# Quantum geometry and bounds on dissipation in slowly driven quantum systems

Iliya Esin  Department of Physics and Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena, California 91125, USA
Étienne Lantagne-Hurtubise  Department of Physics and Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena,
California 91125, USA  Frederik Nathan  Department of Physics and Institute
for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA  Center for Quantum Devices, Niels Bohr
Institute, University of Copenhagen, 2100 Copenhagen, Denmark  Gil Refael
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA

###### Abstract

We show that heat production in slowly driven quantum systems is linked to the
topological structure of the driving protocol through the Fubini-Study tensor.
Analyzing a minimal model of a spin weakly coupled to a heat bath, we find
that dissipation is controlled by the quantum metric and a “quality factor”
characterizing the spin’s precession. Utilizing these findings, we establish
lower bounds on the heating rate in two-tone protocols, such as those employed
in topological frequency converters. Notably, these bounds are determined by
the topology of the protocol, independent of its microscopic details. Our
results bridge topological phenomena and energy dissipation in slowly driven
quantum systems, providing a design principle for optimal driving protocols.

Heating is a ubiquitous non-equilibrium phenomenon that influences a broad
range of systems, including quantum computation platforms, semiconductor
devices, and mesoscopic setups. In closed, driven quantum many-body systems,
heating manifests through the growth of entanglement entropy, resulting in
volume-law entangled states that are akin to infinite temperature states in
equilibrium D’Alessio and Rigol (2014); Lazarides  _et al._ (2014); Kaufman
_et al._ (2016). Typically lacking distinctive features, these states are
generally considered undesirable.

Heating can be suppressed in systems possessing a macroscopic number of local
integrals of motion, such as in the many-body localized phase Lazarides  _et
al._ (2015); Ponte  _et al._ (2015); Nandkishore and Huse (2015); Abanin  _et
al._ (2016); Altman (2018); Abanin  _et al._ (2019), or prepared in a
dynamically obstructed state characterized by quantum many-body scar Lin  _et
al._ (2020); Serbyn  _et al._ (2021); Moudgalya  _et al._ (2022); Chandran
_et al._ (2023). On the other hand, certain types of drives can give rise to
long-lived quasi-steady states, both in high and low frequency regimes Abanin
_et al._ (2015, 2017a, 2017b); Lindner  _et al._ (2017); Esin  _et al._
(2022); Ho  _et al._ (2023). In the quasi-adiabatic regime, heating occurs due
to diabatic corrections Polkovnikov  _et al._ (2011); Kolodrubetz  _et al._
(2017) which can be mitigated by incorporating counter-diabatic terms that
counteract excitations del Campo (2013); Sels and Polkovnikov (2017); Claeys
_et al._ (2019) and through drive-engineering techniques known as shortcuts to
adiabaticity Chen  _et al._ (2010); Guéry-Odelin  _et al._ (2019).

A different approach consists of coupling the system to a low-temperature heat
bath, maintaining the system in a low-entropy steady state. Such steady states
can have exotic features with no equilibrium analogues Iadecola  _et al._
(2015); Dehghani  _et al._ (2015); Seetharam  _et al._ (2015); Rudner and Song
(2019); Sato  _et al._ (2019); Esin  _et al._ (2020); Nuske  _et al._ (2020);
Esin  _et al._ (2021). To stabilize a low-entropy steady state, the bath must
be capable of continuously absorbing the heat generated by the drive.
Therefore, controlling heat production is crucial to maintain its constant
flow. In this work, our primary objective is to evaluate the steady-state
energy flow from a slowly-driven system to the heat bath.

Figure 1: The setup. A spin driven by two fields $\bm{\alpha}_{1}(t)$ and
$\bm{\alpha}_{2}(t)$, slowly oscillating with frequencies $\omega_{1}$ and
$\omega_{2}$, and weakly-coupled to a heat bath (represented by the blue
halo). The black and red curves denote the trajectories of the Bloch-sphere
vector $\hat{\mathbf{d}}(t)$ representing the instantaneous Hamiltonian and
the steady-state vector $\mathbf{S}_{\rm st}(t)$ of the spin. The
instantaneous coordinate system consists of $\hat{\mathbf{d}}$, its normalized
time derivative $\hat{\bm{f}}=\dot{\hat{\mathbf{d}}}/|\dot{\hat{\mathbf{d}}}|$
and $\hat{\mathbf{e}}=\hat{\bm{f}}\times\hat{\mathbf{d}}$. To leading order,
nonadiabatic corrections to the dynamics create an offset of the spin in the
$\hat{\bm{e}}$ direction which gives rise to energy pumping between
$\bm{\alpha}_{1}$ and $\bm{\alpha}_{2}$ with rate $W_{c}$. The lag between the
spin vector and the Hamiltonian (along the $\hat{\bm{f}}$ direction) induced
by the heat bath leads to energy dissipation with rate $W_{d}$.

In the quasi-adiabatic regime, time evolution is governed by the quantum
geometry of the wavefunction Xiao  _et al._ (2010); Avron  _et al._ (2010,
2011, 2012); Kolodrubetz  _et al._ (2013); Ludovico  _et al._ (2016); Bleu
_et al._ (2018); Bukov  _et al._ (2019); Bhandari  _et al._ (2020); Morimoto
_et al._ (2023). The corresponding heating rate can be found by evaluating an
action along a trajectory in parameter space controlled by the “dissipation
metric” Sivak and Crooks (2012); Zulkowski and DeWeese (2015); Scandi and
Perarnau-Llobet (2019); Abiuso  _et al._ (2020); Bhandari  _et al._ (2020);
Loutchko  _et al._ (2022). Here, we establish a connection between the
dissipation metric and the quantum metric Provost and Vallee (1980). We then
exploit inequality relations between the quantum metric and Berry curvature
Provost and Vallee (1980); Ozawa and Mera (2021); Graf and Piéchon (2021) to
establish a “topological” bound on heating rates in two-tone protocols Martin
_et al._ (2017). While heating is affected by the details of the driving
protocol, its lower bound depends only on the topological features of the
protocol and the quality factor of the system-bath coupling.

## Setup.—

We consider a two-level system subject to quasi-adiabatic driving (see a
generalization to $N$ levels in the Supplementary Material 100100100See
Supplemental Material for details.), described by

$H(t)=h_{0}(t)+\mathbf{d}(t)\cdot\bm{\sigma},$ (1)

where $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is a vector of Pauli
matrices spanning the Hilbert space 111The Hamiltonian can depend on
additional (time-independent) parameters, such as crystal momentum in
solids—our analysis generalizes straightforwardly to such cases.. The
Hamiltonian can be diagonalized at any time $t$, defining the instantaneous
eigenbasis spanned by states $\ket{\psi_{\pm}(t)}$ with energies $E_{\pm}(t)$.

The two-level system is then coupled to a macroscopic thermal bath Leggett
_et al._ (1987). We consider the case of weak coupling to the bath, where the
dynamics of the system are effectively Markovian Alicki (1979); Sarandy and
Lidar (2005a); Breuer  _et al._ (2007); Avron  _et al._ (2011); Mozgunov and
Lidar (2020); Nathan and Rudner (2020) and described by a master equation of
the form (taking $\hbar=1$ throughout)

$\dot{\rho}(t)=-i[H(t),\rho(t)]+{\cal D}\\{\rho(t)\\}.$ (2)

Here, ${\cal D}$ is a dissipator that depends on the details of the system,
the thermal bath, and the system-bath coupling. We can rewrite Eq. (2) in the
form of a Bloch equation Bloch (1946) for the vector ${\bf{S}}(t)={\rm
Tr}[\rho(t)\cdot\bm{\sigma}]$,

$\dot{{\bf{S}}}(t)=2\mathbf{d}(t)\times{\bf{S}}(t)-\Gamma(t)[{\bf{S}}(t)-{\bf{S}}_{0}(t)].$
(3)

The matrix $\Gamma$ controls relaxation and is obtained from the dissipator
through $\Gamma_{ij}=-\frac{1}{2}{\rm Tr}\left[\sigma_{i}{\cal
D}\\{\sigma_{j}\\}\right]$, and $\Gamma\mathbf{S}_{0}=\frac{1}{2}{\rm
Tr}\left[{\bm{\sigma}}\cdot{\cal D}\\{\bf 1\\}\right]$ 222The existence of
${\bf{S}}_{0}$ requires $\Gamma$ to be invertible. This is the case whenever
$\tau_{1}$ is finite, due to the inequality $\tau_{2}\leq 2\tau_{1}$ which
occurs in generic situations.

We consider the generic class of dissipators that are isotropic with respect
to $H(t)$. In this case, the most general form of the relaxation matrix reads

$\Gamma_{ij}=\frac{1}{\tau_{1}}\hat{\mathbf{d}}_{i}\hat{\mathbf{d}}_{j}+\frac{1}{\tau_{2}}(\delta_{ij}-\hat{\mathbf{d}}_{i}\hat{\mathbf{d}}_{j})+\delta\epsilon_{ijk}\hat{\mathbf{d}}_{k},$
(4)

where $\hat{\mathbf{d}}=\mathbf{d}/|\mathbf{d}|$ and $\tau_{1}$, $\tau_{2}$
are scalars that can be interpreted as longitudinal and transversal relaxation
times, with $\tau_{2}\leq 2\tau_{1}$ Levitt (2008). The last term in Eq. (4),
can be absorbed in Eq. (3) by rescaling the vector $\mathbf{d}(t)$—we
therefore set $\delta=0$ for simplicity. An isotropic dissipator gives rise to
$\mathbf{S}_{0}=s_{0}\hat{\mathbf{d}}$ for some scalar $s_{0}$. We consider
the case where the dissipator relaxes the system towards instantaneous thermal
equilibrium with the bath, implying $s_{0}=\tanh(\beta\Delta/2)$ with the
spectral gap $\Delta=2|\mathbf{d}|$ and bath temperature $T_{\rm B}=1/\beta$.
Working in the low-temperature limit, $\Delta\beta\gg 1$, we set $s_{0}=1$.

## Steady-state solution.—

An instantaneous steady-state solution to Eq. (3) can be found, up to leading
order in diabatic corrections, by performing a rotating frame transformation
100100footnotemark: 100,

$\mathbf{S}_{\rm
st}=\frac{\left(1+\Delta^{2}\tau_{2}^{2}\right)\hat{\mathbf{d}}+\tau_{2}\dot{\hat{\mathbf{d}}}+\Delta\tau_{2}^{2}\left(\hat{\mathbf{d}}\times\dot{\hat{\mathbf{d}}}\right)}{1+|\dot{\hat{\mathbf{d}}}|^{2}\tau_{1}\tau_{2}+\Delta^{2}\tau_{2}^{2}}+{\cal
O}\left(\frac{|\dot{\hat{\mathbf{d}}}|^{2}}{\Delta^{2}}\right).$ (5)

This result can be visualized with the help of Fig. 1. As the Hamiltonian
varies in time, the direction of the Bloch vector $\hat{\mathbf{d}}$ changes.
When driving is quasi-adiabatic the spin follows the trajectory of
$\hat{\mathbf{d}}(t)$ to remain close to the instantaneous ground state.
Dissipation to the heat bath causes the spin to lag behind the Hamiltonian by
the contribution $\propto\dot{\hat{\mathbf{d}}}$ in Eq. (5). There is also a
correction that pushes the spin in the direction perpendicular to
$\hat{\mathbf{d}}$ and $\dot{\hat{\mathbf{d}}}$, giving rise to Berry
curvature as discussed below.

## Energy dissipation in the steady state.—

The steady-state ${\bf{S}}_{\rm st}(t)$ of the system is characterized by a
balance between the work done by the drive and the heat dissipated to the
bath. The time dependence of the internal energy of the system reads
$W=\partial_{t}[\mathbf{d}\cdot{\bf{S}}_{\rm
st}]=\dot{\mathbf{d}}\cdot{\bf{S}}_{\rm st}+\mathbf{d}\cdot\dot{{\bf{S}}}_{\rm
st}$. The first term on the r.h.s describes the work done by the drive,
$W_{\rm w}=\langle\partial_{t}H(t)\rangle$. The second term is identified as
the rate of energy transfer due to dissipation, $-W_{\rm d}$. Indeed, $W_{\rm
d}=-\mathbf{d}\cdot\dot{{\bf{S}}}_{\rm st}=\mathbf{d}\cdot\Gamma[{\bf{S}}_{\rm
st}-{\bf{S}}_{0}]$ is controlled by the relaxation matrix, using Eq. (3) and
$\mathbf{d}\cdot(\mathbf{d}\times{\bf{S}}_{\rm st})=0$.

Using our steady-state expression [Eq. (5)], we compute the time-averaged
dissipation to leading diabatic order 100100footnotemark: 100. Because the
long-time average of $W$,
$\overline{W}\equiv\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dtW(t)$, vanishes
333This can be shown as follows:
$\overline{W}=\lim_{T\to\infty}\frac{1}{T}({\bf{d}}\cdot{\bf{S}}_{\rm
st}|_{t=T}-{\bf{d}}\cdot{\bf{S}}_{\rm st}|_{t=0})$ which vanishes because
$|{\bf{d}}\cdot{\bf{S}}_{\rm st}|$ is bounded., we find $\overline{W}_{\rm
d}=\overline{W}_{\rm w}$ ($\overline{W}_{\rm d,\rm w}$ are the corresponding
averages of $W_{\rm d,\rm w}$), such that

$\overline{W}_{\rm
d}=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\left[\frac{\frac{1}{2}\tau_{2}\Delta|\dot{\hat{\mathbf{d}}}|^{2}}{1+|\dot{\hat{\mathbf{d}}}|^{2}\tau_{1}\tau_{2}+\Delta^{2}\tau_{2}^{2}}\right].$
(6)

Assuming that $\tau_{1}$ and $\tau_{2}$ are comparable, Eq. (6) can be
simplified by neglecting the contribution $\sim|\dot{\hat{\mathbf{d}}}|^{2}$
in the denominator, yielding
$\overline{W}_{d}=\frac{1}{T}\int_{0}^{T}dt\frac{1}{4}\gamma|\dot{\hat{\mathbf{d}}}|^{2}$,
where $\gamma=\frac{2\tau_{2}\Delta}{1+\Delta^{2}\tau_{2}^{2}}$. In the limit
of weak relaxation $\Delta\tau_{2}\gg 1$, $\gamma$ is proportional to the
number of precession cycles before the pseudospin relaxes, and can thus be
understood as an effective quality factor. For Ohmic baths, which occur in
many realistic situations Breuer  _et al._ (2007), $\tau_{2}$ is inversely
proportional to $\Delta$, giving rise to approximately constant $\gamma$ over
a cycle.

## Two-parameter driving.—

We now specialize to the case where time evolution is generated by two drive
fields $\bm{\alpha}_{1}(t)$ and $\bm{\alpha}_{2}(t)$, such that
$\mathbf{d}(t)=\mathbf{d}(\bm{\alpha}(t))$, where
$\bm{\alpha}=\bm{\alpha}_{1}+\bm{\alpha}_{2}=(\alpha^{x},\alpha^{y},\alpha^{z})$.
This setting was recently discussed in the context of topological frequency
conversion, where classes of driving protocols result in quantized energy
transfer between two driving fields Martin  _et al._ (2017). The corresponding
averaged dissipation reads

$\overline{W}_{\rm d}=\frac{1}{T}\int_{0}^{T}dt\left[\gamma
G_{ij}\right]\dot{\alpha}^{i}\dot{\alpha}^{j},$ (7)

where
$G_{ij}=\frac{1}{4}(\partial_{\alpha^{i}}\hat{\mathbf{d}}\cdot\partial_{\alpha^{j}}\hat{\mathbf{d}})$
is the quantum metric in the parameter space describing the driving fields,
and we assume Einstein summation convention throughout.

Figure 2: Approaching the incommensurate limit. a. Trajectories on the surface
$\mathbf{s}=\frac{1}{2}\sqrt{\gamma}\hat{\mathbf{d}}$, for different values of
$\omega_{2}/\omega_{1}=\frac{n}{n+1}$. b. Normalized average dissipation rate
along the “Lissajous” curve, $\overline{\rm w}_{\rm d}=\overline{W}_{\rm
d}/W_{0}$, as a function of $n$. We use the driven spin model in Eq. 14 and
consider $b_{11}=b_{22}=1$, $b_{12}=b_{21}=0.5$, $\theta=0$, $\tau_{2}=10$,
and three values of $m$ indicated in the bottom-left inset. The top-right
inset shows the deviation of the dissipation rate from its asymptotic value,
$\delta\overline{\rm w}_{\rm d}=(\overline{\rm w}_{\rm d}(n)-\overline{\rm
w}_{\rm d}(\infty))/\overline{\rm w}_{\rm d}(\infty)$. c. Normalized frequency
conversion rate, ${\rm\overline{w}_{c}}=\overline{W}_{\rm c}/W_{0}$ as a
function of $n$ for the same parameters as in b. The inset shows the deviation
of $\overline{\rm w}_{\rm c}$ from its asymptotic value, $\delta\overline{\rm
w}_{\rm c}=\overline{\rm w}_{\rm c}(n)-\overline{\rm w}_{\rm c}(\infty)$.

Interestingly, Eq. (7) is given by the integral of a metric,
$\Lambda_{ij}=\gamma G_{ij}$, over the path traced by time evolution. This
result provides a connection to earlier works on quantum thermodynamics Sivak
and Crooks (2012); Zulkowski and DeWeese (2015); Scandi and Perarnau-Llobet
(2019); Abiuso  _et al._ (2020); Bhandari  _et al._ (2020); Loutchko  _et al._
(2022) that showed how such a geometric interpretation provides an operating
principle to minimize dissipation for a given task: the trajectory spanned by
the fields $\bm{\alpha}(t)$ should follow a geodesic at uniform speed on the
surface defined by $\Lambda_{ij}$. When $\gamma$ is nearly constant, this
surface is described by $\mathbf{s}=\frac{1}{2}\sqrt{\gamma}\hat{\mathbf{d}}$,
such that
$\Lambda_{ij}=\partial_{\alpha^{i}}\mathbf{s}\cdot\partial_{\alpha^{j}}\mathbf{s}$,
see Fig. 2a.

Alternatively, the quantum metric can be expressed as a function of the two
phases $\phi_{a}=\omega_{a}t$ describing the two drives (with $a=1,2$). In
this language the energy dissipation rate is given by

$\overline{W}_{\rm d}=\frac{\omega_{a}\omega_{b}}{T}\int_{0}^{T}dt\gamma
g_{ab},$ (8)

with the quantum metric in the space of the phases $\phi_{a}$,
$g_{ab}=G_{ij}\frac{\partial\alpha^{i}}{\partial{\phi_{a}}}\frac{\partial\alpha^{j}}{\partial{\phi_{b}}}$.
Similarly, the transferred power between modes $1$ and $2$ in the steady state
can be calculated through
$W_{12}=\frac{1}{2}(W_{1}-W_{2})\equiv\frac{1}{2}\left(\frac{\partial\mathbf{d}}{\partial\phi_{1}}\dot{\phi}_{1}-\frac{\partial\mathbf{d}}{\partial\phi_{2}}\dot{\phi}_{2}\right)\cdot\mathbf{S}_{\rm
st}$. This term is anti-symmetric in indices $1,2$, and thus does not
contribute to the net energy dissipation. Using the steady-state vector in Eq.
(5), one obtains two contributions, $W_{12}=W_{\rm ad}+W_{\rm c}$ where
$W_{\rm ad}=\gamma((\dot{\phi}_{1})^{2}g_{11}-(\dot{\phi}_{2})^{2}g_{22})$
denotes the difference in energy dissipation due to each drive, and the (time-
averaged) conversion power

$\overline{W}_{\rm
c}=\frac{\omega_{1}\omega_{2}}{T}\int_{0}^{T}dt\left[\frac{\tau_{2}^{2}\Delta^{2}}{1+\tau_{2}^{2}\Delta^{2}}\right]\Omega_{12},$
(9)

where
$\Omega_{12}=\frac{1}{2}\mathbf{d}\cdot\left(\partial_{\phi_{1}}\hat{\mathbf{d}}\times\partial_{\phi_{2}}\hat{\mathbf{d}}\right)$
is the Berry curvature associated with the phases of the two drives.

## Incommensurate driving.—

For two harmonic fields $\bm{\alpha}_{1}(t)$ and $\bm{\alpha}_{2}(t)$
oscillating with frequencies $\omega_{1}$ and $\omega_{2}$, respectively, the
trajectory of $\mathbf{S}_{\rm st}(t)$ on the Bloch sphere is a “Lissajous”
curve Nathan  _et al._ (2022). For a given ratio
$\omega_{2}/\omega_{1}=\frac{n}{n+1}$, the time to complete one cycle
increases with $n$ as $T=(n+1)\frac{2\pi}{\omega_{1}}$, see Fig. 2a. We
compute the dissipation and frequency conversion rates using Eqs. (8) and (9);
the data are plotted in Figs. 2b,c. Both quantities rapidly converge for
$n\gtrsim 5$ to their respective asymptotic values ($n\to\infty$)
corresponding to incommensurate driving.

When the drive frequencies $\omega_{1}$ and $\omega_{2}$ are incommensurate,
the system explores its entire phase space in the long-time limit. The long-
time averages in Eqs. (8) and (9) can then replaced by an average over the
phases $\phi_{a}=\omega_{a}t$,
$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\to\oiint\frac{d\phi_{1}d\phi_{2}}{4\pi^{2}}$.
In the limit of weak relaxation $\tau_{2}\Delta\gg 1$, the frequency
conversion rate has a clear topological character, $\overline{W}_{\rm
c}=\frac{\omega_{1}\omega_{2}}{2\pi}{\cal C}$, where

${\cal C}=\frac{1}{2\pi}\oiint d\phi_{1}d\phi_{2}\Omega_{12}$ (10)

is the corresponding Chern number, see Fig. 3a.

## Bounds on dissipation and topology.—

We now use the connection between energy dissipation and the quantum metric to
derive lower bounds on dissipation for topological drives. We begin with a
simple scenario in which the system and drives are in a symmetric
configuration 100100footnotemark: 100 with $\overline{g_{12}}\equiv\oiint
d\phi_{1}d\phi_{2}g_{12}=0$ 444Because we are deriving lower bounds on
dissipation, any driving protocol with $\oint g_{12}d\phi_{1}d\phi_{2}>0$ also
works similarly. This condition can equally be interpreted as a bound on
ensembles of two driving protocols where one of the frequencies is reversed,
e.g. $\omega_{1}$ and $-\omega_{1}$.. In this case, the time-averaged
dissipation arises only from the diagonal components of the quantum metric,
proportional to $\omega_{1}^{2}g_{11}+\omega_{2}^{2}g_{22}$. These two terms
can be bounded from below by $2\omega_{1}\omega_{2}\sqrt{g_{11}g_{22}}\geq
2\omega_{1}\omega_{2}\sqrt{\det g}$, using $\det g=g_{11}g_{22}-g_{12}^{2}$.
Next, we use the identity $\sqrt{\det g}\geq|\Omega_{12}|/2$ (saturated for
two-band models Ozawa and Mera (2021)), leading to a geometric lower bound on
dissipation, $\overline{W}_{\rm d}\geq W_{\rm gb}$, where

$W_{\rm
gb}=\omega_{1}\omega_{2}\oiint\frac{d\phi_{1}d\phi_{2}}{4\pi^{2}}\gamma|\Omega_{12}|.$
(11)

A purely topological lower bound, $W_{\rm tb}\leq W_{\rm gb}$, can be obtained
by replacing the $\phi_{a}$-dependent term $\gamma(\phi_{1},\phi_{2})$, by its
minimal value $\gamma_{\rm
min}=\min_{\phi_{1},\phi_{2}}\gamma(\phi_{1},\phi_{2})$, yielding

$W_{\rm tb}=\frac{\omega_{1}\omega_{2}}{2\pi}\gamma_{\rm min}|{\cal C}|.$ (12)

It follows from Eq. (12) that a finite frequency conversion rate ($|{\cal
C}|>0$), implies a minimum dissipation rate in the symmetric drive
configuration. Further, $W_{\rm gb}$ provides a lower bound on dissipation for
topologically-trivial protocols, with ${\cal C}=0$, yet with a finite Berry
curvature.

In the general case, when the configuration is not symmetric, the bound is
more subtle because the term $\oiint g_{12}d\phi_{1}d\phi_{2}$, encoding
cross-correlations of the drives, can be negative. Then, a bound on
dissipation reads 100100footnotemark: 100,

$W_{\rm fb}=\frac{\omega_{1}\omega_{2}\gamma_{\rm min}\Delta_{\rm
min}^{2}{\cal C}^{2}}{8\pi^{2}({\cal P}_{1}+{\cal P}_{2})},$ (13)

where ${\cal
P}_{i}=\oiint\frac{d\phi_{1}d\phi_{2}}{4\pi^{2}}(\partial_{\phi_{i}}\mathbf{d})^{2}$
is a measure of the smoothness of the mapping ${\bf{d}}(\phi)$.

Figure 3:  Frequency conversion and bounds on dissipation . a. Normalized
frequency conversion rate $\overline{\rm w}_{\rm c}=\overline{W}_{\rm
c}/W_{0}$ as a function of $m$ for $\theta=0$, and different values of
$\tau_{2}$ indicated in the inset. In the limit $\tau_{2}\to\infty$,
$\overline{\rm w}_{\rm c}\to{\cal C}$. b. Energy dissipation rate compared to
the bounds on dissipation in Eqs. (11), (12), normalized by $W_{0}$, using the
same parameters as in a and with $\tau_{2}=10$. Inset shows the minimal value
of $\gamma$ as a function of $m$. c. Normalized dissipation rate and bounds
for the fixed-gap Hamiltonian $\tilde{H}(t)$, for the same parameters as in b
and $\Delta_{0}=1$. Inset shows $\gamma$ as a function of $\Delta\tau_{2}$.
The yellow circle indicates the point $\Delta\tau_{2}=10$. d. Normalized
dissipation rate and bounds for $\theta=0.2\pi$, $m=1.2$, and $\tau_{2}=10$ as
a function of $\varphi$ on a log scale. e. The value of
$\overline{g}_{12}=\oiint g_{12}d\phi_{1}d\phi_{2}$ for the same parameters as
in d. f. Normalized frequency conversion rate for the same parameters as in d.
We use parameters $b_{11}=b_{22}=1$ and $b_{12}=b_{21}=0.5$.

## Numerical simulation.—

As an application of our results, we consider the classical problem of a spin
coupled to two elliptically-polarized magnetic fields Martin  _et al._ (2017),
$\mathbf{B}_{1}(t)=\left(0,b_{11}\sin\omega_{1}t,b_{12}\cos\omega_{1}t\right)$
and
$\mathbf{B}_{2}(t)=\left(0,b_{21}\sin\omega_{2}t,b_{22}\cos\omega_{2}t\right)$.
The Hamiltonian reads

$H(t)={\bf{m}}(\theta,\varphi)\cdot\bm{\sigma}+\mathbf{B}(t)\cdot\bm{\sigma},$
(14)

where $\mathbf{B}(t)=\mathbf{B}_{1}(t)+\mathbf{B}_{2}(t)$ is the total
magnetic field and
${\bf{m}}=m(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$ is a
time-independent Zeeman coupling parameterized by the spherical coordinates
$m$, $\theta$ and $\varphi$. This model exhibits distinct topological regimes
characterized by ${\cal C}=\\{-1,0,1\\}$, as a function of ${\bf{m}}$. For
simplicity we take $\tau_{2}$ to be time-independent.

Fig. 3b shows the dissipation in the different topological regimes of Eq.
(14), controlled by $m$, calculated numerically using Eq. (8) for $\theta=0$.
For this value of $\theta$, $\oiint g_{12}d\phi_{1}d\phi_{2}=0$; therefore,
the energy dissipation rate is bounded by $W_{\rm gb}$ and $W_{\rm tb}$ [Eqs.
(11) and (12)]. In the topological regimes (${\cal C}\neq 0$), there is an
obstruction to construct drives with arbitrarily low dissipation (for a fixed
$\tau_{2}$). The geometric bound $W_{\rm gb}$ tracks the actual dissipation
even near the topological phase transition points (at $m=\pm 0.5,\pm 1.5$).
Notice that the topological bound $W_{\rm tb}$ is not tight due to large
variations of $\gamma$ as a function of $\phi_{1}$ and $\phi_{2}$ for a time-
independent $\tau_{2}$. For Ohmic baths, we expect $\gamma$ to be
approximately constant Breuer  _et al._ (2007). We therefore consider a model
with a fixed value of $\gamma$, fixing $\Delta(t)=\Delta_{0}$, leading to
markedly improved bound $W_{\rm tb}$ (Fig. 3c) compared to the time-dependent
gap problem.

Finally, we test the case $\overline{g}_{12}=\oiint
g_{12}d\phi_{1}d\phi_{2}\neq 0$. To this end, we fix $\theta=0.2\pi$, $m=1.2$,
and vary $\varphi$. The value and sign of $\overline{g}_{12}$ vary as a
function of $\varphi$, see Fig. 3e. Fig. 3d shows the dissipation and the
bounds appearing in Eqs. (12) and (13). While formally, the topological bound
$W_{\rm tb}$ is not valid when $\overline{g}_{12}<0$, it provides a natural
scale for the dissipation rate which is not reached in the model we used.
Finally, the frequency conversion rate in Fig. 3f shows that the system
switches between different topological phases as a function of $\varphi$.

## Summary and outlook.—

In this work, we investigated heat transfer from a quasi-adiabatically driven
spin to a cold heat bath. We found that the heating rate is controlled by the
trajectory of the spin in the phase-space determined by quantum geometry and
the “quality factor” of the spin-bath coupling [see Eq. (7)]. Remarkably, in a
topological regime, the system inevitably generates heat at a rate no lower
than that given by Eqs. (11) and (12), for a symmetric configuration of the
drives. Interestingly, for an asymmetric configuration, the theoretically
minimal heating rate is reduced due to interference between the two drives
[see Eq. (13)]. However, despite our best efforts, we were not able to
identify configurations that result in heating rates below the topological
bound. Investigating drives that undershoot the topological lower bound offers
an exciting direction for designing low-dissipation driving protocols.

The implications of our findings extend to a range of experimental platforms,
such as nuclear spins Jones  _et al._ (2000), solids Nathan  _et al._ (2022),
superconducting qubits Malz and Smith (2021); Gümüş  _et al._ (2023), and
cavity QED Long  _et al._ (2022), emphasizing the importance of considering
heating effects in practical applications. To provide a concrete estimate of
the heating rate, let us consider an isolated cloud of $N$ atoms carrying
spin-1/2 subjected to a time-dependent magnetic field Zhang  _et al._ (2017);
Monroe  _et al._ (2021); Feng  _et al._ (2022). Each atom weakly interacts
with other identical atoms in its surroundings, acting as an effective heat
bath. The energy dissipated by $N$ atoms leads to an increase in the cloud’s
temperature $T_{\rm c}$, with the rate $\frac{dT_{\rm
c}}{dt}=N\overline{W}_{\rm d}/C_{\rm c}$, where $C_{\rm c}$ is the heat
capacity of the cloud and $\overline{W}_{\rm d}$ the heating rate per spin.
Assuming a magnetic field amplitude $|\mathbf{B}|\sim 2\pi\times 10\rm\,KHz$,
frequencies $\omega_{1},\omega_{2}\sim 2\pi\times 1\,\rm{KHz}$, and relaxation
time $\tau_{2}\sim 1\,\rm ms$, we estimate $\gamma\approx 0.03$ and
$\overline{W}_{\rm d}\approx 2\times 10^{-29}\,\rm{Watt}$, using Eq. (12).
Finally, approximating $C_{\rm c}\approx Nk_{\rm B}$, we find $\frac{dT_{\rm
c}}{dt}\approx 1.4\,\rm{\mu K/s}$.

In this study, we focused on a specific class of isotropic dissipators. In
general, heat baths derived from microscopic models Breuer  _et al._ (2007);
Nathan and Rudner (2020); Mozgunov and Lidar (2020) may contain non-isotropic
corrections whose examination is left for future work. An intriguing direction
for further study is the analysis of dissipation in multi-qubit adiabatic
computational protocols Sarandy and Lidar (2005b); Albash and Lidar (2018),
where the occurrence of heating can result in an increased error rate Albash
and Lidar (2015). Moreover, higher-dimensional systems (such as quantum pumps
Xiao  _et al._ (2010); Bustos-Marún  _et al._ (2013); Citro and Aidelsburger
(2023)) constitute an intriguing platform where driving can induce heat
currents. Our findings align with recent studies emphasizing the crucial role
of the quantum metric in determining the electronic properties of solids
Neupert  _et al._ (2013); Peotta and Törmä (2015); Julku  _et al._ (2016);
Liang  _et al._ (2017); Mitscherling (2020); Rossi (2021); Takayoshi  _et al._
(2021); Mitscherling and Holder (2022); Törmä  _et al._ (2022); Komissarov
_et al._ (2023).

###### Acknowledgements.

## Acknowledgments.—

We thank Jason Alicea, Israel Klich, Nandagopal Manoj, Johannes Mitscherling,
Valerio Peri, and Mark Rudner for insightful discussions. É. L.-H. was
supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant
GBMF8682. F.N. was supported by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences under award DE-SC0019166, the Simons Foundation
under award 623768, and the Carlsberg Foundation, grant CF22-0727. G.R. and
I.E. are grateful for support from the Simons Foundation and the Institute of
Quantum Information and Matter. G.R. is grateful for support from the ARO MURI
grant FA9550-22-1-0339. This work was performed in part at Aspen Center for
Physics, which is supported by National Science Foundation grant PHY-1607611.

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# LSTMs with Attention for Aggression Detection

Nishant Nikhil
IIT Kharagpur
Kharagpur India
<EMAIL_ADDRESS>
Ramit Pahwa
IIT Kharagpur
Kharagpur India
<EMAIL_ADDRESS>
Mehul Kumar Nirala
IIT Kharagpur
Kharagpur India
<EMAIL_ADDRESS>
Rohan Khilnani
IIT Kharagpur
Kharagpur India
<EMAIL_ADDRESS>
###### Abstract

In this paper, we describe the system submitted for the shared task on
Aggression Identification in Facebook posts and comments by the team Nishnik.
Previous works demonstrate that LSTMs have achieved remarkable performance in
natural language processing tasks. We deploy an LSTM model with an attention
unit over it. Our system ranks 6th and 4th in the Hindi subtask for Facebook
comments and subtask for generalized social media data respectively. And it
ranks 17th and 10th in the corresponding English subtasks.

##  1 Introduction

00footnotetext:  This work is licensed under a Creative Commons Attribution
4.0 International License. License details:
http://creativecommons.org/licenses/by/4.0/

In recent years, there has been a rapid growth in social media usage.
Interactions over the web and social media have seen an exponential increase.
While usage of social media helps users stay connected; incidents of
aggression, trolling, cyberbullying, flaming, and hate speech are more
prevalent now than ever.

Recent works on aggression classification include the use of logistic
regression classifier [Davidson et al. (2017]. They create a bunch of hand-
crafted features like binary and count indicators for hashtags, lexicon based
sentiment scores for each tweet, unigram, bigram, and trigram features. They
use two logistic regression models, the first one to reduce dimensionality of
the features and the second one to make classification. ?) train a binary
classifier to label tweets into ‘racist’ and ‘non-racist’. They deploy Naive
Bayes classifier on unigram features. Neural language model was used in ?).
First they learn embedding of the text passages using paragraph2vec [Le and
Mikolov (2014]. Then, they train a logistic regression classifier over those
embeddings to classify into hateful and clean comments. ?) surveys the recent
development in this field.

The first shared task on aggression identification [Kumar et al. (2018a] was
held at the first workshop on Trolling, Aggression and Cyberbullying (TRAC).
The goal was to classify social media posts into one of three labels (Overtly
aggressive, Covertly aggressive, Non-aggressive).

The major contribution of the work can be summarized as a neural network based
model which has LSTM units followed by an attention unit to embed the given
social media post and training a classifier to detect aggression. We discuss
our methods in section 2. Section 3 contains the details about the experiments
and training data. In Section 4, we discuss the results and Section 5
concludes the paper with closing remarks.

##  2 Methodology

We hypothesize that aggression identification requires processing of the words
of a sentence in a sequential manner. The positioning of a particular word at
different places can alter the aggressiveness of the sentence. Example:

These aliens are filthy, but they live in a good neighbourhood. (Aggressive)
These aliens are good, but they live in a filthy neighbourhood. (Less
aggressive)

Recurrent Neural Networks [Mikolov et al. (2010] are good at handling
sequential data and have achieved good results in natural language processing
tasks. As RNNs share parameters across time, they are capable of conditioning
the model on all previous words of a sentence. Although theoretically it is
correct that RNNs can retain information from all previous words of a
sentence, but practically they fail at handling long-term dependencies. Also,
RNNs are prone to the vanishing and exploding gradient problems when dealing
with long sequences. Long Short-Term Memory networks [Hochreiter and
Schmidhuber (1997], a special kind of RNN architecture, were designed to
address these problems.

###  2.1 Long Short-Term Memory networks

LSTMs use special units in addition to standard RNN units. These units include
a ‘memory cell’ which can maintain its state for long periods of time. A set
of non-linear gates is used to control when information enters the
memory(Input gate), when it’s outputted (Output gate), and when it’s forgotten
(Forget gate). The equations for the LSTM memory blocks are given as follows:

$f_{t}=\sigma_{g}(W_{f}x_{t}+U_{f}h_{t-1}+b_{f})\\\ $ (1)
$i_{t}=\sigma_{g}(W_{i}x_{t}+U_{i}h_{t-1}+b_{i})\\\ $ (2)
$o_{t}=\sigma_{g}(W_{o}x_{t}+U_{o}h_{t-1}+b_{o})\\\ $ (3) $c_{t}=f_{t}\circ
c_{t-1}+i_{t}\circ\sigma_{c}(W_{c}x_{t}+U_{c}h_{t-1}+b_{c})\\\ $ (4)
$h_{t}=o_{t}\circ\sigma_{h}(c_{t})$ (5)

In these equations, $x_{t}$ is the input vector to the LSTM unit, $f_{t}$ is
the forget gate’s activation vector, $i_{t}$ is the input gate’s activation
vector, $o_{t}$ is the output gate’s activation vector, $h_{t}$ is the output
vector of the LSTM unit and $c_{t}$ is the cell state vector. $w,u,B$ are the
parameters of weight matrices and bias vectors which are learned during the
training.

###  2.2 Attention

Here, the attention module is inspired by [Bahdanau et al. (2014]. We deploy
it after the LSTM unit. It helps the model decide the importance of each word
for the classification task. It scales the representation of the words by a
learned weighing factor, as determined by these equations:

$e_{t}=h_{t}w_{a}\\\ $ (6)
$a_{t}=\frac{exp(e_{t})}{\sum_{i=1}^{T}exp(e_{i})}\\\ $ (7)
$v=\sum_{i=1}^{T}a_{i}h_{i}$ (8)

In these equations, $h_{t}$ is the hidden representation of a word at a time
step $t$, $w_{a}$ is the weight matrix for the attention layer, $a_{t}$ is the
attention score for the word at time $t$, and $v$ is the final representation
of the sentence obtained by taking a weighted summation over all time steps.

##  3 Experiments

###  3.1 Datasets

The training datasets for the English and Hindi sub-tasks are constructed by
11,999 and 12,000 Facebook posts and comments, respectively. The testing set
includes 3,001 and 3,000 respectively. These are collected and manually
annotated by the organizers. Most of the datum has an id and is classified
into one of the three classes: OAG (Overtly aggressive), CAG (Covertly
aggressive), NAG (Non-aggressive). The distribution of the classes in the
training dataset for English and Hindi are shown in Table 1. The organizers
released a modified version of the data where they remove the rows without
specific id. As the id is not important for prediction, we decided to work on
the initially released dataset. The data collection methods used to compile
the dataset for the shared task are described in ?).

Class | Train (English) | Test (English) | Train (Hindi) | Test (Hindi)
---|---|---|---|---
Non-aggressive | 5,051 | 1,233 | 2,275 | 538
Covertly aggressive | 4,240 | 1,057 | 4,869 | 1,246
Overtly aggressive | 2,708 | 711 | 4,856 | 1,217
Table 1: Class distribution in train and test sets

###  3.2 Preprocessing

Before feeding the Facebook comments to the LSTM classifier, we performed the
following operations on the text:

  1. 1.

We used the ekphrasis toolkit [Baziotis et al. (2017] for normalizing the
occurrence of the following in the comments: URL, E-mail, percent, money,
phone, user, time, date, and number. For example, URLs are replaced by <url>,
and all occurrences of @someone are replaced by <user>.

  2. 2.

We then passed the normalized text through the Social tokenizer. Unlike normal
tokenizers, the Social tokenizer is specifically aimed at the unstructured
social media content. It understands and parses complex emoticons, emojis and
other unstructured expressions like dates, times, phone numbers etc.

  3. 3.

Then, we removed the punctuations and used ekphrasis’s inbuilt spell corrector
on the text.

  4. 4.

Lastly, we used NLTK’s WordNet lemmatizer [Loper and Bird (2002] to lemmatize
the words to their roots.

###  3.3 Parameters

Our model uses an embedding layer of 100 dimensions to project each word into
a vector space. We place a dropout [Srivastava et al. (2014] layer after this.
To capture the context of the words passed from the dropout layer we use an
LSTM layer having 100 hidden dimensions. As the LSTM cells already have non-
linear activation functions, it helps the model capture non-linear semantics
from the data. The output from the LSTM is then passed through an attention
module. The attention module helps the model determine which word to give more
importance to. The weighted output from attention module is passed through a
fully-connected layer. To get the probabilities of each class, softmax
function is applied to the output. We use the cross-entropy function to
calculate the loss between the predicted and the target value. Adam optimizer
is used with a learning rate of 0.001 to learn the weights of the model. The
dropout rate was either 0.2 or 0.3 and is discussed in the Results section.

Although many machine learning classifiers like Naive Bayes, Decision Tree,
Support Vector Machine or Random Forest could be used as a baseline classifier
for this task. Due to constraint of time we have only used a Random Forest
classifier. We train the classifier on a set of hand-crafted features. The
features used are as follows:

  1. 1.

Number of words with positive sentiment.

  2. 2.

Number of words with negative sentiment.

  3. 3.

Number of punctuations.

  4. 4.

Total number of words.

  5. 5.

Inverse of the 2nd feature.

  6. 6.

Natural logarithm of the 2nd feature.

We use the lists made available by ?) for extracting positive and negative
words.

##  4 Results

Due to a mistake on our side, we first submitted a model which considered only
the first 45 words of the post/comment and used a dropout rate of 0.2, we
denote this model as Eng-A in the tables. In Eng-B, we use dropout rate of 0.3
and considered all the words. RF baseline is the random forest classifier
based model baseline of hand-crafted features.

System | F1 (weighted)
---|---
Random Baseline | 0.3535
EF-A | 0.5533
EF-B | 0.5746
Table 2: Results for the English (Facebook) task. System | F1 (weighted)
---|---
Random Baseline | 0.3477
RF Baseline | 0.3888
Eng-A | 0.5304
Eng-B | 0.5548
Table 3: Results for the English (Social Media) task.

For both the Hindi sub-tasks, we used the LSTM classifier with dropout
probability of 0.3. We denote the model as Hi-A in the tables.

System | F1 (weighted)
---|---
Random Baseline | 0.3571
Hi-A | 0.6032
Table 4: Results for the Hindi (Facebook) task. System | F1 (weighted)
---|---
Random Baseline | 0.3206
Hi-A | 0.4703
Table 5: Results for the Hindi (Social Media) task.

Looking at the confusion matrices of the English subtasks, it is clear that
the model is performing well at classifying the non-aggressive comments from
the aggressive or covertly aggressive comments. But it performs poorly and
classifies a lot of over-aggressive and non-aggressive comments to covertly
aggressive. The results in ?) also convey the same message.

Figure 1: Confusion matrix for English (Facebook) task.

##  5 Conclusion

In this paper, we present an LSTM network with an attention based classifier
for aggression detection. It gives competitive results while relying only on
the dataset provided. The performance reported in this paper could be further
boosted by utilizing transfer learning methods from larger datasets, like
using pre-trained word embeddings. Furthermore, the model tends to over-fit on
the training data. Better generalization techniques, like the use of an
increased dropout rate, might help in increasing the performance of the model.

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