FERMILAB-PUB-22-767-T EXCEED-DM Extended Calculation of Electronic Excitations for Direct Detection of Dark Matter

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FERMILAB-PUB-22-767-T

EXCEED-DM: Extended Calculation of Electronic Excitations for Direct

Detection of Dark Matter

Tanner Trickle∗

Theory Division, Fermi National Accelerator Laboratory, Batavia, Illinois, 60510, USA

(Dated: March 16, 2023)

Direct detection experiments utilizing electronic excitations are spearheading the search for

light, sub-GeV, dark matter (DM). It is thus crucial to have accurate predictions for any DM-

electron interaction rate in this regime. EXCEED-DM (EXtended Calculation of Electronic

Excitations for Direct detection of Dark Matter) computes DM-electron interaction rates

with inputs from a variety of ab initio electronic structure calculations. The purpose of this

manuscript is two-fold: to familiarize the user with the formalism and inputs of EXCEED-DM,

and perform novel calculations to showcase what EXCEED-DM is capable of. We perform four

calculations which extend previous results: the scattering rate in the dark photon model,

screened with the numerically computed dielectric function, the scattering rate with an

interaction potential dependent on the electron velocity, an extended absorption calculation

for scalar, pseudoscalar, and vector DM, and the annual modulation of the scattering rate

in the dark photon model.

∗ttrickle@fnal.gov

arXiv:2210.14917v2 [hep-ph] 15 Mar 2023

CONTENTS

I. Introduction 3

II. Formalism 7

A. Electronic State Conﬁguration 7

B. Transition Matrix Elements 9

C. Calculations 10

1. Binned Scatter Rate 11

2. Absorption Rate 13

3. Dielectric Function 14

III. Inputs to EXCEED-DM 15

A. Input File 16

B. Electronic Conﬁguration File 19

IV. Applications 23

A. DM-Electron Scattering Rate Screened with Numeric Dielectric Function 24

B. DM-Electron Scattering Rate with Electron Velocity Dependent Operator 31

C. Extended Absorption Calculation 32

D. Annual Modulation of DM-Electron Scattering Rate 34

V. Summary and Future Development 36

Acknowledgments 39

A. Installation of EXCEED-DM 39

B. Eﬀective Lagrangian to Scattering Form Factor 40

C. Transition Matrix Elements for Bloch States 44

References 45

I. INTRODUCTION

Discovering the nature of dark matter (DM) is one of the most important goals in all of physics.

Constituting 26.5% of the universe’s energy density [1], DM is roughly ﬁve times more abundant

than ordinary matter; yet any understanding of its particle content eludes us. One class of ongoing

experiments, which search for DM-Standard Model interactions in a laboratory, are known as direct

detection experiments. The canonical example of a direct detection experiment looks for nuclear

recoils induced by a scattered DM particle. Many of these experiments have been performed,

or are proposed, e.g., LUX [2], SuperCDMS [3], ANAIS [4], CRESST [5], SABRE [6], DAMA [7],

PandaX [8], DAMIC [9], NEWS-G [10], LZ [2], XENON10/100/1T [11–13], KIMS [14], DM-Ice [15],

DarkSide [16], and the absence of any signal has led to stringent bounds on a wide variety of DM

models. However, all of these experiments share a common kinematics problem when looking for

DM with mass lighter than a GeV; when the DM mass drops below the target nucleus mass, the

energy deposited in a scattering event rapidly falls below the detection threshold.

This is an unfortunate problem because many well-motivated DM models can possess a light,

sub-GeV, DM candidate (see Refs. [17–19] for recent reviews). This problem has been remedied by

an extraordinary inﬂux of new detection ideas for covering this region of DM parameter space (see

Refs. [19–22] for recent reviews). Currently, the preeminent method for searching for sub-GeV DM

is via electronic excitations across band gaps in crystal targets [23–61]. With band gaps of O(eV),

DM with masses as light as O(MeV)can be searched for via a scattering event, and O(eV)masses

can be searched for via absorption. Experiments such as SuperCDMS [62–64], DAMIC [65–69],

EDELWEISS [70–72], SENSEI [73–75], and CDEX [76] are currently in operation searching for

DM induced electronic excitations with a combination of Silicon (Si) and Germanium (Ge) targets.

Similar targets utilizing electronic excitations with smaller, O(meV), band gaps such as spin-orbit

coupled materials [24,77], Dirac materials [46,56,57], and superconductors [54,55,58] have also

been proposed.

The calculation of DM-electron interaction rates in these targets is more involved than the

standard nuclear recoil calculation. This is because electrons in crystal targets cannot be treated

as free states. The crystal creates a lattice potential, distorting the electronic wave functions.

Therefore, estimates of DM-electron interaction rates are crucially dependent on how the electronic

wave functions, and band structure, are modeled. With experiments already being performed, and

the nature of DM still unknown, it is crucial to have the tools to make predictions for the broadest

set of DM models [39,78]. The earliest calculations used analytic approximations of the wave

functions, along with the measured density of states to account for the band structure [51,61].

Then the QEDark [50] program was introduced, and was the ﬁrst to incorporate electronic wave

functions, and band structures, computed with density functional theory (DFT) in calculations

of the DM-electron scattering rate. This was a monumental step forward, creating a connection

between ﬁrst principles condensed matter calculations and particle physics observables which exists

today [20,22]. Additionally, an extension of QEDark,QEDark-EFT [38,39], has been introduced

to generalize the set of DM models for which DM-electron scattering can be computed. Recently

it was shown that, in some simple DM models, the DM-electron interaction rate could be related

to the (q, ω dependent) dielectric function [40,43]. This rewriting rigorously included screening

eﬀects, which had previously been assumed to be small. While projections for the DM-electron

interaction rate in these models were made with DFT calculations and simpliﬁed models of the

dielectric function, the longer term vision is to use the experimentally measured dielectric function

to circumvent the uncertainties associated with DFT calculations. The program DarkELF [42] was

introduced to compute DM-electron interaction rates from input dielectric functions.

In parallel with these developments, EXCEED-DM (EXtended Calculation of Electronic

Excitations for Direct detection of Dark Matter) [44,79] has also been developed. Similar to

QEDark,EXCEED-DM can utilize ab initio DFT calculations of the target electronic structure.

Previous versions of EXCEED-DM have been used to compute DM-electron scattering in a variety

of target materials [23,44], DM-electron absorption for a variety of DM models [34], as well as

DM-electron interaction rates in more novel, spin-orbit coupled targets [24]. With similar goals

as the previously discussed programs, it is important to distinguish how EXCEED-DM, speciﬁcally

the newest version, v1.0.0, diﬀerentiates itself. EXCEED-DM has three main advantages relative to

QEDark, the current standard for these calculations:

•Independent of Electronic Structure Calculator.QEDark is intrinsically linked to

Quantum Espresso [80,81], a speciﬁc program for performing DFT calculations of the elec-

tronic structure, i.e., the electronic wave functions and band structure. Therefore, the ac-

curacy of the electronic structure is limited to the methods used within Quantum Espresso.

EXCEED-DM is not linked to a speciﬁc electronic structure calculator; if the output wave func-

tions are in a format supported by EXCEED-DM (discussed more in the next bullet point,

Sec. II A, and in the documentation ) any electronic structure calculator can be used. For

example, EXCEED-DM has been used with electronic wave functions in the plane wave (PW)

basis computed with Quantum Espresso and VASP [82–85].

•Variety of electronic state approximations.QEDark performs calculations with the

initial and ﬁnal electronic states expanded in a PW basis. This basis works well for states

close to the Fermi surface. For states farther away from the Fermi surface this approach

is suboptimal since larger momentum cutoﬀs are needed, especially for deeply bound, core,

states [44]. Therefore it is beneﬁcial to represent electronic states in a variety of bases. As

of v1.0.0,EXCEED-DM supports three diﬀerent bases, the PW basis, a Slater type orbital

(STO) basis, and a single PW, see Sec. II A for more detail. Any of these bases can be used

to approximate the initial or ﬁnal electronic states; moreover a combination of bases can

be used within the set of initial and ﬁnal states. These bases were chosen since they were

previously used to extend the description of the electronic states in Si and Ge targets, see

Ref. [44] for more details. Any electronic structure calculator, or combination of them, which

can output wave functions in these bases can be used as input to EXCEED-DM. While v1.0.0

only includes a few bases, these are meant to serve as examples for future versions, which

will expand the number of bases included.1

•Additional calculations. In addition to DM-electron scattering rate calculations, EXCEED-

DM can perform DM-electron absorption calculations. While for some DM models, absorption

on electrons can be related to the long wavelength dielectric function, ε(0, ω), for other models

this is not possible [24,34]. Moreover, within DM-electron scattering rate calculations,

EXCEED-DM can straightforwardly compute the daily modulation rate [23]; an important

feature for discriminating against potential backgrounds in anisotropic targets.

Additionally, there are miscellaneous technical advantages, e.g., EXCEED-DM is parallelized with

OpenMPI, versus OpenMP used in QEDark. This allows for calculations to utilize the many memory

independent cores available in, for example, supercomputers, thereby increasing parallelization.

Given enough cores, this will lead to faster DM-electron interaction rate calculations. In principle,

all previous QEDark calculations can be performed with EXCEED-DM with appropriate conversion

of Quantum Espresso output to EXCEED-DM input. This may be useful when comparing the results

of previous calculations.

The QEDark-EFT extension to QEDark can compute DM-electron scattering rates for a wider

variety of DM models, under the same approximations about the electronic states. Therefore the

main advantages of EXCEED-DM are still present relative to QEDark-EFT. While EXCEED-DM does

1Until the basis sets are included in a versioned release it is up to the user to implement these in EXCEED-DM.

However, an experienced Fortran programmer should be able to add new bases with relative ease following the

currently implementations as examples.

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FERMILAB-PUB-22-767-TEXCEED-DM:ExtendedCalculationofElectronicExcitationsforDirectDetectionofDarkMatterTannerTrickleTheoryDivision,FermiNationalAcceleratorLaboratory,Batavia,Illinois,60510,USA(Dated:March16,2023)Directdetectionexperimentsutilizingelectronicexcitationsarespearheadingthesearchforligh...

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