# Many-body quantum thermal machines in a Lieb-kagome Hubbard model

Saikat Sur  Department of Chemical and Biological Physics & AMOS, Weizmann
Institute of Science, Rehovot 7610001, Israel  Pritam Chattopadhyay
Department of Chemical and Biological Physics & AMOS, Weizmann Institute of
Science, Rehovot 7610001, Israel  Madhuparna Karmakar<EMAIL_ADDRESS>Department of Physics and Nanotechnology, College of Engineering and
Technology, SRM Institute of Science and Technology, Kattankulathur,
Chennai-603203, India  Avijit Misra<EMAIL_ADDRESS>Centre for
Quantum Engineering, Research and Education (CQuERE), TCG CREST, Salt Lake,
Sector 5, Kolkata 700091, India

(August 28, 2024)

###### Abstract

Quantum many-body systems serve as a suitable working medium for realizing
quantum thermal machines (QTMs) by offering distinct advantages such as
cooperative many-body effects, and performance boost at the quantum critical
points. However, the bulk of the existing literature exploring the criticality
of many-body systems in the context of QTMs involves models sans the
electronic interactions, which are non-trivial to deal with and require
sophisticated numerical techniques. Here we adopt the prototypical Hubbard
model in two dimensions (2D) in the framework of the line graph Lieb-kagome
lattice for the working medium of a multi-functional QTM. We resort to a non-
perturbative, static path approximated (SPA) Monte Carlo technique to deal
with the repulsive Hubbard model. We observe that in a Stirling cycle, in both
the interacting and non-interacting limits, the heat engine function dominates
and its performance gets better when the strain is induced from the kagome to
the Lieb limit, while for the reverse the refrigeration action is preferred.
Further, we show that the QTM performs better when the difference between the
temperatures of the two baths is lower and the QTM reaches the Carnot limit in
this regime. Further, we extensively study the performance of the QTM in the
repulsive Hubbard interacting regime where the magnetic orders come into the
picture. We explore the performance of the QTM along the quantum critical
points and in the large interaction limit.

##  I Introduction

With recent advancements towards miniaturized devices at the nanoscale, there
has been an upsurge of activities in modeling and designing quantum thermal
devices including quantum heat engines Scovil and Schulz-DuBois (1959); Alicki
(1979); Mukherjee and Divakaran (2021); Geva and Kosloff (1996); Cangemi  _et
al._ (2023); Gelbwaser-Klimovsky  _et al._ (2015, 2013); Opatrnỳ  _et al._
(2023); Misra  _et al._ (2022); Opatrný  _et al._ (2021); Myers  _et al._
(2022); Bhattacharjee and Dutta (2021); Pandit  _et al._ (2021); Chattopadhyay
_et al._ (2021a); Saha  _et al._ (2023); Sur and Ghosh (2023); Kurizki and
Kofman (2022), refrigerators Levy and Kosloff (2012); Linden  _et al._ (2010);
Mukhopadhyay  _et al._ (2018); Das  _et al._ (2019); Ray  _et al._ (2023),
diodes and transistors Naseem  _et al._ (2020); Karg ı  _et al._ (2019); Ghosh
_et al._ (2021); Joulain  _et al._ (2016), transformers Maity  _et al._
(2024). Though the theoretical explorations have been much ahead, there have
been successful experimental demonstrations of quantum heat engines (QHE) in
trapped ions Roßnagel  _et al._ (2016); Maslennikov  _et al._ (2019); von
Lindenfels  _et al._ (2019), nuclear magnetic resonance Peterson  _et al._
(2019), and nitrogen-vacancy centers in diamonds Klatzow  _et al._ (2019)
based setups. One of the main motivations of these explorations is to harness
quantumness for better functioning of the QHE which includes exploiting non-
thermal and squeezed baths Scully (2002); Roßnagel  _et al._ (2014); Gardas
and Deffner (2015); Niedenzu  _et al._ (2018), presence of quantum coherence
in the working medium (WM) Watanabe  _et al._ (2017); Camati  _et al._ (2019),
quantum measurements Elouard  _et al._ (2017a, b); Cottet  _et al._ (2017);
Santos and Chattopadhyay (2023).

Primarily the study of QHE has dealt with a single particle or a few Kosloff
and Rezek (2017). Considering many-body (MB) quantum systems as WM of a QHE is
relatively new. Quantum MB systems have been employed to harness MB quantum
correlation Jaramillo  _et al._ (2016), quantum criticality Polettini  _et
al._ (2015); Fadaie  _et al._ (2018); Campisi and Fazio (2016), MB
localization Halpern  _et al._ (2019), super-radiance Hardal and
Müstecaplıoğlu (2015), minimizing friction Deng  _et al._ (2013); Campo  _et
al._ (2014); Beau  _et al._ (2016); Binder  _et al._ (2018), and investigate
the effect of quantum statistics Bengtsson  _et al._ (2018) in QHE. Quantum
criticality has been proven to be a useful resource in MB-QHE Campisi and
Fazio (2016); Chen  _et al._ (2019).

The bulk of the existing literature exploring the criticality of MB in the
context of QHE involves models sans the electronic interactions, which are
non-trivial to deal with and require sophisticated numerical techniques. At
the same time, electronic interactions in conjunction with competing
correlations and/or geometric frustration of lattices are known to bring forth
rich quantum phases and phase transitions Dagotto (2005); Pesin and Balents
(2010); Basov  _et al._ (2011); Imai and Kawakami (2002); Parcollet  _et al._
(2004); Kyung and Tremblay (2006); Ohashi  _et al._ (2008); Wietek  _et al._
(2021). The aim of this work is to analyze such an interplay between quantum
correlations and geometric frustration in the light of quantum thermodynamics,
thereby proposing its prospective application as a quantum thermal machine
(QTM).

Our starting Hamiltonian corresponds to the prototypical Hubbard model in two
dimensions (2D) in the framework of the line graph Lieb-kagome lattice
Kaufmann  _et al._ (2021); Medeiros-Silva  _et al._ (2023). The unprecedented
control over the engineering and subsequent tuning of these lattices in
ultracold atomic gases Taie  _et al._ (2015); Ozawa  _et al._ (2017),
optically induced photonic systems Vicencio  _et al._ (2015); Xia  _et al._
(2016); Mukherjee and Thomson (2015), artificial lattices engineered through
lithography and atomic manipulations Drost  _et al._ (2017); Slot  _et al._
(2017); Qiu  _et al._ (2016) and more recently in a metal-organic framework
(MOF) Ni  _et al._ (2022) have provided the required impetus to the search for
their potential applications. In the non-interacting limit, this lattice with
a three-site unit cell hosts three electronic dispersion bands as two
dispersive and one flat band. The position of the electronic bands and the
corresponding underlying Brillouin zone provides the Lieb and the kagome
lattices with their distinct properties that have been investigated
extensively. The impact of the inclusion of electronic interactions in this
system shows up as two important observations viz. $(i)$ beyond a critical
interaction, magnetic correlation sets in, demarcating the low-temperature
phases to be magnetically disordered and ordered, $(ii)$ the electronic
transport, controlled by the underlying band structure quantifies the
different interaction regimes of the system as metallic and insulating Kunwar
and Karmakar (2024). Being a line-graph lattice the band structure between the
Lieb and kagome limits can be smoothly evolved by suitable strain engineering,
thereby tuning the thermodynamic, spectroscopic, and transport properties of
this system.

In the next few sections, we discuss the impact of this strain engineering on
electronic dispersion and explore its implications on the function of a QTM.
Our primary inference from this work is that by suitable strain engineering
one can achieve four working domains such as quantum heat engine (QHE),
quantum refrigerator (QR), quantum accelerator (QA), and quantum heater (QH)
by running a Stirling cycle Thomas  _et al._ (2019); Hamedani Raja  _et al._
(2021); Chattopadhyay and Paul (2019); Chattopadhyay  _et al._ (2021b);
Chattopadhyay (2020). The QHE domain is more feasible and its performance gets
better when we engineer the strain from the kagome limit to the Lieb limit in
the line graph lattice. Conversely, for QR, it is better, if the strain is
engineered in the reverse direction. The performance of both QHE and QR gets
better when the temperature difference between the two operating baths, hot
and cold namely, is lower and the Carnot limit is achieved in this region.
This is valid for both the interacting and the non-interacting cases.

For the interacting case, though a low-temperature difference between the
working bath is preferred, unlike the non-interacting case the favorable
working temperatures are not close to the zero temperature. Interestingly, at
the critical point where the system exhibits a magnetic transition from a
paramagnetic metal to an anti-ferromagnetic metal, the QTM performs well
through the entire parameter space of the bath temperature, close to zero
being the most favorable region.

The article is organized as follows. In Sec. II, we analyze the model of the
WM, a 2D line graph lattice (Lieb and kagome) having onsite repulsive Hubbard
interaction. Then we discuss the Stirling cycle and thermodynamic quantifiers
of heat and work and performance bounds of the same in Sec. III. In Sec. IV,
we discuss the results for both the non-interacting (Sec. IV.1) and
interacting (Sec. IV.2) cases. Finally, we conclude in Sec. V.

##  II Model

Figure 1: Schematic diagram showing the strain-induced evolution of the
lattice structure between the Lieb and the kagome limits.

The prototypical repulsive Hubbard Hamiltonian on a 2D line graph Lieb-kagome
lattice reads as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\mu\sum_{i,\sigma}c^{\dagger}_{i,\sigma}c_{i,\sigma}+\sum_{\langle
ij\rangle,\sigma}t_{ij}c^{\dagger}_{i,\sigma}c_{j,\sigma}+\sum_{\langle\langle
ij\rangle\rangle,\sigma}t^{\prime}_{ij}c^{\dagger}_{i,\sigma}c_{j,\sigma}$ (1)
$\displaystyle+\mbox{H.c.}+U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}.$

where, $t_{ij}$ corresponds to the nearest neighbor hopping and $t_{ij}=t=1$
sets the reference energy scale of the model. The applied strain is quantified
in terms of $t_{ij}^{\prime}=t^{\prime}$. $U>0$ corresponds to the repulsive
Hubbard interaction. We work at the half-filling and the chemical potential
$\mu$ is adjusted to achieve the same. The strain-induced excursion of the
lattice and therefore the corresponding electronic band structure between the
Lieb and the kagome limits is represented in Fig. 1.

In the presence of interactions ($U\neq 0$), the model is made numerically
tractable via Hubbard-Stratonovich (HS) decomposition of the four-fermion term
Hubbard (1959); Schulz (1990), introducing a vector ${\bf m}_{i}(\tau)$ and a
scalar $\phi_{i}(\tau)$ (bosonic) auxiliary fields at each site, which couples
to the spin and the charge channels, respectively. The problem is addressed
via the static path approximated (SPA) Monte Carlo technique wherein the model
is envisaged as an effective spin-fermion model with the random, fluctuating,
"classical" background of the auxiliary fields coupled to the free fermions.
The $\phi_{i}$ field is treated at the saddle point level as
$\phi_{i}\rightarrow\langle\phi_{i}\rangle=\langle n_{i}\rangle U/2$ (where,
$\langle n_{i}\rangle$ is the number density of the fermions), while the
complete spatial fluctuations of ${\bf m}_{i}$ are retained. The quantum
thermodynamic properties are analyzed based on the proper identification of
heat and work in the quantum domain Misra  _et al._ (2015); Skrzypczyk  _et
al._ (2014). Thereby, we analyze the performance of a multifunctional (see
Table. 1) QTM. The results presented in this paper correspond to a system size
of $3\times L^{2}$, with $L=16$, and are verified to be robust against finite
system size effects. The details of the numerical technique are presented in
the appendix.

We begin the analysis of our results with the non-interacting limit by setting
$U=0$ in Eq. (1). To evaluate the energy spectrum, we transform the
Hamiltonian in the momentum space as
$\mathscr{H}=\sum_{k}\Phi_{k}^{\dagger}H_{k}\Phi_{k}$ with
$\Phi_{k}^{\dagger}=\begin{pmatrix}c^{\dagger}_{\textbf{Ak}}&c^{\dagger}_{\textbf{Bk}}&c^{\dagger}_{\textbf{Ck}}\end{pmatrix}$,
where

$\displaystyle
H_{\textbf{k}}=\begin{pmatrix}0&A_{\textbf{k}}&B_{\textbf{k}}\\\
{A_{\textbf{k}}}&0&C_{\textbf{k}}\\\
{B_{\textbf{k}}}&C_{\textbf{k}}&0\end{pmatrix}.$ (2)

Here $A_{\textbf{k}}=-2\cos\frac{k_{x}}{2}$,
$B_{\textbf{k}}=-2\cos(-\frac{k_{x}}{2}\cos\theta+\frac{k_{y}}{2}\sin\theta)$,
and
$C_{\textbf{k}}=-t^{1}_{AC}\cos(\frac{k_{x}}{2}(1+\cos\theta)-\frac{k_{y}}{2}\sin\theta)-t^{2}_{AC}\cos(\frac{k_{x}}{2}(1-\cos\theta)+\frac{k_{y}}{2}\sin\theta)$.
The strain is quantified in terms of the parameters $t^{1}_{AC}$ and
$t^{2}_{AC}$ between the sites $A$ and $C$ as Jiang  _et al._ (2019a)

$\displaystyle t^{1}_{AC}=\exp[\eta(1-2\cos\frac{\theta}{2})],$ $\displaystyle
t^{2}_{AC}=\exp[\eta(1-2\sin\frac{\theta}{2})],$ (3)

where the parameter $\eta$ governs the rate at which the strain diminishes
with distance. A higher $\eta$ results in a more gradual decrease in the band
curvature, leading to a flatter band. Conversely, a smaller $\eta$ is more
adept at capturing the transition of the band structure from Lieb to kagome.
For $\eta=8$, we achieve relatively flat bands and smooth transitions between
the two lattices Jiang  _et al._ (2019b). The angle $\theta$ is tuned within
the range of $[\frac{\pi}{2},\frac{2\pi}{3}]$. The lower and the upper bounds
correspond to the Lieb and the kagome lattice, respectively. In the Lieb limit
($\theta=\pi/2$), with the strain being,
$t^{1}_{AC}=t^{2}_{AC}=\exp[\eta(1-\sqrt{2})]$. On the other hand, in the
kagome limit ($\theta=2\pi/3$), with the corresponding strain being
parametrized as, $t^{1}_{AC}=t=1$ and $t^{2}_{AC}=\exp[\eta(1-\sqrt{3})]$. The
transition between the Lieb and kagome lattices is observed at
$\theta^{*}=7\pi/12$.

Now diagonalizing $H_{k}$, we get the three respective eigenvalues

$\displaystyle\varepsilon_{\mathbf{k},j}=\sqrt{\frac{2}{3}}(A^{2}_{\textbf{k}}+{B_{\textbf{k}}}^{2}+{C_{\textbf{k}}}^{2})^{1/2}\times$
$\displaystyle\cos\left[\frac{1}{3}\cos^{-1}\big{(}\frac{3\sqrt{3}A_{\textbf{k}}B_{\textbf{k}}C_{\textbf{k}}}{(A^{2}_{\textbf{k}}+{B_{\textbf{k}}}^{2}+{C_{\textbf{k}}}^{2})^{3/2}}\big{)}-\frac{2\pi(j-1)}{3}\right],$
$\displaystyle{\leavevmode\nobreak\ \leavevmode\nobreak\ }j=1,2,3.$ (4)

##  III Stirling cycle and thermodynamic quantities

Figure 2: Schematic representation of the Stirling cycle. $A\rightarrow B$ and
$C\rightarrow D$ are the two isothermal steps where the Hamiltonian parameter
$\theta$ is changed when the WM is in contact with the heat baths at
temperatures $T_{h}$ and $T_{c}$ respectively. $B\rightarrow C$ and
$D\rightarrow A$ are the two isochoric thermalization steps in contact with
the heat baths at temperatures $T_{h}$ and $T_{c}$ respectively, while the
Hamiltonian parameter $\theta$ is fixed.

A quantum Stirling cycle Thomas  _et al._ (2019); Hamedani Raja  _et al._
(2021); Chattopadhyay and Paul (2019); Chattopadhyay  _et al._ (2021b);
Chattopadhyay (2020) consists of four steps (Fig. 2). First $(A\rightarrow
B)$, the WM is in equilibrium with the heat bath at temperature $T_{h}$, while
the Hamiltonian parameter is isothermally changed,
$\theta_{1}\rightarrow\theta_{2}$ (Eq. (II)). In the second step
$(B\rightarrow C)$, the WM is detached from the hot bath and connected with a
cold bath at a temperature $T_{c}$ with which it thermalizes isochorically
without performing any mechanical work. Next $(C\rightarrow D)$, the WM is
brought back to its initial Hamiltonian configuration isothermally
($\theta_{2}\rightarrow\theta_{1}$) while keeping the system connected to the
cold bath. In the final step $(D\rightarrow A)$, the system is detached from
the cold bath and connected to the hot bath where it undergoes an isochoric
thermalization process without performing any mechanical work. Thus, the WM
fully recovers to its initial state and the cycle is completed.

The total heat transferred to the system from the bath in the isothermal
processes $(A\rightarrow B)$ and $(C\rightarrow D)$, while keeping the system
in equilibrium with the bath at temperature $T_{h}$ and $T_{c}$ respectively,
are combinations of the mechanical works performed due to deformation of the
potential and the change of the internal energies. Thus the heat transferred
to the bath from the system during two isothermal processes is given by

$\displaystyle
Q_{AB}=-(U^{\theta_{2}}_{T_{h}}-U^{\theta_{1}}_{T_{h}}+k_{B}T_{h}\ln
Z^{\theta_{2}}_{T_{h}}-k_{B}T_{h}\ln Z^{\theta_{1}}_{T_{h}}),$ (5)

and

$\displaystyle
Q_{CD}=-(U^{\theta_{1}}_{T_{c}}-U^{\theta_{2}}_{T_{c}}+k_{B}T_{c}\ln
Z^{\theta_{1}}_{T_{c}}-k_{B}T_{c}\ln Z^{\theta_{2}}_{T_{c}}),$ (6)

respectively. Here, $Z^{\theta}_{T}$ is the partition function evaluated as

$\displaystyle
Z^{\theta}_{T}=\sum_{\mathbf{k}}\sum_{j}e^{-\frac{\varepsilon^{\theta}_{\mathbf{k},j}}{T}}=\frac{1}{4\pi^{2}}\sum_{j}\int\int
dk_{x}dk_{y}e^{-\frac{\varepsilon^{\theta}_{\mathbf{k},j}}{T}}$ (7)

and the internal energy $U^{\theta}_{T}$ is evaluated as
$T^{2}\frac{\partial}{\partial T}\ln Z^{\theta}_{T}.$ On the other hand, the
heat transferred to the system during the isochoric thermalization processes
$(B\rightarrow C)$ and $(D\rightarrow A)$ are the differences between the
average energies of the initial and the final configurations as no mechanical
work has been considered. Therefore, we have

$\displaystyle
Q_{BC}=-(U^{\theta_{2}}_{T_{c}}-U^{\theta_{2}}_{T_{h}}),\;\;\text{and}\;\;Q_{DA}=-(U^{\theta_{1}}_{T_{h}}-U^{\theta_{1}}_{T_{c}}).$
(8)

Provided all the processes involved in the cycle are reversible and no leakage
takes place, the expressions for the net work done on the system $W$ and the
heat transferred to the system with the hot and cold baths, $Q_{h}$ and
$Q_{c}$ after completion of one cycle are given as follow:

$\displaystyle W$ $\displaystyle=$
$\displaystyle(Q_{AB}+Q_{BC}+Q_{CD}+Q_{DA})$
$\displaystyle=-T_{h}\ln\frac{Z^{\theta_{2}}_{T_{h}}}{Z^{\theta_{1}}_{T_{h}}}+T_{c}\ln\frac{Z^{\theta_{2}}_{T_{c}}}{Z^{\theta_{1}}_{T_{c}}},$
$\displaystyle Q_{h}$ $\displaystyle=$
$\displaystyle-(Q_{AB}+Q_{DA})=T_{h}\ln\frac{Z^{\theta_{2}}_{T_{h}}}{Z^{\theta_{1}}_{T_{h}}}+U^{\theta_{2}}_{T_{h}}-U^{\theta_{1}}_{T_{c}},$
$\displaystyle Q_{c}$ $\displaystyle=$
$\displaystyle-(Q_{BC}+Q_{CD})=-T_{c}\ln\frac{Z^{\theta_{2}}_{T_{c}}}{Z^{\theta_{1}}_{T_{c}}}-U^{\theta_{2}}_{T_{h}}+U^{\theta_{1}}_{T_{c}}.$

By the principle of conservation of energy or the first law of thermodynamics,
one can check that $Q_{h}+Q_{c}+W=0$. Importantly, it has been established
that the maximal isothermal work extraction also in the quantum domain is
given by the change in free energy $\Delta F$ Misra  _et al._ (2015);
Skrzypczyk  _et al._ (2014), where

$\displaystyle F=U-TS.$ (10)

Here, $S$ is the von Neuman entropy of the WM $\rho$ as quantified by
$S(\rho)=-\mbox{Tr}[\rho\log\rho]$ and $T$ is the temperature of the heat
bath. One can check that work $W$ in Eq. (LABEL:eq:w) follows it. If the
quantities $Q_{h}$ or $Q_{c}$ are positive, the heat is flowing into the
system, similarly, if work $W$ is positive, work is done on the system. In
conformity with the second law of thermodynamics, only four modes of operation
are possible. The four possible modes can be identified by the signs of $W$,
$Q_{h}$, and $Q_{c}$, as depicted in Table 1.

Modes of operation |  $Q_{h}$ |  $Q_{c}$ |  $W$
---|---|---|---
Engine |  $>0$ |  $<0$ |  $<0$
Refrigerator |  $<0$ |  $>0$ |  $>0$
Accelerator |  $>0$ |  $<0$ |  $>0$
Heater |  $<0$ |  $<0$ |  $>0$
Table 1: Different modes of operation of a quantum thermodynamic cycle.

The efficiency $\eta$ of the engine is given by

$\eta=\frac{|W|}{Q_{h}},$ (11)

and the coefficient of the performance (COP) of the refrigerator is given as

$\displaystyle\eta_{R}=\frac{Q_{c}}{|W|}.$ (12)

Given a pair of thermal baths, the maximum efficiency or coefficient of
performance is obtained for the Carnot cycle, a theoretical proposition with
only reversible cycles. We present the efficiency of the engine and
coefficient of performance of the refrigerator scaled by the same as its
Carnot counterpart in this article. The efficiency of a Carnot engine and the
coefficient of performance of the Carnot refrigerator are given respectively
as

$\displaystyle\eta^{max}=1-\frac{T_{c}}{T_{h}},\mbox{\leavevmode\nobreak\
\leavevmode\nobreak\ and \leavevmode\nobreak\ \leavevmode\nobreak\
}\eta^{max}_{R}=\frac{T_{c}}{T_{h}-T_{c}}.$ (13)

##  IV Performance of the many-body quantum thermal machine

Figure 3: Demonstration of multi-functionality of the Stirling cycle by tuning
the WM strain $(\theta_{1},\theta_{2})$ without the onsite repulsive
interaction, with different sets of bath temperatures: (a)
$T_{h}=0.02,T_{c}=0.01$; (b) $T_{h}=0.03,T_{c}=0.01$; (c)
$T_{h}=0.2,T_{c}=0.01$; (d) $T_{h}=0.05,T_{c}=0.04$. The refrigerator, the
accelerator, and the heater regions are marked in blue, grey, and green
respectively. The engine region is plotted with color density where the color
denotes the engine efficiency.

In this section, we investigate and analyze the thermodynamic performance of
the many-body quantum thermal machine based on the Lieb-kagome Hubbard model.
Interestingly, we observe that all four operational regimes are mentioned in
the Table. 1, i.e. QHE, QR, QA, and QH can be realized in our model by either
tuning the strain, the bath temperatures, or the onsite repulsive Hubbard
interaction. Thus, the model works as a multifunctional quantum thermal
device. For brevity, we present the analysis for the non-interacting ($U=0$)
and interacting ($U\neq 0$) WM in separate subsections.

###  IV.1 For non-interacting working medium

Figure 4: Function of the Stirling cycle with two fixed strain configurations
(a) $\theta_{1}=2\pi/3,\theta_{2}=\pi/2$; (b)
$\theta_{1}=\pi/2,\theta_{2}=2\pi/3$ as a function of the thermal bath
temperatures. The refrigerator, the accelerator, and the heater regions are
marked in blue, grey, and green respectively. The engine and refrigerator
region is plotted with color density where the color denotes the efficiency
and coefficient of performance (COP) of the engine and refrigerator
respectively.

We begin with the Stirling cycle by tuning the strain of the WM
$(\theta_{1},\theta_{2})$. The performance of the QHE and other functional
regions QR, QA, and QH have been shown in Fig. 3. It shows that we get the QHE
performance better when we start the cycle from the Kagome limit and then move
to the Lieb limit and the opposite for QR. The maximum efficiency of the QHE
reaches the Carnot bound for the Lieb-kagome configuration, specifically when
$\theta_{1}=2\pi/3$ and $\theta_{2}=\pi/2$. In the low-temperature limit, the
performance of the QHE and feasible region (in the strain parameter space) for
it and QR increases when the difference between the two bath temperatures is
lower. Moreover, we observe distinct signatures of all the functions of the
proposed QTM. However, as the temperature difference increases, the
characteristic features of the QR and QH diminish.

We further look into the role of temperature difference in Fig. 4 to
substantiate our observation. It clearly shows that the efficiency or the
coefficient of performance (COP) of the QHE or QR are higher near the
$T_{h}=T_{c}$ line and for the configuration with $T_{c}\rightarrow 0$,
$T_{h}\rightarrow\epsilon$ (where $\epsilon$ is close to zero) with fixed
strain configuration.

###  IV.2 For interacting working medium

Figure 5: Low-temperature phase diagram (at $T=0.01$) of the kagome Hubbard
model at half filling mapping out the various thermodynamic phases as (i) Flat
band Insulator (FI), (ii) Paramagnetic Metal (PM), (iii) Antiferromagnetic
Metal (AFM) and (iv) Antiferromagnetic Mott Insulator (AF-MI)Kunwar and
Karmakar (2024).

Let us now explore the interacting WM. The interacting domain hosts several
critical points. In the kagome limit the ground state phase diagram hosts
several quantum critical points, as shown in Fig. 5. Here, the QPT $U_{CM}$
corresponds to a magnetic transition between a paramagnetic and an
antiferromagnetic metal. $U_{c1}$ shows an insulator-metal transition (IMT),
while $U_{c2}$ corresponds to a metal-Mott insulator transition (MIT). The
phases and transitions are determined based on various thermodynamics,
spectroscopic, and electrical/optical transport properties, viz. magnetic
correlations, structure factor, single-particle density of states, optical and
electrical conductivity, etc.

These phases and phase transitions can also be captured based on the quantum
information perspective based on indicators such as entanglement entropy,
mutual information, etc. and such studies have been reported for a square
lattice that has a relatively simple phase diagram Walsh  _et al._ (2021,
2020, 2019a, 2019b); Qin  _et al._ (2023). The several standard Green’s
function-based techniques are used to calculate the relevant quantities viz.
determinant quantum Monte Carlo, dynamical mean field theory, static path
approximated quantum Monte Carlo, etc.

We demonstrate the behavior of the Stirling cycle with different values of the
Hubbard interaction parameter computed from numerical analysis in Fig 6 (a-j)
(see App. B for details regarding the analysis). We have chosen some
representative values of $U$ to display the changing of QHE, QR, QA, and QH
and their relationship with the quantum phases in the model. Although
introducing the interaction in the model does not drastically change the
behavior of the cycle for a given set of strain parameters (cf Fig. 4), it is
observed that the regions corresponding to different modes do change non-
trivially showing the signature of various phases of the system.

For a cycle with the strain parameters $(\theta_{1}=2\pi/3,\theta_{2}=\pi/2)$,
the regions of QH, QR and QA (where $W>0$) exist in the mode-diagrams for
$U=1.0,3.6$ and $U=4.6$ (see Fig 6 (a), (b) and (d)) but missing for $U=4.6$
and $9.0$ when the bath temperatures are low. On the other hand, for a cycle
with the strain parameters $(\theta_{1}=\pi/2,\theta_{2}=2\pi/3)$, a small
region of QHE appears in the mode diagram for $U=1.0,3.6$ and $U=4.6$ when the
bath temperatures are low (see Fig 6 (f), (g) and (i)). Otherwise, in these
cases, most of the mode-diagram is still dominated by the QHE mode (where
$W<0$) and QH, QA, QR modes (where $W>0$) for the set of strain parameters
$(\theta_{1}=3\pi/2,\theta_{2}=\pi/2)$ and
$(\theta_{1}=\pi/2,\theta_{2}=2\pi/3)$ respectively, as already shown for the
noninteracting counterpart.

This non-triviality of the results with the interacting model emerges as an
artifact of the non-monotonic change in the energy levels of the system in
both strain configurations. In other words, the behavior of the cycle will be
determined by the phase diagrams of the model with two different strain
configurations. In general, we observe that introducing the interaction in the
model non-trivially changes the efficiency of the engine for a given pair of
bath temperatures as shown in Fig. 7. The efficiency first increases with the
interaction, then decreases and increases further. It is observed to be
minimal near the regions of phase transitions.

##  V Conclusions

Quantum many-body (MB) systems have been proven to be useful resources for
realizing quantum thermal machines (QTMs) as they offer several benefits such
as cooperative performance boost, simultaneous achievements of high efficiency
and power at the quantum critical points, minimizing frictional effects, and
many more. However, the bulk of the existing literature exploring the
criticality of MB in the context of QTMs involves models sans the electronic
interactions, which are non-trivial to deal with and require sophisticated
numerical techniques. At the same time, electronic interactions in conjunction
with competing correlations and/or geometric frustration of lattices are known
to bring forth rich quantum phases and phase transitions. To harness quantum
criticality for QTMs, here we resort to the prototypical Hubbard model in two
dimensions (2D) in the framework of the line graph Lieb-kagome lattice as the
working medium of a multi-functional QTM. We resort to a non-perturbative,
static path approximated (SPA) Monte Carlo technique to deal with the
repulsive Hubbard model. We demonstrate that with this model operating between
the hot and cold baths one can achieve multiple thermal actions like quantum
heat engines, refrigerators, accelerators, and heaters in a quantum Stirling
cycle by either engineering the system strain or tuning the bath temperatures.

Furthermore, we show that the heat engine action gets better when the strain
is engineered from the kagome limit to the Lieb limit and the refrigeration
gets better for the converse. For both cases, in the low-temperature limit, we
show that the proposed QTM reaches the ultimate Carnot limit when the
temperature difference between the baths is low. The introduction of the
onsite Hubbard repulsion allows one to have the Carnot efficiency even in the
relatively high-temperature limit while the difference between the two bath
temperatures is still close to zero. It indicates that a relatively higher
temperature than the absolute zero is preferred for intermediate interaction
strength $U$ while operating a heat engine by engineering the strain from the
kagome to the Lieb limit and the converse while operating a refrigerator, near
the interaction strength ($U=4.0t$) for which we get a paramagnetic metal to
an anti-ferromagnetic metal, the low-temperature regimes is preferred for both
the thermal actions just like the non-interacting case ($U=0$).

In conclusion, we have modeled a multifunctional QTM, based on a 2D line graph
lattice having onsite Hubbard repulsion in the framework of the Lieb and the
kagome lattices, which is very efficient. The unprecedented control over the
engineering and subsequent tuning of these lattices in ultracold atomic gases,
optically induced photonic systems, artificial lattices engineered through
lithography and atomic manipulations, and more recently in a metal-organic
framework (MOF) make the proposed QTM experimentally realizable in the near
future. Therefore, our study can be a stepping stone towards exploring the
role of strong electronic interaction, which offers rich phases and phase
transitions involving magnetic orders, on QTMs.

Figure 6: The diagram of modes for a Stirling cycle operating in the
refrigerator, the accelerator, and the heater regions are marked in blue,
grey, and green respectively. The set of subfigures (a-e) corresponds to the
Stirling cycle with strain parameters $(\theta_{1}=2\pi/3,\theta_{2}=\pi/2)$
and the set (f-j) correspond to the same with strain parameters
$(\theta_{1}=\pi/2,\theta_{2}=2\pi/3)$. The Hubbard interaction parameter
assumes the following values: $U=1.0$ in (a) and (f); $U=3.6$ in (b) and (g);
$U=4.2$ in (c) and (h); $U=4.6$ in (d) and (i); $U=9.0$ in (e) and (j). The
engine region is plotted with color density where the color denotes the engine
efficiency. Figure 7:  Efficiency ($\theta_{1}=2\pi/3,\theta_{2}=\pi/2$) as a
function of $U$ for the set of bath temperatures $T_{h}=0.5,T_{c}=0.3$.

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##  Appendix A Working Medium: Tight-binding calculation

The electron field operator in the momentum space can be expressed in the
quasiparticle basis via Bogoliubov canonical transformation as,

$\displaystyle c_{i\sigma,k}=\sum_{n}u_{i\sigma}^{n}\gamma_{n,k}$ (14)
$\displaystyle
c_{i\sigma,k}^{\dagger}=\sum_{n}u_{i\sigma}^{n*}\gamma_{n,k}^{\dagger}$ (15)

where, $\sigma=\pm 1$ corresponds to the spin label and $n$ is the energy
index. The operators create (annihilate) a Bogoliubov quasiparticle at state
$n$ in the momentum space. The quasiparticle operators satisfy the anti-
commutation relations:

$\displaystyle\\{\gamma_{n,k},\gamma_{m,k}^{\dagger}\\}=\delta_{nm}$ (16)
$\displaystyle\\{\gamma_{n,k},\gamma_{m,k}\\}=\\{\gamma_{n,k}^{\dagger},\gamma_{m,k}^{\dagger}\\}=0$
(17)

These relations also guarantee the anti-commutation relations among the
original electronic field operators. With the above canonical transformation,
the Hamiltonian is diagonalized in the following form:

$\displaystyle H_{k}^{\text{diag}}$ $\displaystyle=$
$\displaystyle\sum_{n}E_{n}\gamma_{n,k}^{\dagger}\gamma_{n,k}+E_{const}$ (18)

The above Hamiltonian is diagonal in the quasiparticle basis in the momentum
space. So, the eigenvalues of the Hamiltonian for a $k$ mode given in Eq. 2
are given by Viéte’s formula Nickalls (2006),

$\displaystyle\varepsilon^{\theta}_{\mathbf{k},j}=\sqrt{\frac{2}{3}}(A^{2}_{\textbf{k}}+{B^{\theta}_{\textbf{k}}}^{2}+{C^{\theta}_{\textbf{k}}}^{2})^{1/2}\times$
$\displaystyle\cos\left[\frac{1}{3}\cos^{-1}\big{(}\frac{3\sqrt{3}A_{\textbf{k}}B^{\theta}_{\textbf{k}}C^{\theta}_{\textbf{k}}}{(A^{2}_{\textbf{k}}+{B^{\theta}_{\textbf{k}}}^{2}+{C^{\theta}_{\textbf{k}}}^{2})^{3/2}}\big{)}-\frac{2\pi(j-1)}{3}\right],$
$\displaystyle{\leavevmode\nobreak\ \leavevmode\nobreak\ }j=1,2,3.$ (19)

##  Appendix B Numerical scheme for interacting Hamiltonian

The repulsive Hubbard Hamiltonian on a 2D line graph Lieb-kagome lattice reads
as,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\mu\sum_{i,\sigma}c^{\dagger}_{i,\sigma}c_{i,\sigma}+\sum_{\langle
ij\rangle,\sigma}t_{ij}c^{\dagger}_{i,\sigma}c_{j,\sigma}+\mbox{H.
c.}+\sum_{\langle\langle
ij\rangle\rangle,\sigma}t^{\prime}_{ij}c^{\dagger}_{i,\sigma}c_{j,\sigma}+\mbox{H.
c.}$ (20) $\displaystyle+U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}.$

where, $t_{ij}$ corresponds to the nearest neighbor hopping and $t_{ij}=t=1$
sets the reference energy scale of the model. The applied strain is quantified
in terms of $t_{ij}^{\prime}=t^{\prime}$. $U>0$ corresponds to the repulsive
Hubbard interaction. We work at the half-filling and the chemical potential
$\mu$ is adjusted to achieve the same. In order to make the model numerically
tractable we decompose the interaction term using Hubbard Stratonovich (HS)
decomposition and thereby introduce two (bosonic) auxiliary fields viz. a
vector field ${\bf m}_{i}(\tau)$ and a scalar field $\phi_{i}(\tau)$, which
couples to the spin and charge densities, respectively. The introduction of
these auxiliary fields aids in capturing the Hartree-Fock theory at the saddle
point and retains the spin rotation invariance and the Goldstone modes.

In terms of the Grassmann fields $\psi_{i\sigma}(\tau)$, we have,

$\displaystyle\exp[U\sum_{i}\bar{\psi}_{i\uparrow}\psi_{i\uparrow}\bar{\psi}_{i\downarrow}\psi_{i\downarrow}]$
$\displaystyle=$ $\displaystyle\int{\bf\Pi}_{i}\frac{d\phi_{i}d{\bf
m}_{i}}{4\pi^{2}U}{\exp}[\frac{\phi_{i}^{2}}{U}+i\phi_{i}\rho_{i}+\frac{m_{i}^{2}}{U}-2{\bf
m}_{i}.{\bf s}_{i}]$ (21)

where, the charge and spin densities are defined as,
$\rho_{i}=\sum_{\sigma}\bar{\psi}_{i\sigma}\psi_{i\sigma}$ and ${\bf
s}_{i}=(1/2)\sum_{a,b}\bar{\psi}_{ia}{\bf\sigma}_{ab}\psi_{ib}$, respectively.
The corresponding partition function thus takes the form,

$\displaystyle{\cal Z}$ $\displaystyle=$
$\displaystyle\int{\bf\Pi}_{i}\frac{d\bar{\psi}_{i\sigma}d\psi_{i\sigma}d\phi_{i}d{\bf
m}_{i}}{4\pi^{2}U}\exp[-\int_{0}^{\beta}{\cal L}(\tau)]$ (22)

where, the Lagrangian ${\cal L}$ is defined as,

$\displaystyle{\cal L}(\tau)$ $\displaystyle=$
$\displaystyle\sum_{i\sigma}\bar{\psi}_{i\sigma}(\tau)\partial_{\tau}\psi_{i\sigma}(\tau)+H_{0}(\tau)+\sum_{i}[\frac{\phi_{i}(\tau)^{2}}{U}+(i\phi_{i}(\tau)-\mu)\rho_{i}(\tau)+\frac{m_{i}(\tau)^{2}}{U}-2{\bf
m}_{i}(\tau).{\bf s}_{i}(\tau)]$ (23)

where, $H_{0}$ is the kinetic energy contribution. The $\psi$ integral is now
quadratic but at the cost of an additional integration over the fields ${\bf
m}_{i}(\tau)$ and $\phi_{i}(\tau)$. The weight factor for the ${\bf m}_{i}$
and $\phi_{i}$ configurations can be determined by integrating out the $\psi$
and $\bar{\psi}$, and using these weighted configurations one goes back and
computes the fermionic properties. Formally,

$\displaystyle{\cal Z}$ $\displaystyle=$ $\displaystyle\int{\cal D}{\bf
m}{\cal D}{\phi}e^{-S_{eff}\\{{\bf m},\phi\\}}$ (24) $\displaystyle S_{eff}$
$\displaystyle=$ $\displaystyle\log Det[{\cal G}^{-1}\\{{\bf
m},\phi\\}]+\frac{\phi_{i}^{2}}{U}+\frac{m_{i}^{2}}{U}$ (25)

where, ${\cal G}$ is the electron Green’s function in a $\\{{\bf
m}_{i},\phi_{i}\\}$ background.

The weight factor for an arbitrary space-time configuration $\\{{\bf
m}_{i}(\tau),\phi_{i}(\tau)\\}$ involves computation of the fermionic
determinant in that background. The auxiliary field quantum Monte Carlo with
static path approximation (SPA) retains the full spatial dependence in ${\bf
m}_{i}$ and $\phi_{i}$ but keeps only the $\Omega_{n}=0$ mode. It thus
includes classical fluctuations of arbitrary magnitudes but no quantum
($\Omega_{n}\neq 0$) fluctuations. The resulting model can be thought of as
fermions coupled to the spatially fluctuating random background of the
classical field ${\bf m}_{i}$. With these approximations, the effective
Hamiltonian corresponds to a coupled spin-fermion model, which reads as

$\displaystyle H_{eff}$ $\displaystyle=$ $\displaystyle\sum_{\langle
ij\rangle,\sigma}t_{ij}[c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.]+\sum_{\langle\langle
ij\rangle\rangle,\sigma}t_{ij}^{\prime}[c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.]+\sum_{i\sigma}(\frac{U}{2}-\mu)\hat{n}_{i\sigma}$
(26) $\displaystyle-\frac{U}{2}\sum_{i}{\bf
m}_{i}.{\bf\sigma}_{i}+\frac{U}{4}\sum_{i}m_{i}^{2}$

where the last term corresponds to the stiffness cost associated with the now
classical field ${\bf m}_{i}$ and
${\bf\sigma}_{i}=\sum_{a,b}c_{ia}^{\dagger}{\bf\sigma}_{ab}c_{ib}={\bf
s}_{i}$.

The random background configurations of $\\{{\bf m}_{i}\\}$ are generated
numerically via Monte Carlo simulation and obey the Boltzmann distribution,

$\displaystyle P\\{{\bf m}_{i}\\}\propto Tr_{c,c^{\dagger}}e^{-\beta H_{eff}}$
(27)

For large and random configurations the trace is computed numerically, wherein
we diagonalize $H_{eff}$ for each attempted update of ${\bf m}_{i}$ and
converge to the equilibrium configuration using Metropolis algorithm.
Evidently, the process is numerically expensive and involves an ${\cal
O}(N^{3})$ computational cost per update (where $N=3\times L\times L$
corresponds to the system size), thus the cost per MC sweep is $N^{4}$. We cut
down on the computation by using the traveling cluster algorithm, wherein
instead of diagonalizing the entire lattice for each attempted update of ${\bf
m}_{i}$ we diagonalize a smaller cluster surrounding the update site. The
computation cost now scales as ${\cal O}(NN_{c}^{3})$ (where $N_{c}$ is the
size of a smaller cluster surrounding the update site), which is linear in
lattice size $N$. This allows us to access large system sizes, which is
essential to capture the inhomogeneity of the underlying magnetic phase.


# Leveraging Text Repetitions and Denoising Autoencoders in OCR Post-
correction

Kai Hakala1,2 , Aleksi Vesanto111footnotemark: 1 , Niko Miekka1,
Tapio Salakoski1 and Filip Ginter1
1Turku NLP Group, Department of Future Technologies, University of Turku,
Finland
2University of Turku Graduate School, University of Turku, Finland
<EMAIL_ADDRESS>
These authors contributed equally.

###### Abstract

A common approach for improving OCR quality is a post-processing step based on
models correcting misdetected characters and tokens. These models are
typically trained on aligned pairs of OCR read text and their manually
corrected counterparts. In this paper we show that the requirement of manually
corrected training data can be alleviated by estimating the OCR errors from
repeating text spans found in large OCR read text corpora and generating
synthetic training examples following this error distribution. We use the
generated data for training a character-level neural seq2seq model and
evaluate the performance of the suggested model on a manually corrected corpus
of Finnish newspapers mostly from the 19th century. The results show that a
clear improvement over the underlying OCR system as well as previously
suggested models utilizing uniformly generated noise can be achieved. ††A
Preprint

##  1 Introduction

Analyzing data from scanned documents is a common task in both scientific and
industrial applications. This, however is often hindered by the errors
produced by the OCR software, which can have major influence on the task at
hand. Therefore correcting the errors is a necessity before conducting many of
the analyses. OCR improvements are usually separated into three categories:
enhancing the OCR software itself, manipulating the scanned images so that the
OCR software has an easier job, and lastly correcting the errors afterwards
(Bieniecki et al., 2007; Smith, 2007; Schmaltz et al., 2017). In this paper we
focus on the the last of the three. Various different methods have been
proposed to achieve this: Schulz and Kuhn (2017) use a modular system that
combines statistical machine translation with multiple vocabulary level checks
to vote for a corrected word. The downside of such systems is the requirement
of gold standard training data, i.e. erroneous OCR output and the
corresponding manual corrections.

Recent studies have shown that the need for gold standard data can be
circumvented by synthesizing training data by replacing or adding characters
with uniform probability and training recurrent neural models to remove the
added noise (D’hondt et al., 2017). Older studies on the other hand rely on
replacing the given OCR tokens with their most similar hits from a known
vocabulary, leading to both misrecognized and other lexical variants being
replaced with a unified form (Reynaert, 2011).

In this paper we build upon both of these ideas: we train a neural sequence-
to-sequence (seq2seq) model (Cho et al., 2014) able to receive a text span
containing OCR errors as an input and producing corrected text as output. To
train the model we generate synthetic data by imposing errors on clean text
data. Instead of relying on uniformly added noise, we attempt to model the
characteristics and systematic nature of the errors produced by a real OCR
system. These errors are estimated by aligning similar text spans found from a
large OCR read newspaper corpus.

##  2 Data

The main data we attempt to correct is the National Library of Finland’s (NLF)
111https://www.kansalliskirjasto.fi/ publicly available collection of
approximately three million scanned issues of newspapers and journals ranging
from years 1771 to 1910. These scanned documents have been OCR read, but due
to many factors, such as old age, bad scans, ink leakage and the hard to read
font, Fraktur, the quality is at times very low. NLF has recently published a
corpus of 479 manually corrected issues sampled from the fore-mentioned data.
To compare our noise generation approach to a model trained with gold standard
data, we split the data on issue level to training, validation and test sets
with 60% / 20% / 20% division. The training and validation sets are only
utilized for a gold standard model, whereas the test set is used as the final
evaluation for all compared models. This results in training and test sets of
approximately 300K and 110K tokens respectively.

In addition to the NLF data, we also use two clean corpora roughly from the
same time period: Finnish literature from the Gutenberg project222Project
Gutenberg. Retrieved February 2018, from www.gutenberg.org. and the early
modern Finnish corpus provided by the Institute for the Languages of Finland
(Kotus) 333http://kaino.kotus.fi/korpus/1800/meta/1800_coll_rdf.xml. Although
the Kotus corpus aims at covering a large variety of Finnish texts from years
1809 to 1899, we only use the subset concentrating on Finnish newspapers.
Whereas there are plenty of novels in the Gutenberg project data, their style
and vocabulary are distinctly different from newspapers. The selected Kotus
dataset on the other hand shares both the style and vocabulary of the data we
wish to correct, but lacks the size of the Gutenberg corpus. The purpose for
selecting these two datasets is to assess how closely related the used corpus
has to be to the target text. Even though Gutenberg corpus could benefit from
its larger size, in this study we wish to analyze the influence of the used
text genre and thus sample 1.5M tokens from both corpora for training.

Figure 1: A sample from a single cluster. The most common character for each
position is chosen as the representative word and is subsequently aligned
against the others to get the replacements. The span can be translated as
_”Any laws and regulations could be imposed on Finnish people”._

##  3 Methods

Our approach to correcting OCR errors is to train a neural seq2seq model that
takes in a broken word as input and outputs the corrected word a single
character at a time. To this end, we require corrected and aligned OCR texts
as training data, which are expensive to manually annotate. To circumvent
this, we use clean text and generate noise on top of it. This allows us to
generate practically limitless training dataset, given that a clean text
corpus is available. In this section we first go through how a realistic noise
generation is achieved and then describe how this noise is utilized in
training the OCR post-correction model.

###  3.1 Noise generation

We experiment with generating two types of noise for our data. One with
uniform random noise, where characters in a word are randomly replaced by an
ASCII letter or a number, following the approach by D’hondt et al. (2017), and
a second with noise that attempts to mimic the actual OCR noise in the data.
To this end, we calculate a character replacement distribution that tells us
how often a given character is replaced by another character. Rather than
calculating this from our gold standard documents, we leverage text reuse, a
common phenomenon in newspapers (Smith et al., 2013). We run the OCR data
through a text reuse detection method Vesanto et al. (2017), originally
developed for analyzing how Finnish news spread over time. The tool finds
repetition pairs in the text collection and then clusters them. The method is
based on sequence alignment algorithms Altschul et al. (1990), originally
developed for gene and protein sequences, and can find text reuse clusters
even when there is a significant amount of noise in the data. Each cluster
consists of the text spans where the reuse occurs.

We only look at clusters with at least 20 text spans, as the intuition is that
if we have enough repetitions of the same piece, we can align them and
statistically determine the correct character at each position. Figure 1 shows
an example of such alignment. For each cluster we first tokenize the spans and
group every word in a span with other similar words found in the other spans
of the same cluster. Word similarity is measured with Levenshtein distance. We
then align all the words in a group with others and calculate the most
frequent character for every position. Only words that are common in the reuse
cluster are aligned and subsequently used to calculate the distribution. We
calculate a replacement occurrence for every character and the distributions
are then simply collected by counting the occurrences of such replacements
from all clusters.

With the character replacement distribution we generate noise the same way as
with the uniform noise. The only difference being that rather than using
random ASCII character replacements, we pick a random character from the
replacement distribution for the given character. This kind of noise is closer
to the actual OCR noise and therefore should make the model a better fit for
the task. To distinguish this approach from the uniform noise generation and
the manually corrected gold standard data, we refer to it as _realistic_ noise
in this paper. Examples of the generated noise are shown in Table 1.

We also calculate the average character error rate from the distribution and
use that as the uniform noise level, i.e. how much noise to generate on top of
the clean text. The amount of noise should be set close to the actual noise,
or the model’s accuracy will suffer D’hondt et al. (2017). This type of noise
rate is not needed for the realistic noise as each character has a probability
of being replaced by itself.

Type | Text
---|---
GOLD | kirkonkylän
REAL | tirkonkvlln
UNI | kzrkonkblän
Table 1: Examples of realistic and uniform noise compared to the Finnish word
_kirkonkylän_ , genetive form of _”parish”_.

###  3.2 Denoising Autoencoder

For correcting the OCR output we have chosen an RNN and attention based
seq2seq model as implemented in the Pytorch version of the OpenNMT software
(Paszke et al., 2017; Klein et al., ; Cho et al., 2014). This model uses a
stacked LSTM (Hochreiter and Schmidhuber, 1997) encoder to read the input
sequence and a similar decoder together with a global attention (Luong et al.,
2015) over the encoder states to form the output predictions. Beam search is
used to select the final output sequence from multiple plausible outputs.

The noisified text is fed as an input to the model and the clean version of
the corresponding text is used as the desired output. Thus the approach can be
seen as a denoising autoencoder model (Vincent et al., 2008). In the case of
the gold standard model the original OCR data is used as the input and the
manually corrected texts as output.

We use learning rate decay of 0.75 and start the decay only after 25 epochs.
Otherwise the default hyperparameters of OpenNMT are used. The training is
stopped when the model performance on the validation set is no longer
increasing. Note that only the gold standard model uses a manually corrected
validation set for this early stopping, whereas all the models based on
artificial noise rely on the same noise in the corresponding validation sets.
Thus there are no hyperparameters which are optimized for the characteristics
of the actual target data.

##  4 Results and Discussion

We evaluate all the tested models against the manually corrected newspaper
texts using character and word error rates (CER/WER) as the measurements. The
noise generation approaches suggested in this paper are evaluated against the
original OCR output as well as a seq2seq model trained on manually corrected
data. The model trained on gold standard data gives us an intuition on how
well the selected neural network architecture is able to learn the task to
begin with. This is crucial as the model performance is used as a proxy to
evaluate the quality of the generated noise.

Four models with generated noise are evaluated in total: one with uniform
noise and another with realistic noise for both Gutenberg and Kotus corpora.
With both corpora even the uniformly generated noise can be utilized to
improve the OCR system as the models trained on Gutenberg and Kotus corpora
achieve +3.79pp and +3.6pp increase over the original OCR measured in WER (see
Table 2). This supports the findings of D’hondt et al. (2017), although the
improvements are not as noticeable on our data as in their experiments.

On both used corpora the models trained with realistic noise achieve clearly
better performance than the models trained on uniform noise, achieving +5.90pp
and +5.69pp in WER over original OCR, for Gutenberg and Kotus respectively.
This suggests that our original assumption of uniformly generated noise not
being optimal for OCR post-correction is correct and that the OCR produced
noise can be estimated efficiently by aligning the repeating text spans found
in large text corpora. As both corpora result in fairly similar performance,
we conclude that in our current setting selecting a corpus with the same text
type and genre as the target data is not critical as long as the texts are
roughly from the same time period.

Although the denoising model based on artificial noise leads to clear
improvements in OCR quality, the model trained on manually corrected data
achieves far better results with +12.38pp in WER over original OCR. We
speculate that this difference is for a large part caused by the fact that in
this initial study we have simplified the noise generation to one-to-one
character replacements whereas in reality many of the OCR errors are of more
complex nature, which can be learned from the manually corrected data. Thus,
as a future work, we should look into noise generator models with one-to-many
character replacements. As the found text repetitions are aligned on character
level by introducing necessary caps, these caps will also reveal the locations
where a single character has been replaced by several and our current noise
generation model should thus be easily extendable to such error types.

On character level the performance difference between the original OCR and the
trained models is more modest, even for the gold standard model. This is
mostly due to the fact that the post-processing is focusing on words with only
a few incorrect characters.One possible reason for this is that we are trying
to fix independent tokens unaware of their context, a task which is
challenging even for humans. The currently used model, however, is not in
anyway restricted to single token inputs and we have briefly assessed the
plausibility of correcting full sentences at once. The initial observations
from this experiment are conflicting as the model trained on gold standard
data resulted in abysmal 23.40pp decrease in WER, i.e. the model introduced a
plethora of new errors instead of correcting the input. However, a model
trained with realistic noise on 2.5M sentences sampled from Gutenberg data
resulted in +6.17pp over original OCR, a slight improvement over a model
trained with individual tokens. This suggest that a much larger dataset is
required for training such model, even if manually corrected data is used. On
the other hand this highlights the importance of artificially generated
training data, as manually correcting a dataset an order of magnitude larger
than the current one would require tremendous resources, whereas artificial
data can be generated infinitely by introducing varying noise on top of the
same sentences.

Another future work direction would be to look into generative adversarial
models (Goodfellow et al., 2014). In this setting there would be no need for
aligned texts spans, but instead a generative model could be used to produce
noisified texts. An adversarial network would estimate whether the noisified
text is artificially generated or a real OCR output. In this approach similar
noising and denoising neural models could be utilized, essentially reversing
each other, in a cycle-consistent adversarial setting (Zhu et al., 2017).

Model | CER | WER
---|---|---
GUT-UNI | 0.0943 | 0.2535
GUT-REAL | 0.0876 | 0.2324
KOT-UNI | 0.0941 | 0.2554
KOT-REAL | 0.0902 | 0.2345
OCR | 0.0978 | 0.2914
GOLD | 0.0680 | 0.1676
Table 2: Comparison of the tested models and baselines. The performance is
measured in character error rate (CER) and word error rate (WER). OCR = FNL
OCR without post-processing, GOLD = model trained with gold standard data, GUT
= Gutenberg corpus, KOT = Kotus corpus, UNI = uniform noise, REAL = realistic
noise.

##  5 Conclusions

In this study we show that OCR errors can be estimated by leveraging text
repetitions found from large OCR read corpora. The quantity and quality of
these repetitions is such that a reliable ground truth can be untangled to the
extent that measuring the differences between this ground truth and all the
corresponding aligned text repetitions can be utilized to approximate the
natural error distribution of the underlying OCR system.

We demonstrate that introducing artificial noise on top of clean text
following this discovered error distribution can be used as synthetic training
data for neural OCR post-correction models. Our study shows that models
trained on this type of noise outperform models trained with uniformly
generated noise.

The paper also mentions initial results from further experiments with context-
aware models and suggests various future work directions.

## Acknowledgments

This work was supported by Academy of Finland. Computational resources were
provided by CSC - IT Center For Science Ltd., Espoo, Finland.

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