent. Considering the full elec-

trostatic interaction with a regularization in

ﬁnite systems leads to a free energy per

electron that is asymptotically independent

of the chemical potential in the inﬁnite-size

limit. The electrostatic terms are strong

and do not allow a ﬁnite-order perturba-

tive treatment, as discussed in Subsection

2.1 on ﬁxed orbitals. Although related to

it, this work does not aim at solving the

Kohn–Luttinger conundrum. It may very

well be that the long-ranged Coulomb inter-

action causes a discontinuity at inﬁnite-size

and zero-temperature and the result may de-

pend on which limit is taken ﬁrst.

2 Methods

Let us now develop the regularization ap-

proach for the prototypical warm-dense sys-

tem: the warm uniform electron gas (UEG).

submitted manuscript

The UEG is a model of a metal, where the

positive ions of the lattice are replaced by

a static homogeneous positive background

charge. It has a vanishing band gap in

the inﬁnite-size limit and thus qualiﬁes for

a warm-dense system at all non-zero temper-

atures. All properties of the warm UEG de-

pend only on the thermodynamic state, speci-

ﬁed by its density and temperature. The den-

sity is usually given in terms of the Wigner–

Seitz radius rsin atomic units, such that the

volume of a sphere with radius rscorresponds

to the average volume per electron. It is

also convenient to specify the temperature in

terms of the dimensionless ratio Θ = kBT/εF,

where kBTis the average thermal energy and

εF=k2

F/2is the Fermi energy of a free non-

spin-polarized, inﬁnite electron gas at the cor-

responding density and at zero temperature

with k3

F= 9π/4r3

s. This deﬁnes a natural

temperature scale where diﬀerent densities

can be compared to each other more directly.

The UEG is modeled by a ﬁnite cubic box

of length Lunder periodic boundary condi-

tions having the volume V=L3= 4πr3

sN/3.

It contains a homogeneous positive charge

density with a total charge of Nelementary

charges, which is considered ﬁxed. In the

grand-canonical ensemble the number of elec-

trons in the system is not ﬁxed but rather

ﬂuctuates around its expectation value which

depends on the chemical potential µ. Later,

µwill be chosen such that the expected num-

ber of electrons N:= hˆ

Niequals, or is close

to, the number of positive charges N. To

treat the diverging electrostatic interaction

the Spencer–Alavi truncation of the electro-

static interaction is used. It is given by

the usual Coulomb interaction 1/r12 for the

distance between two electronic coordinates

r12 <Rand zero otherwise with the trun-

cation radius R=rsN1/3. The kernel of

this interaction within a sum over momenta

is V(q) = 4π(1 −cos qR)/Vq2. For ﬁnite N

the kernel is also ﬁnite at q= 0 and evaluates

to 2πR2/V. With this choice the interaction

“sees” on average Nelectrons and it reduces

to the usual electrostatic interaction in the

limit N → ∞.9

All states are expanded in anti-

symmetrized products of one-electron wave-

functions that are eigenfunctions of the

single-electron kinetic operator −∇2/2un-

der periodic boundary conditions. The nor-

malized eigenfunctions are the plane waves

commensurate with the box length

ψkσ(r, τ) = 1

√Ve−ik·rδστ ,(1)

with k∈2πZ3/L and where σ, τ ∈ {↑,↓} de-

note the spin coordinate of the wavefunction

and the electron, respectively. With the op-

erator ˆc†

k,σ creating an electron in the state

k, σ and ˆck,σ annihilating it, the electronic

Hamiltonian of the modeled UEG reads

H=ˆ

T+ˆ

Vext +ˆ

k,σ

2ˆc†

k,σ ˆck,σ −X

k,σ

V(0) Nˆc†

k,σ ˆck,σ

k,σ,k0,σ0,q

V(|q|) ˆc†

k+q,σ ˆc†

k0−q,σ0ˆck0,σ0ˆck,σ.

(2)

It consists of three terms: the kinetic term ˆ

the electron–background interaction ˆ

Vext and

the electron–electron interaction ˆ

V, respec-

tively. Exact diagonalization of the Hamilto-

nian is infeasible except for very limited sys-

tem sizes. This work shall also employ the ap-

proximation approach of computational ma-

terials science, where one ﬁrst performs a self-

consistent ﬁeld (SCF) calculation, followed

by a perturbative approximation of the corre-

lation based on the SCF result. In accordance

with the common workﬂow of Random Phase

Approximation (RPA) calculations for low-

band-gap systems, the SCF only employs the

Hartree approximation rather than Hartree–

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submittedmanuscriptOnthechemicalpotentialofmany-bodyperturbationtheoryinextendedsystemsFelixHummelInstituteforTheoreticalPhysics,TUWien,WiednerHauptstraÿe8-10/136,1040Vienna,AustriaE-mail:felix.hummel@tuwien.ac.atAbstractManymethodsforcomputingelectroniccorrelationeectsatnitetemperaturearerelated...

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submitted manuscriptOn the chemical potential of many-body perturbation theory in extended systems Felix Hummel

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