# Beyond-DFT ab initio Calculations for Accurate Prediction of Sub-GeV Dark
Matter Experimental Reach

Elizabeth A. Peterson  Theoretical Division, Los Alamos National Laboratory,
Los Alamos, NM 87545, USA  Samuel L. Watkins  Physics Division, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  Christopher Lane  Theoretical
Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA  Jian-Xin
Zhu  Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM
87545, USA  Center for Integrated Nanotechnologies, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA

###### Abstract

As the search space for light dark matter (DM) has shifted to sub-GeV DM
candidate particles, increasing attention has turned to solid state detectors
built from quantum materials. While traditional solid state detector targets
(e.g. Si or Ge) have been utilized in searches for dark matter (DM) for
decades, more complex, anisotropic materials with narrow band gaps are
desirable for detecting sub-MeV dark matter through DM-electron scattering and
absorption channels. In order to determine if a novel target material can
expand the search space for light DM it is necessary to determine the
projected reach of a dark matter search conducted with that material in the DM
mass – DM-electron scattering cross-section parameter space. The DM-electron
scattering rate can be calculated from first-principles with knowledge of the
loss function, however the accuracy of these predictions is limited by the
first-principles level of theory used to calculate the dielectric function.
Here we perform a case study on silicon, a well-studied semiconducting
material, to demonstrate that traditional Kohn-Sham density functional theory
(DFT) calculations erroneously overestimate projected experimental reach. We
show that for silicon this can be remedied by the incorporation of self-energy
corrections as implemented in the GW approximation. Moreover, we emphasize the
care that must taken in selecting the appropriate level of theory for
predicting experimental reach of next-generation complex DM detector
materials.

##  I Introduction

As the search for dark matter (DM) particles turns towards efforts to detect
light DM in the sub-GeV mass range, solid-state detectors have increasingly
become detector targets of interest Essig  _et al._ (2016); Hochberg  _et al._
(2016a, b, c, 2018); Griffin  _et al._ (2018); Geilhufe  _et al._ (2020);
Griffin  _et al._ (2021a); Hochberg  _et al._ (2021); Knapen  _et al._ (2021);
Coskuner  _et al._ (2021); Kahn and Lin (2022); Huang  _et al._ (2023). Recent
experimental and theoretical efforts have focused on absorption and scattering
of bosonic and fermionic light DM particles with nucleons, electrons, and
bosonic quasiparticles like phonons in detector materials Hochberg  _et al._
(2016a, b, c, 2018); Griffin  _et al._ (2018); Trickle  _et al._ (2020a);
Griffin  _et al._ (2021a); Hochberg  _et al._ (2021); Knapen  _et al._ (2021);
Coskuner  _et al._ (2021). As nuclear recoil experiments have struggled to
detect GeV-scale DM candidate particles, attention has turned to electron
recoil experiments in semiconductor targets, that promise to enable detection
of even lighter DM due to their meV-eV electronic excitation energies Essig
_et al._ (2016); Hochberg  _et al._ (2018); Geilhufe  _et al._ (2020); Kahn
and Lin (2022).

Next generation DM detectors that can probe even lighter DM will require more
complex materials such as topological materials, superconductors, etc. which
host desirable properties such as narrow band gaps and anisotropies that
capture the daily modulation in the DM wind impinging on the Earth Hochberg
_et al._ (2016a, b, c, 2018); Geilhufe  _et al._ (2020); Rosa  _et al._
(2020); Hochberg  _et al._ (2021); Coskuner  _et al._ (2021). The novelty and
complexity of these materials, however, may limit the experimental ease of
rapid characterization and screening for desirable properties. As such,
leveraging first-principles methods that handle many-body effects and strong
correlations will become increasingly important in predicting projected
experimental reach of novel DM detector materials. Moreover, the calculated
DM-electron scattering rates may be strongly affected by the first-principles
approximations employed with certain, generally low-cost, approximations
producing results that may significantly misrepresent the parameter space that
is excluded.

In a semiconducting detector, dielectric screening plays a significant role in
determining the measured signals from the absorption and scattering of DM
particles with electrons Hochberg  _et al._ (2021); Knapen  _et al._ (2021);
Coskuner  _et al._ (2021). The loss function, i.e. the imaginary part of the
inverse dielectric function, can be utilized to predict expected DM-electron
scattering rates Nozières and Pines (1959); Hochberg  _et al._ (2021), in
analogy to electron energy loss spectroscopy (EELS). As such, the complex
dielectric function is a central quantity of interest.

From a first-principles perspective, calculation of the dielectric function is
a standard procedure Baroni and Resta (1986); Gajdos̆  _et al._ (2006). From
density functional theory (DFT) calculations, the dielectric function in the
random phase approximation (RPA), neglecting local field effects, or exchange-
correlation effects, is readily calculable from DFT wavefunctions and
eigenvalues. Other methods for calculating the electronic structure or the
dielectric function that can capture exchange and (strong) correlation effects
include dynamical mean-field theory (DMFT) Metzner and Vollhardt (1989), many-
body perturbation theory (MBPT) Hedin (1965), density functional perturbation
theory (DFPT) Gonze and Lee (1997), and time-dependent density functional
theory (TD-DFT) Runge and Gross (1984). In practice, the methods employed by
existing open-source software packages for calculating dielectric functions
from first-principles are oriented towards understanding optical properties of
materials, in particular absorption of photons in the visible spectrum, and
hence restricted to calculations in the long wavelength limit with negligible
momentum transfer $\mathbf{q}$. Impinging DM particles may impart both finite
momentum $\mathbf{q}$ and finite energy $\omega$ when scattering off an
electron, necessitating methods for calculating the dielectric function at
finite momentum transfer $\mathbf{q}$.

Response functions like the dielectric function describe the reaction of a
material to a perturbation, making them fundamentally excited state properties
which require a careful consideration of many-body effects to accurately
calculate. In a first-principles framework, in order to accurately capture
many-body effects it is necessary to work beyond the standard first-principles
DFT formalism. Green’s function approaches have been shown to significantly
improve upon the accuracy of the computed energy eigenvalues by implementing
self-energy corrections Hedin (1965); Hybertsen and Louie (1986); Zhang  _et
al._ (1989). The workhorse method of Green’s function approximations is the GW
approximation Hedin (1965); Hybertsen and Louie (1986) which approximates the
self-energy as a product of the single-particle Green’s function ($G$) and the
screened Coulomb potential ($W$), or $\Sigma=iGW$, neglecting vertex
corrections. The GW approximation is typically implemented as a correction
scheme to the DFT eigenvalues and wavefunctions to compute quasiparticle-
corrected single-particle energy eigenvalues. Beyond improving upon the
accuracy of the electronic structure energy eigenvalues, the GW approximation
is a convenient method for calculations of the dielectric function. This is
because, in GW, calculation of the inverse dielectric function at finite
$\mathbf{q}$ is a necessary prerequisite for calculating the screened Coulomb
potential ($W$). A number of GW approximation software packages exist
Hybertsen and Louie (1986); Deslippe  _et al._ (2012); Marini  _et al._
(2009); Sangalli  _et al._ (2019); Govoni and Galli (2015); Gonze  _et al._
(2020); Bruneval  _et al._ (2006); Shishkin and Kresse (2006) and it has been
shown that the GW quasiparticle-correction scheme can immensely improve the
accuracy of calculated dielectric functions Hybertsen and Louie (1986); Zhang
_et al._ (1989).

There are existing packages, such as QEDark Essig  _et al._ (2016), DarkELF
Knapen  _et al._ (2022), and EXCEED-DM Griffin  _et al._ (2021b); Trickle
(2023), that enable the calculation of DM-electron scattering rates for
various common semiconducting materials used as DM detectors (e.g. Si and Ge).
While QEDark does not include in-medium screening effects, DarkELF and EXCEED-
DM calculate the scattering based on the dielectric function, thereby
including these effects. However, these packages base their dielectric
function calculations on DFT outputs, which may not provide the most accurate
energy eigenvalues, as noted above. These packages have been frequently used
in high-impact results in the field of light DM Agnese  _et al._ (2018);
Amaral  _et al._ (2020); Crisler  _et al._ (2018); Abramoff  _et al._ (2019);
Barak  _et al._ (2020); Aguilar-Arevalo  _et al._ (2019); Arnquist  _et al._
(2023); Zhang  _et al._ (2022); Arnaud  _et al._ (2020), suggesting that if
there is a significant difference in the DM-electron scattering rate when
including, for example, the GW-corrected energy eigenvalues, then these
results may be misrepresenting the parameter space that is excluded.

Here we illustrate and emphasize the significance of incorporating many-body
effects into calculations of the dielectric function for prediction of
projected experimental reach of novel DM detector materials. We present a case
study of the well-studied semiconductor silicon (Si), demonstrating that even
for a material that does not host substantial relativistic effects or strong
correlations, the incorporation of many-body effects is essential for accurate
prediction of the projected experimental reach obtained via first-principles
calculations. We leave calculation of more complex materials to future work.

##  II Theoretical Framework

The loss function
$L(\mathbf{q},\omega)=\operatorname{Im}\left[-\epsilon(\mathbf{q},\omega)^{-1}\right]$
is the central quantity describing how a dielectric medium responds to an
impinging particle. To predict the experimental reach of a detector material,
we calculate the electronic structure in order to calculate the complex
dielectric function and from there the DM-electron scattering rate and
projected reach.

We begin with a DFT calculation using a plane-wave basis to obtain the Kohn-
Sham energy eigenvalues and wavefunctions using the Kohn-Sham equations Kohn
and Sham (1965)

$H^{KS}\psi_{n\mathbf{k}}^{KS}(\mathbf{r})=\left[-\frac{1}{2}\nabla^{2}+v_{ext}(\mathbf{r})+\int\frac{\rho(\mathbf{r^{\prime})}}{|\mathbf{r}-\mathbf{r^{\prime}}|}d\mathbf{r^{\prime}}+v_{XC}[\rho]\right]\psi_{n\mathbf{k}}^{KS}(\mathbf{r})=\varepsilon_{n\mathbf{k}}^{KS}\psi_{n\mathbf{k}}^{KS}(\mathbf{r})\;,$
(1)

Here energy is measured in atomic units, $\psi_{n\mathbf{k}}^{KS}(\mathbf{r})$
and $\varepsilon_{n\mathbf{k}}^{KS}$ are Kohn-Sham wavefunctions and
eigenvalues parametrized by band index $n$ and crystal momentum $\mathbf{k}$,
$\rho(\mathbf{r})=\sum_{n\mathbf{k}}|\psi_{n\mathbf{k}}(\mathbf{r})|^{2}$ is
the charge density, $v_{ext}(\mathbf{r})$ is the external potential from the
ionic background, and $v_{XC}[\rho]$ is the exchange-correlation potential
encompassing all many-body effects not captured by the single-particle
framework.

We employ the GW approximation as a correction scheme to the Kohn-Sham DFT
energy eigenvalues by replacing the exchange-correlation potential with the
self-energy operator as in Refs. Hedin (1965); Hybertsen and Louie (1986):

$\left[-\frac{1}{2}\nabla^{2}+v_{ext}(\mathbf{r})+\int\frac{\rho(\mathbf{r^{\prime})}}{|\mathbf{r}-\mathbf{r^{\prime}}|}d\mathbf{r^{\prime}}+\Sigma(\varepsilon_{n\mathbf{k}}^{GW})\right]\psi_{n\mathbf{k}}^{GW}(\mathbf{r})=\varepsilon_{n\mathbf{k}}^{GW}\psi_{n\mathbf{k}}^{GW}(\mathbf{r})\;,$
(2)

Specifically, we perform a single-shot $G_{0}W_{0}$ calculation, using the
Kohn-Sham DFT eigenvalues and wavefunctions to produce the self-energy
operator and then solve for the GW energy eigenvalues
$\varepsilon_{n\mathbf{k}}^{GW}$ as

$H^{KS}\psi_{n\mathbf{k}}^{KS}(\mathbf{r})-\left[\Sigma(\varepsilon_{n\mathbf{k}}^{KS})-v_{XC}[\rho]\right]\psi_{n\mathbf{k}}^{KS}(\mathbf{r})=\varepsilon_{n\mathbf{k}}^{GW}\psi_{n\mathbf{k}}^{KS}(\mathbf{r})\;.$
(3)

With the Kohn-Sham DFT wavefunctions $\psi_{n\mathbf{k}}^{KS}$ and GW energy
eigenvalues $\varepsilon_{n\mathbf{k}}^{GW}$ we may construct the complex
dielectric function.

Most generally, the dielectric tensor is a linear response function that
describes how a displacement field in a dielectric medium is produced by
application of a perturbing electric field as

$D_{\alpha}(\mathbf{q},\omega)=\sum_{\beta}\epsilon_{\alpha\beta}(\mathbf{q},\omega)E_{\beta}(\mathbf{q},\omega)\;.$
(4)

where $\alpha$ and $\beta$ are Cartesian directions. Several symmetries can be
leveraged to simplify the dielectric tensor. In an isotropic material, the
longitudinal (diagonal) and transverse (off-diagonal) components of the
dielectric tensor can be strictly separated. In a material with cubic
symmetry, the diagonal components of the dielectric tensor reduce to a single
scalar function, significantly simplifying the problem of calculating the
dielectric function.

We follow the standard procedure to calculate the scalar dielectric function
in the random phase approximation (RPA) in reciprocal space as:

$\epsilon_{\mathbf{GG^{\prime}}}(\mathbf{q},\omega)=\delta_{\mathbf{GG^{\prime}}}-v(\mathbf{q+G})P_{\mathbf{GG^{\prime}}}(\mathbf{q},\omega)\;,$
(5)

where $v$ is the bare Coulomb potential, $P$ is the polarizability,
$\mathbf{q}$ is a wave vector confined to the first Brillouin zone (BZ), and
$\mathbf{G}$ is a reciprocal lattice vector. For an isotropic material in the
RPA, the polarizability $P$ can be expressed as the independent particle
polarizability in terms of matrix elements of the density operator, where
$\rho_{\mathbf{q}}=\sum_{i}e^{-i\mathbf{q}\cdot\mathbf{x}_{i}}$ is the density
operator for a system of electrons at positions ${\mathbf{x}_{i}}$ in the
presence of a momentum transfer $\mathbf{q}$ from a fast impinging charged
particle Nozières and Pines (1959). The polarizability is then written as

$P_{\mathbf{GG^{\prime}}}(\mathbf{q},\omega)=\operatorname{lim}_{\eta\rightarrow
0}\sum_{nn^{\prime}\mathbf{k}}\braket{n\mathbf{k}}{e^{-i(\mathbf{q}+\mathbf{G})\cdot\mathbf{r}}}{n^{\prime}\mathbf{k+q}}\braket{n^{\prime}\mathbf{k+q}}{e^{i(\mathbf{q}+\mathbf{G^{\prime}})\cdot\mathbf{r}}}{n\mathbf{k}}\frac{f(\varepsilon_{n^{\prime}\mathbf{k+q}})-f(\varepsilon_{n\mathbf{k}})}{\varepsilon_{n^{\prime}\mathbf{k+q}}-\varepsilon_{n\mathbf{k}}-\hbar\omega-i\hbar\eta}\;,$
(6)

where $n$ is a band index of an electronic state, $\mathbf{k}$ is the crystal
momentum of an electronic state, $\mathbf{q}$ is the momentum transfer (here
restricted to the first Brillouin zone), $\mathbf{G}$ is a reciprocal lattice
vector, $f(\varepsilon)$ is the Fermi-Dirac distribution function, and $\eta$
is an infinitesimal positive quantity.

Figure 1: The loss function, the negative imaginary part of the inverse
dielectric function, for Si calculated with (a) DFT and (b) the GW
approximation plotted as a function of energy transfer $\omega$ and momentum
transfer $\mathbf{q}$, restricted to the first BZ, in units of energy. The
difference between the two loss functions indicates a blue shift in the
quasiparticle-corrected GW loss function relative to that calculated using DFT
as seen over (c) all the $\mathbf{q}$ points sampled along the $q_{z}$ axis of
the first BZ and in (d) the $\mathbf{q}\rightarrow 0$ cut. Figure 2: The real
part of the inverse dielectric function for Si calculated with (a) DFT and (b)
the GW approximation plotted as a function of energy transfer $\omega$ and
momentum transfer $\mathbf{q}$, restricted to the first BZ, in units of
energy. The difference between the two inverse dielectric functions indicates
a blue shift in the quasiparticle-corrected GW dielectric function relative to
that calculated using DFT as seen over (c) all the $\mathbf{q}$ points sampled
along the $q_{z}$ axis of the first BZ and in (d) the $\mathbf{q}\rightarrow
0$ cut.

The DM-electron scattering rate can be written in terms of the loss function
$L(\mathbf{q},\omega)=\operatorname{Im}\left[-\epsilon(\mathbf{q},\omega)^{-1}\right]$
Hochberg  _et al._ (2021); Knapen  _et al._ (2021), or in some formalisms the
dynamic structure factor, a related quantity Griffin  _et al._ (2020); Trickle
_et al._ (2020b); Sturm (1991). For example, for an impinging DM particle
$\chi$ with velocity $\mathbf{v}_{\chi}$ and an arbitrary DM-electron
interaction potential $V(\mathbf{q})$, Ref. Hochberg  _et al._ (2021) outlines
how the transition rate can be expressed as

$\Gamma(\mathbf{v}_{\chi})=\int\frac{d^{3}q}{(2\pi)^{3}}|V(\mathbf{q})|^{2}\left[2\frac{q^{2}}{e^{2}}\operatorname{Im}\left(-\frac{1}{\epsilon(\mathbf{q},\omega)}\right)\right]\;.$
(7)

From the first-principles calculated inverse dielectric function, we estimate
the expected sensitivity of a detector to DM-electron scattering following the
formalism outlined in Refs. Trickle  _et al._ (2020b); Geilhufe  _et al._
(2020); Hochberg  _et al._ (2021); Knapen  _et al._ (2021); Kahn and Lin
(2022). The total rate per target mass at time $t$ can be expressed as

$R(t)=\frac{1}{\rho_{T}}\frac{\rho_{\chi}}{m_{\chi}}\int
d^{3}\mathbf{v}_{\chi}f_{\chi}(\mathbf{v}_{\chi},t)\Gamma(\mathbf{v}_{\chi}),$
(8)

where the transition rate $\Gamma(\mathbf{v}_{\chi})$ has been defined in Eq.
(7), $f_{\chi}(\mathbf{v}_{\chi},t)$ is the DM velocity distribution at time
$t$, $\rho_{\chi}=0.3\,\mathrm{GeV}/\mathrm{cm}^{3}$ is the local DM density,
$m_{\chi}$ is the DM mass, and $\rho_{T}$ is the mass density of the target
material.

In DM-electron scattering, the scattering rate is usually parameterized in the
literature with

$\displaystyle V(\mathbf{q})$
$\displaystyle=\frac{g_{\chi}g_{e}}{q^{2}+m_{V/\phi}^{2}},$ (9)
$\displaystyle\bar{\sigma}_{e}$ $\displaystyle\equiv\frac{\mu_{\chi
e}^{2}}{\pi}\left|V(q_{0})\right|^{2},$ (10) $\displaystyle
F_{\mathrm{DM}}(q)$
$\displaystyle\equiv\frac{q_{0}^{2}+m_{V/\phi}^{2}}{q^{2}+m_{V/\phi}^{2}},$
(11)

where the DM is coupled to the Standard Model through some scalar ($\phi$) or
vector ($V$) mediator with mass $m_{V/\phi}$ with coupling $g_{\chi}$ to DM
and coupling $g_{e}$ to electrons. Here, $\bar{\sigma}_{e}$ is a reference
cross section for DM-electron scattering, $\mu_{\chi e}$ is the DM-electron
reduced mass $\mu_{\chi e}=\frac{m_{e}m_{\chi}}{m_{e}+m_{\chi}}$,
$q_{0}\equiv\alpha m_{e}$ is a reference momentum, and $F_{\mathrm{DM}}(q)$ is
the DM form factor. With Eqs. (7) and (8), we arrive at the following
scattering rate per target mass at time $t$

$R(t)=\frac{\bar{\sigma}_{e}}{\rho_{T}}\frac{\rho_{\chi}}{m_{\chi}}\frac{\pi}{(2\pi)^{4}\alpha\mu_{\chi
e}^{2}}\int
d\omega\,d^{3}\mathbf{q}\,q\,F_{\mathrm{DM}}^{2}(q)\operatorname{Im}\left[-\frac{1}{\varepsilon(\mathbf{q},\omega_{\mathbf{q}})}\right]\tilde{g}(v_{\mathrm{min}},\psi,t)\;,$
(12)

where we have defined

$\tilde{g}(v_{\mathrm{min}},\psi,t)=q\int
d^{3}\mathbf{v}_{\chi}f_{\chi}(\mathbf{v}_{\chi},t)\delta(E_{f}-E_{i})\;,$
(13)

similarly to Ref. Geilhufe  _et al._ (2020).

To calculate the expected sensitivity of a Si detector using our DFT and GW
calculations, we assume a background-free search with a kg-year exposure, and
set limits at the 90% C.L. We assume the Standard Halo Model Drukier  _et al._
(1986); Evans  _et al._ (2019) and average the time-dependent rate over the
full exposure (a small effect for an isotropic crystal such as Si).

Two caveats of this formalism are immediately apparent. The first is that it
is highly dependent on the accuracy of the first-principles wavefunctions and
energy eigenvalues used to compute the polarizability. For highly accurate
predictions of scattering rates this requires using first-principles methods
beyond traditional density functional theory (DFT). The second is that it is
restricted to calculation of scalar dielectric functions, limiting its ability
to capture screening effects in highly anisotropic materials. For this work we
address the first of these caveats, leaving discussion of anisotropic
materials to forthcoming work.

##  III Calculation Details

To calculate the finite $\mathbf{q}$ RPA dielectric function of Si, we first
perform density functional theory calculations using Quantum Espresso Gianozzi
_et al._ (2009, 2017, 2020). We use the BerkeleyGW package Hybertsen and Louie
(1986); Deslippe  _et al._ (2012) to calculate the full frequency dielectric
function using DFT-level energy eigenvalues and wavefunctions. We next
calculate the self-energy operator and quasiparticle corrections in the GW
approximation and recalculate the full frequency dielectric function with the
quasiparticle-corrected GW energy eigenvalues.

DFT calculations are performed with a plane-wave basis in the generalized-
gradient approximation (GGA) as implemented by Perdew, Burke, and Ernzerhof
(PBE) Perdew  _et al._ (1996) using a 100 Ry energy cut-off on a $10\,\times
10\,\times 10$ $\mathbf{k}$-grid. A scalar relativistic norm-conserving
pseudopotential including 4 valence electrons per Si atom from the Pseudo-Dojo
project is used Hamann (2013).

The DFT-level and quasiparticle-corrected GW full frequency dielectric
functions are calculated using a 4.0 Ry $\mathbf{G}$-vector cut-off and 100
meV broadening for a frequency range from 0-30 eV on a $10\,\times 10\,\times
10$ $\mathbf{q}$-grid. The self-energy operator is calculated using 4 filled
and 96 empty bands.

Figure 3: Sensitivity curves (90% C.L.) for DM-electron scattering with a Si
target, assuming a kg-yr exposure and zero background for a (a) heavy mediator
and (b) light mediator. In both panels, we compare the sensitivity obtained
using the loss function obtained from DFT and that calculated using the GW
approximation, where we see a significant difference in the expected
sensitivity and (c) relative electron-DM scattering cross-section.

##  IV Results

We calculate the RPA dielectric function of Si using DFT and GW eigenvalues
for a broad range of relevant energy transfers $\omega$ and momentum transfers
$\mathbf{q}$ from impinging particles. We calculate the dielectric function
for momentum transfer $\mathbf{q}$ up to a 4.0 Ry (54.4 eV) kinetic energy
cut-off for plane waves with momentum $\mathbf{q+G}$ and kinetic energy
$\frac{\hbar^{2}|\mathbf{q+G}|^{2}}{2m_{e}}$ with $\mathbf{q}$ and
$\mathbf{G}$ defined as in Eq.5 above. We consider energy transfers $\omega$
up to 30 eV. The static dielectric constant $\epsilon(\mathbf{q\rightarrow
0,\omega=0})$ calculated with GW, 11.46, agrees much better with the
experimental value, 11.7 Dunlap and Watters (1953), than that calculated using
DFT, 12.86.

In Fig. 1 we plot the RPA loss function, restricting ourselves to $\mathbf{q}$
vectors along the high symmetry $q_{z}$ line in the first BZ for visual
brevity. The momentum transfer $\mathbf{q}$ is plotted in units of energy. The
loss function is qualitatively similar for the DFT (Fig. 1a) and GW (Fig. 1b)
eigenvalues. By plotting the difference between the GW and DFT loss functions
(Fig. 1c,d), we see that the GW loss function is blue shifted to higher energy
relative to the DFT loss function, by $\sim$ 1 eV, a substantial energy
difference relative to the DM-electron interaction energy-scales of interest.
Additionally, the peak of the loss function calculated from GW is slightly
larger than that calculated from DFT due to quasiparticle self-energy effects.
The blue shift of the GW loss function is a result of the larger band gap
calculated via the GW approximation (0.97 eV) relative to that calculated
using DFT (0.58 eV) as a result of quasiparticle self-energy corrections. This
band gap increase shifts the imaginary part of the dielectric function to
higher energy because the smallest energy difference for a transition between
a filled and unfilled band shift to higher energy.

In Fig. 2 we plot the corresponding real part of the RPA inverse dielectric
function. Again, the real part of the RPA inverse dielectric function is
qualitatively similar for the DFT (Fig. 2a) and GW (Fig. 2b) eigenvalues. The
difference between the real parts of the GW and DFT inverse dielectric
functions (Fig. 2c) reveals again that the GW results are blue shifted to
higher energy.

In Fig. 3, we show the results of the sensitivity calculation for MeV-scale DM
masses for the limits of a heavy vector mediator ($m_{V}\to\infty$) (Fig.
3(a)) and a light vector mediator ($m_{V}\to 0$) (Fig. 3(b)). The limits
calculated with the GW and DFT loss functions are markedly different,
emphasizing the importance of accounting for many-body effects in first-
principles calculations of the dielectric function for predicting experimental
reach. In fact, the bare DFT calculation tends to overestimate DM sensitivity.
As shown in Fig. 3(c), the electron scattering cross sections corresponding to
equivalent DM masses are consistently larger when calculated with GW as
compared to DFT. The effect is most significant for heavy mediators, where the
electron scattering cross sections can be up to $\sim$4.5 times larger. This
corresponds to a substantial overestimation of the parameter space that can be
probed by Si when the reach estimation is performed using only DFT. For more
complicated, anisotropic crystal structures, we expect this discrepancy to
increase and leave those calculations to future work.

##  V Summary

Here we have used Si as a case study to demonstrate the importance of
incorporating beyond-DFT, many-body effects in first-principles calculations
of the dielectric function for predicting the expected scattering rates and
the experimental reach of detectors utilizing DM-electron scattering for sub-
GeV DM detection. We have calculated the RPA dielectric function of Si using
DFT eigenvalues and GW-corrected eigenvalues obtained through a single-shot
$G_{0}W_{0}$ calculation. By using the loss function to model the response of
a Si detector to impinging particles, we have calculated the projected
experimental reach of a Si detector. Crucially, we have found that DFT
overestimates the experimental reach. By using the GW approximation, we have
been able to correct for this overestimation to produce a more realistic
projected reach. As recent DM search results often rely on DFT to calculate
limits on DM-electron scattering, these limits are likely overestimating their
exclusion regions. We suggest to revisit these results with energy eigenvalues
that have been calculated at a higher level of theory, such as the GW
approximation. The novel detector materials that will be at the forefront of
the search for sub-GeV DM are expected to host much more complex many-body and
correlation effects than Si. We note that the GW approximation is only one of
many ways, such as DMFT, to account for many-body effects and that different
treatments will be more appropriate for different materials. Ultimately
though, our work underscores the limitations of DFT and the importance of
utilizing beyond-DFT methods in first-principles calculations of the
experimental reach of both current and next-generation sub-GeV DM detectors.

##  VI Acknowledgements

This work was supported by the U.S. DOE NNSA under Contract No.
89233218CNA000001. E.A.P., C. L. and J.-X.Z. acknowledge support by the LANL
LDRD Program through project number 20220135DR. S.L.W. acknowledges support
from the LANL Director’s Postdoctoral Fellowship award 20230782PRD1. This work
was supported in part by the Center for Integrated Nanotechnologies, a DOE
Office of Science user facility, in partnership with the LANL Institutional
Computing Program for computational resources. Additional computations were
performed at the National Energy Research Scientific Computing Center (NERSC),
a U.S. Department of Energy Office of Science User Facility located at
Lawrence Berkeley National Laboratory, operated under Contract No. DE-
AC02-05CH11231 using NERSC award ERCAP0020494.

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11institutetext:  Technische Hochschule Ingolstadt, Ingolstadt, Germany
Medical Department IV, Pulmonary Medicine and Thoracic Oncology, Klinikum
Ingolstadt, Ingolstadt, Germany
Institute of Pathology, University of Regensburg, Regensburg, Germany
Department of Internal Medicine II, University Hospital Regensburg,
Regensburg, Germany
Department Artificial Intelligence in Biomedical Engineering, Friedrich-
Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany

# Attention-based Multiple Instance Learning for Survival Prediction on Lung
Cancer Tissue Microarrays

Jonas Ammeling 11 Lars-Henning Schmidt 22 Jonathan Ganz 11 Tanja Niedermair 33
Christoph Brochhausen-Delius 33 Christian Schulz 44 Katharina Breininger 55
Marc Aubreville 111122334455<EMAIL_ADDRESS>
###### Abstract

Attention-based multiple instance learning (AMIL) algorithms have proven to be
successful in utilizing gigapixel whole-slide images (WSIs) for a variety of
different computational pathology tasks such as outcome prediction and cancer
subtyping problems. We extended an AMIL approach to the task of survival
prediction by utilizing the classical Cox partial likelihood as a loss
function, converting the AMIL model into a nonlinear proportional hazards
model. We applied the model to tissue microarray (TMA) slides of 330 lung
cancer patients. The results show that AMIL approaches can handle very small
amounts of tissue from a TMA and reach similar C-index performance compared to
established survival prediction methods trained with highly discriminative
clinical factors such as age, cancer grade, and cancer stage.

##  1 Introduction

Despite advances in medicine over the past two decades, lung cancer remains
the leading cause of cancer-related deaths worldwide [1]. Survival prediction
methods are used for predicting time-to-event outcomes such as cancer
recurrence or death and play an important role in clinical decision making in
oncology. Recent survival prediction methods used tissue samples from whole
sections [2, 3] to make predictions about a patient’s prognosis. However, with
the trend toward minimally invasive biopsy techniques in the treatment of lung
cancer, approaches that can process smaller amounts of tissue are needed [4].
TMAs provide the opportunity to explore such methods because they typically
contain only a very small amount of tissue for each patient from the original
tumor sample [5]. Bychkov et al. [6] used a combination of convolutional and
recurrent architectures to predict five-year survival as a binary
classification problem from TMA cores of colon cancer patients. However,
recurrent architectures are more complex than attention-based approaches,
which have been shown to be successful for predicting patient prognosis on
whole-slide images (WSI) [2, 7]. To investigate the applicability of
attention-based methods to TMA slides, we extended an AMIL approach originally
developed for weakly supervised classification to the survival prediction task
similar to [2] by converting the classification problem into a regression
problem by optimizing the modified partial Cox likelihood [8] as our loss
function. We applied the model to TMA slides from patients with non-small cell
lung cancer (NSCLC) and compared performance with established survival
prediction methods trained on prognostically discriminative tabular data. The
full code is available online111https://github.com/DeepPathology/Cox_AMIL.

##  2 Materials and Methods

###  2.1 Data

Tissue samples, clinicopathologic features, and follow-up information were
collected and examined from 379 NSCLC patients in the thoracic department of
St. Georgs Klinikum in Ostercappeln, Germany. Clinical TNM staging (including
clinical examination, CT scans, sonography, endoscopy, MRI, bone scan) was
performed based on UICC/AJCC recommendations. The definite tumor staging was
carried out post-surgically by pathological exploration. The histologic grade
was determined according to WHO criteria on the original whole section tumor
sample. The follow-up time was computed from the date of histological
diagnosis to death or censored at the date of last contact. The TMAs were
constructed by sampling with a 0.6 mm core needle from the most representative
parts of the original tumor block. Three cores per patient were punched from
the paraffin-embedded tumor block and assembled into TMA blocks. Multiple (at
most 3x) four-micrometer-thick sections were cut from the TMA blocks, stained
with hematoxylin and eosin (H&E), and digitized with a whole-slide scanner
(Pannoramic P1000, 3D HISTECH Ltd., Budapest, Hungary) at a resolution of
$0.24$ $\mu$m/pixel. The TMA cores were linked to a unique patient identifier
and extracted from the whole-slide TMA image. Due to missing values in the
tabular data and loss of tissue sections from the TMA blocks, 49 cases were
excluded from the analysis. Thus, data from 330 NSCLC patients was included
for further analysis.

###  2.2 Image Processing

To compare the use of the patient-level label for different tissue amounts,
all cores from the same patient were processed once individually and stitched
together once horizontally to create a new patient-level TMA (Fig. 1). All
images were processed using the public CLAM [9] repository for WSI analysis.
After tissue segmentation, non-overlapping patches of size $256\times 256$
were extracted at full resolution. Then, a ResNet50 model, pre-trained on
ImageNet, was used to convert each patch into a $1024$-dimensional feature
vector by spatial average pooling after the third residual block.

###  2.3 Attention-based Multiple Instance Learning

In the context of multiple instance learning, each image is viewed as a
collection of $M$ patches or instances, known as a bag, with a corresponding
bag-level label associated with it. After image processing, each bag is
represented by the patch-level embeddings $\mathbf{H}\in\mathbb{R}^{M\times
C}$, where $C$ is the feature dimension from the ResNet50 model. The model
consists of three components as shown in Figure 1: the projection layer
$f_{\text{proj}}$, the attention module $f_{\text{attn}}$, and the prediction
layer $f_{\text{pred}}$. The projection layer $f_{\text{proj}}$ is a fully-
connected layer with weights $\mathbf{W}_{\text{proj}}\in\mathbb{R}^{512\times
C}$ (all bias terms are implied for notational convenience) that maps the
patch-level embedding into a more compact, dataset-specific 512-dimensional
feature space. The attention module $f_{\text{attn}}$ learns to assign a score
to each patch based on its contribution to the patient’s predicted prognosis.
In particular, $f_{\text{attn}}$ consists of three fully connected layers with
weights $\mathbf{U}\in\mathbb{R}^{256\times 512}$,
$\mathbf{V}\in\mathbb{R}^{256\times 512}$, and
$\mathbf{Z}\in\mathbb{R}^{1\times 256}$. After projection, the attention score
$a_{m}$ for the $m$-th patch embedding $\mathbf{h}_{m}\in\mathbb{R}^{512}$, is
computed by [9]:

$a_{m}=\frac{\exp\bigl{\\{}\mathbf{Z}\left(\tanh\left(\mathbf{V}\mathbf{h}_{m}^{\text{T}}\right)\odot\text{sigm}\left(\mathbf{U}\mathbf{h}_{m}^{\text{T}}\right)\right)\bigr{\\}}}{\sum_{m=1}^{M}\exp\bigl{\\{}\mathbf{Z}\left(\tanh\left(\mathbf{V}\mathbf{h}_{m}^{\text{T}}\right)\odot\text{sigm}\left(\mathbf{U}\mathbf{h}_{m}^{\text{T}}\right)\right)\bigr{\\}}}$
(1)

The computed attention scores for each patch are then used as weight
coefficients to aggregate the patch-level embeddings into the bag
representation $\mathbf{h}_{\text{bag}}\in\mathbb{R}^{512}$ via attention-
pooling [9] by:

$\mathbf{h}_{\text{bag}}=\sum_{m=1}^{M}a_{m}\mathbf{h}_{m}$ (2)

The final prediction layer $f_{\text{pred}}$ is a single fully-connected layer
with weights $\mathbf{W}_{\text{pred}}\in\mathbb{R}^{1\times 512}$ with a
single output node and linear activation. Let $\theta$ represent all weights
of the network, then the model can be described as a function
$h_{\theta}:\mathbb{R}^{M\times C}\mapsto\mathbb{R}$, where the output is a
patient’s hazard log risk score, described in more detail in the next section.

Figure 1: Data stitching and overall architecture. Left: tissue-cores of the
same patient stitched together horizontally. Right: overview of the attention-
based multiple instance learning algorithm for survival prediction.

###  2.4 Loss Function

The right-censored patient-level survival data for the $i$-th patient consist
of the triple $(t_{i},\delta_{i},x_{i})$ and represent the observed time,
binary censoring status ($\delta=0$, death not observed), and image data,
respectively. Censoring is assumed to be non-informative, such that for a
given $x_{i}$, survival time and censoring are independent. Let
$t_{1}<t_{2}<t_{D}$ be the ordered event times. The risk set
$\mathcal{R}(t_{i})$ is defined as the set of all individuals still in the
study at a time immediately preceding $t_{i}$. The loss function is derived
from the common Cox proportional hazards model where the hazard function of
the form $\lambda(t\mid x)=\lambda_{0}(t)\cdot e^{h(x)}$ is composed of the
hazard baseline function $\lambda_{0}(t)$, and a risk score $r(x)=e^{h(x)}$.
Our proposed model estimates a patient’s log-risk score $\hat{h}_{\theta}(x)$
parameterized by the weights of the network $\theta$ such that the modified
Cox partial likelihood becomes [8]:

$L(\theta)=\prod_{i:\delta=1}\frac{\hat{r}_{\theta}(x_{i})}{\sum_{j\in\mathcal{R}(t_{i})}\hat{r}_{\theta}(x_{j})}=\prod_{i:\delta=1}\frac{\exp\left(\hat{h}_{\theta}(x_{i})\right)}{\sum_{j\in\mathcal{R}(t_{i})}\exp\left(\hat{h}_{\theta}(x_{j})\right)}$
(3)

The loss function which is minimized by the network is then obtained from the
average of the negative partial log likelihood with regularization as shown
below:

$l(\theta)=-\frac{1}{N}\sum_{i}\delta_{i}\left(\hat{h}_{\theta}(x_{i})-\log\sum_{j\in\mathcal{R}(t_{i})}\exp\left(\hat{h}_{\theta}(x_{j})\right)\right)+\lambda\cdot\|\theta\|_{1},$
(4)

where $N$ is the number of patients with an observable event and $\lambda$ is
the $l_{1}$ regularization parameter. Intuitively, the loss function penalizes
discordance between the scores of higher-risk and lower-risk patients. Similar
loss functions have been used previously by other authors [10, 7]. Details
about the training and hyperparameter settings are online1.

###  2.5 Baselines

The performance of the proposed model was compared against three established
survival analysis methods based on tabular data. The tabular data used to
train these baseline methods consist of patient characteristics (age, sex, and
smoking status) and clinical characteristics (cancer stage and grade). It
should be noted that the latter require elaborate determination and are
commonly accepted to be highly prognostic. The first baseline method was a
classical linear Cox proportional hazards (CPH) model [8]. The second was a
random survival forest (RSF) [11], a non-proportional, flexible and robust
alternative to the classical CPH model. The third baseline method was DeepSurv
[10], a modern nonlinear, deep learning-based CPH model. Moreover, the
proposed model was compared with a MIL method using classical max-pooling, and
with the performance of an AMIL model trained on each patient core
individually, using the respective patient-level label.

###  2.6 Evaluation

To evaluate and compare the predictive performance on the survival data, we
performed a 10-fold cross-validation and measured Harrell’s concordance index
(C-index) [12]. The C-index indicates how well a model predicts the ranking of
patients’ death times, where large values of $\hat{h}_{\theta}(x_{i})$ should
be associated with small values of $t_{i}$ and vice versa. A value of $C=0.5$
corresponds to the average C-index of a random model, whereas $C=1$
corresponds to a perfect association. Patient stratification was assessed by
assigning patients to either a high-risk or a low-risk group based on the
median of the predicted risk score. Kaplan-Meier curves were constructed by
pooling the risk predictions of the test folds and plotting them against their
survival time. Logrank tests were performed to check for statistically
significant differences (P-value $<0.05$) between the two survival
distributions.

##  3 Results

The cross-validated C-index values and the Kaplan-Meier curves for the patient
stratification results are shown in Figure 2. The classical CPH model achieved
the overall best performance with an average C-index of $0.61\pm 0.07$
(P-value: $7.98\times 10^{-4}$) and a median C-index of $0.59$. Closely
similar, the RSF model achieved an average C-index of $0.60\pm 0.06$ (P-value:
$1.61\times 10^{-3}$) and a median C-index of $0.60$. The DeepSurv method
performed the worst among all methods with an average C-index of $0.50\pm 0.1$
(P-value: $2.20\times 10^{-1}$) and a median C-index of $0.53$. Among the
image-based methods, the classical MIL model with max-pooling performs the
worst with an average C-index of $0.54\pm 0.08$ (P-value: $2.90\times
10^{-1}$) and a median C-index of $0.53$. The AMIL model performed the best
among the image-based methods with an average C-index of $0.61\pm 0.07$
(P-value: $1.69\times 10^{-5}$) and a median C-index of $0.63$. The AMIL model
trained on individual cores performed worse than the AMIL model trained using
the patient label for all cores together, with an average C-index of $0.56\pm
0.06$ (P-value: $5.72\times 10^{-11}$) and a median C-index of $0.58$.

Figure 2: a) Cross-validated C-index performance. + indicates mean value. b)
Kaplan-Meier curves for patient stratification results on the pooled
validation splits.

##  4 Discussion

The results show that the proposed model performed similarly well to
established survival prediction methods based on tabular data, such as CPH and
RSF, by extracting features directly from the small amount of tissue provided
by the TMA. Attention pooling proved to be very effective in this regard for
aggregating these features to estimate the patient’s risk score. However, the
experiments also showed that the prognostically relevant information is not
necessarily evenly distributed across a patient’s tissue cores when selected
from the original sample, as indicated by lower average C-index when trained
on individual cores. Thus, performance is highly dependent on tissue core
selection, and it is recommended to use all available tissue cores.
Nevertheless, the results show that attention-based MIL approaches are
applicable to TMA slides where only a small amount of tissue per patient is
available to perform reliable patient stratification, as shown by the
statistically significant logrank test results. Therefore, in the context of
the trend toward minimally invasive biopsy procedures in lung cancer
treatment, these methods could become a part of reliable decision support
systems in the future.

J. A. acknowledges funding by the Bavarian Institute for Digital
Transformation (Project ReGInA).

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# It’s anyon’s game: the race to quantum computation

Jainendra K. Jain  Physics Department, 104 Davey Lab, Pennsylvania State
University, University Park, Pennsylvania 16802, USA

###### Abstract

In 1924, Satyendra Nath Bose dispatched a manuscript introducing the concept
now known as Bose statistics to Albert Einstein. Bose could hardly have
imagined that the exotic statistics of certain emergent particles of quantum
matter would one day suggest a route to fault-tolerant quantum computation.
This non-technical Commentary on “anyons,” namely particles whose statistics
is intermediate between Bose and Fermi, aims to convey the underlying concept
as well as its experimental manifestations to the uninitiated.

Fractional statistics: The quantum mechanical state of a collection of
particles is described by a complex valued wave function
$\Psi(\\{{\textbf{{r}}}_{j}\\})$, with $|\Psi(\\{{\textbf{{r}}}_{j}\\})|^{2}$
giving the probability that the particles are located at positions
$({\textbf{{r}}}_{1},{\textbf{{r}}}_{2},\cdots{\textbf{{r}}}_{N})$. An
exchange of two identical particles does not produce a new state, and in
particular, leaves the probability $|\Psi(\\{{\textbf{{r}}}_{j}\\})|^{2}$
unchanged. This imposes certain symmetry constraints on the allowed wave
functions. Two possibilities are $\Psi\rightarrow+\Psi$ or
$\Psi\rightarrow-\Psi$ under exchange of two particles. Particles obeying the
former relation are called bosons, and the latter fermions. We speak of Bose
and Fermi statistics because the property of particles under exchange affects
the counting of distinct microscopic configurations and thus their statistical
mechanics and thermodynamics.

Examples of fermions are electrons, quarks, protons, neutrons and 3He atoms;
examples of bosons are the Higgs bosons, 4He atoms and photons. Even though
particle statistics may appear an abstract concept unconnected to our daily
lives, it is central to our existence. The Fermi statistics of electrons is
responsible for the structure of atoms and molecules, and thus permeates
atomic physics, chemistry and biology; it governs, at the most fundamental
level, the properties of all matter around and within us. The Bose statistics
is responsible for dramatic phenomena at low temperatures, such as Bose-
Einstein condensation, superconductivity and 4He superfluidity.

Figure 1: (a) Exchange of two particles produces a phase factor $e^{i\theta}$.
(b) Particle exchange is equivalent to half a loop of one particle around
another. (c) A full loop equals two exchanges. (d) In two dimensions the
orange and green paths are topologically distinct, but in higher dimensions
they are not.

It was recognized four decades ago Leinaas77 ; Wilczek82  that, in two space
dimensions, quantum mechanics also admits particles that behave under exchange
as $\Psi\rightarrow e^{i\theta}\Psi$ with arbitrary $\theta$. To see the
essential idea, let us imagine exchanging two particles along a
counterclockwise path shown in Fig. 1 (a), which produces a phase factor
$e^{i\theta}$. (A clockwise exchange will produce $e^{-i\theta}$.) From the
reference frame of one of the particles, this is a half loop [Fig. 1 (b)]. A
full loop [Fig. 1 (c)], called a winding or braiding, represents two
exchanges. The phase $\theta$ is called the (braiding) statistical phase. For
bosons (fermions) we have $\theta=m\pi$ where $m$ is an even (odd) integer.
Particles for which $\theta$ is not an integer multiple of $\pi$ are referred
to as anyons, which stands for particles with any statistics. It is stressed
that $\theta$ is a topological quantity, i.e. it does not depend on the size
and shape of the exchange path or the closed loop. Anyons can be defined only
in two space dimensions, because in higher dimensions the notion of a
counterclockwise / clockwise exchange or of a particle going around another is
not well defined. In three dimensions, by lifting the paths off the page, one
can continuously deform the counterclockwise exchange into a clockwise one in
Fig. 1 (a), or the orange loop into the green loop in Fig. 1 (d); this implies
$e^{i\theta}=e^{-i\theta}$ and hence $\theta=m\pi$.

Of course, just because anyons can be defined does not mean that they exist.
Our world is three dimensional, and there certainly are no anyons in the list
of particles in a particle physics text book. Fortunately, this list is
incomplete. Interacting quantum matter generates its own emergent particles,
which can be rather exotic, and no principle of physics precludes the
possibility that some of them might obey fractional braiding statistics. A
quantum state supporting such particles would have to be a rather exotic state
in two dimensions. Soon after the concept of fractional statistics was
proposed, nature produced a promising candidate for its realization Halperin84
; Arovas84 , namely the fractional quantum Hall effect.

Fractional quantum Hall effect: Typically, electric current flows in the
direction of an applied voltage. In the presence of a magnetic field $B$,
however, it flows at an angle; that is, there is a voltage in the direction of
the current flow as well as across it. The Hall resistance $R_{H}$ is the
ratio of transverse voltage to current. The laws of classical electrodynamics
tell us that $R_{H}$ is proportional to $B$, as seen routinely by experiments.

Dramatic quantum phenomena are revealed, however, when electrons are confined
to two dimensions, cooled to near zero Kelvin, and subjected to a strong
magnetic field $B$. The primary observation is that as $B$ is varied, the Hall
resistance exhibits a series of plateaus precisely quantized at $R_{H}=h/(\nu
e^{2})$, where $h$ is the Planck’s constant, $e$ is electron’s charge, and
$\nu$ is either an integer or a fraction. The appearance of such simple and
universal values that are utterly oblivious to all of the complexities of the
sample is an amazing result that has fascinated physicists for decades.

The observation of integer and fractional values of $\nu$ are referred to as
the integer quantum Hall effect Klitzing80  (IQHE) and fractional quantum Hall
effect Tsui82  (FQHE), respectively. The IQHE can be understood using standard
methods, because it occurs in a theoretical model of noninteracting electrons,
and one can make convincing arguments that the interaction does not destroy
it. In contrast, the FQHE results fundamentally due to the interaction between
electrons, which produces certain complex liquids of strongly correlated
electrons. Close to one hundred such FQH liquids, distinguished by $\nu$, have
been observed so far in a variety of two-dimensional materials, such as
semiconductor quantum wells and graphene.

Figure 2: A light-hearted depiction of how interacting electrons in a magnetic
field (upper row) capture magnetic flux quanta to turn into noninteracting
composite fermions (lower row). Illustration by Kwon Park.

Composite fermions: There are several ways of viewing the FQHE. The simplest
understanding of its origin is achieved in terms of an emergent particle
called the composite fermion Jain07 . The composite fermion is an unusual
particle: it is often visualized as the bound state of an electron and an even
integer number ($2p$) of magnetic flux quanta, where a flux quantum is defined
as $\phi_{0}=h/e$. (This picture of composite fermion is not to be taken
literally, but is sufficient for many purposes.) Tremendous simplification
occurs because while electrons are strongly interacting, composite fermions
are, to a good approximation, noninteracting [Fig. 2]. In other words, the
primary role of the interaction between electrons in the FQH regime is simply
to create composite fermions. (How many flux quanta composite fermions capture
depends on $B$.) Many predictions of composite fermions have been confirmed.
In particular, the IQHE of composite fermions appears as the FQHE of electrons
at fractions $\nu=n/(2pn\pm 1)$, $n$ and $p$ being integers, which explains
almost all of the observed fractions. Composite fermions have been directly
observed, and provide a natural framework for understanding various properties
of the FQH liquids.

Figure 3: Extracting the statistical phase for quasiparticles, i.e. excited
composite fermions, shown in blue. The composite fermions belonging to the
ground state are shown in green. The statistical phase for the quasiparticles
is given by the phase associated with loop on the left, which encloses another
quasiparticle, minus the phase associated with the loop on the right, which
does not.  Figure 4: Experimental realization of the scheme in Fig. 3, using
interference of quasiparticles moving along the sample edges. Quasiparticles
are injected at the upper left edge, and measured at the lower left edge. The
two relevant paths are shown in red and orange, artificially displaced in the
region where they coincide. The vertical dotted lines indicate quantum
mechanical tunneling across a barrier. Composite fermions belonging to the
ground state are not shown to avoid clutter.

What about fractional statistics? The excitations of the FQH liquids are
nothing but excited composite fermions, referred to as quasiparticles. It
turns out that these quasiparticles obey fractional braiding statistics. The
gist of the idea is as follows. Consider a closed loop of a quasiparticle,
shown in blue in Fig. 3, winding around another quasiparticle. The phase
associated with the loop can be calculated, but it is complicated: in addition
to the statistical phase $2\theta$, it also involves a contribution from the
composite fermions in the ground state (shown in green) as well as a phase
(called the Aharonov-Bohm phase) that a charged particle acquires when it
moves in the presence of a magnetic field. Fortunately, these other path-
dependent contributions can be eliminated by subtracting the phase for an
identical loop that does not enclose any quasiparticle (see Fig. 3). A
microscopic calculation Jain07  gives $2\theta=4\pi p/(2pn\pm 1)$ modulo
$2\pi$ for the quasiparticles of the $\nu=n/(2pn\pm 1)$ FQH state, consistent
with earlier results Halperin84 ; Arovas84 .

Experimental evidence: To measure the statistical phase experimentally, one
first needs to produce well defined paths. Here another property of quantum
Hall effect comes in handy: the current flows along the edges of the sample,
and moreover, on a given edge it flows only in one direction. This suggests
the geometry shown in Fig. 4. A composite fermion injected into the sample at
the upper left edge has several choices. The most probable outcome is that it
continues to ride the upper edge and escapes to the right. However, it can
also tunnel into the lower edge where the two edges come close. The
probability of its appearing at the lower left edge depends, according to the
laws of quantum mechanics, on the phase difference between the red and the
orange paths, which is precisely the phase associated with the loop encircling
the inner island. The change in this phase when a quasiparticle is added to
the interior will give the statistical phase $2\theta$, as discussed above.

Impressive progress has been made by several experimental groups toward
detecting fractional braiding statistics using this geometry Willett09 ; An11
; Mcclure12 ; Nakamura19 . One complicating factor is that if the paths are
too long, the composite fermion forgets its phase along the way, whereas if
they are too short, another phenomenon called Coulomb blockade comes into
play. Overcoming these obstacles with a clever sample design, a recent
experiment Nakamura20  has reported the most convincing observation so far of
discrete phase slips of $-2\pi/3$ at $\nu=1/3$, precisely as expected from
fractional braiding statistics for the quasiparticles of the $\nu=1/3$ FQH
state.

Quantum computation with anyons? As physicists work further to better
understand, confirm, and probe fractional statistics for the excitations of
various FQH states, and search for other phenomena arising from them, an
outsider may wonder: How will fractional statistics affect my life? While it
would be unwise to make predictions, because new ideas often surprise us by
opening unanticipated directions, a dream application would be topological
quantum computation Nayak08 . This would actually require more complex anyons
than the ones discussed above, called “non-Abelian anyons.” For such anyons
the outcome of a sequence of windings depends on the order in which they are
performed. A special case of such anyons, called Majorana particles, are
theoretically believed to occur in the $\nu=5/2$ FQH state wherein composite
fermions pair up to form a superconductor Read00 . This state has also been
investigated in interference experiments Willett09 ; An11 ; Willett13 .
Intense search for Majorana particles is also ongoing in contexts outside of
the FQHE. How can they help with quantum computation? A fundamental impediment
to quantum computation is decoherence due to interaction with the environment.
Here, one can imagine a topological qubit (quantum bit) in which information
is stored non-locally over two distant Majoranas, and thus immune to
decoherence by local fluctuations in the environment. The realization of a
quantum computer based on non-Abelian anyons of the FQHE would be a fitting
legacy of fractional statistics.

I thank Yayun Hu for discussions. Financial support from the US Department of
Energy, Office of Basic Energy Sciences, under Grant No. DE-SC0005042 is
gratefully acknowledged.

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