# Non-Hermitian Aubry-André model with Power-Law Hopping

Zhihao Xu  Institute of Theoretical Physics and State Key Laboratory of
Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006,
China  Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
Xu Xia<EMAIL_ADDRESS>Chern Institute of Mathematics and LPMC,
Nankai University, Tianjin 300071, China  Shu Chen<EMAIL_ADDRESS>Beijing
National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China

###### Abstract

We study a non-Hermitian AA model with long-range hopping, $1/r^{a}$, and
different choices of quasiperiodic parameters $\beta$ to be a member of the
metallic mean family. We find that when the power-law exponent is in the $a<1$
regime, the system displays a delocalized-to-multifractal (DM) edge in its
eigenstate spectrum. For the $a>1$ case, a delocalized-to-localized (DL) edge
exists, also called the mobility edge. While a striking feature of the
Hermitian AA model with long-range hopping is that the fraction of delocalized
states can be obtained from a general sequence manifesting a mathematical
feature of the metallic mean family, we find that the DM or DL edge for the
non-Hermitian cases is independent of the mathematical feature of the metallic
mean family. To understand this difference, we consider a specific case of the
non-Hermitian long-range AA model with $a=2$, for which we can apply the
Sarnak method to analytically derive its localization transition points and
the exact expression of the DL edge. Our analytical result clearly
demonstrates that the mobility edge is independent of the quasi-periodic
parameter $\beta$, which confirms our numerical result. Finally, an optical
setup is proposed to realize the non-Hermitian long-range AA model.

##  I Introduction

Quasicrystals exhibit an intermediate localization feature between fully
periodic systems and fully disordered media. For the Anderson model with fully
disordered media, an infinitesimal random potential results in localization in
both one- and two-dimensional (1D and 2D) systems, whereas the mobility edges
exist in 3D cases, which separate extended and localized single-particle
states in the energy spectra Anderson ; Anderson79 . The situation is changed
in quasi-periodic cases. A paradigmatic example of a 1D quasicrystal system is
the Aubry-André (AA) model with nearest-neighbor hopping, which has been
experimentally realized by the ultracold atomic technique in bichromatic
optical lattices Roati ; Deissler ; Schreiber ; Bordia . A typical feature of
the AA model is that above a finite critical quasi-periodic amplitude, all the
eigenstates change from extended to localized, which is determined by its
self-dual property Aubry ; Suslov ; Wilkinson .

Beyond nearest-neighbor hopping, breaking the self-duality of the AA model
leads to the emergence of energy-dependent mobility edges, such as the system
in shallow lattices Scherg ; Hepeng ; Diener ; Ancilotto ; Holthaus  or with
exponential decay hopping Biddle . Self-duality is also lost in the
quasicrystals with power-law hoppings or interactions ($\propto 1/r^{a}$)
Biddle1 , which have been particularly interesting since power-law
interactions emerge in many systems, such as dipole-dipole interactions
$(\propto 1/r^{3})$ in polar molecules Gorceix ; Moses , Rydberg atoms Saffman
, nitrogen-vacancy centers Waldherr , and nuclear spins in condensed-matter
systems Kaiser . The tunable power-law interactions can be realized in laser-
driven ions ($0<a<3$) Richerme ; Jurcevic , which induce long-range exchange
between the synthetic lattices resulting in power-law hoppings. The effect of
power-law hoppings in the quasi-periodic AA model has been studied recently
Shlyapnikov ; NRoy . One has shown the localization properties of the long-
range AA model are characterized by the coexistence of localized
(multifractal) states with delocalized states for $a>1$ ($a<1$) and by the
emergence of a quasiperiodic parameter-dependent ladder of intermediate
regimes in which the eigenstate blocks become localized or multifractal
Shlyapnikov ; NRoy . By choosing a broader class of irrational Diophantine
numbers, referred to as the metallic mean family, one shows the relation of
the fraction of delocalized eigenstates and the irrational Diophantine numbers
in the intermediate regimes NRoy .

On the other hand, non-Hermitian systems sparked a great interest both in
experimental and theoretical fields Bender ; Bender1 ; Hatano1 ; Hatano2 ;
Hatano3 ; SLongi1 ; SLongi2 ; SLongi3 ; HuiJiang ; HuitaoShen ; Shunyu1 ;
Shunyu2 ; Zongping ; Lee1 ; Lee2 ; Harari ; Parto ; Schomerus18 ; Schomerus13
; Schomerus15 ; Schomerus152 ; Schomerus2020 ; Schomerus2021 ; LiuZhouChen ;
HengyunZhou ; Ruter ; LiangFeng ; Regensburger ; Joglekar ; XiangZhan ;
LeiXiao ; Zeuner ; YongXu ; Okuma ; LinhuLi ; Kunst ; Takata ; MPan ; Nakagawa
; Hamazaki ; Yamamoto ; Ashida ; Kawabata ; Kawabata2 ; LihongZhou ; Xuepeng1
; Xuepeng2 ; Xuepeng3 ; XuePeng4 ; XuePeng5 ; XuePeng6 ; ZhihaoXu1 ; ZhihaoXu2
; Linhu1 ; Longwen1 . Striking feathers are the failure of the bulk-boundary
correspondence HuitaoShen ; Shunyu1 ; Shunyu2 ; Zongping ; Lee1 ; Takata , the
non-Hermitian skin effect SLongi1 ; HuiJiang ; Shunyu1 ; Shunyu2 ; Zongping ,
the sensitivity of the spectra on boundary conditions Hatano2 ; Kunst , and
the non-Hermitian-induced topology Takata ; MPan . Recently, the interplay of
non-Hermiticity and disorder has brought a new perspective of the localization
properties Hatano1 ; Hatano2 ; Hatano3 ; SLongi2 ; SLongi3 ; HuiJiang ;
LiuZhouChen ; Zongping ; ZhihaoXu4 ; ZhihaoXu3 ; SLonghi4 ; Longwen ; YLiu2020
; XuXia ; YanxiaoLiu2021 ; YuCheng2021 ; TongLiu1 ; Longwen1 . The famous
Hatano-Helson model describing the interplay of the random disorder and the
nonreciprocal hopping in 1D lattices displays a finite metal-insulator
transition Hatano1 ; Hatano2 ; Hatano3 ; Zongping . One found that the
coincidence of the metal-insulator phase transition point with the
$\mathcal{PT}$ symmetry breaking point is of topological nature for the
$\mathcal{PT}$ symmetrical extension of the AA models both in 1D SLongi2 ;
SLongi3 ; HuiJiang ; ZhihaoXu4 ; LiuZhouChen  and 2D systems ZhihaoXu3 . The
non-Hermitian Maryland model shows a localization-delocalization phase
transition via topological mobility edges in the complex energy plane SLonghi4
; Longwen , while there are not extended states in its Hermitian version
Grempel ; Simon . These surprising results in non-Hermitian disorder fields
inspire us to study the non-Hermitian effect in long-range AA models.

In this paper, we study a non-Hermitian AA model with power-law hopping, and
its quasi-periodic parameter is set to be a member of the metallic mean
family, with special attention on the golden mean, silver mean, and bronze
mean. Similar to the Hermitian cases Shlyapnikov ; NRoy , the non-Hermitian
long-range AA model possesses the mixed regime where the multifractal
(localized) states coexist with the delocalized ones for $a<1$ ($a>1$).
However, our numerical results imply that the delocalized-to-multifractal (DM)
and delocalized-to-localized (DL) edges are independent of the choice of
irrational Diophantine numbers. We especially consider the non-Hermitian long-
range AA model with $a=2$, for which we can apply the Sarnak method to obtain
the exact expression of the DL edge. The analytical result for the DL edge
clearly indicates that the mobility edge is independent of the quasi-periodic
parameter $\beta$, which shows obviously different features from the Hermitian
long-range AA. Such a non-Hermitian AA model with power-law hopping can be
realized by an optical setup.

This paper is organized as follows. In Sec. II, we describe the Hamiltonian of
the non-Hermitian AA model with power-law hopping and metallic means. In Sec.
III, we apply the fractal dimension to study the localization features of the
eigenstates both for $a<1$ and $a>1$ regimes. We numerically and analytically
display the distinct properties for non-Hermitian cases that the fraction of
delocalized eigenstates is independent of the quasi-periodic parameter
$\beta$. In Sec. IV, we propose an experimental scheme to realize such a non-
Hermitian long-range model. Then we conclude in Sec. V.

##  II Model and Hamiltonian

We consider a non-Hermitian AA model with power-law hopping, which can be
described by the Hamiltonian

$\hat{H}=-J\sum_{j\neq i}\frac{1}{|j-i|^{a}}|j\rangle\langle
i|+\sum_{j}\lambda_{j}|j\rangle\langle j|,$ (1)

with

$\lambda_{j}=\lambda e^{i2\pi\beta j},$ (2)

where $|j\rangle$ denotes the state in which the excitation is localized at
the $j$th lattice site, and the hopping amplitude between sites $j$ and $j$ is
$J/|j-i|^{a}$. We set $J=1$ as an energy unit. The complex on-site potential
is characterized by its amplitude $\lambda$, and the incommensurabilty $\beta$
chosen to be a irrational Diophantine number. The potential is the
complexification of $\lambda_{j}=2\lambda^{\prime}\cos(2\pi\beta+\varphi)$
taking $\varphi=ih$, and the limits $\lambda^{\prime}\to 0$, $h\to\infty$,
keeping $\lambda^{\prime}e^{h}=\lambda$ finite SLongi2 ; SLongi3 . Notice that
due to $\lambda_{-j}=\lambda_{j}^{*}$, the non-Hermitian model displays
$\mathcal{PT}$ symmetry. Supposing that the eigenstate of a single particle in
our system is given by $|\psi\rangle=\sum_{j}\psi_{j}|j\rangle$, we can obtain
the eigenvalue equation as follows:

$-\sum_{n=1}^{\infty}\frac{1}{n^{a}}(\psi_{j-n}+\psi_{j+n})+\lambda_{j}\psi_{j}=E\psi_{j},$
(3)

where $\psi_{j}$ is the amplitude of the particle wave function at the $j$-th
site, $n=|j-i|$, and $E$ is the single-particle eigenvalue.

The solution of Eq. (3) is closely related to $a$ and the structure of the
potential $\lambda_{j}$ for both non-Hermitian and Hermitian cases. For non-
Hermitian cases, in the $a\gg 1$ limit SLongi3 , it has been verified that the
metal-insulator transition emerges at $\lambda=J$, which also corresponds to
$\mathcal{PT}$ symmetry breaking. When $\lambda<J$, all the eigenstates are
delocalized, and its energy spectra are real numbers, while for $\lambda>J$,
all the eigenstates are localized, and its energies become complex. For the
Hermitian case, in addition to the fully localized and delocalized phases, one
displays a coexistence of localized (multifractal) states with delocalized
states for $a>1$ ($a<1$), where the fraction of delocalized states can be
obtained from a general sequence manifesting the mathematical properties of
the metallic mean family of the irrational Diophantine numbers Shlyapnikov ;
NRoy .

To obtain the metallic mean family of the irrational Diophantine number, it is
useful to consider a generalized $\kappa$-Fibonacci sequence,

$F_{\nu+1}=\kappa F_{\nu}+F_{\nu-1},$ (4)

with $F_{0}=0$ and $F_{1}=1$. The irrational number $\beta$ can be obtained by
the limit $\beta=\lim_{\nu\to\infty}F_{\nu-1}/F_{\nu}$ with
$\kappa=1,2,3,\cdots$, yielding the metallic mean family, such as the golden
mean ($\beta_{g}=(\sqrt{5}-1)/2$) for $\kappa=1$, the silver mean
($\beta_{s}=\sqrt{2}-1$) for $\kappa=2$, the bronze mean
($\beta_{b}=(\sqrt{13}-3)/2$) for $\kappa=3$, and so on. In this paper, we
take the system size $L=F_{\nu}$ and the rational approximation
$\beta=F_{\nu-1}/F_{\nu}$ under periodic boundary conditions when numerically
diagonalizing the non-Hermitian long-range quasicrystal model defined in Eq.
(1).

##  III Localization transition in non-Hermtiain long-range AA model

Figure 1: (Color online) Fractal dimension $D_{2}$ of different eigenstates as
a function of the index $\mu$ of the corresponding real part of the
eigenvalues and the quasi-periodic potential amplitude $\lambda$ with
(a1)–(a3) $a=0.5$ and (b1)–(b3) $a=2.0$, respectively. The real part of the
energies is ordered in ascending order. The top row corresponds to $L=2584$,
$\beta_{g}=1597/2584$, and $l=4$, the center row is for $L=2378$,
$\beta_{s}=985/2378$ and $l=2$, and the bottom row is for $L=3927$,
$\beta_{b}=1189/3927$, $l=3$, respectively. In (b1)–(b3), the blue and yellow
dashed lines, respectively, represent $\lambda_{c1}\approx 0.6679$ and
$\lambda_{c2}\approx 2.6714$, and the black solid lines denote the DL edges
$E_{c}$ given by Eq. (12). Figure 2: (Color online) (a1)–(b1) $D_{2}$ versus
the index $\mu/L$ of the real part of the eigenenergies for $\lambda=1.0$. The
real part of the energies is ordered in ascending order. The dashed lines
represent the energy indexes of the localization transitions $\mu_{c}/L$.
(a2)–(b2) $\overline{D_{f}}$ as a function of $f$ for different $L$ and
$\lambda=1.0$. $\overline{D_{f}}$ is calculated by averaging over the fraction
of delocalized and the fraction of multifractal/localized eigenstates. The
left column is for $a=0.5$ and the right one is for $a=2.0$, respectively.
Here, for $L=2584$, we choose $\beta_{g}=1597/2584$ and $l=4$, and for
$L=6765$, we choose $\beta_{g}=4181/6765$, $l=5$, respectively.

One way of discerning between extended, localized, and multifractal states is
given by the analysis of the eigenstates’ fractal dimensions $D_{f}$ Chhabra ;
Janssen ; Huckestein ; Cuevas . For an extended (localized) state, $D_{f}=0$
($D_{f}=1$), whereas for a multifractal state, $D_{f}\in(0,1)$ and exhibits a
dependence of $f$. To determine the fractal dimension, one divides the system
of $L$ sites into $L_{s}=L/l$ boxes, and each box has $l$ sites. The fractal
dimension is defined as

$D_{f}=\lim_{L_{s}\to\infty}\frac{1}{1-f}\frac{\ln{\left[\sum_{p=1}^{L_{s}}(\mathcal{I}_{p})^{f}\right]}}{\ln{L_{s}}},$
(5)

where $\mathcal{I}_{p}=\sum_{j\in p}|\psi^{(\mu)}_{j}|^{2}$ corresponding to
the probability of detecting inside the $p$th box for the $\mu$th normalized
eigenstate $|\psi^{(\mu)}\rangle$ with $\sum_{j=1}^{L}|\psi_{j}^{(\mu)}|=1$,
and the corresponding eigenvalue $E_{\mu}$. Figures 1(a1) and 1(b1) show the
fractal dimensions $D_{2}$ of the non-Hermitian long-range AA model as a
function of $\lambda$ for all the eigenstates with the real part of
eigenvalues in ascending order, when the non-Hermitian parameter is fixed at
$\beta_{g}=1597/2584$ for $a=0.5$ and $2.0$, respectively. As seen from Fig.
1(a1) with $a=0.5$, it displays a DM edge in the energy spectrum, where the
fraction of extended states decreases and the multifractal states emerge with
the increase of $\lambda$. Figure 2(a1) shows the fractal dimension $D_{2}$
for different eigenstates $|\psi^{(\mu)}\rangle$ with different $L$,
$\lambda=1.0$, and $a=0.5$, and the real part of the corresponding eigenvalues
$E_{\mu}$ being ordered in ascending order. The DM edge emerges at
$\mu_{c}/L\approx 520/2584\approx 0.2$ for $\lambda=1.0$. When $\mu_{c}<520$,
the corresponding eigenstates are extended with $D_{2}\approx 1$. For
$\mu_{c}>520$ for $\lambda=1.0$, the corresponding eigenstates show the
multifractal feature with $D_{2}$ being finite values. However, for $a>1$, one
can observe a DL edge, also called a mobility edge, in its energy spectrum.
Figure 1(b1) with $a=2.0$ shows the appearance of the block of localized
states with $D_{2}\to 0$ with the increase of $\lambda$. Also shown in Fig.
2(b1), the fractal dimension $D_{2}$ with different $L$, $\lambda=1.0$, and
$a=2.0$ displays a jump from $D_{2}\to 1$ to $D_{2}\to 0$ at $\mu_{c}/L\approx
0.388$. As shown in Figs. 2(a1) and 2(b1), we can see the fractal dimensions
$D_{2}$ are independent on the system size $L$.

To further clarify the existence of mulitifractality in the regime $a<1$, we
numerically calculate the average fractal dimension $\overline{D_{f}}$ for the
regions with different localization features as a function of $f$ with
$\lambda=1.0$ and the modulation parameter being $\beta_{g}$ for different $L$
shown in Figs. 2(a2) ($a=0.5$) and 2(b2) ($a=2.0$). Here, $\overline{D_{f}}$
represents $D_{f}$ averaged over the fraction of delocalized and the fraction
of nondelocalized eigenstates. In Figs. 2(a2) and 2(b2), $\overline{D_{f}}$
averaged over the fraction of eigenstates with the indexes being less than
$\mu_{c}$ are close to $1$ for different $f$, which implies these states are
delocalized. $\overline{D_{f}}$ averaged over the fraction of eigenstates with
the indexes being larger than $\mu_{c}$ are finite values and show a
nontrivial dependence on $f$ for $a=0.5$, whereas $\overline{D_{f}}$ approach
to $0$ and display almost no dependence on $f$ for $a=2.0$. It indicates that
these states are multifractal for $a=0.5$ and localized for $a=2.0$. Also, we
find $\overline{D_{f}}$ for different system sizes exhibit similar behaviors.
Our numerical results imply that similar to the Hermitian cases, the non-
Hermitian cases show a DM edge for $a<1$ and a DL edge for $a>1$.

We also apply the Simon-Spencer theorem BSimon ; book  to analytically discuss
the absence of localized states for $a<1$ (for some details, see Appendix A).
By performing the Fourier transformation,

$f(\theta)=\frac{1}{\sqrt{L}}\sum_{j}\psi_{j}e^{i2\pi j\theta},$ (6)

we can obtain the dual equation of Eq. (3),

$\lambda
f(\theta+\tilde{\omega})=\left(E+\sum_{m}\frac{2}{m^{a}}\cos{m\theta}\right)f(\theta),$
(7)

with $\tilde{\omega}=2\pi\beta$, $\theta=2\pi\beta\tilde{j}$, $\tilde{j}$
being the site index in the dual space, and the potential of the system
$\tilde{\lambda}_{\tilde{j}}=-\sum_{m}2/m^{a}\cos{(2\pi\beta m\tilde{j})}$ in
its dual space. The dual-Hamiltonian matrix in the thermodynamic limit can be
written as:

$\tilde{H}=\begin{bmatrix}\ddots&\ddots&&&&&&\\\
&\tilde{\lambda}_{-\tilde{j}_{1}}&\lambda&&&&&\\\
&&\tilde{\lambda}_{-\tilde{j}_{1}+1}&\lambda&&&&\\\ &&&\ddots&\ddots&&&\\\
&&&&\tilde{\lambda}_{\tilde{j}_{2}-1}&\lambda&&\\\
&&&&&\tilde{\lambda}_{\tilde{j}_{2}}&\lambda&\\\ &&&&&&\ddots&\ddots&\\\
\end{bmatrix}.$ (8)

The Simon-Spencer theorem says that if there are two series of monotonically
increasing positive numbers, $\\{\tilde{k}_{s}\\}_{s=1}^{\infty}$ and
$\\{\tilde{k}^{\prime}_{s}\\}_{s=1}^{\infty}$ to make
$\sum_{s=1}^{\infty}\frac{1}{|\tilde{\lambda}_{-\tilde{k}_{s}}|}<\infty$ and
$\sum_{s=1}^{\infty}\frac{1}{|\tilde{\lambda}_{\tilde{k}^{\prime}_{s}}|}<\infty$.
Then the new block-diagonal dual-Hamiltonian

$\tilde{H}^{\prime}=\begin{bmatrix}\ddots&\ddots&&&&&&\\\
&\tilde{\lambda}_{-\tilde{k}_{s}}&0&&&&&\\\
&&\tilde{\lambda}_{-\tilde{k}_{s}+1}&\lambda&&&&\\\ &&&\ddots&\ddots&&&\\\
&&&&\tilde{\lambda}_{\tilde{k}^{\prime}_{s}-1}&0&&\\\
&&&&&\tilde{\lambda}_{\tilde{k}^{\prime}_{s}}&\lambda&\\\
&&&&&&\ddots&\ddots&\\\ \end{bmatrix}$ (9)

and the original dual-Hamiltonian $\tilde{H}$ possess the same absolutely
continuous spectrum, whereas the existence of
$\\{\tilde{k}_{s}\\}_{s=1}^{\infty}$ and
$\\{\tilde{k}^{\prime}_{s}\\}_{s=1}^{\infty}$ ensures the absence of the
absolutely continuous spectrum for $\tilde{H}^{\prime}$. For $0<a<1$, the dual
potential $\tilde{\lambda}_{\tilde{j}}$ is unbounded, and one can always find
out the two series $\\{\tilde{k}_{s}\\}_{s=1}^{\infty}$ and
$\\{\tilde{k}^{\prime}_{s}\\}_{s=1}^{\infty}$ to keep
$\sum_{s=1}^{\infty}\frac{1}{|\tilde{\lambda}_{-\tilde{k}_{s}}|}<\infty$ and
$\sum_{s=1}^{\infty}\frac{1}{|\tilde{\lambda}_{\tilde{k}^{\prime}_{s}}|}<\infty$.
Thus, our dual-Hamiltonian $\tilde{H}$ does not have the absolutely continuous
spectrum, which means there are no localized states in the real space for our
model in the $0<a<1$ regime. This proof is also suited for the Hermitian
cases.

Figure 3: (Color online) Energy spectrum $E_{\mu}$ of $\hat{H}$ for
$\beta_{g}=1597/2584$ and $a=2$ with (a) $\lambda=0.5$, (b) $\lambda=1.0$, and
(c) $\lambda=3.0$, respectively. The dashed line in (b) denotes the mobility
edge $E_{c}\approx-0.2091$ corresponding to $\mu_{c}=1002$ for $\lambda=1.0$
and $L=2584$.

In the Hermitian cases, the fraction of the delocalized part depends on the
choice of quasi-periodic parameter $\beta$. However, as shown in Fig. 1 for
$\beta$ being chosen as the different metallic mean family of irrational
numbers, one can compare the numerical results with different quasi-periodic
parameters $\beta$ for the same $a$, and easily find that, unlike the
Hermitian case, the fraction of delocalized eigenstates is independent of
$\beta$ in the non-Hermitian cases. To analytically verify the property, as a
concrete example, we study the localization features of the non-Hermitian
long-range AA model with $a=2$. The dual equation for $a=2$ is given as
follows:

$\lambda
f(\theta+\tilde{\omega})=(E+\frac{\theta^{2}}{2}-\pi\theta+\frac{\pi^{2}}{3})f(\theta).$
(10)

According to the Sarnak method Sarnak ; ZhihaoXu3  (for some details on the
Sarnak method, see Appendix B), we define a characteristic function as
follows:

$\displaystyle G(E)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\ln{\left|E+\frac{\theta^{2}}{2}-\pi\theta+\frac{\pi^{2}}{3}\right|}$
$\displaystyle=-2-\ln{2}+\frac{u_{E1}\ln{u_{E1}}+u_{E2}\ln{u_{E2}}}{\pi},$
(11)

with $u_{E1}=\pi+\sqrt{\pi^{2}/3-2E}$ and $u_{E2}=\pi-\sqrt{\pi^{2}/3-2E}$.
The Sanark method says (i) when $G(E)=\ln{|\lambda|}$, the spectrum is dense
with the localized eigenstates, and the corresponding eigenvalue $E$ is a
complex value and (ii) when $\\{G(E)>\ln{|\lambda|}\\}\cap U_{E}$, it
corresponds to a dense set of $E$ with the extended eigenstates, where
$U_{E}=[-\pi^{2}/3,\pi^{2}/6]$ is the set of the spectrum such that
$E+\theta^{2}/2-\pi\theta+\pi^{2}/3=0$ for some $\theta$, and $E$ is a real
value. For the eigenenergies in the region $U_{E}$, we find
$G(E)\in[2\ln{\pi}-\ln{2}-2,2\ln{\pi}+\ln{2}-2]$. Hence, two delocalization-
localization transition points exist at
$\lambda_{c1}=\exp{(2\ln{\pi}-\ln{2}-2)}\approx 0.6679$ and
$\lambda_{c2}=\exp{(2\ln{\pi}+\ln{2}-2)}\approx 2.6714$, which are labeled by
blue and yellow dashed lines, respectively, shown in Figs. 1(b1)–1(b3). When
$\lambda<\lambda_{c1}$, all the eigenstates are extended, and the
corresponding energy spectrum is entirely real, which is shown in Fig. 3(a)
with $\lambda=0.5$ and $\beta_{g}=1597/2584$. For $\lambda>\lambda_{c2}$, all
the eigenstates are localized, and the corresponding energy spectrum becomes
totally complex shown in Fig. 3(c) with $\lambda=3.0$ and
$\beta_{g}=1597/2584$. The mobility edge emerges in the region
$\lambda\in[\lambda_{c1},\lambda_{c2}]$. In this intermediate regime, the DL
edge $E_{c}$ is a real number that satisfies

$G(E_{c})=\ln{\lambda},$ (12)

which is marked by the black solid lines in Figs. 1(b1)–1(b3). As shown in
Fig. 3(b) with $\lambda=1.0$ and $\beta_{g}=1597/2584$, when
$\mathrm{Re}(E_{\mu})<E_{c}\approx-0.2091$, the eigenenergies are real values,
while when $\mathrm{Re}(E_{\mu})>E_{c}$, the eigenenergies become complex
values. Here, $E_{c}\approx-0.2091$ for $\lambda=1$ corresponds to
$\mu_{c}=1002$, which is marked by the red dashed line in Fig. 3(b). According
to the numerical and analytical results, we find $\lambda_{c1}$ is also the
$\mathcal{PT}$ symmetry broken point corresponding to the point emerging in
the complex eigenenergies, and the DL edges are independent on $\beta$. In
Appendix C, we show the relation of $\mathcal{PT}$-symmetry transition and the
DM transition. Our results imply that when the quasi-periodic parameter
$\beta$ is chosen as the different metallic mean family of irrational
Diophantine numbers, the fraction of delocalized states shall display the same
feature.

##  IV Optical realization of non-Hermitian long-range AA model

Figure 4: (Color online) (a) Schematic of the FM multi-mode laser. (b) The
axial spectral modes of the laser cavity spaced by $\Delta\omega_{a}$ realized
a 1D non-Hermitian lattice with the long-range hopping. The full width at half
maximum of the gain line is $\Delta\omega_{g}$ with
$\Delta\omega_{g}\gg\Delta\omega_{a}$.

In the experiment, one can implement the non-Hermitian long-range AA model in
an optical setup by applying the frequency-modulated (FM) multi-mode laser
Kuizenga1 ; Kuizenga2  shown in Fig. 4(a). The optical setup is done with a
standard Fabry-Perot laser cavity with the axial modes spaced by
$\Delta\omega_{a}$ undergoing the FM modulator, the gain medium, and the
etalon. The light circulates back and forth between the two end mirrors of the
cavity, and the round-trip number in the cavity is marked by $t$, which can be
normalized to physical time. When the light finishes a round trip, the
spectral components $\psi_{j}$ of cavity axial modes can be given Haus

$\displaystyle\psi_{j}(t+1)$
$\displaystyle=\psi_{j}(t)+\\{\delta\psi_{j}(t)\\}_{\mathrm{loss}}+\\{\delta\psi_{j}(t)\\}_{\mathrm{modul}}$
$\displaystyle+\\{\delta\psi_{j}(t)\\}_{\mathrm{gain}}+\\{\delta\psi_{j}(t)\\}_{\mathrm{etalon}}.$
(13)

Here, $\\{\delta\psi_{j}(t)\\}_{\mathrm{loss}}=-\gamma_{c}\psi_{j}(t)$
represents the losses of the cavity with loss rate $\gamma_{c}$ per round-
trip. $\\{\delta\psi_{j}(t)\\}_{\mathrm{modul}}$ is the change from the FM
modulator, which introduces a time-dependent phase shift
$\Delta\varphi(t)=\Delta_{\mathrm{FM}}\sum_{n=1}\cos{(n\omega_{m}t)}/n^{a}$ to
the incident light field with the amplitude $\Delta_{\mathrm{FM}}\ll 1$ and
the frequency $\omega_{m}=\Delta\omega_{a}$. One has

$\\{\delta\psi_{j}(t)\\}_{\mathrm{modul}}=i\frac{\Delta_{\mathrm{FM}}}{2}\sum_{n=1}\frac{1}{n^{a}}\left[\psi_{j}(t)e^{in\omega_{m}t}+\psi_{j}(t)e^{-in\omega_{m}t}\right].$
(14)

When the round-trip passes through the gain medium, one hasHaus

$\\{\delta\psi_{j}(t)\\}_{\mathrm{gain}}\approx\frac{g}{1+4j^{2}\omega_{m}^{2}/\Delta\omega_{g}^{2}}\psi_{j}(t),$
(15)

which is the small change of $\psi_{j}$ for a homogeneously broadened active
material with a slow relaxation dynamics and wide gain bandwidth
$\Delta\omega_{g}\gg\Delta\omega_{a}$ Kuizenga1  shown in Fig. 4(b). Here, $g$
is the saturated gain. The change arising from etalon transmission is
represented by $\\{\delta\psi_{j}(t)\\}_{\mathrm{etalon}}$. We assume the
etalon with the refractive index $\bar{n}$ and the thickness $\bar{L}$ is
placed inside the cavity and slightly tilted from the normal incidence. For
near-normal incidence, the spectral transmission of the etalon
$T(\omega)=(1-R)/[1-R\exp{(i2\bar{n}\bar{L}\omega/c+i\phi)}]$ with the
reflectance $R\approx(\frac{\bar{n}-1}{\bar{n}+1})^{2}$ of etalon facets, the
speed of light in vacuum $c$, and the additional phase shift $\phi$ adjusted
by slight etalon tilting. For a glass-air interface, $R\ll 1$, we have
$T(\omega)\approx 1-R+R\exp{(i2\bar{n}\bar{L}\omega/c+i\phi)}$, and

$\displaystyle\\{\delta\psi_{j}(t)\\}_{\mathrm{etalon}}$
$\displaystyle=[T(j\omega_{m})-1]\psi_{j}(t)$
$\displaystyle=\left[-R+Re^{i(2\pi\beta j+\phi)}\right]\psi_{j}(t),$ (16)

where $\beta=\omega_{m}/\Delta\omega_{et}$ and $\Delta\omega_{et}=\pi
c/(\bar{n}\bar{L})$. After setting
$\psi_{j}(t+1)-\psi_{j}(t)\approx(\partial\psi_{j}/\partial t)$ and
$\psi_{j}(t)=\psi_{j}\exp{(-i\omega_{m}t)}$, one can obtain from Eq. (IV) Haus
; Gordon

$i\frac{\partial\psi_{j}}{\partial
t}=-J\sum_{n=1}\frac{1}{n^{a}}(\psi_{j-n}+\psi_{j+n})+\lambda e^{i2\pi\beta
j}\psi_{j}+i\mathcal{L}_{j}\psi_{j},$ (17)

where $J=\Delta_{FM}/2$, $\lambda=R$,
$\mathcal{L}_{j}=-\gamma_{c}-R+g/(1+4j^{2}\omega_{m}^{2}/\Delta\omega_{g}^{2})$,
and $\phi=\pi/2$. In the broad gain-line limit,
$\omega_{m}/\Delta\omega_{g}\to 0$, and setting $\gamma_{c}+R\approx g$, the
spectral mode dynamics can emulate the non-Hermitian long-range AA model.

##  V Conclusions

The study of long-range quasicrystals in Hermitian systems exhibits rich
localization phenomena. Recent fruitful achievements on the interplay of non-
Hermiticity and quasicrystals inspire us to consider the non-Hermitian AA
model with power-law hopping and the quasiperiodic parameter chosen to be a
set of the metallic mean family. In this paper, we find in addition to the
fully delocalized and localized regime, a coexistence of multifractal
(localized) eigenstates with delocalized eigenstates for $a<1$ ($a>1$), which
is similar to the Hermitian cases. Unlike the Hermitian long-range AA model,
we analytically and numerically verify that the fraction of delocalized
eigenstates is independent of the quasi-periodic parameter in the intermediate
regime. We propose an optical setup by using the FM laser to realize the non-
Hermitian long-range AA model. We believe that our study will motivate further
studies on the exploration of localization properties in non-Hermitian long-
range quasicrystals.

## Appendix A: Some details on Simon-Spencer theorem

The Simon-Spencer theorem is based on the trace class perturbation method book
, which is used to transform an infinite-dimensional matrix
($\hat{\mathcal{A}}$) to a direct sum of a series of finite-dimensional
matrices ($\hat{\mathcal{B}}=\bigoplus_{k}\hat{b}_{k}$) by a series of
perturbation operators. Under some kinds of trace class perturbation, one can
guarantee such two matrices $\hat{\mathcal{A}}$ and $\hat{\mathcal{B}}$ have
the same absolutely continuous spectrum if the trace class norm can be
effectively controlled.

Applied to our case for $a<1$, a trace class perturbation corresponds to
changing the left hopping amplitude from $\lambda$ to $0$ at some positions
$\tilde{k}_{s}$ for the infinite-dimension matrix $\tilde{H}$ described by Eq.
(8) in the main text, and the trace class norm is then controlled by
$\frac{1}{|\tilde{\lambda}_{\tilde{k}_{s}}|}$. By performing a series of trace
class perturbations, the infinite system can be decomposed into an infinite
number of small closed systems described by $\tilde{H}^{\prime}$ in the main
text, naturally in which there are no extended states. Due to the unbounded
dual potential $\tilde{\lambda}_{\tilde{j}}$, the trace class norms can be
controlled by
$\sum_{s=1}^{\infty}\frac{1}{|\tilde{\lambda}_{\tilde{k}_{s}}|}<\infty$ and
$\sum_{s=1}^{\infty}\frac{1}{|\tilde{\lambda}_{-\tilde{k}^{\prime}_{s}}|}<\infty$
for two series of monotonically increasing positive numbers
$\\{\tilde{k}_{s}\\}_{s=1}^{\infty}$ and
$\\{\tilde{k}^{\prime}_{s}\\}_{s=1}^{\infty}$. According to the Simon-Spencer
theorem, we can say that $\tilde{H}$ and $\tilde{H}^{\prime}$ have the same
absolutely continuous spectrum. Hence, the dual equation $\tilde{H}$ does not
have extended states.

## Appendix B: Some details on the Sanark method

The Sanark method can solve a class problems for quasicrystals with the
eigenvalue equation described by:

$f(\theta+\tilde{\omega})=g(E,\theta)f(\theta),$ (18)

where $E$ is a given value, $\theta=2\pi\beta j$, $\tilde{\omega}=2\pi\beta$,
and $j$ is the site index. One can define the Lyapunov exponent as

$\displaystyle\mathcal{L}(E)$
$\displaystyle=\lim_{m\to\infty}\frac{1}{m}\ln{\left|\frac{f(\theta+m\tilde{\omega})}{f(\theta)}\right|}$
$\displaystyle=\lim_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m-1}\ln{\left|g(E,\theta+k\tilde{\omega})\right|}.$
(19)

In the main text, $G(E)=\mathcal{L}(E)+\ln{\lambda}$. Then if the Lyapunov
exponent $\mathcal{L}(E)=0$, the given value $E$ is the eigenvalue of Eq.
(seq1) and the corresponding eigenstate is an extended state. If
$\mathcal{L}(E)\neq 0$ and $g(E,\theta)\neq 0$ are always true, according to
the ergodic theory that such $E$ is not the solution of the equation. For some
$\theta$, $g(E,\theta)=0$, and when $\mathcal{L}(E)>0$, the given $E$ is the
eigenvalue corresponding to a localized state.

## Appendix C: $\mathcal{PT}$ symmetry transition for $a<1$

Figure 5: (Color online) (a) The behavior of the maximum value of
$|\mathrm{Im}(E)|$ and the fractal dimension of the $L$-th eigenstate
$D_{2}^{(L)}$ as the functions of $\lambda$ for $a=0.5$ and
$\beta_{g}=1597/2584$. The dashed line in (a) denotes the $\mathcal{PT}$
symmetry breaking point and the DM transition point. Energy spectrum $E_{\mu}$
of $\hat{H}$ for $a=0.5$ and $\beta_{g}=1597/2584$ with (b) $\lambda=0.1$ and
(c) $\lambda=1.0$, respectively. The dashed line in (c) corresponds the DM
edge with $\mu_{c}\approx 520$ for $\lambda=1.0$ and $L=2584$.

In this appendix, we consider the $\mathcal{PT}$ symmetry transition in the
$a<1$ region. Figure 5(a) shows the behavior of the maximum value of
$|\mathrm{Im}(E)|$ and the fractal dimension of the $L$-th eigenstate
$D_{2}^{(L)}$ as the functions of $\lambda$ for $a=0.5$ and
$\beta_{g}=1597/2584$. Here, the real part of the eigenvalues is ordered in
ascending order. According to the numerical calculation of fractal dimension
for $a=0.5$ and $\beta_{g}=1597/2584$ shown in Fig. 1(a1) in the main text,
one can see that the DM transition point is denoted by the eigenstate with the
maximum real part of the energy. As seen in Fig. 5(a), the $\mathcal{PT}$
symmetry breaking point coincides with the DM phase transition point at
$\lambda_{c}\approx 0.24$ for $a=0.5$. The energy spectrum for $a=0.5$ and
$\lambda<\lambda_{c}$ is shown in Fig. 5(b) ($\lambda=0.1$), where all the
eigenvalues are real. When $\lambda>\lambda_{c}$, the complex energies emerge.
For the case of $\lambda=1.0$ and $a=0.5$ [Fig. 5(c)], the real-complex
transition of the energy spectrum emerges at $\mu_{c}\approx 520$ which
corresponds to the DM edge shown in the main text. In conclusion, similar to
the case of $a>1$, we numerically verify the coincidence of the $\mathcal{PT}$
symmetry breaking point with the DM phase transition point for the long-range
quasicrystals ($a<1$).

###### Acknowledgements.

Z.X. thanks Linjie Zhang for helpful discussions. Z. X. is supported by the
NSFC (Grants No. 11604188 and No. 12047571), Beijing National Laboratory for
Condensed Matter Physics, and STIP of Higher Education Institutions in Shanxi
under Grant No. 2019L0097. X.X. is supported by NanKai Zhide Foundation. S.C.
is supported by NSFC under Grant No. 11974413 and the Strategic Priority
Research Program of Chinese Academy of Sciences under Grant No. XDB33000000.
This work is also supported by NSF for Shanxi Province Grant No. 1331KSC.

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# QUARC: Quaternion Multi-Modal Fusion Architecture For Hate Speech
Classification
††thanks: 1 Authors have equal contribution

Deepak Kumar12, Nalin Kumar13 and Subhankar Mishra4  School of Computer
Sciences, NISER, Bhubaneswar $752050$, India
Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai $400094$, India
Email: [2deepak.kumar, 3nalin.kumar<EMAIL_ADDRESS>
###### Abstract

Hate speech, quite common in the age of social media, at times harmless but
can also cause mental trauma to someone or even riots in communities. Image of
a religious symbol with derogatory comment or video of a man abusing a
particular community, all become hate speech with its every modality (such as
text, image, and audio) contributing towards it. Models based on a particular
modality of hate speech post on social media are not useful, rather, we need
models like multi-modal fusion models that consider both image and text while
classifying hate speech. Text-image fusion models are heavily parameterized,
hence we propose a quaternion neural network-based model having additional
fusion components for each pair of modalities. The Model is tested on the
MMHS150K twitter dataset for hate speech classification. The model shows an
almost $75\%$ reduction in parameters and also benefits us in terms of storage
space and training time while being at par in terms of performance as compared
to its real counterpart.

###### Index Terms:

Quaterion algebra, Text-image fusion, MMHS150K, Hate speech, Octonion

##  I Introduction

Internet, social media, in particular, make a large and integral part of our
society. Laws of civilized society apply there too but a certain sense of
anonymity on social media brings the worst in many people. This results in
incidents of hate speech in terms of online posts on social media. Hence, some
moderation system is required to identify what can be classified as hate
speech. To name a few websites doing such moderation are Facebook and Reddit.
Due to volume, variety, and velocity of social media content [13], it renders
human moderators useless, leading to the need for machine moderators (using
machine learning).

Existing hate speech detection models in the text have achieved significant
accuracy using standard architectures like SVM, MLP, DNN [14]. However,
indirect comments with some context can also be considered as hate speech.
Also, with the rise of memes and emojis, hate speeches have become quite
diverse and with it, the need for investigation in machine learning-based
models for hate speech detection in images [15] arises. Hence, one needs to
consider all aspects of a social media post. However, such models could easily
have parameters in millions [4], which poses its own difficulties in terms of
practical deployment. This brings us to quaternion algebra, an extension of
complex algebra. Recently its use in deep neural network has shown faster
convergence and reduced parameters [3]. This rekindled interest in its
possible application in machine learning. Quaternion neural networks have
recently been considered in natural language processing (NLP) [2] tasks
especially because of their ability to reduce parameters by $75\%$.

This paper introduces multi-modal fusion model of image, text, and text of
image for hate speech classification on social media posts. We present a
quaternion convolution neural network (QCNN) based fusion model and its
simpler variants to do sentiment analysis for hate speech detection. Our model
is inspired from existing symmetric gated fusion model [4]. We apply QCNN on
text, image, and image text separately to get three vectors. We then find
additional fusion vectors which are quaternion attention on pairs of three
modalities followed by their concatenation. The vector we finally get is used
for hate speech classification. The model achieves expected $75\%$ parameter
reduction and faster model training and also achieves better or similar
precision scores than its real counterpart.

The main contributions of our paper are as follows:

  * •

Novel quaternion multi-modal fusion model for hate speech classification.

  * •

Almost $75\%$ reduction in parameter size (hence, requires lesser storage
space) of multi-modal fusion model as compared to current state of art.

  * •

At par performance of quaternion multi-modal fusion model in comparison to its
real counterpart.

##  II Related Works

Hate speech classification has been of interest for almost three decades. One
of the earliest works in the hate speech domain was Ellen Spertus’s Smokey
[5], which classified private messages between two parties using a tree-based
classifier. This paper used a rigid set of rules to create the feature vector.
This model had several limitations, like, it could not handle unusual
typography, grammatical mistakes or sarcasm. Though its performance was at par
with that of human annotators’ but what constitutes hate speech was also quite
restrictive there. [6] provides the hate speech classification as an NLP task
on Web$2.0$ (The social media) using SVM classifier public forum chats. This
model, in addition to the rule-based features as in Smokey, also has sentiment
and context features. [7] provides an extensive review of hate speech
classification literature in terms of the NLP task and presents different
data-sets available & different approaches to the problem.

With the ease of sharing different types of content over social media the
modalities of hate speech also increased. One prominent example can be memes
and emojis which contain image, text in the image, whose investigation
requires a new approach as presented in [15] which simply concatenated the
different modalities. Another interesting work in multi-modal area is the
tensor fusion network, a novel model for multi-modal sentiment analysis given
by [8] exploring inter and intra-modality associations. This model observed
video data giving text, image, and audio modalities and fused them using
vector product of all three. Further improvement was the inter-modal sentiment
analysis using attention architecture over the tensor fusion approach by [9].
In state-of-the-art [4], the authors proposed a series of multi-modal fusion
models based on CNN, ResNet, and attention for hate speech classification on
the Facebook data set. They claimed that an additional fusion vector improved
accuracy. However, the number of parameters involved in the fusion model was
too high.

Quaternion neural networks have been in computer science for quite some time
in fields like signal processing and computer vision. Recently, using the
quaternion transformer model [2], the authors showed that for almost all NLP
tasks, quaternion model reduced $75\%$ parameters without any significant loss
in accuracy. The reason was claimed to be the Hamilton product in quaternion
algebra which is the counterpart of the dot product in real algebra. [3]
proposed QCNN, quaternion weight initialization, and batch normalization
scheme. They experimented over the image dataset and showed that the
quaternion model performed at par with the real model while reducing
parameters and converging faster.

Figure 1:  Quaternion fusion model

##  III Methodology

We propose QUARC-a QUaternion multi-modal fusion ARChitecture, which uses the
QCNN [3] as the baseline architecture for all text, image and fusion part of
model. Overall structure of our model is shown in Figure $1$.

###  III-A Text

This text model is inspired by [10], where CNN was used for sentence
classification. There are two types of text, tweet text and image text. Text
model is same for both but are trained separately as follows.

  1. 1.

Firstly, to process the text before generating the word embeddings, we proceed
as follows:

     * •

We removed URLs and associated symbols.

     * •

as it contain considerable amount of emojis, they are changed in text.

  2. 2.

Then we retrieve pre-trained embeddings from GloVe[11] for tweet as well as
image texts. Words not present in GloVe embeddings are set randomly. As data-
set is not large enough, we don’t train GloVe word embedding model[1]. We get
vectors $v_{t\\_t}$ & $v_{t\\_p}$ for tweet text and image text respectively
after proper padding with word limit to be 150 per input text.

  3. 3.

Then we apply quaternion $1$-D convolution layer with same padding to keep
input and output length same for applying quaternion attention in additional
fusion.

  4. 4.

After that we apply max-pooling, ReLu and flatten it.

  5. 5.

Lastly, the resulting vector representation $T_{t}$ (tweet text) & $T_{p}$
(image text) (see Fig.1) obtained after applying quaternion multi-layer
perceptron (Q-MLP) and a dropout layer, is used in final concatenation.

Hyper-parameters | Values
---|---
Dropout rate | $0.35$
Batch Normalization | Quaternion Batch Normalization
Kernel Initializer | Glorot
Optimizer | Adam
Loss Function | Binary Cross-entropy
Epochs | $20$
Batch Size | $128$
TABLE I: Hyper-parameters

###  III-B Image

The image model is trained as follows:

  1. 1.

First we resize the images in $32\times 32$ pixels. Now we make sure all
images are 3-channeled, if not, we convert them into it by adding null
channels.

  2. 2.

These input images are passed through pre-trained ResNet50 without its output
layer.

(a) “What separates humans from animals?”

(b) “It’s so hard being a nigga”

(c) “Hey…. fucck raghead”

Figure 2: Sample tweets with caption as their tweet texts. [1]

  3. 3.

The vector ($p`$) finally obtained after passing the max pooled output ($p$)
through quaternion dense layer and dropout layer is used in final
concatenation.

###  III-C Fusion

####  III-C1 Simple fusion

Here we simply concatenate $T_{t}$, $T_{p}$ and $p`$. Then we apply dropout,
MLP and softmax on $concat$($T_{t}$, $T_{p}$, $p`$) for the final hate speech
classification.

####  III-C2 Symmetric gated fusion

We follow the symmetric gated summation approach similar to [12] and [4]. In
$gated\\_sum(a,p)$, we linearly transform the input vectors $a$ & $p$ such
that they both have the same dimension. Then, we calculate
$\beta_{a},\beta_{p}$ & $m$ as given in the equ. $(\ref{eq 1})$, $(\ref{eq
2})$ & $(\ref{eq 3})$, where the weight & bias vectors
($W_{a},U_{a},W_{p},U_{p},W_{m},U_{m},B_{a},B_{p},B_{m}$) are to be learned by
the model. The $visual\;modulation\;gate$ ($f$), as referred by [12], can
dynamically control the combination of text and image information.

$a^{\prime}=W_{a}\cdot a+b_{a}$ $p^{\prime}=W_{p}\cdot p+b_{p}$
$\beta_{a}=\sigma(W_{a}\cdot a^{\prime}+U_{a}\cdot p^{\prime}+B_{a})$ (1)
$\beta_{p}=\sigma(W_{p}\cdot a^{\prime}+U_{p}\cdot p^{\prime}+B_{p})$ (2)
$m=\tanh{(W_{m}\ast a^{\prime}+U_{m}\ast p^{\prime}+B_{m})}$ (3)
$gated\\_sum(a,p)=f=\beta_{a}\ast a^{\prime}+\beta_{p}\ast m$ (4)

Using the quaternion attention architecture ([2]), we try to find how much the
given photo information (query vector) is related to the certain part of the
text (context vectors). The explicit equation for the attention is given in
the equ. $(\ref{eq 5})$. The weighted sum $a_{c}$ of the text vectors, using
the photo information, is then passed through the above $gated.sum()$ function
along with the image vector $p$.

$s_{c_{i}}=softmax(c^{T}_{i}\cdot W_{a}\otimes p^{\prime})\;\;\;\;i=1,...,n$
(5) $a_{c}=\sum(s_{c_{i}}\ast c_{i})$ (6)

where $c=[c_{1},..,c_{n}]\;$ is the text ($t_{p}$ = image text / $t_{t}$=
tweet text) vector obtained in 3.1.3.

The resultant vectors $a_{t_{t}}$, $a_{t_{p}}$, $a_{t_{w}}$ are then passed
through the $gated\\_sum()$ along with the image vector $p$ (where,
$a_{t_{w}}=a_{t_{t}}+a_{t_{p}}$). We now concatenate the vectors $T_{t}$,
$T_{p}$, $p`$, $gated\\_sum(a_{t_{t}},p)$, $gated\\_sum(a_{t_{p}},p)$&
$gated\\_sum(a_{t_{w}},p)$ and apply QMLP, dropout & softmax for hate speech
classification.

###  III-D Models

So in summary, here are the fusion models with left side of equality denoting
model with its input and right side of equality giving concatenation layer
with its input which is then followed by dropout, quaternion dense layer and
softmax operations for the final hate speech classification.

$\displaystyle\textbf{Model\; 1}:$ $\displaystyle
F(v_{t_{t}},v_{t_{p}},v_{p})=C(T_{t},T_{p},p`,GS(a_{t_{t}},p),GS(a_{t_{p}},p),GS(a_{t_{w}},p))$
$\displaystyle\textbf{Model\;
2}:F(v_{t_{t}},v_{p})=C(T_{t},T_{p},p`,GS(a_{t_{t}},p))$
$\displaystyle\textbf{Model\;
3}:F(v_{t_{p}},v_{p})=C(T_{t},T_{p},p`,GS(a_{t_{p}},p)$
$\displaystyle\textbf{Model\;
4}:F(sum(v_{t_{t}},v_{t_{p}}),v_{p})=C(T_{t},T_{p},p`,GS(a_{t_{w}},p))$
$\displaystyle\textbf{Model\;
5}:simpleconcat(v_{t_{t}},v_{t_{p}},v_{p})=C(T_{t},T_{p},p`)$
$\displaystyle\textbf{Model\; 6}:v_{t_{t}}=C(T_{t})$
$\displaystyle\textbf{Model\; 7}:v_{p}=C(p`)$

Here F represents fusion, C represents concat, GS represents gated_sum

Model No. | Model | # Trainable Param. | ROC-AUC | Avg Precision Score
---|---|---|---|---
|  | $\rm I\\!R$ | $\rm I\\!H$ | $\rm I\\!R$ | $\rm I\\!H$ | $\rm I\\!R$ | $\rm I\\!H$
1. | $fusion(v_{t_{t}},v_{t_{p}},p)$ | $2,810,573$ | $719,837$ | 70.28 | 71.49 | $0.8664$ | $0.8763$
2. | $fusion(v_{t_{t}},p)$ | $1,241,545$ | $311,353$ | 70.39 | 71.47 | $0.8754$ | $0.8764$
3. | $fusion(v_{t_{p}},p)$ | $1,241,545$ | $311,353$ | 70.46 | 70.80 | $0.8717$ | $0.8746$
4. | $fusion(sum(v_{t_{t}},v_{t_{p}}),p)$ | $2,752,305$ | $701,553$ | 70.03 | 70.79 | $0.8679$ | $0.8775$
5. | $simpleconcat(v_{t_{t}},v_{t_{p}},p)$ | $1,219,894$ | $301,361$ | 70.84 | 70.87 | $0.8671$ | $0.8748$
6. | $v_{t_{t}}$ | $530,229$ | $132,729$ | 70.80 | 71.38 | $0.8729$ | $0.8738$
7. | $v_{p}$ | $131,201$ | $32,897$ | 50.34 | 51.28 | $0.7436$ | $0.7474$
TABLE II: Performance of the proposed models

##  IV Experiments

###  IV-A Data

We did our experiment on the only publicly available dataset having both image
and text that we could gather, which is MMHS150K [1] twitter data-set for hate
speech. It is a manually annotated data comprising $150,000$ twitter tweets,
each containing a tweet text component (may contain emojis, and URL), a tweet
image component and may also contain image text component (text extracted from
tweet image using Google Vision API). Figure $2$, shows few sample images with
corresponding tweet text as its caption. Out of this validation set contains
$5,000$ and test set $10,000$ tweets. Our code is available at
https://github.com/smlab-niser/quaternionFusion.

###  IV-B Hyper-parameters

We use 100-dimensional twitter GloVe embeddings, which are trained on $2$B
tweets, making it suitable for our experiment. Maximum word length is set to
$150$ for each input. We use $1$D-convolution windows of size $5$ with $128$
filters for both image and tweet text while $2$D-convolution windows of

(a) Trainable Parameters

(b) Avg. Precision Score

Figure 3: Performance of models. For model no. see Sec. III.D

size $2\times 2$ & $3\times 3$ for image training. Dimensions of $T_{t}$,
$T_{p}$ & $p$ is set to $64$. Other hyper-parameters are given in Table $1$.

###  IV-C Results & Analysis

Results of our experiments are given in Table II. In this table, we have real
and quaternion version of each model. The table shows number of trainable
parameters, ROC-AUC score and average precision score for each fusion model.

In the Table II, we can see that quaternion version for each fusion model
shows almost $75\%$ reduction in trainable parameters as compared to the real
one. Despite this reduction in parameters, we can see comparable performance
(ROC-AUC score and average precision score) between quaternion and real. The
quaternion version of $fusion(v_{t_{t}},v_{t_{p}},p)$ model shows best ROC-AUC
score and the quaternion version of $fusion(sum(v_{t_{t}},v_{t_{p}}),p)$ model
shows best average precision score among all models.

The ROC-AUC score of $v_{t_{t}}$ model is slightly better than some of the
fusion models, which is against our expectation. However, similar trend is
seen in [1]. Also, we traded off performance by resizing the images in
$32\times 32$ pixels (while [1] considered images of 500 pixels) because
otherwise matrices corresponding to images were getting too big to be handled
by our current setup. We suspect this might be the reason for information loss
in image, hence, contributing less in the fusion models.

From the results of [1], we can observe that performance isn’t improving much
from only text ($T_{t}$) to fusion of text and image text ($T_{p}$) and
finally fusion of all three, regarding which [1] suspects that fusion model is
not able to extract information other that the text modality. However, we
suspect that this can be asserted to the unique nature of this dataset and its
noise because of extracted image text and the subjectivity of annotator (also
mentioned by [1]). But, the improvement of performance with fusion can be seen
in experiments of [4], which is done over Facebook dataset (not public).

##  V Conclusion

Social media is an important platform to bring our society together with
healthy discourse. But, because of some bad apples, even this place is
becoming toxic, unsafe and, a propaganda tool. Hate speech nowadays comes in
the combination of image, video, audio and text. So multi-modal models become
essential for hate speech classification. However, these models are having
high number of parameters. In this paper we conclude that quaternion based
multi-modal fusion model not only reduces parameters in comparison to its real
counterpart but also achieves comparable performance because of the ability of
quaternion to handle images better. These results motivate us to further look
into similar model based on another such hyper-complex space, octonion
algebra, an extension of quaternion algebra. Also, improvements can be made by
considering video as part of posts. Further we are working on developing Qu-
BERT architecture for multi-modal fusion.

## References

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$\textstyle{\Phi^{C_{p}}\operatorname{THH}^{(p^{n})}(A;|\operatorname{Bar}_{\bullet}(M;B;L)|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Phi^{C_{p}}\sigma^{n}_{A,B}}$$\textstyle{\Phi^{C_{p}}\operatorname{THH}^{(p^{n})}(B;|\operatorname{Bar}_{\bullet}(L;A;M)|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{THH}^{(p^{n-1})}(A;|\operatorname{Bar}_{\bullet}(M;B;L)|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{n-1}_{A,B}}$$\textstyle{\operatorname{THH}^{(p^{n-1})}(B;|\operatorname{Bar}_{\bullet}(L;A;M)|)}$

It follows that the isomorphisms $\tau^{n}_{A,B}$ respect the restriction
maps, and therefore induce equivalences

$\theta_{A,B}:\textnormal{TR}(A;|\operatorname{Bar}_{\bullet}(M;B;L)|)\to\textnormal{TR}(B;|\operatorname{Bar}_{\bullet}(L;A;M)|)$

as required. ∎

In particular, for every $k$, $\pi_{k}\textnormal{TR}(R;-)$ extends to a
shadow on the bicategory $\scr{R}_{\mathsf{S}p}$, taking values in
$\mathsf{Ab}$.

######  Proposition 5.2.3.

The Eilenberg–MacLane spectrum functor $H:s\mathsf{Ab}\to\mathsf{Sp}$ defines
a lax shadow functor between the shadows $\pi_{k}\textnormal{TR}$ and
$tr_{k}$. In particular, for any ring $A$ there is a natural linearization map

$\pi_{k}\textnormal{TR}(HA)\to tr_{k}(A).$

###### Proof.

By Proposition 5.1.6, the equivariant Eilenberg–MacLane spectrum functor
defines a lax shadow functor
$\pi_{k}\operatorname{THH}_{C_{p^{n}}}(-;-)^{C_{p^{n}}}\to\underline{\operatorname{HH}}^{C_{p^{n}}}(-;-)_{k}(C_{p^{n}}/C_{p^{n}})$.
Precomposing with the norm $N^{C_{p^{n}}}_{e}:s\mathsf{Ab}\to
s\mathsf{Mack}_{C_{p^{n}}}$, we obtain a lax shadow functor

$\pi_{k}\operatorname{THH}_{C_{p^{n}}}(HN^{C_{p^{n}}}_{e}A;HN^{C_{p^{n}}}_{e}M)^{C_{p^{n}}}\to\underline{\operatorname{HH}}_{k}^{C_{p^{n}}}(N^{C_{p^{n}}}_{e}A;N^{C_{p^{n}}}_{e}M)(C_{p^{n}}/C_{p^{n}})$

for each simplicial ring $A$ and bimodule $M$. For a Mackey functor
$\underline{M}$,
$N_{e}^{C_{p^{n}}}\underline{M}=\underline{\pi}_{0}^{C_{p^{n}}}(N_{e}^{C_{p^{n}}}H\underline{M})$.
If we denote $N_{e}^{C_{p^{n}}}H\underline{M}$ by $R$, the linearization map
$R\to H\underline{\pi}_{0}^{C_{p^{n}}}(R)$ therefore results in a natural
transformation $N^{C_{p^{n}}}_{e}H\to HN^{C_{p^{n}}}_{e}$. Thus we obtain a
lax shadow functor

$\pi_{k}\operatorname{THH}_{C_{p^{n}}}(N^{C_{p^{n}}}_{e}HA;N^{C_{p^{n}}}_{e}HM)^{C_{p^{n}}}\to\underline{\operatorname{HH}}_{k}^{C_{p^{n}}}(N^{C_{p^{n}}}_{e}A;N^{C_{p^{n}}}_{e}M)(C_{p^{n}}/C_{p^{n}}),$

or, simplifying notation,

$\pi_{k}\operatorname{THH}^{(p^{n})}(HA;HM)^{C_{p^{n}}}\to\underline{\operatorname{HH}}_{e}^{C_{p^{n}}}(A;M)_{k}(C_{p^{n}}/C_{p^{n}}).$

Since these maps are compatible with the restriction maps defined on both
sides, there is an induced lax shadow functor

$\pi_{k}\operatorname*{holim}_{\begin{subarray}{c}\longleftarrow\\\
R\end{subarray}}\operatorname{THH}^{(p^{n})}(HA;HM)^{C_{p^{n}}}\to\lim_{\begin{subarray}{c}\longleftarrow\\\
R\end{subarray}}\pi_{k}\operatorname{THH}^{(p^{n})}(HA;HM)^{C_{p^{n}}}\to\lim_{\begin{subarray}{c}\longleftarrow\\\
r\end{subarray}}\underline{\operatorname{HH}}_{e}^{C_{p^{n}}}(A;M)_{k}(C_{p^{n}}/C_{p^{n}}),$

i.e., a lax shadow functor

$\pi_{k}\textnormal{TR}(HA;HM)\to tr_{k}(A,M)$

as desired. ∎

It follows from the proposition above that there is a linearization map
relating TR and its algebraic analogue, $tr$, for any $(-1)$-connected ring
spectrum.

######  Corollary 5.2.4.

There is a linearization map

$\pi_{k}\textnormal{TR}(R)\to tr_{k}(\pi_{0}(R))$

for $(-1)$-connected ring spectra $R$.

###### Proof.

For $R$ a $(-1)$-connected ring spectrum, there is a map of ring spectra $R\to
H\pi_{0}(R)$. We then compose with the lax shadow functor of Proposition 5.2.3
to obtain

$\pi_{k}\textnormal{TR}(R)\to\pi_{k}\textnormal{TR}(H\pi_{0}(R))\to
tr_{k}(\pi_{0}(R)).$

∎

In [10], the authors compute the algebraic $tr$ of $\mathbb{F}_{p}$, proving
that

$tr_{k}(\mathbb{F}_{p};p)=\begin{cases}\mathbb{Z}_{p}&:k=0\\\
0&:\text{otherwise},\end{cases}$

which agrees with $\pi_{k}(\textnormal{TR}(\mathbb{F}_{p};p))$. In general,
one would like to understand how closely algebraic restriction homology
approximates the analogous topological theory, TR. The highly structured
linearization map we construct in Proposition 5.2.3 is useful in pursuing this
question. Below, we prove that the linearization map is often an isomorphism
in degree 0.

######  Proposition 5.2.5.

Let $A$ be a ring, and let $R^{1}\lim$ denote the first right derived functor
of the limit functor on abelian groups. If

$R^{1}\lim_{\begin{subarray}{c}\longleftarrow\\\
R\end{subarray}}\pi_{1}\operatorname{THH}^{(p^{n})}(HA)^{C_{p^{n}}}=0,$

then

$\pi_{0}\textnormal{TR}(HA)\cong tr_{0}(A).$

###### Proof.

For a ring $A$, by Theorem 5.1 of [10], the map

$\underline{\pi}_{0}^{C_{p^{n}}}\operatorname{THH}_{C_{p^{n}}}(N^{C_{p^{n}}}_{e}HA)\to\underline{\operatorname{HH}}^{C_{p^{n}}}(N^{C_{p^{n}}}_{e}A)_{0}$

is an isomorphism. Therefore the map

$\lim_{\begin{subarray}{c}\longleftarrow\\\
R\end{subarray}}\pi_{0}\operatorname{THH}^{(p^{n})}(HA)^{C_{p^{n}}}\to\lim_{\begin{subarray}{c}\longleftarrow\\\
r\end{subarray}}\underline{\operatorname{HH}}_{e}^{C_{p^{n}}}(A)_{0}(C_{p^{n}}/C_{p^{n}})$

is an isomorphism. If

$R^{1}\lim_{\begin{subarray}{c}\longleftarrow\\\
R\end{subarray}}\pi_{1}\operatorname{THH}^{(p^{n})}(HA)^{C_{p^{n}}}=0,$

then

$\pi_{0}\textnormal{TR}(HA)=\pi_{0}\operatorname*{holim}_{\begin{subarray}{c}\longleftarrow\\\
R\end{subarray}}\operatorname{THH}^{(p^{n})}(HA)^{C_{p^{n}}}=\lim_{\begin{subarray}{c}\longleftarrow\\\
R\end{subarray}}\pi_{0}\operatorname{THH}^{(p^{n})}(HA)^{C_{p^{n}}},$

and the conclusion follows. ∎

Proposition 5.2.5 is analogous to the classical result that the linearization
map

$\pi_{k}(\operatorname{THH}(A))\to\operatorname{HH}_{k}(A)$

is an isomorphism in degree 0. Classically, the linearization map on THH is
also an isomorphism in degree 1. It would be interesting to consider under
what conditions the linearization map on TR is similarly an isomorphism in
degree 1, although we will not pursue that here.

##  6\. Generalized Dennis traces

In this section, we show that there is a Dennis trace map whose target is
$\operatorname{THH}_{C_{n}}(R)$. Furthermore, we show that this map factors
through $\textnormal{TR}_{C_{n}}(R)$.

###  6.1. The twisted Dennis trace.

Let $(\mathsf{V},\otimes,S)$ be a symmetric monoidal, simplicial model
category. As in section 4, let
$(\mathsf{V}_{\mathrm{aut}},\otimes,(S,\textup{Id}_{S}))$ denote the symmetric
monoidal category with objects $(V,\varphi_{V})$, where $V$ is an object of
$\mathsf{V}$, and $\varphi_{V}$ is an automorphism of $V$, while morphisms in
$\mathsf{V}_{\mathrm{aut}}$ are morphisms in $\mathsf{V}$ that commute with
the automorphism.

Let $(R,\varphi)$ be a monoid in $\mathsf{V}_{\mathrm{aut}}$, and let
$\mathsf{Perf}_{(R,\varphi)}$ denote its category of perfect left modules in
$\mathsf{V}_{\mathrm{aut}}$. The objects in this category are perfect
$R$-modules $P$ equipped with an automorphism $\gamma:P\to P$ such that the
following diagram commutes:

$\textstyle{R\otimes
P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{act}$$\scriptstyle{\varphi\otimes\gamma}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{R\otimes
P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{act}$$\textstyle{P.}$

That is, $\gamma$ is an isomorphism of left modules from $P$ to
${}^{\varphi}P$. Here ${}^{\varphi}P$ indicates that the left action on $P$ is
twisted by $\varphi:R\to R$, as in Definition 4.1.3. Let
$\textnormal{K}(R,\varphi)$ denote
$\textnormal{K}(\mathsf{Perf}_{(R,\varphi)})$.

Let $M$ be an $R$-bimodule. Consider the category of $M$-parametrized
endomorphisms of perfect $R$-modules, denoted by Betley [9] as
$\mathsf{End}(R,M)$. Its objects are maps of left modules $\gamma:P\to
M\otimes_{R}P$, where $P$ is a perfect left $R$-module, and its morphisms are
maps of left modules $P\to Q$ that make the relevant diagram commute.

######  Theorem 6.1.1.

There is a trace map
$\textnormal{K}(R,\varphi)\to\operatorname{HH}(R,{}^{\varphi}R)$, given by the
composite

$\textnormal{K}(\mathsf{Perf}_{(R,\varphi)})=\textnormal{K}(\mathsf{Aut}(R,{}^{\varphi}R))\to\textnormal{K}(\mathsf{End}(R,{}^{\varphi}R))\to\operatorname{HH}(R,{}^{\varphi}R).$

When $R$ is a $C_{n}$-ring spectrum and $\varphi$ is given by the action of a
chosen generator $g\in C_{n}$, we obtain a trace map

$\textnormal{K}(\mathsf{Perf}_{(R,g)})\to\operatorname{THH}_{C_{n}}(R).$

###### Proof.

Observe that $\mathsf{End}(R,{}^{\varphi}R)$ is very similar to
$\mathsf{Perf}_{(R,\varphi)}$, except that in $\mathsf{Perf}_{(R,\varphi)}$,
$\gamma$ is required to be an isomorphism. We may therefore denote
$\mathsf{Perf}_{(R,\varphi)}$ by $\mathsf{Aut}(R,{}^{\varphi}R)$, and it
includes naturally into the category $\mathsf{End}(R,{}^{\varphi}R)$.

The K-theory of $\mathsf{End}(R,M)$, denoted $\textnormal{K}(R,M)$, admits a
trace map to $\operatorname{HH}(R,M)$ (see, e.g., Section 3 of [16]), which
can be considered as a twisted version of the trace
$\textnormal{K}(\mathsf{End}(R,R))\to\operatorname{HH}(R)$, through which the
Dennis trace $\textnormal{K}(R)\to\operatorname{HH}(R)$ factors. We consider
this trace map when $M={}^{\varphi}R$. The composite of this trace with the
map from $\textnormal{K}(\mathsf{Perf}_{(R,\varphi)})$ to
$\textnormal{K}(\mathsf{End}(R,{}^{\varphi}R))$ yields a twisted Dennis trace
from $\textnormal{K}(R,\varphi)$ to $\operatorname{HH}(R,{}^{\varphi}R).$

When $R$ is a $C_{n}$-ring spectrum, and $\varphi$ is given by the action of a
chosen generator $g\in C_{n}$, we therefore obtain a trace map

$\textnormal{K}(\mathsf{Perf}_{(R,g)})\to\operatorname{THH}(R;{}^{g}R)\cong\operatorname{THH}_{C_{n}}(R).$

∎

If $R$ is a $C_{n}$-ring spectrum, let $\mathsf{Perf}^{C_{n}}_{R}$ denote the
category of perfect left modules over $R$ in $C_{n}$-spectra. Objects in this
category are modules over $R$ in $C_{n}$-spectra that are perfect as
$R$-modules.

######  Corollary 6.1.2.

There is a trace map
$\textnormal{K}(\mathsf{Perf}^{C_{n}}_{R})\to\operatorname{THH}_{C_{n}}(R)$.

###### Proof.

Let $g$ denote a chosen generator of $C_{n}$. We can include
$\mathsf{Perf}^{C_{n}}_{R}$ into $\mathsf{Perf}_{(R,g)}$, by sending a module
$P$ to $(P,g)\in\mathsf{Perf}_{(R,g)}$. This is a twisted version of the
functor $\mathsf{Perf}_{R}\to\mathsf{End}(R,R)$ that sends $P$ to $(P,id)$ and
induces the map $\textnormal{K}(R)\to\textnormal{K}(\mathsf{End}(R,R))$. We
thus obtain a trace map

$\textnormal{K}(\mathsf{Perf}^{C_{n}}_{R})\to\operatorname{THH}_{C_{n}}(R)$

given by the composite

$\textnormal{K}(\mathsf{Perf}^{C_{n}}_{R})\to\textnormal{K}(\mathsf{Perf}_{(R,g)})=\textnormal{K}(\mathsf{Aut}(R,{}^{g}R))\to\textnormal{K}(\mathsf{End}(R,{}^{g}R))\to\operatorname{THH}(R;{}^{g}R).$

∎

######  Remark 6.1.3.

When $R$ is an ordinary ring, we can identify the trace
$\textnormal{K}_{0}(R,\varphi)\to\operatorname{HH}_{0}(R;{}^{\varphi}R)$ as
sending $\gamma:P\to{}^{\varphi}P$ to its Hattori-Stallings trace, i.e., its
bicategorical trace in the category of rings, bimodules, and bimodule maps.
Consider the shadow $\operatorname{HH}_{0}$ on this bicategory. The trace of a
2-cell $\gamma:P\otimes\mathbb{Z}\to{}^{\varphi}R\otimes_{R}P$, where we
regard $P$ as an $R-\mathbb{Z}$ bimodule, is a map

$\operatorname{HH}_{0}(\mathbb{Z})\to\operatorname{HH}_{0}(R;{}^{\varphi}R)$

and thus an element of $\operatorname{HH}_{0}(R;{}^{\varphi}R)$. For more
details on the Hattori-Stallings trace as a bicategorical trace, see Section 5
of [33]. More generally, Theorem 1.2 of [13] states that on $\pi_{0}$, this
twisted Dennis trace takes an endomorphism $f:P\to{}^{g}P$ to its
bicategorical trace
$\mathbb{S}\to\operatorname{THH}(R;{}^{g}R)\cong\operatorname{THH}_{C_{n}}(R)$.

This twisted Dennis trace factors through the $C_{n}$-twisted TR of [4],
$\textnormal{TR}_{C_{n}}$. The following lemma will be useful in the proof.

######  Lemma 6.1.4.

Let $R$ be a $C_{n}$-ring spectrum, and $g\in C_{n}$ a generator. Let $p$ be a
prime that does not divide $n$, and $k$ an integer such that $p^{k}\equiv 1$
(mod $n$). Then

$\operatorname{THH}^{(p^{k})}(R;{}^{g}R)\simeq\operatorname{THH}_{C_{n}}(R)$

as $C_{p^{k}}$-spectra.

###### Proof.

Recall from [4] that
$\operatorname{THH}_{C_{n}}(R)\simeq\operatorname{THH}_{C_{np^{k}}}(N_{C_{n}}^{C_{np^{k}}}R)$
as $S^{1}$-spectra. Since $p$ does not divide $n$, $C_{np^{k}}\cong
C_{n}\times C_{p^{k}}$; let $g^{\prime}$ denote a generator of $C_{p^{k}}$, so
that $g^{\prime}g$ is a generator of $C_{np^{k}}$. As $C_{np^{k}}$-spectra,

$\operatorname{THH}_{C_{np^{k}}}(N_{C_{n}}^{C_{np^{k}}}R)\simeq\operatorname{THH}(N_{C_{n}}^{C_{np^{k}}}R;{}^{g^{\prime}g}N_{C_{n}}^{C_{np^{k}}}R)\simeq\operatorname{THH}(N_{C_{n}}^{C_{np^{k}}}R;{}^{g^{\prime}g^{p^{k}}}N_{C_{n}}^{C_{np^{k}}}R).$

The second equivalence holds because $p^{k}\equiv 1$ (mod $n$). Note that the
last term is equivalent to
$\operatorname{THH}(N_{C_{n}}^{C_{np^{k}}}R;{}^{g^{\prime}}N_{C_{n}}^{C_{np^{k}}}({}^{g}R))$
as a $C_{np^{k}}$-spectrum. We now restrict to $C_{p^{k}}$-spectra. The
restriction functor respects smash products and geometric realization, and
therefore commutes with $\operatorname{THH}$. In addition, if $K,H$ are finite
groups and $X$ is a $K$-spectrum, examining the indexed smash product defining
the norm yields $\iota_{e\times H}N^{K\times H}_{K\times e}X\cong
N_{e}^{H}\iota_{e}X$ as $H$-spectra. Therefore we obtain

$\iota_{C_{p^{k}}}\operatorname{THH}(N_{C_{n}}^{C_{np^{k}}}R;{}^{g^{\prime}}N_{C_{n}}^{C_{np^{k}}}({}^{g}R))\simeq\operatorname{THH}(N_{e}^{C_{p^{k}}}\iota_{e}R;{}^{g^{\prime}}N_{e}^{C_{p^{k}}}\iota_{e}({}^{g}R)),$

which is equivalent as a $C_{p^{k}}$-spectrum to

$\operatorname{THH}_{C_{p^{k}}}(N_{e}^{C_{p^{k}}}\iota_{e}R;N_{e}^{C_{p^{k}}}\iota_{e}({}^{g}R))=\operatorname{THH}^{(p^{k})}(R;{}^{g}R)$

as required. ∎

######  Proposition 6.1.5.

The twisted Dennis trace
$\textnormal{K}(\mathsf{Perf}_{(R,g)})\to\operatorname{THH}_{C_{n}}(R)$
factors through $\textnormal{TR}_{C_{n}}(R;p)$ for $p$ coprime to $n$.

###### Proof.

By Theorem 1.3 of [13], the twisted Dennis trace
$\textnormal{K}(R,{}^{g}R)\to\operatorname{THH}(R,{}^{g}R)$ factors through TR
with coefficients, $\textnormal{K}(R,{}^{g}R)\to\textnormal{TR}(R,{}^{g}R)$.
On $\pi_{0}$, this takes an endomorphism $f:P\to{}^{g}P$ to the bicategorical
trace of its $k$-fold composite $f^{k}:P\to{}^{g^{k}}P$ for all
$k\in\mathbb{N}$.

We claim that when $p$ is coprime to $n$,
$\textnormal{TR}(R,{}^{g}R;p)\simeq\textnormal{TR}_{C_{n}}(R;p)$. We prove
this by comparing the inverse systems that form both spectra. Recall that
$\textnormal{TR}(R,{}^{g}R;p)$ is the inverse limit along the restriction maps

$R:\operatorname{THH}^{(p^{k})}(R;{}^{g}R)^{C_{p^{k}}}\to\operatorname{THH}^{(p^{k-1})}(R;{}^{g}R)^{C_{p^{k-1}}}.$

Note that by Fermat’s little theorem, for $p$ coprime to $n$,
$p^{\varphi(n)}\equiv 1$ (mod $n$), where $\varphi(n)$ denotes Euler’s totient
function. We can therefore look instead at the composition of $\varphi(n)$
restriction maps at a time,

$\operatorname{THH}^{(p^{j\varphi(n)})}(R;{}^{g}R)^{C_{p^{j\varphi(n)}}}\to\operatorname{THH}^{(p^{(j-1)\varphi(n)})}(R;{}^{g}R)^{C_{p^{(j-1)\varphi(n)}}}.$

By Lemma 6.1.4 above, this agrees with the system

$\operatorname{THH}_{C_{n}}(R)^{C_{p^{j\varphi(n)}}}\to\operatorname{THH}_{C_{n}}(R)^{C_{p^{(j-1)\varphi(n)}}}$

and the limit of this system agrees with the limit of the system

$\operatorname{THH}_{C_{n}}(R)^{C_{p^{k+1}}}\to\operatorname{THH}_{C_{n}}(R)^{C_{p^{k}}}$

which is $\textnormal{TR}_{C_{n}}(R;p)$. ∎

######  Remark 6.1.6.

This suggests that $\textnormal{TR}(R,{}^{g}R)$ is a good generalization for
$\textnormal{TR}_{C_{n}}(R)$, which also works at primes that divide $n$.

###  6.2. Relationship to equivariant algebraic K-theory

We have shown above that there is a twisted Dennis trace map

$\textnormal{K}(\mathsf{Perf}_{(R,g)})\to\operatorname{THH}_{C_{n}}(R)$

that factors through $\textnormal{TR}_{C_{n}}(R)$. A natural question to ask
is whether twisted topological Hochschild homology also receives a trace map
from equivariant algebraic $K$-theory. Merling [28] defined a genuine
equivariant algebraic K-theory $G$-spectrum, $\textnormal{K}_{G}(R)$, for a
$G$-ring $R$. There is related work on equivariant algebraic $K$-theory due to
Barwick and Barwick-Glasman-Shah [5, 6] and Malkiewich–Merling [26].

The equivariant algebraic $K$-theory $\textnormal{K}_{G}(R)$ of a $G$-ring $R$
is roughly the K-theory of the category of perfect $R$-modules, which admits
an action of $G$ sending $P$ to ${}^{g}P$. Building on Merling’s work [28],
Malkiewich and Merling showed in Theorem 1.3 of [26] that for a $G$-ring
spectrum $R$, the fixed point spectrum $\textnormal{K}_{G}(R)^{G}$ is
equivalent to the $K$-theory of a category of perfect $R$-modules on which $G$
acts semilinearly. In particular, if $G$ is $C_{n}$ with generator $g$, then
$\textnormal{K}_{C_{n}}(R)^{C_{n}}\simeq\textnormal{K}(\mathsf{Perf}_{(R,g)})$.
Therefore we conclude the following as a corollary of Theorem 6.1.1.

######  Corollary 6.2.1.

For $R$ a $C_{n}$-ring spectrum, the twisted Dennis trace is a map

$\textnormal{K}_{C_{n}}(R)^{C_{n}}\to\operatorname{THH}_{C_{n}}(R).$

Work of Horev [20] shows that twisted topological Hochschild homology can be
viewed through the lens of equivariant factorization homology. In particular,
Horev proves in Proposition 7.2.2 of [20] that for a $C_{n}$-ring spectrum
$R$,

$\operatorname{THH}_{C_{n}}(R)\simeq\Phi^{C_{n}}\int_{S^{1}_{rot}}R.$

Here $\int_{S^{1}_{rot}}R$ denotes the $C_{n}$-equivariant factorization
homology of $S^{1}$ (with the rotation action of $C_{n}$) with coefficients in
$R$. The $C_{n}$-spectrum $\int_{S^{1}_{rot}}R$ is non-equivariantly
equivalent to $\operatorname{THH}(R)$, but has a “diagonal” action of $C_{n}$–
that is, $C_{n}$ acts on both $S^{1}$ and $R$. This perspective on twisted
topological Hochschild homology leads to the following conjecture.

######  Conjecture 6.2.2.

The twisted Dennis trace of Corollary 6.2.1 arises from a $C_{n}$-equivariant
map $\textnormal{K}_{C_{n}}(R)\to\int_{S^{1}_{rot}}R$.

Work in progress of Angelini-Knoll, Merling, and Péroux [3] may shed light on
this conjecture, as they expand the definition of norms for compact Lie
groups. Take $R$ a $C_{n}$-ring spectrum; using such norms, one would like to
construct a Dennis trace map $\textnormal{K}_{C_{n}}(R)\to N_{e\times
C_{n}}^{S^{1}\times C_{n}}R$ which is $S^{1}\times C_{n}$ equivariant. Here we
extend the $C_{n}$-action on $K_{C_{n}}(R)$ to an $S^{1}\times C_{n}$ action
using the trivial $S^{1}$-action. Let $H\cong C_{n}$ denote the diagonal
subgroup of $C_{n}\times C_{n}\leq S^{1}\times C_{n}$. An equivariant Dennis
trace of this form would produce a map

$\textnormal{K}_{C_{n}}(R)^{C_{n}}=\textnormal{K}_{C_{n}}(R)^{H}\to(N_{e\times
C_{n}}^{S^{1}\times C_{n}}R)^{H}\to\Phi^{H}N_{e\times C_{n}}^{S^{1}\times
C_{n}}R.$

If the diagonal formulas of Proposition 2.19 of [4] still hold for this
generalized norm, so that

$\Phi^{H}N_{e\times C_{n}}^{S^{1}\times C_{n}}R\cong N_{e\times C_{n}/(e\times
C_{n}\cap H)}^{S^{1}\times C_{n}/H}\Phi^{e\times C_{n}\cap H}R\cong
N_{C_{n}}^{S^{1}}R,$

then a twisted Dennis trace

$\textnormal{K}_{C_{n}}(R)^{C_{n}}\to\operatorname{THH}_{C_{n}}(R)$

would indeed arise from an equivariant Dennis trace

$\textnormal{K}_{C_{n}}(R)\to N_{e\times C_{n}}^{S^{1}\times C_{n}}R.$

This norm, $N_{e\times C_{n}}^{S^{1}\times C_{n}}R$, can be thought of as
$\operatorname{THH}(R)$ taken in the category of $C_{n}$-spectra. That is, it
has a $C_{n}$-action coming from the fact that $R$ is a $C_{n}$-ring spectrum,
and an $S^{1}$-action coming from the cyclic bar construction. The restriction
of $N_{e\times C_{n}}^{S^{1}\times C_{n}}R$ to a $C_{n}$-spectrum (via the
diagonal subgroup $H$) should agree with $\int_{S^{1}_{rot}}R$.

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