# Strongly correlated ladders in K-doped $p$-terphenyl crystals

John Sous<EMAIL_ADDRESS>Department of Physics, Columbia University, New
York, New York 10027, USA  Natalia A. Gadjieva  Department of Chemistry,
Columbia University, New York, New York 10027, USA  Colin Nuckolls  Department
of Chemistry, Columbia University, New York, New York 10027, USA  David R.
Reichman<EMAIL_ADDRESS>Department of Chemistry, Columbia University,
New York, New York 10027, USA  Andrew J. Millis<EMAIL_ADDRESS>Department of Physics, Columbia University, New York, New York 10027, USA
Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue,
New York, NY 10010

###### Abstract

Potassium-doped terphenyl has recently attracted attention as a potential host
for high-transition-temperature superconductivity. Here, we elucidate the
many-body electronic structure of recently synthesized potassium-doped
terphenyl crystals. We show that this system may be understood as a set of
weakly coupled one-dimensional ladders. Depending on the strength of the
inter-ladder coupling the system may exhibit spin-gapped valence-bond solid or
antiferromagnetic phases, both of which upon hole doping may give rise to
superconductivity. This terphenyl-based ladder material serves as a new
platform for investigating the fate of ladder phases in presence of three-
dimensional coupling as well as for novel superconductivity.

_Introduction_.— Recent reports Wang  _et al._ (2017); Pinto  _et al._ (2020)
of temperature-dependent gaps suggestive of superconductivity with a
transition temperature of up to $\sim$ 100 K in powders formed from annealed
alkali-doped terphenyl molecules have generated both excitement and questions.
Terphenyl molecules are organic compounds based on linked aromatic (benzene)
rings that exhibit strong electronic correlations, but until recently could
not be doped to levels relevant for strongly correlated electronic and
magnetic phases. The materials on which the recent reports are based are
however annealed powders not single crystals, which, from a synthetic and
structural point of view, cannot be fully well characterized Rickhaus  _et
al._ (2010). Progress requires convergence of experiment and theory based on
well characterized materials with specific crystal structures. A recent in-
solution synthetic approach presents an alternative path to the synthesis of
doped $p$-terphenyl crystals with well characterized structures Gadjieva  _et
al._ (2020), opening a door to the systematic study of correlation-driven
physics in these systems 111Note that well characterized crystals of alkali-
doped terphenyl were also reported in the 1980s Noordik  _et al._ (1981)..

Figure 1: Structure of [K($\mathbf{222}$)]2[$p$-terphenyl]3 crystal. Upper
panel: Crystal structure in two different orientations corresponding
approximately (tilde) to planes defined by the body-centered cubic (BCC)
lattice axes introduced in the text. In the left image, corresponding to the
$y_{\rm BCC}$-$\tilde{z}_{\rm BCC}$ plane, note visible channels of the (222)
(cage) chelated to four K+ ions (violet dots), surrounded by $p$-terphenyl
molecules that are either doped (blue) or neutral (red). In the right image,
corresponding to the $\tilde{x}_{\rm BCC}$-$\tilde{z}_{\rm BCC}$ plane, note
the chain pattern of terphenyl molecules along $\tilde{x}_{\rm BCC}$. Lower
panel: Schematic representation of the tight-binding model corresponding to
the BCC lattice of coupled ladders with $t_{0}$ (black solid lines) and
$t_{1}$ (heavy and light solid blue lines) hoppings within individual ladders
and $t_{2}$ hopping (dotted gray lines) coupling different ladders. Note
$t_{1}$ hopping within a ladder connects orbitals in different unit cells. We
employ different thickness of the $t_{1}$ bonds as a guide to the eye. Thin
vertical lines represent the edges of the BCC lattice.

In this work, we present a theory of electronic structure and correlations in
the potassium-doped terphenyl crystals reported in Ref. Gadjieva  _et al._
(2020). We find that this system can be accurately described in terms of a
model of coupled ladders at half filling with an inter-ladder hopping
amplitude about half as large as the intra-ladder hopping amplitudes. This
class of models exhibits a transition as a function of the inter-ladder
coupling strength from a spin-gapped paramagnet to a gapless
antiferromagnetically ordered state Dagotto and Moreo (1988); Gopalan  _et
al._ (1994). Our computed value of the inter-ladder coupling places this
material near to the transition on the antiferromagnetic side of the phase
diagram. Using literature results on related models, we sketch the physical
behavior, including potential $d$-wave superconductivity, expected upon doping
the parent compound with holes and propose experiments. Our results establish
terphenyl-based supramolecular materials whose structure can be controllably
manipulated synthetically as a new family of ladder materials within which to
explore correlation-driven emergent electronic and magnetic behavior.

Crystal structure.— The recently synthesized crystals of potassium-doped para-
terphenyl, [K($\mathbf{222}$)]2[$p$-terphenyl]3 Gadjieva  _et al._ (2020),
consists of two potassium (K) atoms, each within a cryptand ($\mathbf{222}$),
for every three $p$-terphenyl (C18H14) molecules, see Fig. 1). The cryptand is
a polydentate ligand formed of nitrogen, oxygen, carbon and hydrogen atoms
used in a solution-based approach to hold the dopant K atoms in place within
the crystal structure upon crystallization. The synthesized crystals are
monoclinic, with a primitive unit cell formed of two units of
[K($\mathbf{222}$)]2[C18H14]3, containing four K atoms, six terphenyl
molecules and a total of 444 atoms per unit cell. The cell parameters:
$\vec{a}=\\{0,13.0237,0.0\\}$$\mathrm{\SIUnitSymbolAngstrom}$,
$\vec{b}=\\{13.694074,6.51185,8.698852\\}$$\mathrm{\SIUnitSymbolAngstrom}$ and
$\vec{c}=\\{13.694074,6.51185,-13.976448\\}$$\mathrm{\SIUnitSymbolAngstrom}$
in Cartesian $x,y,z$ coordinates. Note $\vec{a}$ is purely along one of the
principal axes of the system, which implies that a two-dimensional (2D)
structure may be possible to engineer, for example by exfoliation or
adsorption of a solution to a surface. We also note that this system differs
significantly from optimized structures obtained in simulations of doped
$p$-terphenyl crystals without cryptand ligands Zhong  _et al._ (2018). The
cryptand leads to larger inter-terphenyl distances than those simulated in
Ref. Zhong  _et al._ (2018), and enables a different level of doping. The
compound we study corresponds to a level of doping of less than one K atom per
terphenyl molecule (or of one K atom per _active_ terphenyl molecule, see
below), while the structure studied in Ref. Zhong  _et al._ (2018) considers a
range of doping levels from one to three K atoms per terphenyl molecule.

Electronic structure.— We perform ab-initio density-functional theory (DFT)
calculations using the Quantum ESPRESSO software package Giannozzi  _et al._
(2009), utilizing the Perdew, Burke and Ernzerhof (PBE) exchange correlation
functional Perdew  _et al._ (1996), see Supplementary Information for details.
We note that prior spin-polarized DFT calculations revealed a stable three-
dimensional (3D) antiferromagnet formed from the terphenyl molecules in this
crystal structure Gadjieva  _et al._ (2020). These calculations are in effect
a Hartree-Fock approximation that cannot capture the spin-gapped state of
isolated ladders and may in general overestimate magnetism. The ground state
and exchange coupling constants were reported in this work but not the band
structure. Here, we aim to characterize the system’s spin-unconstrained
electronic band structure and indicate how to incorporate electron-electron
and electron-lattice correlations in order to understand correlation-driven
behavior and emergent phases characteristic of this class of organic systems.

Figure 2 summarizes our main result concerning the electronic structure of
this system. Our DFT band structure calculations (Fig. 2, upper panel; left
column) reveal six electronic bands near the Fermi level $\mathrm{E_{F}}$,
with four lower-energy bands whose bandwidth ranges in the interval $\sim 50$
\- $75\mathrm{meV}$ roughly symmetric about $\mathrm{E_{F}}$, and two higher-
energy bands whose bandwidth lies in the $\sim 10\mathrm{meV}$ range located
above $\mathrm{E_{F}}$. This doped crystalline structure exhibits much
narrower bands than those found in the theoretically generated structures of
Ref. Zhong  _et al._ (2018), likely due to the extended space between
terphenyl molecules occupied by the cryptand molecules in this crystal, absent
in simulated optimized structures studied in Ref. Zhong  _et al._ (2018). To
understand the bands, we review the electronic structure of undoped terphenyl
molecules. Puschnig and Drexl Puschnig and Ambrosch-Draxl (1999) used DFT
calculations for single oligophenyl molecules (e.g. terphenyl) isolated in
space to argue for a Hückel molecular orbital picture in which the lowest
unoccupied molecular orbital (LUMO) is localized on the molecule’s aromatic
rings. For the structure considered in this work, this means that the six
terphenyl molecules in the unit cell contribute six states, giving rise to six
bands near the Fermi level. We also expect that each of the four dopant K
atoms donates a single unit of charge to the terphenyl molecules. Single-
crystal X-ray diffraction experiments reveal four equivalent terphenyl
molecules with relatively shorter inter-phenyl bond lengths (shown in blue in
Fig. 1, upper panel) and two others with a relatively longer bond length
(shown in red in Fig. 1, upper panel) within each unit of
[K($\mathbf{222}$)]4[$p$-terphenyl]6 Gadjieva  _et al._ (2020). We associate
the four dispersive bands with the LUMO states of the four short-bond
molecules and argue that these contain the four electrons donated by the K
atoms.

Figure 2: Electronic band structure of [K($\mathbf{222}$)]2[$p$-terphenyl]3
crystal. Upper panel: Electronic band structure from DFT calculations and
Wannier fits (left column) and from the minimal model with all Hamiltonian
matrix elements smaller in magnitude than than 0.01eV neglected (right column)
for $t_{2}\neq 0$ and $t_{2}=0$. Lower panel: Electronic band dispersion from
the minimal model along
$\vec{v}_{\vec{b}+\vec{c}-\vec{a}}=(\vec{b}+\vec{c}-\vec{a})/|\vec{b}+\vec{c}-\vec{a}|^{2}$
(left) and its two orthogonal directions
$\vec{v}_{\perp_{1}}=\vec{a}/|\vec{a}|^{2}$ (center) and
$\vec{v}_{\perp_{2}}={\mathcal{R}_{x\rightarrow z,y\rightarrow
y,z\rightarrow-x}}[\vec{v}_{\vec{b}+\vec{c}-\vec{a}}]$ (right) for $t_{2}\neq
0$ (top) and $t_{2}=0$ (bottom). Here we interpolate the dispersion uniformly
along $[-\vec{v},\vec{v}]$ through the $\Gamma$ point.
${\mathcal{R}_{x\rightarrow z,z\rightarrow-x}}=\begin{pmatrix}0&0&-1\\\
0&1&0\\\ 1&0&0\end{pmatrix}$ denotes a rotation matrix that takes
$x\rightarrow z$, $y\rightarrow y$, $z\rightarrow-x$.

Minimal model.— We use the wannier90 software package Mostofi  _et al._ (2014)
to obtain a tight-binding representation of the near-$\mathrm{E_{F}}$ bands
Marzari and Vanderbilt (1997), see Supplementary Information for details. We
obtain Wannier fits of six- (not shown) and four-band (Fig. 2, upper panel;
left column) models in excellent agreement with the DFT bands. We study the
electronic behavior in the crystal using the constructed four-band Wannier
model. Projecting the DFT Hamiltonian onto Wannier states yields a minimal
model containing four orbitals in each unit cell $\psi_{\nu}$ with
$\nu\in\\{1,2,3,4\\}$. These are coupled in real space via four types of
matrix elements:

  * •

Intra-unit cell hopping $t_{0}=0.022$eV between $\psi_{1}$ and $\psi_{2}$ and
also between $\psi_{3}$ and $\psi_{4}$.

  * •

Inter-unit cell hopping $t_{1}=0.028$eV from $\psi_{1}$ in unit cell $\vec{i}$
to $\psi_{4}$ in unit cell $\vec{i}+\vec{a}$ and to $\psi_{4}$ in unit cell
$\vec{i}+\vec{b}+\vec{c}$, and similarly from $\psi_{2}$ in unit cell
$\vec{i}$ to $\psi_{3}$ in unit cell $\vec{i}+\vec{a}$ and to $\psi_{3}$ in
unit cell $\vec{i}+\vec{b}+\vec{c}$, and their Hermitian conjugates.

  * •

Inter-unit cell hopping $t_{2}=0.013$eV from $\psi_{1}$ in unit cell $\vec{i}$
to $\psi_{3}$ in unit cell $\vec{i}+\vec{b}$ and from $\psi_{2}$ in unit cell
$\vec{i}$ to $\psi_{4}$ in unit cell $\vec{i}+\vec{c}$, and their Hermitian
conjugates.

  * •

Very small further-neighbor hoppings with magnitude $<0.01$eV that have a
negligible effect on the physics which we ignore in the following, see Fig. 2,
upper panel; right column.

To better understand the minimal model, we consider the $t_{2}=0$ limit (see
Supplementary Information for more details). We contrast the electronic
behavior of the model in this artificial $t_{2}=0$ limit against the nearly
exact $t_{2}\neq 0$ limit in Fig. 2, upper (right column) and lower panels.
Unlike in the $t_{2}\neq 0$ case (Fig. 2, lower panel; top), in the $t_{2}=0$
limit we find perfectly dispersionless bands along the two directions defined
by the vectors orthogonal to the direction of $\vec{b}+\vec{c}-\vec{a}$ (Fig.
2, lower panel; bottom). The band structure becomes flat along the segments in
$k$-space corresponding to these orthogonal directions (Fig. 2, upper panel;
right column). Thus, the vector $\vec{b}+\vec{c}-\vec{a}$ defines a special
one-dimensional (1D) connectivity between Wannier orbitals in the minimal
model. We can understand this 1D limit as follows. First setting $t_{0}=0$, we
obtain two uncoupled 1D zigzag chains: one where an electron hops with
amplitude $t_{1}$ from $\psi_{1}$ to $\psi_{4}$ one cell further in the
$\vec{b}+\vec{c}$ direction, then again to $\psi_{1}$ one more cell further in
the $-\vec{a}$ direction, and a second involving hopping between $\psi_{2}$
and $\psi_{3}$ in similar directions. Thus we see that electrons disperse
along the $\vec{b}+\vec{c}-\vec{a}$ direction. The zigzag pattern reflects the
spatial orientation of the Wannier orbitals across different unit cells. A
finite $t_{0}$ hopping amplitude couples the two zigzag chains via hopping
between $\psi_{1}$ ($\psi_{4}$) on one chain and $\psi_{2}$ ($\psi_{3}$) on
the other. This connectivity corresponds naturally to a ladder geometry with
hopping $t_{0}$ along the rungs and with hopping $t_{1}$ along the legs, see
Fig. 1, lower panel. At $t_{2}=0$, the bands are exactly particle-hole
symmetric (Fig. 2, lower panel; bottom) and exhibit a doubled Brillouin zone
or, equivalently, a halved unit cell, as expected in a ladder structure whose
primitive unit cell contains two orbitals. The coupled-ladder structure
present in this system was not discussed in the terphenyl structures obtained
theoretically in Ref. Zhong  _et al._ (2018).

We can now easily understand the $t_{2}\neq 0$ limit of the minimal model. We
see that $t_{2}$ couples different legs of pairs of ladders, either in the
$\vec{b}$ or in the $\vec{c}$ direction, see Fig. 1, lower panel. In this
coupled-ladder construction, an electron can hop for example from $\psi_{1}$
on one given ladder to $\psi_{3}$ on another in the $\vec{b}$ direction, and
then move on the ladder arbitrarily until it arrives at $\psi_{4}$, at which
point it can hop to $\psi_{2}$ on a third ladder in the $-\vec{c}$ direction,
or alternatively until it arrives at $\psi_{2}$, at which point it can hop to
a different third ladder in $\vec{c}$ direction. The former process implies
electrons disperse along the $\vec{b}-\vec{c}$ direction and the latter one
tells us they also disperse along the $\vec{a}$ direction (the $y$-component
of $\vec{b}+\vec{c}$). This explains the full anisotropic electronic band
dispersion in all three space and momentum directions shown in Fig. 2. This 3D
network of weakly coupled ladders must be described by the full unit cell of
four orbitals, giving rise to the enlarged zone in Fig. 2, lower panel; top.

Making use of this information, we derive a Hamiltonian for this system in a
more convenient coordinate system described by a body-centered cubic (BCC)
lattice geometry. We introduce
$\vec{x}_{\mathrm{BCC}}=\vec{b}+\vec{c}-\vec{a}$,
$\vec{y}_{\mathrm{BCC}}=\vec{a}$ and $\vec{z}_{\mathrm{BCC}}=\vec{b}-\vec{c}$
(where
$\vec{x}_{\mathrm{BCC}}\cdot\vec{y}_{\mathrm{BCC}}=\vec{y}_{\mathrm{BCC}}\cdot\vec{z}_{\mathrm{BCC}}=0$),
which define the principal axes along which electrons in the minimal model
disperse. We write primitive lattice vectors for the BCC lattice in terms of
these vectors: $\vec{a}_{1}^{\rm
P}=\frac{1}{2}(\vec{x}_{\mathrm{BCC}}+\vec{y}_{\mathrm{BCC}}+\vec{z}_{\mathrm{BCC}})$,
$\vec{a}_{2}^{\rm
P}=\frac{1}{2}(\vec{x}_{\mathrm{BCC}}-\vec{y}_{\mathrm{BCC}}+\vec{z}_{\mathrm{BCC}})$
and $\vec{a}_{3}^{\rm
P}=\frac{1}{2}(\vec{x}_{\mathrm{BCC}}+\vec{y}_{\mathrm{BCC}}-\vec{z}_{\mathrm{BCC}})$.
In this coordinate system the minimal tight-binding model is given by

$\displaystyle\quad\hat{\mathcal{H}}_{\mathrm{t.b.}}=\hat{\mathcal{H}}_{\mathrm{L}}+\hat{\mathcal{H}}_{\mathrm{L\mbox{-}L}},$
(1) $\displaystyle\hat{\mathcal{H}}_{\mathrm{L}}$ $\displaystyle=$
$\displaystyle-
t_{0}\sum_{\vec{i},\sigma}\Big{(}\hat{c}^{\dagger}_{\vec{i},2,\sigma}\hat{c}_{\vec{i},1,\sigma}+\hat{c}^{\dagger}_{\vec{i},4,\sigma}\hat{c}_{\vec{i},3,\sigma}+{\rm
h.c.}\Big{)}$ $\displaystyle-
t_{1}\sum_{\vec{i},\sigma}\Bigg{\\{}\Big{(}\hat{c}^{\dagger}_{\vec{i}+\vec{a}_{1}^{\rm
P}+\vec{a}_{3}^{\rm P},4,\sigma}+\hat{c}^{\dagger}_{\vec{i}+\vec{a}_{1}^{\rm
P}-\vec{a}_{2}^{\rm P},4,\sigma}\Big{)}\hat{c}_{\vec{i},1,\sigma}$
$\displaystyle\quad\quad+\Big{(}\hat{c}^{\dagger}_{\vec{i}+\vec{a}_{1}^{\rm
P}+\vec{a}_{3}^{\rm P},3,\sigma}+\hat{c}^{\dagger}_{\vec{i}+\vec{a}_{1}^{\rm
P}-\vec{a}_{2}^{\rm P},3,\sigma}\Big{)}\hat{c}_{\vec{i},2,\sigma}+{\rm
h.c.}\Bigg{\\}},$ $\displaystyle\hat{\mathcal{H}}_{\mathrm{L\mbox{-}L}}$
$\displaystyle=$ $\displaystyle-
t_{2}\sum_{\vec{i},\sigma}\Big{(}\hat{c}^{\dagger}_{\vec{i}+\vec{a}_{1}^{\rm
P},3,\sigma}\hat{c}_{\vec{i},1,\sigma}+\hat{c}^{\dagger}_{\vec{i}+\vec{a}_{3}^{\rm
P},4,\sigma}\hat{c}_{\vec{i},2,\sigma}+{\rm h.c.}\Big{)},$ (3)

where the operator $\hat{c}^{\dagger}_{\vec{j},\nu,\sigma}$
($\hat{c}_{\vec{j},\nu,\sigma}$) creates (annihilates) a spin-$1/2$ electron
with spin $\sigma\in\\{\uparrow,\downarrow\\}$ on $\psi_{\nu}$ with orbital
index $\nu\in\\{1,2,3,4\\}$ located in a unit cell whose location is given by
$\vec{j}$. This Hamiltonian describes a system of 1D ladders coupled in a BCC
lattice geometry in 3D space, as depicted in Fig. 1, lower panel. The level of
K doping found in this compound studied in experiment corresponds to half
filling of the electronic bands. This type of model system of half-filled
coupled ladders provides fertile ground for exploring emergent collective
electronic and magnetic phases due to an interplay between correlations and
dimensionality, as we discuss below.

Correlations at strong coupling.— Given the large size of the molecular
orbitals found in this system, it is reasonable to assume electron-electron
correlations are predominantly local and proceed within a Hubbard Hubbard
(1963, 1964) or Pariser-Parr-Pople Pariser and Parr (1953); Pople (1953)
approach. We thus study correlation-driven physics using an interaction
Hamiltonian that takes the form:
$\hat{\mathcal{H}}_{\mathrm{t.b.}}+U\sum_{\vec{i},\nu}\hat{n}_{\vec{i},\nu\uparrow}\hat{n}_{\vec{i},\nu,\downarrow}$.
Based on existing literature on aromatic compounds we expect $U\sim 0.5$ -
$2\mathrm{eV}\gg t_{0},t_{1}$ and $t_{2}$ for our system. For these large
$U/t$ (where $t\in\\{t_{0},t_{1},t_{2}\\}$) the electronic correlation opens a
charge gap at half filling, inducing insulating behavior as also seen in
experiment Gadjieva  _et al._ (2020). To unveil the nature of correlations in
the half-filled state, we perform a strong-coupling expansion to leading order
in ${t_{0}^{2}}/{U}$, ${t_{1}^{2}}/{U}$ and ${t_{2}^{2}}/{U}$, and obtain a
magnetic spin model with Heisenberg interactions:

$\displaystyle\hat{\mathcal{H}}_{\mathrm{t.b.};\mathrm{M}}$ $\displaystyle=$
$\displaystyle\hat{\mathcal{H}}_{\mathrm{L};\mathrm{M}}+\hat{\mathcal{H}}_{\mathrm{L}\mbox{-}\mathrm{L};\mathrm{M}},$
(4) $\displaystyle\hat{\mathcal{H}}_{\mathrm{L};\mathrm{M}}$ $\displaystyle=$
$\displaystyle\sum_{\vec{i},\vec{j},\nu,\mu}\mathcal{J}_{\vec{i},\vec{j}}^{\nu,\mu}\Big{(}\vec{S}_{\vec{i}}^{\nu}\vec{S}_{\vec{j}}^{\mu}-\frac{\hat{n}_{\vec{i},\nu}\hat{n}_{\vec{j},\mu}}{4}\Big{)},$
(5) $\displaystyle\hat{\mathcal{H}}_{\mathrm{L}\mbox{-}\mathrm{L};\mathrm{M}}$
$\displaystyle=$ $\displaystyle
J_{2}\sum_{\vec{i}}\Big{(}\vec{S}_{\vec{i}}^{1}\vec{S}_{\vec{i}+\vec{a}_{1}^{\rm
P}}^{3}-\frac{\hat{n}_{\vec{i},1}\hat{n}_{\vec{i}+\vec{a}_{1}^{\rm P},3}}{4}$
(6)
$\displaystyle\quad\quad\quad+\vec{S}_{\vec{i}}^{2}\vec{S}_{\vec{i}+\vec{a}_{3}^{\rm
P}}^{4}-\frac{\hat{n}_{\vec{i},2}\hat{n}_{\vec{i}+\vec{a}_{3}^{\rm
P},4}}{4}\Big{)},$

where
$\mathcal{J}_{\vec{i},\vec{j}}^{\nu,\mu}=\frac{1}{2}J_{0}\delta_{\vec{i},\vec{j}}(\delta_{\nu,1}\delta_{\mu,2}+\delta_{\nu,2}\delta_{\mu,1}+\delta_{\nu,3}\delta_{\mu,4}+\delta_{\nu,4}\delta_{\mu,3})+\frac{1}{2}J_{1}\big{(}\delta_{\vec{j}-\vec{i},\vec{a}_{1}^{\rm
P}+\vec{a}_{3}^{\rm P}}+\delta_{\vec{j}-\vec{i},\vec{a}_{2}^{\rm
P}+\vec{a}_{3}^{\rm
P}}\big{)}(\delta_{\nu,1}\delta_{\mu,4}+\delta_{\nu,4}\delta_{\mu,1}+\delta_{\nu,2}\delta_{\mu,3}+\delta_{\nu,3}\delta_{\mu,2})$
and the magnetic moment of an electron in orbital $\nu$ at unit cell $\vec{j}$
is given by
$\vec{S}_{\vec{j}}^{\nu}=\sum_{\alpha,\beta}\hat{c}^{\dagger}_{\vec{j},\nu,\alpha}\frac{\hbar}{2}\vec{\sigma}_{\alpha,\beta}\hat{c}_{\vec{j},\nu,\beta}$,
where
$\vec{\sigma}=\\{\sigma_{\mathcal{X}},\sigma_{\mathcal{Y}},\sigma_{\mathcal{Z}}\\}$
is a vector of spin-$1/2$ Pauli matrices. Here $J_{0}=4t_{0}^{2}/U$,
$J_{1}=4t_{1}^{2}/U$ and $J_{2}=4t_{2}^{2}/U$, where the $J_{0}$, $J_{1}$ and
$J_{2}$ bonds correspond, respectively, to the $t_{0}$, $t_{1}$ and $t_{2}$
bonds in Fig. 1, lower panel. The couplings $J_{0}$, $J_{1}$ and $J_{2}$
respectively correspond to the couplings $J_{b}$, $J_{a}$ and $J_{c}$ obtained
in Ref. Gadjieva  _et al._ (2020) by fitting energy differences of
magnetically ordered states obtained in spin-polarized DFT calculations. The
ratios between the couplings are similar, and the reported magnitudes of
$J_{b}$, $J_{a}$ and $J_{c}$ in Ref. Gadjieva  _et al._ (2020) would
correspond to a $U\sim 0.4$eV.

Magnetic behavior.— In the limit $J_{2}=0$, the system reduces to a set of
uncoupled quasi-1D ladders, as discussed above. The magnetic behavior of two-
leg ladder systems is by now fairly well understood Strong and Millis (1992);
Noack  _et al._ (1994); White  _et al._ (1994); White (1996); Shelton  _et
al._ (1996); Noack  _et al._ (1996). A ladder is formed of two coupled
spin-$1/2$ chains, and thus supports a dimer phase formed of singlets
localized over the ladder rungs. The ground state of this phase is a spin-
gapped paramagnet. A non-zero value of $J_{2}$ couples the ladders, mixing
spin-triplet states into the ground state, and a sufficiently large $J_{2}$
eventually induces magnetic ordering Dagotto and Moreo (1988); Gopalan  _et
al._ (1994); Bouillot  _et al._ (2011). The value of $J_{2}$ required to
induce a transition may exhibit non-monotonic behavior with temperature $T$
because the magnetically ordered state has gapless (spin wave) excitations and
so has larger entropy than the spin-gapped paramagnetic state Villain  _et
al._ (1980), see the proposed phase diagram in Fig. 3. At higher $T$ thermal
fluctuations ultimately destroy ordering, and thus a larger $J_{2}$ is
required to induce a transition Sachdev (2011). In our system
$J_{2}/J_{0}\approx J_{2}/J_{1}\approx 1/4$, which is likely sufficient to
induce ordering into a 3D antiferromagnetic state at zero $T$, see the
location of this compound on the phase diagram of the magnetic model shown in
Fig. 3. A precise determination of the exact behavior of this system in the
$T$-$J_{2}$ plane necessitates development of new exact methods suitable for
the study of coupled ladders in 3D space, a current challenge to theory and
numerical approaches. Studies of a few weakly coupled half-filled ladders in
cylindrical geometries may be accessible to tensor network methods White
(1992), a direction we leave to the future.

Figure 3: Schematic phase diagram of coupled Heisenberg ladders in
[K($\mathbf{222}$)]2[$p$-terphenyl]3 in the ladder-ladder coupling
($J_{2}$)-temperature ($T$) plane. The dot denotes the proposed position of
the compound studied in experiment Gadjieva  _et al._ (2020) on the phase
diagram at $T=0$.

Light doping with holes.— The crystal can be lightly doped with holes via
gating or by using an oxidizing agent placed near the crystal surface. This
situation gives rise to a $t$-$J_{0}$-$J_{1}$-$J_{2}$ model given by
$\hat{\mathcal{H}}_{t\mbox{-}J_{0}\mbox{-}J_{1}\mbox{-}J_{2}}=\hat{\mathcal{P}}\Big{[}\hat{\mathcal{H}}_{\mathrm{t.b.}}+\hat{\mathcal{H}}_{\mathrm{t.b.};\mathrm{M}}\Big{]}\hat{\mathcal{P}}$,
where $\hat{\mathcal{P}}$ implements a constraint of no double occupancy that
reflects the large onsite Coulomb repulsion of the strong-coupling limit.
Consider again the $J_{2}=0$ limit. Here, the problem reduces to the
extensively studied doped $t$-$J$ ladder (with $J=J_{0}\approx J_{1}$) in
which the magnetic background mediates $d$-wave pairing of holes with
concentration $\delta$, resulting in a phase with power-law superconducting
correlations in the presence of a spin gap $\sim J_{0}$ Dagotto  _et al._
(1992); Troyer  _et al._ (1996); Noack  _et al._ (1995, 1997); Dagotto (1999).
In this 1D limit, the superfluid stiffness $\rho_{\rm s}$ is a fraction of
$t\delta$, where $t=t_{0}\approx t_{1}$. At sufficiently large hole
concentrations, the superconducting phase gives way to regimes with Fermi
liquid-like behavior Troyer  _et al._ (1996). For $J_{2}\neq 0$, the ladders
are coupled, and the ultimate fate of the $d$-wave superconducting phase is an
open problem whose solution in 2D and 3D space constitutes a major challenge
to theory Dagotto and Rice (1996); Scalapino (2012). In the absence of
definitive results, we base our analysis on existing results for
inhomogeneously coupled quasi-1D Arrigoni  _et al._ (2004); Yao  _et al._
(2007); Tsai  _et al._ (2008); Karakonstantakis  _et al._ (2011); Jiang and
Kivelson (2021) and finite-width (e.g. four- Jiang  _et al._ (2018) and eight-
Jiang  _et al._ (2018) leg) ladders, and provide a qualitative discussion of
the expected behavior in the $J_{2}\mbox{-}\delta$ plane in Fig. 4. We expect
a superconducting region in the phase diagram, reflecting the survival of
ladder $d$-wave superconducting behavior for a range of optimal $\delta$ and
$J_{2}$ Arrigoni  _et al._ (2004); Jiang and Kivelson (2021). This phase is
stable for $T<T_{\rm c}$ and $T_{\rm c}\sim\rho_{\rm s}$ for small $\delta$.
From this we estimate that this compound may exhibit superconductivity on
hole-doping with a low $T_{\rm c}$ on the order of a few degrees Kelvin. A
sufficiently large $J_{2}\sim J_{0},J_{1}$, corresponding to homogeneously
coupled ladders, or, a 3D network of spins, at zero $\delta$ gives rise to
antiferromagnetic ordering, which becomes increasingly unstable to
superconductivity and ultimately to Fermi liquid phases with increasing
$\delta$, and thus a larger $J_{2}$ is needed in order stabilize the magnetic
order. From this we suggest that the parent anitferromagnetic compound
obtained in recent experiments Gadjieva  _et al._ (2020) may give rise to
superconductivity upon doping with holes for $\delta\sim 0.1$, see Fig. 4.

Figure 4: Schematic $T=0$ phase diagram of lightly doped coupled Heisenberg
ladders in [K($\mathbf{222}$)]2[$p$-terphenyl]3 in the hole concentration
($\delta$)-ladder-ladder coupling ($J_{2}$) plane. The thick vertical line at
$\delta=0$ denotes the non-superconducting spin-gapped zero doping phase, see
Fig. 3. The arrow depicts hole doping of the parent antiferromagnetic compound
obtained in experiment Gadjieva  _et al._ (2020), which becomes
superconducting for $\delta\sim 0.1>\delta_{\rm c}$. We do not show here
regions corresponding to charge-density wave order potentially emergent at
much larger, commensurate dopings such as quarter filling White  _et al._
(2002).

Conclusions.— The K-doped terphenyl ladder materials open a new chapter in
strongly correlated-electron behavior in the organics, potentially providing
access to unconventional magnetic and superconducting phases. Recent studies
Wang  _et al._ (2017); Pinto  _et al._ (2020) report temperature-dependent
gaps in annealed terphenyl-based powders whose exact structural composition
remains unsettled. Our theoretical analysis provides a clear picture of the
many-body electronic band structure and correlations in the experimentally
well-characterized, recently synthesized doped terphenyl crystals Gadjieva
_et al._ (2020). We show using DFT calculations and Wannier fitting that this
complex material whose unit cell contains 444 atoms can be accurately
described by a model of ladders sparsely coupled in 3D space with hopping
matrix elements in the $\sim 10$ \- $30\mathrm{meV}$ range. We develop a
strong-coupling approach in the limit of large Coulomb repulsion from which we
argue that this half-filled system is likely an antiferromagnetic insulator.
Upon hole doping this narrow-band system, we expect a regime of $d$-wave
superconductivity whose signatures in experiment are of major interest.
Because of the small values of the hopping matrix elements, ladder
superconductivity in this system should occur at a low $T_{\rm c}$, on the
order of a few degrees Kelvin.

From the perspective of materials science, understanding the difference in
electronic structure of this system in relation to optimized doped terphenyl
structures Zhong  _et al._ (2018) represents an important question for
research. On the theory side, the discovery of this ladder system calls for a
systematic study of the physics of weakly coupled ladders in 3D space.
Electron-phonon coupling is sometimes important in organic materials. Based on
arguments for similar organic crystalline systems Girlando  _et al._ (2011);
Chang  _et al._ (2021), we expect that polaron (or at least small polaron)
formation, and thus charge localization, to be unlikely, and that coupling to
Peierls Marchand  _et al._ (2010) phonon modes related to terphenyl-terphenyl
bond distortions exerts the largest effect in transport, possibly leading to
unconventional electronic behavior Sous  _et al._ (2018). A quantitative
treatment of this problem however, requires inclusion of the coupling of
electrons to both Holstein Holstein (1959a, b) and Peierls modes, a direction
we plan to address in the future.

Future work will be devoted to a full theoretical characterization of the
behavior of coupled ladder systems via tensor network and quantum Monte Carlo
techniques in order to precisely pinpoint the doping range needed for inducing
superconductivity, and to the experimental implementation of the conditions
that will enable testing of the full range of predictions made in this work,
thus opening a door to the observation of emergent electronic and magnetic
phases in these novel and controllable organic crystalline systems.

_Acknowledgements_.— We acknowledge useful discussions with S. Kivelson, X.
Roy, M. L. Steigerwald and S. White. J. S. thanks A. B. Georgescu, J. Lee and
especially J. Bonini for assistance with DFT calculations, and L. Reading-
Ikkanda for assistance with creating schematics used in the figures. J. S., N.
A. G., C. N., D. R. R. and A. J. M. acknowledge support from the National
Science Foundation (NSF) Materials Research Science and Engineering Centers
(MRSEC) program through Columbia University in the Center for Precision
Assembly of Superstratic and Superatomic Solids under Grant No. DMR-1420634
and Grant No. DMR-2011738. C. N. thanks Sheldon and Dorothea Buckler for their
generous support. J. S. also acknowledges the hospitality of the Center for
Computational Quantum Physics (CCQ) at the Flatiron Institute. The Flatiron
Institute is a division of the Simons Foundation.

## References

  * Wang  _et al._ (2017) R.-S. Wang, Y. Gao, Z.-B. Huang, and X.-J. Chen, “Superconductivity above 120 Kelvin in a chain link molecule,” arXiv preprint arXiv:1703.06641 (2017).
  * Pinto  _et al._ (2020) N. Pinto, C. Di Nicola, A. Trapananti, M. Minicucci, A. Di Cicco, A. Marcelli, A. Bianconi, F. Marchetti, C. Pettinari, and A. Perali, “Potassium-doped para-terphenyl: Structure, electrical transport properties and possible signatures of a superconducting transition,” Condensed Matter 5 (2020).
  * Rickhaus  _et al._ (2010) M. Rickhaus, A. P. Belanger, H. A. Wegner, and L. T. Scott, “An oxidation induced by potassium metal. Studies on the anionic cyclodehydrogenation of 1,1’-binaphthyl to perylene,” J. Org. Chem. 75, 7358 (2010).
  * Gadjieva  _et al._ (2020) N. A. Gadjieva, P. Szirmai, O. Sági, P. Alemany, A. K. Bartholomew, I. Stone, S. Conejeros, D. W. Paley, R. Hernández Sánchez, B. Fowler, S. R. Peurifoy, B. Náfrádi, L. Forró, X. Roy, P. Batail, E. Canadell, M. L. Steigerwald, and C. Nuckolls, “Intermolecular resonance correlates electron pairs down a supermolecular chain: Antiferromagnetism in K-doped p-terphenyl,” J. Am. Chem. Soc. 142, 20624 (2020).
  * Note (1) Note that well characterized crystals of alkali-doped terphenyl were also reported in the 1980s Noordik  _et al._ (1981).
  * Dagotto and Moreo (1988) E. Dagotto and A. Moreo, “Zero-temperature properties of the two-dimensional Heisenberg antiferromagnet: A numerical study,” Phys. Rev. B 38, 5087 (1988).
  * Gopalan  _et al._ (1994) S. Gopalan, T. M. Rice, and M. Sigrist, “Spin ladders with spin gaps: A description of a class of cuprates,” Phys. Rev. B 49, 8901 (1994).
  * Zhong  _et al._ (2018) G.-H. Zhong, X.-H. Wang, R.-S. Wang, J.-X. Han, C. Zhang, X.-J. Chen, and H.-Q. Lin, “Structural and bonding characteristics of potassium-doped $p$-terphenyl superconductors,” J. Phys. Chem. C 122, 3801 (2018).
  * Giannozzi  _et al._ (2009) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, “QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials,” J. Condens. Matter Phys. 21, 395502 (2009).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865 (1996).
  * Puschnig and Ambrosch-Draxl (1999) P. Puschnig and C. Ambrosch-Draxl, “Density-functional study for the oligomers of poly(para-phenylene): Band structures and dielectric tensors,” Phys. Rev. B 60, 7891 (1999).
  * Mostofi  _et al._ (2014) A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, “An updated version of wannier90: A tool for obtaining maximally-localised Wannier functions,” Comput. Phys. Commun. 185, 2309 (2014).
  * Marzari and Vanderbilt (1997) N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847 (1997).
  * Hubbard (1963) J. Hubbard, “Electron correlations in narrow energy bands,” Proc. R. Soc. London, Ser. A 276, 238 (1963).
  * Hubbard (1964) J. Hubbard, “Electron correlations in narrow energy bands III. An improved solution,” Proc. R. Soc. London, Ser. A 281, 401 (1964).
  * Pariser and Parr (1953) R. Pariser and R. G. Parr, “A semi‐empirical theory of the electronic spectra and electronic structure of complex unsaturated molecules. II,” J. Chem. Phys. 21, 767 (1953).
  * Pople (1953) J. A Pople, “Electron interaction in unsaturated hydrocarbons,” Trans. Faraday Soc. 49, 1375 (1953).
  * Strong and Millis (1992) S. P. Strong and A. J. Millis, “Competition between singlet formation and magnetic ordering in one-dimensional spin systems,” Phys. Rev. Lett. 69, 2419 (1992).
  * Noack  _et al._ (1994) R. M. Noack, S. R. White, and D. J. Scalapino, “Correlations in a two-chain Hubbard model,” Phys. Rev. Lett. 73, 882 (1994).
  * White  _et al._ (1994) S. R. White, R. M. Noack, and D. J. Scalapino, “Resonating valence bond theory of coupled Heisenberg chains,” Phys. Rev. Lett. 73, 886 (1994).
  * White (1996) Steven R. White, “Equivalence of the antiferromagnetic Heisenberg ladder to a single ${S}=1$ chain,” Phys. Rev. B 53, 52 (1996).
  * Shelton  _et al._ (1996) D. G. Shelton, A. A. Nersesyan, and A. M. Tsvelik, “Antiferromagnetic spin ladders: Crossover between spin ${S}=1/2$ and ${S}=1$ chains,” Phys. Rev. B 53, 8521 (1996).
  * Noack  _et al._ (1996) R. M. Noack, S. R. White, and D. J. Scalapino, “The ground state of the two-leg Hubbard ladder: A density-matrix renormalization group study,” Physica C 270, 281 (1996).
  * Bouillot  _et al._ (2011) P. Bouillot, C. Kollath, A. M. Läuchli, M. Zvonarev, B. Thielemann, C. Rüegg, E. Orignac, R. Citro, M. Klanjšek, C. Berthier, M. Horvatić, and T. Giamarchi, “Statics and dynamics of weakly coupled antiferromagnetic spin-$\frac{1}{2}$ ladders in a magnetic field,” Phys. Rev. B 83, 054407 (2011).
  * Villain  _et al._ (1980) J. Villain, R. Bidaux, J.-P. Carton, and R. Conte, “Order as an effect of disorder,” J. Phys. (France) 41, 1263 (1980).
  * Sachdev (2011) S. Sachdev,  _Quantum phase transitions_ (Cambridge university press, 2011).
  * White (1992) S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863 (1992).
  * Dagotto  _et al._ (1992) E. Dagotto, J. Riera, and D. Scalapino, “Superconductivity in ladders and coupled planes,” Phys. Rev. B 45, 5744 (1992).
  * Troyer  _et al._ (1996) M. Troyer, H. Tsunetsugu, and T. M. Rice, “Properties of lightly doped $t-{J}$ two-leg ladders,” Phys. Rev. B 53, 251 (1996).
  * Noack  _et al._ (1995) R. M. Noack, S. R. White, and D. J. Scalapino, “The doped two-chain Hubbard model,” Europhys. Lett. 30, 163 (1995).
  * Noack  _et al._ (1997) R. M. Noack, N. Bulut, D. J. Scalapino, and M. G. Zacher, “Enhanced ${d}_{{x}^{2}-{y}^{2}}$ pairing correlations in the two-leg Hubbard ladder,” Phys. Rev. B 56, 7162 (1997).
  * Dagotto (1999) E. Dagotto, “Experiments on ladders reveal a complex interplay between a spin-gapped normal state and superconductivity,” Rep. Prog. Phys. 62, 1525 (1999).
  * Dagotto and Rice (1996) E. Dagotto and T. M. Rice, “Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials,” Science 271, 618 (1996).
  * Scalapino (2012) D. J. Scalapino, “A common thread: The pairing interaction for unconventional superconductors,” Rev. Mod. Phys. 84, 1383 (2012).
  * Arrigoni  _et al._ (2004) E. Arrigoni, E. Fradkin, and S. A. Kivelson, “Mechanism of high-temperature superconductivity in a striped Hubbard model,” Phys. Rev. B 69, 214519 (2004).
  * Yao  _et al._ (2007) H. Yao, W.-F. Tsai, and S. A. Kivelson, “Myriad phases of the checkerboard Hubbard model,” Phys. Rev. B 76, 161104 (2007).
  * Tsai  _et al._ (2008) W.-F. Tsai, H. Yao, A. Läuchli, and S. A. Kivelson, “Optimal inhomogeneity for superconductivity: Finite-size studies,” Phys. Rev. B 77, 214502 (2008).
  * Karakonstantakis  _et al._ (2011) G. Karakonstantakis, E. Berg, S. R. White, and S. A. Kivelson, “Enhanced pairing in the checkerboard hubbard ladder,” Phys. Rev. B 83, 054508 (2011).
  * Jiang and Kivelson (2021) H.-C. Jiang and S. A Kivelson, “Stripe order enhanced superconductivity in the Hubbard model,” arXiv preprint arXiv:2105.07048 (2021).
  * Jiang  _et al._ (2018) H.-C. Jiang, Z.-Y. Weng, and S. A. Kivelson, “Superconductivity in the doped $t-{J}$ model: Results for four-leg cylinders,” Phys. Rev. B 98, 140505 (2018).
  * White  _et al._ (2002) S. R. White, I. Affleck, and D. J. Scalapino, “Friedel oscillations and charge density waves in chains and ladders,” Phys. Rev. B 65, 165122 (2002).
  * Girlando  _et al._ (2011) A. Girlando, L. Grisanti, M. Masino, A. Brillante, R. G. Della Valle, and E. Venuti, “Interaction of charge carriers with lattice and molecular phonons in crystalline pentacene,” J. Chem. Phys. 135, 084701 (2011).
  * Chang  _et al._ (2021) B. K. Chang, J.-J. Zhou, N.-E. Lee, and M. Bernardi, “Intermediate polaronic charge transport in organic crystals from first principles,” arXiv preprint arXiv:2106.09810 (2021).
  * Marchand  _et al._ (2010) D. J. J. Marchand, G. De Filippis, V. Cataudella, M. Berciu, N. Nagaosa, N. V. Prokof’ev, A. S. Mishchenko, and P. C. E. Stamp, “Sharp transition for single polarons in the one-dimensional Su-Schrieffer-Heeger model,” Phys. Rev. Lett. 105, 266605 (2010).
  * Sous  _et al._ (2018) J. Sous, M. Chakraborty, R. V. Krems, and M. Berciu, “Light bipolarons stabilized by peierls electron-phonon coupling,” Phys. Rev. Lett. 121, 247001 (2018).
  * Holstein (1959a) T. Holstein, “Studies of polaron motion: Part I. The molecular-crystal model,” Ann. Phys. (NY) 8, 325–342 (1959a).
  * Holstein (1959b) T. Holstein, “Studies of polaron motion: Part II. The “small” polaron,” Ann. Phys. (NY) 8, 343 (1959b).
  * Noordik  _et al._ (1981) J. H. Noordik, H. M. Doesburg, and P. A. J. Prick, “Structures of the sodium–p-terphenyl ion pairs: disodium terphenylide–tetrahydrofuran (1/6) and disodium diterphenylide terphenyl–1,2-dimethoxyethane (1/6),” Acta Crystallogr. 37, 1659 (1981).
  * Ong  _et al._ (2013) S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson, and G. Ceder, “Python materials genomics (pymatgen): A robust, open-source python library for materials analysis,” Comput. Mater. Sci. 68, 314 (2013).
  * Dal Corso (2014) A. Dal Corso, “Pseudopotentials periodic table: From H to Pu,” Comput. Mater. Sci. 95, 337 (2014).
  * Marzari  _et al._ (1999) N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, “Thermal contraction and disordering of the Al(110) surface,” Phys. Rev. Lett. 82, 3296 (1999).

SUPPLEMENTARY INFORMATION
for
“Phonon-induced disorder in dynamics of optically pumped metals from
non-linear electron-phonon coupling”

##  I Crystal structure

#### Chemical Stoichiometry.

We study the recently synthesized K-doped $p$-terphenyl compound Gadjieva  _et
al._ (2020) with the chemical formula: [K($\mathbf{222}$)]2[$p$-terphenyl]3
corresponding to 2 K atoms, each within a $\mathbf{222}$ cryptand, for every 3
$p$-terphenyl (C18H14) molecules. The cryptand is a polydentate ligand formed
from 2 nitrogen, 6 oxygen, 18 carbon, and 36 hydrogen atoms with 62 atoms in
total. $p$-terphenyl consists of three linked benzene rings with a total of 32
atoms. Each unit of [K($\mathbf{222}$)]2[$p$-terphenyl]3 thus consists of a
total of 222 atoms.

#### Primitive unit cell.

We use the pymatgen software library Ong  _et al._ (2013) to generate input
files of the crystal data of our structure for density-functional theory (DFT)
calculations, see details below. We identify using pymatgen a primitive unit
cell for the crystal with 444 atoms corresponding to two units of
[K($\mathbf{222}$)]2[$p$-terphenyl]3 per unit cell, with cell parameters:
$\vec{a}=\\{0,13.0237,0.0\\}$$\mathrm{\SIUnitSymbolAngstrom}$,
$\vec{b}=\\{13.694074,6.51185,8.698852\\}$$\mathrm{\SIUnitSymbolAngstrom}$ and
$\vec{c}=\\{13.694074,6.51185,-13.976448\\}$$\mathrm{\SIUnitSymbolAngstrom}$
in Cartesian coordinates. We obtain high-symmetry $k$ points in the Brillouin
zone corresponding to this unit cell using pymatgen.

#### Crystal symmetry.

This system possesses a $C$2/$c$ monoclinic group lattice structure. As we
show in the main text, the system’s electronic structure is accurately
described by a minimal tight-binding model with a connectivity that is
isomorphic to one in a (distorted) body-centered cubic (BCC) lattice, see
below.

##  II Methods

###  II.1 Density-functional theory (DFT) calculations

We use the Quantum ESPRESSO softwards package Giannozzi  _et al._ (2009) to
perform density-functional theory (DFT) calculations within the generalized
gradient approximation for the exchange-correlation functional parametrized by
Perdew, Burke, and Ernzerhof (PBE) Perdew  _et al._ (1996), utilizing scalar
relativistic ultrasoft pseudopotentials obtained from the PSlibrary Dal Corso
(2014) for the atoms in the crystal. We use a kinetic energy cutoff of
$60\mathrm{Ry}$ for the wavefunctions and of $480\mathrm{Ry}$ for the charge
density and potential. We employ the Marzari-Vanderbilt-DeVita-Payne “cold
smearing” approach Marzari  _et al._ (1999) for occupations with a Gaussian
spreading of $1.5\times 10^{-2}$. We employ a convergence threshold for self-
consistency of $10^{-9}$. We self-consistently obtain electronic densities on
a shifted Monkhrost-Pack 3 $\times$ 3 $\times$ 3 $k$-point grid, which we use
to compute the electronic band structure along various $k$ paths suggested by
pymatgen. We verify that a 3 $\times$ 3 $\times$ 3 Monkhrost-Pack grid
suffices to achieve convergence of the interpolated band structure.

This system exhibits weak variation in the bond lengths between a certain
subsets of pairs of $p$-terphenyl molecules along one particular orientation
of the crystal, see Ref. Gadjieva  _et al._ (2020) for more details and
discussions. Here we use in our calculations a structure with averaged bond
lengths, ignoring the split short-long bond pattern. Weak disorder effects we
ignore will only lead to modest changes in the hopping parameters. In
particular, disorder leads to a slight reduction in the hopping magnitude in
the one-dimensional (1D) limit Gadjieva  _et al._ (2020). The cif file of this
structure is available upon request.

###  II.2 Wannier localization

We use the Wannier90 software package Mostofi  _et al._ (2014) to obtain a
tight-binding representation of the DFT bands. We first use the electron
densities computed self-consistently in DFT calculations to generate in a non-
self-consistent manner the densities at symmetry-equivalent $k$-points (this
step inherits the same level of accuracy as that of the self-consistent
calculation and incurs no greater error). We then use Wannier90 to obtain
maximally localized Wannier orbitals and a tight-binding Hamiltonian from the
$k$-space wavefunctions Marzari and Vanderbilt (1997). We find excellent
agreement between the DFT bands and the tight-binding fits of four- and six-
band (shown in the main text) models.

##  III Details of the interpretation of the electronic band structure

Our approximate minimal tight-binding model in which we ignore matrix elements
$<0.01\mathrm{eV}$ is given by the following Hamiltonian written in the
Wannier basis:

$\displaystyle\hat{\mathcal{H}}_{\mathrm{t.b.}}=\hat{\mathcal{T}}_{0}+\hat{\mathcal{T}}_{1}+\hat{\mathcal{T}}_{2},$
(1) $\displaystyle\hat{\mathcal{T}}_{0}$
$\displaystyle=-t_{0}\sum_{\vec{i},\sigma}\Big{(}\hat{c}^{\dagger}_{\vec{i},2,\sigma}\hat{c}_{\vec{i},1,\sigma}+\hat{c}^{\dagger}_{\vec{i},4,\sigma}\hat{c}_{\vec{i},3,\sigma}+{\rm
h.c.}\Big{)},$ (3) $\displaystyle\hat{\mathcal{T}}_{1}$
$\displaystyle=-t_{1}\sum_{\vec{i},\sigma}\Bigg{\\{}\Big{(}\hat{c}^{\dagger}_{\vec{i}+\vec{a},4,\sigma}+\hat{c}^{\dagger}_{\vec{i}+\vec{b}+\vec{c},4,\sigma}\Big{)}\hat{c}_{\vec{i},1,\sigma}$
$\displaystyle\quad\quad\quad\quad+\Big{(}\hat{c}^{\dagger}_{\vec{i}+\vec{a},3,\sigma}+\hat{c}^{\dagger}_{\vec{i}+\vec{b}+\vec{c},3,\sigma}\Big{)}\hat{c}_{\vec{i},2,\sigma}+{\rm
h.c.}\Bigg{\\}},$ $\displaystyle\hat{\mathcal{T}}_{2}$
$\displaystyle=-t_{2}\sum_{\vec{i},\sigma}\Big{(}\hat{c}^{\dagger}_{\vec{i}+\vec{b},3,\sigma}\hat{c}_{\vec{i},1,\sigma}+\hat{c}^{\dagger}_{\vec{i}+\vec{c},4,\sigma}\hat{c}_{\vec{i},2,\sigma}+{\rm
h.c.}\Big{)},$ (4)

where the operator $\hat{c}^{\dagger}_{\vec{j},\nu,\sigma}$
($\hat{c}_{\vec{j},\nu,\sigma}$) creates (annihilates) a spin-$1/2$ electron
with spin $\sigma\in\\{\uparrow,\downarrow\\}$ on a Wannier orbital
$\psi_{\nu}$ with orbital index $\nu\in\\{1,2,3,4\\}$ located in a unit cell
whose location is given by $\vec{j}$, and $t_{0}=0.022\mathrm{eV}$,
$t_{1}=0.028\mathrm{eV}$ and $t_{2}=0.013\mathrm{eV}$.

To understand the physics operative here we consider the hopping connecting
orbitals across different unit cells. Consider the location of the Wannier
orbitals within a unit cell given by
$\vec{v}_{\psi_{1}}=\\{6.127,5.55,2.223\\}$$\mathrm{\SIUnitSymbolAngstrom}$,
$\vec{v}_{\psi_{2}}=\\{7.567,5.55,-4.862\\}$$\mathrm{\SIUnitSymbolAngstrom}$,
$\vec{v}_{\psi_{3}}=\\{-6.127,-5.55,-2.223\\}$$\mathrm{\SIUnitSymbolAngstrom}$
and
$\vec{v}_{\psi_{4}}=\\{-7.567,-5.55,4.862\\}$$\mathrm{\SIUnitSymbolAngstrom}$.
The large sizes of these Wannier orbitals imply that one must consider the
specific orbital-orbital distance across different unit cells rather than
cell-to-cell distance to understand the spatial structure of the network of
hopping between orbitals in the crystal. The specific orientation in space of
these orbitals reveals that within a unit cell there are a pair of orbitals
$\\{\psi_{1},\psi_{2}\\}$ related by inversion
$x\rightarrow-x,y\rightarrow-y,z\rightarrow-z$ to $\\{\psi_{3},\psi_{4}\\}$,
and within a pair, orbitals share roughly the same $x$ and $y$ coordinates and
are slightly displaced apart in $z$. We can thus visualize the large unit cell
as composed of two symmetry-related pairs of orbitals, one pair in the $+y$
direction (which we later identify as $+y_{\mathrm{BCC}}$) contains
$\\{\psi_{1},\psi_{2}\\}$ with $\psi_{1}$ elevated in $z$ relative to
$\psi_{2}$, and the other in the $-y$ direction ($-y_{\mathrm{BCC}}$) contains
$\\{\psi_{3},\psi_{4}\\}$ with $\psi_{4}$ elevated in $z$ relative to
$\psi_{3}$. $\hat{\mathcal{T}}_{0}$ (Eq. (3)) connects orbitals within a pair
in the unit cell. To understand the network of hopping in
$\hat{\mathcal{T}}_{1}$ (Eq. (3)), consider first the terms that mediate
hopping of an electron between $\psi_{1}$ and $\psi_{4}$ across different unit
cells (first line of Eq. (3)). $\hat{\mathcal{T}}_{1}$ contains a term that
moves an electron from $\psi_{1}$ at $\vec{i}$ with position
$\\{6.127,5.55,2.223\\}$$\mathrm{\SIUnitSymbolAngstrom}$ to $\psi_{4}$ at
$\vec{i}+\vec{a}$ in the neighboring cell with position
$\\{-7.567,7.4737,4.862\\}$$\mathrm{\SIUnitSymbolAngstrom}$; this hopping
predominantly moves the electron in the $-x$ direction. A second matrix
element in $\hat{\mathcal{T}}_{1}$ connects $\psi_{1}$ at $\vec{i}$ with
position $\\{6.127,5.55,2.223\\}$$\mathrm{\SIUnitSymbolAngstrom}$ to orbital
$\psi_{4}$ at $\vec{i}+\vec{b}+\vec{c}$ with position
$\\{19.8211,7.4737,-0.4156\\}$$\mathrm{\SIUnitSymbolAngstrom}$; this hopping
predominantly moves the electron in the $+x$ direction. $\psi_{4}$ at
$\vec{i}+\vec{a}$ and $\psi_{4}$ at $\vec{i}+\vec{b}+\vec{c}$ do in fact share
the same $y$-coordinate and are only relatively slightly apart in $z$. Since
$\psi_{1}$ and $\psi_{4}$ are symmetry related to $\psi_{2}$ and $\psi_{3}$,
we can understand the terms in $\hat{\mathcal{T}}_{1}$ mixing $\psi_{2}$ and
$\psi_{3}$ in a similar fashion. The 1D limit is now apparent. We observe a
zigzag pattern of connections between $\psi_{1}$ and $\psi_{4}$ ($\psi_{2}$
and $\psi_{3}$) across different unit cells and the two zigzag patterns
intersect. These connections due to $\hat{\mathcal{T}}_{0}$ (Eq. (3)) and
$\hat{\mathcal{T}}_{1}$ (Eq. (3)) in the $t_{2}=0$ limit map perfectly onto a
network described by a ladder geometry with hopping of amplitude $t_{0}$
within a rung between $\psi_{1}$ and $\psi_{2}$ and similarly between
$\psi_{3}$ and $\psi_{4}$ and hopping of amplitude $t_{1}$ along the legs
between $\psi_{1}$ and $\psi_{4}$ and identically between $\psi_{2}$ and
$\psi_{3}$, see Fig. 1, lower panel. This lattice geometry has a primitive
unit cell of two orbitals, explaining the doubled spectrum in $k$-space.

$t_{2}\neq 0$ results in ladder-ladder hopping in all three space directions.
To unveil the orientation of couplings between ladders, we consider the
hopping terms in $\hat{\mathcal{T}}_{2}$ (Eq. (4)). One term in
$\hat{\mathcal{T}}_{2}$ couples $\psi_{1}$ in unit cell $\vec{i}$ at
$\\{6.127,5.55,2.223\\}$$\mathrm{\SIUnitSymbolAngstrom}$ to $\psi_{3}$ in unit
cell $\vec{i}+\vec{b}$ at
$\\{7.5671,0.9619,6.4759\\}$$\mathrm{\SIUnitSymbolAngstrom}$; this hopping
moves an electron predominantly in both the $-y$ and $+z$ directions from one
leg of a given ladder to a different leg of a second ladder. The other term in
$\hat{\mathcal{T}}_{2}$ couples $\psi_{2}$ in unit cell $\vec{i}$ at
$\\{7.567,5.55,-4.862\\}$$\mathrm{\SIUnitSymbolAngstrom}$ to $\psi_{4}$ in
unit cell $\vec{i}+\vec{c}$ at
$\\{6.1271,0.9619,-9.1144\\}$$\mathrm{\SIUnitSymbolAngstrom}$; this hopping
moves an electron predominantly in both the $-y$ and $-z$ directions from the
second leg of the original ladder to the first of a third ladder. We see that
the original ladder is sparsely coupled to two others, both located at about
the same position in $x$ and $y$ but apart in $z$, see Fig. 1, lower panel.
Since different legs of the different ladders are coupled by $t_{2}$,
$\hat{\mathcal{H}}_{\mathrm{t.b.}}$ in Eq. (1) eventually allows electrons to
traverse the network of all ladders spanning 3D space, giving rise to full,
albeit anisotropic, dispersion in all three space/ momentum directions, as
shown in Fig. 2. This orientation of couplings between ladders actually
reflects a distorted body-centered cubic (BCC) lattice geometry (see Fig. 1,
lower panel for a sketch of an idealized BCC structure of this system) in
which we can now recast the minimal tight-binding model, see main text for
details and extended discussion.


# White-Box Attacks on Hate-speech BERT Classifiers in German with Explicit
and Implicit Character Level Defense

Shahrukh Khan
<EMAIL_ADDRESS>
Mahnoor Shahid
<EMAIL_ADDRESS>
Navdeeppal Singh
<EMAIL_ADDRESS>
https://github.com/shahrukhx01/adversarial-bert-german-attacks-defense

(February 2022)

###### Abstract

In this work, we evaluate the adversarial robustness of BERT[1] models trained
on German Hate Speech datasets. We also complement our evaluation with two
novel white-box character and word level attacks thereby contributing to the
range of attacks available. Furthermore, we also perform a comparison of two
novel character level defense strategies and evaluate their robustness with
one another.

##  1 Introduction

Natural language processing has achieved tremendous progress in surpassing
human-level baselines in a plethora of language tasks with the help of
Attention based neural architectures [2]. However, recent studies[3, 4, 5]
show that such neural models trained via transfer learning are susceptible to
adversarial noise. Moreover, to the best of our knowledge, most work
concentrates on English language corpora. Hence, in this study, we evaluate
our attack and defense methods on BERT based models trained on German language
hate-speech datasets.

##  2 Background

Hsieh et al.[3] proposed using self attention scores for computing token
importances in order to rank potential candidate tokens for perturbation.
However, one potential shortcoming of their idea is they replace the potential
token candidate with random tokens from vocabulary which may result in
changing the semantic meaning of perturbed sample. Garg et al.[4] proposed
BERT-based Adversarial Examples for Text Classification in which they employ
Mask Language Modelling for generating potential word replacements in a black-
box setting. Finally, Pruthi et al.[5] showed susceptibility of BERT[1] based
models to character-level miss-spellings also in a black-box setting. In our
study, we employ both character-level and word-level attacks in a white-box
setting.

##  3 Experimental Setting

###  3.1 Undefended Models

####  3.1.1 Datasets

We present our work based on HASOC 2019 (German Language) [6] and GermEval
2021 [7] sub-task 1 respectively. Both of the sub-tasks are binary
classification tasks where the positive labels correspond to hate-speech and
negative labels correspond to non-hate-speech examples.

Dataset | Train | Validation | Test
---|---|---|---
HASOC 2019 | 3054 | 765 | 850
GermEval 2021 | 2594 | 650 | 944
Table 1: Dataset Statistics

####  3.1.2 Training

For training, the undefended models, we fine-tune the GBERT[8] language model
for German language which employs training strategies namely _Whole Word
Masking_ (WWM) and evaluation driven training and currently achieves SoTA
performance for document classification task for German language. We obtain
the following accuracy scores for each dataset respectively.

Dataset | Accuracy(%)
---|---
HASOC 2019 | 84
GermEval 2021 | 69
Table 2: Undefended Models

###  3.2 Attacks

####  3.2.1 Baseline Word-level White-Box Attack

The baseline word-level attack is composed by enhancing Hsieh et. al[3] which
prominently replaces tokens sorted in order of their attention scores with
random tokens from vocabulary which may lead to perturbed sequence being
semantically dissimilar to the source sequence. In the baseline attack, we
address this potential shortcoming by using a language model using _Masked
Language Modeling_ (MLM) to generate potential candidate for each token ranked
in the order of attention scores. Furthermore, instead of just performing the
replacement operation, we employ the perturbation scheme as proposed by Garg
et al.[4] we insert generated tokens to left/right of the target token where
the candidate tokens are generated via MLM.

####  3.2.2 Word-level White-Box Attack

The main motivation behind this attack is based on the fact using only
language models to ensure semantic correctness in the adversarial sequences is
not enough. Since it highly depends on the vocabulary of the pre-trained
language model. We improve the baseline attack for the preserving more
semantic and syntactic correctness of the source sequence by introducing
further constraints on the generated sequence by the baseline attack. Firstly,
we compute the document-level embeddings for both perturbed and source
sequence and then compute cosine similarity with a minimum acceptance
threshold of 0.9363 as originally suggested by Jin et al.[9], since Garg et
al.[4] developed their work using the same threshold value. Finally, we
further add another constraint that Part of Speech (POS) tag of both candidate
and target token should be same.

####  3.2.3 Character-level White-Box attack

In this white-box character level attack, similar to earlier white-box word-
level attacks attention scores are obtained in order to get the word
importance. Then, by ordering the word importance in the order of higher to
lower we employ the character perturbation scheme employed by Pruthi et al.[5]
since they evaluated this in the black-box setting only, we perform character
level perturbation within a target token by token modification of character
(swap, insert, delete etc) applied to cause perturbations such adversarial
examples are utilized to maximize the change in model’s original prediction
confidence with limited numbers of modifications. However, these modifications
prove to be significantly effective as outlined in the results section.

###  3.3 Defenses

####  3.3.1 Abstain Based Training

In several past evaluations and benchmarks of defenses against adversarial
examples [10, 11, 12, 13, 14, 15], adversarial training [16] has been found to
be one of the best ways of conferring robustness. However, it is
computationally expensive due to the need of creating adversarial examples
during training. Thus, we chose to employ a detection based defense, which we
call abstain based training. Although, detection based defenses are known to
be not as effective as adversarial training [11, 15], we still believe our
method will deliver insights into the capability of BERT models in recognizing
adversarial examples similar to adversarial training due the way it works. In
contrast to other detection based defenses in the literature [17, 18, 19, 20,
21], the approach is much simpler. It works as follows.

Let $C$ be the trained undefended classifier. We create a new (untrained)
classifier $C^{\prime}$ from $C$ by extending the number of classes it is able
to predict by one. The new class is labeled ‘ABSTAIN’, representing that the
classifier abstains from making a prediction. Using $C$ we create the
adversarial examples. We mix these with the normal examples from the dataset
(of $C$), where the adversarial examples have the abstain label, to create a
new dataset. We then simply train on this dataset. We applied this defense
strategy on the models from Sec. 3.1.2 and present the results in Tab. 4. We
also show the classification attributions in Fig. 1 to try to interpret the
models’ behaviour.

---
Figure 1: Visualization of the classification attributions of the abstain
based trained models, which correctly classify the examples. The _perturbed_
examples shown above fool the normally trained models. We observe that the
attributions are much more spread out when models encounters a perturbed
example. (Words were split by the tokenizer, thus a single word can have
different sub-attributions.)

####  3.3.2 Explicit Character-Level Defense

Abstain based training defense achieves high success in defending against the
adversarial character-level perturbed inputs. However, this results in
degraded system utility since the model does not make any useful prediction
when the input is perturbed at character-level. To overcome this drawback, we
propose the explicit character-level defense which is an unsupervised approach
which makes an assumption that

${\forall}t\in T_{input}:\quad t\in V_{train}\ .$

Here, $V_{train}$ is the set of all tokens present in the training set.
However, replacing this set with set of words in the given language i.e., set
of all words in German language etc. would result in better results.
$T_{input}$ refers to set of tokens present in the input sequence and we
assume the worst case which means $T_{input}$ is perturbed with character-
level noise.

In this defense method, we firstly re-purpose the Sentence-BERT[22]
architecture which originally trained sentence pairs to compute semantic
vector representations and achieved SoTA results on multiple Information
retrieval datasets. However, we change input to character level by inputting
word pairs to the network. Concretely, we labelled the Birkbeck spelling error
corpus[23] which has word pairs with one correct and the other misspelled word
and we label the each pair based on the Levenshtein distance between each
pair. The schematics of our neural approach are given in Fig. 2.

Figure 2: Sentence-BERT for character-level similarity.

The main idea behind using the neural approach is to project similarly spelled
words close to each other in the vector space. Algorithm 1 outlines main idea
of our approach for explicit character-level defense.

Algorithm 1 Explicit Character Level Defense

$V_{train}\leftarrow t_{1}\dots t_{m}$ $\triangleright$ Set of tokens in
vocabulary

$E_{v}\leftarrow\vec{e_{1}}\dots\vec{e_{m}}$ $\triangleright$ Embeddings of
vocabulary
$T_{input}\leftarrow t_{1}\dots t_{j}$ $\triangleright$ Set of tokens in input

for $k\leftarrow 1$ to $j$ do

$\vec{e_{k}}\leftarrow v_{1}\dots v_{n}$ $\triangleright$ Get embedding of
input token k

$\vec{scores}\leftarrow cos(E_{v},\vec{e_{k}})$ $\triangleright$ Cosine
similarity with vocabulary embeddings

if $\max\vec{scores}\geq 0.7\quad and\quad\max\vec{scores}<1.0\quad$ then
$vocab_{index}\leftarrow\operatorname*{arg\,max}{\vec{scores}}$; $\quad
T_{input}[k]\leftarrow V_{train}[vocab_{index}]$

end for

##  4 Results

###  4.1 Attack Results

As shown in Tab. 3 character-level attacks prove to be most effective on both
models.

Dataset | Attack | Success rate(%)
---|---|---
HASOC 2019 | Baseline | 8.49
GermEval 2021 | 60.3
HASOC 2019 | Word-level | 4.03
GermEval 2021 | 49.8
HASOC 2019 | Character-level | 73.1
GermEval 2021 | 93.5
Table 3: Attacks Result on Undefended Models.

|
---|---
|

(a)  (b)

(c)  (d)

Figure 3:
Fig. 3a illustrates how number of queries required per sample for a successful
attack depends on the dataset and attack type, we further show in Fig. 3b that
both word-level attacks require more queries for a longer sequence as compared
to character-level attack which is slightly agnostic to the sequence length.
Fig. 3c shows that the character level attack require minimal amount of
perturbation since the changes are at word level, moreover from Fig. 3d it can
be concluded that character-level attack also makes the highest difference in
model prediction confidence in case of a successful attack.

###  4.2 Defense Results

Dataset | Defense | Attack success rate(%)
---|---|---
HASOC 2019 | Explicit Character Level | 9.5
GermEval 2021 | 5.3
HASOC 2019 | Implicit Abstain-based | 1
GermEval 2021 | 11.1
Table 4: Character-level Attack on Defended Models

##  5 Conclusion

We show that self-attentive models are more susceptible to character-level
adversarial attacks than word-level attacks on text classification NLP task.
We provide two potential ways to defend against character-level attacks.
Future work can be done to enhance the explicit character-level defense using
supervised sequence-to-sequence neural approaches, since as shown in Fig. 4
current approach enhance the jaccard similarity of defended sequences with
original sequences when compared to jaccard similarity between original
sequence and perturbed sequence in case of GermEval 2021. However, for HASOC
2019 dataset because of abundance of Out of Vocabulary tokens in the unseen
test set the defense degrades the quality of defended sequences. However, even
then the defense proves to be quiet robust against character-level adversarial
examples as shown in Tab. 4

Figure 4: Jaccard similarity between original and perturbed text vs. the
original and defended text.

## References

  * [1] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.  BERT: pre-training of deep bidirectional transformers for language understanding.  CoRR, abs/1810.04805, 2018.
  * [2] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin.  Attention is all you need, 2017.
  * [3] Yu-Lun Hsieh, Minhao Cheng, Da-Cheng Juan, Wei Wei, Wen-Lian Hsu, and Cho-Jui Hsieh.  On the robustness of self-attentive models.  In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 1520–1529, Florence, Italy, July 2019. Association for Computational Linguistics.
  * [4] Siddhant Garg and Goutham Ramakrishnan.  BAE: bert-based adversarial examples for text classification.  CoRR, abs/2004.01970, 2020.
  * [5] Danish Pruthi, Bhuwan Dhingra, and Zachary C. Lipton.  Combating adversarial misspellings with robust word recognition.  CoRR, abs/1905.11268, 2019.
  * [6] Thomas Mandl, Sandip Modha, Prasenjit Majumder, Daksh Patel, Mohana Dave, Chintak Mandlia, and Aditya Patel.  Overview of the hasoc track at fire 2019: Hate speech and offensive content identification in indo-european languages.  In Proceedings of the 11th Forum for Information Retrieval Evaluation, FIRE ’19, page 14–17, New York, NY, USA, 2019. Association for Computing Machinery.
  * [7] Julian Risch, Anke Stoll, Lena Wilms, and Michael Wiegand.  Overview of the GermEval 2021 shared task on the identification of toxic, engaging, and fact-claiming comments.  In Proceedings of the GermEval 2021 Shared Task on the Identification of Toxic, Engaging, and Fact-Claiming Comments, pages 1–12. Association for Computational Linguistics, September 2021.
  * [8] Branden Chan, Stefan Schweter, and Timo Möller.  German’s next language model.  CoRR, abs/2010.10906, 2020.
  * [9] Di Jin, Zhijing Jin, Joey Tianyi Zhou, and Peter Szolovits.  Is BERT really robust? natural language attack on text classification and entailment.  CoRR, abs/1907.11932, 2019.
  * [10] Anish Athalye, Nicholas Carlini, and David A. Wagner.  Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples.  In ICML, 2018.
  * [11] Nicholas Carlini and David A. Wagner.  Adversarial examples are not easily detected: Bypassing ten detection methods.  Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security, 2017.
  * [12] Nicholas Carlini and David A. Wagner.  Towards evaluating the robustness of neural networks.  2017 IEEE Symposium on Security and Privacy (SP), pages 39–57, 2017\.
  * [13] Francesco Croce and Matthias Hein.  Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks.  In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pages 2206–2216. PMLR, 2020.
  * [14] Francesco Croce, Maksym Andriushchenko, Vikash Sehwag, Nicolas Flammarion, Mung Chiang, Prateek Mittal, and Matthias Hein.  Robustbench: a standardized adversarial robustness benchmark.  CoRR, abs/2010.09670, 2020.
  * [15] Oliver Bryniarski, Nabeel Hingun, Pedro Pachuca, Vincent Wang, and Nicholas Carlini.  Evading adversarial example detection defenses with orthogonal projected gradient descent.  CoRR, abs/2106.15023, 2021.
  * [16] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu.  Towards deep learning models resistant to adversarial attacks.  ArXiv, abs/1706.06083, 2018.
  * [17] Kathrin Grosse, Praveen Manoharan, Nicolas Papernot, Michael Backes, and Patrick D. McDaniel.  On the (statistical) detection of adversarial examples.  CoRR, abs/1702.06280, 2017.
  * [18] Zhitao Gong, Wenlu Wang, and Wei-Shinn Ku.  Adversarial and clean data are not twins.  CoRR, abs/1704.04960, 2017.
  * [19] Abhijit Bendale and Terrance E. Boult.  Towards open set deep networks.  In 2016 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2016, Las Vegas, NV, USA, June 27-30, 2016, pages 1563–1572. IEEE Computer Society, 2016.
  * [20] Angelo Sotgiu, Ambra Demontis, Marco Melis, Battista Biggio, Giorgio Fumera, Xiaoyi Feng, and Fabio Roli.  Deep neural rejection against adversarial examples.  EURASIP Journal on Information Security, 2020:1–10, 2020.
  * [21] Jan Hendrik Metzen, Tim Genewein, Volker Fischer, and Bastian Bischoff.  On detecting adversarial perturbations.  In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
  * [22] Nils Reimers and Iryna Gurevych.  Sentence-bert: Sentence embeddings using siamese bert-networks.  In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing. Association for Computational Linguistics, 11 2019\.
  * [23] Birkbeck spelling error corpus / roger mitton.  Oxford Text Archive.