# COVID-19 Diagnosis from Cough Acoustics using ConvNets and Data Augmentation

Saranga Kingkor Mahanta1, Darsh Kaushik2, Shubham Jain3, Hoang Van Truong4 and
Koushik Guha1

1Electronics and Communication Engineering Department, National Institute of
Technology, Silchar, India 788010
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>2Computer Science
and Engineering Department, National Institute of Technology,Silchar, India
788010
Email<EMAIL_ADDRESS>3Software Engineering Department, Paytm,
Noida, India 110096
Email<EMAIL_ADDRESS>4Mathematics in Computer Science,
University of Science of HCMC, Ho Chi Minh City, Vietnam 700000
Email<EMAIL_ADDRESS>
###### Abstract

With the alarming rise of the COVID-19 and countries being inflicted by waves
of decimation due to it, an An efficient, economic, and easy effortless
diagnosis procedure for the disease had COVID-19 has been the utmost need of
the hour. This task is to detect whether a person is positive for The COVID-19
based on their cough. Such individuals may differ from healthy positive
individuals may even be asymptomatic making the diagnosis even difficult. But
amongst the infected subjects, the asymptomatic ones need not be entirely free
of symptoms caused due to the virus. They might not show any observable
symptoms but, like the symptomatic subjects, they may differ from uninfected
ones in the way they cough. The These differences in the coughing sounds are
minute and indiscernible to the human ear. However, , however, these can be
captured using Artificial Intelligencemachine learning based statistical
models. In this paper, we present a deep learning approach to analyze the
acoustic dataset provided in Track 1 of the DiCOVA 2021 Challenge containing
cough sound recordings belonging to both COVID-19 positive and negative
examples. To perform the classification on the sound recordings as belonging
to a COVID-19 positive or negative examples, we propose a ConvNet model. Our
model achieved an AUC score percentage of 72.23 on the blind test set provided
by the same for an unbiased evaluation of the models. The ConvNet model
incorporated with Data Augmentation further increased the AUC-ROC percentage
from 72.23 to 87.07. It also outperformed the DiCOVA 2021 Challenge’s baseline
model by 23% thus, claiming the top position on the DiCOVA 2021 Challenge
leaderboard. This paper proposes the use of Mel frequency cepstral
coefficients as the feature input for the proposed model.

###### Index Terms:

COVID-19, Speech processing, Acoustics, Deep Learning, Healthcare, CNN, Audio
Data Augmentation

##  I Introduction

The novel Coronavirus or COVID-19 has disrupted life across most countries.
The losses are compounding on a daily basis with a sporadic rise in cases.
Quite recently, new and stronger variants of the virus are being reported from
across the globe, thus making herd-immunity a thing of the distant future.
Different vaccination procedures having their intricacies are being carried
out across various countries, however, no absolute cure has been discovered
yet.

The current testing methods include residual transcription-polymerase chain
reaction (RT-PCR) and rapid antigen test (RAT). RT-PCR test involves detection
of SARS-CoV-2 like nucleic acid and proteins in the nasal or throat of a
person through the use of chemical reagents [2]. Although this is an accurate
method, the setup for testing is both time and cost-consuming and hence, not
scalable given the large number of cases emerging every day. RAT is a quick
way of testing, but its reliability is questionable due to the high false
negatives. Thus, there is an urgent requirement for economic, efficient,
reliable, and methods of early diagnosis that avoid physical contact between
the examiner and the patients.

One of the most common and quickest methods today is temperature check and
contact tracing which relies on the fact that the predominant symptom of the
COVID-19 is high-temperature fever, in about 89% of cases [3]. However, since
the temperature measurement of the human body is dependent on various external
factors, the false positive rate is very high.

About 67% of positive COVID-19 cases have shown symptoms of dry cough along
with phlegm production in the remaining ones [3]. Features of coughs and other
vocal sound signals hold information regarding pulmonary health that can be
used for COVID-19 infection diagnosis. Detection of discernible symptoms can
be performed using machine learning networks trained on recorded cough sounds
from healthy and infected subjects. The DiCOVA 2021
Challenge111https://dicova2021.github.io/[4] was organized to encourage
research efforts by creating and releasing an acoustic signal dataset
comprising of cough sounds pertaining to both COVID-positive and negative
subjects and encouraging researchers to develop detection models and report
performance on a blind test set [25]. This paper is the result of our
participation in the challenge . We use where We used a Convolutional Neural
Network (CNN) architecture [5] trained on the provided dataset along with some
augmentations data augmentation that gives an area-under-the-curve (AUC) score
percentage of 87.07 on the blind test set.

##  II Related Works

Researchers around the world have shown that it is possible to detect COVID-19
through coughs and cough recordings from mobile phones using artificial
intelligence (AI) with an accuracy of over 90% [6].

Substantial experiments on cough data have been carried out since the COVID-19
outbreak. In [7] six features were extracted from data (namely Spectral
Centroid, Spectral Rolloff, Zero-Crossing Rate, MFCC, $\Delta$ MFCC, and
$\Delta^{2}$ MFCC) and fed to a Long Short-Term Memory (LSTM) network to
obtain an accuracy of 97% for cough sounds, but more importantly have shown
that MFCCs were the most important feature based on system accuracy for cough
sounds. Another CNN-based approach was followed by Bagad et.al. in [8] for
cough samples where a ResNet18 architecture (pre-trained on ImageNet) with
short-term magnitude spectrogram as input was able to attain an AUC of 72%.

App-based methods for both collections of cough audio samples and detection of
COVID-19 were followed in several works. For instance, in [9] an Artificial
Intelligence-powered smartphone app is developed to provide a preliminary
screening for COVID-19 using cough samples. Using a ConvNet, with Mel-
spectrogram images as input reported a sensitivity of 94% and specificity of
91%. Crowd-sourced data of cough and breathing sounds were used in [10], from
which various frame and segment level features were extracted and combined
with features extracted from a pre-trained VGGish network, to be then passed
through a logistic regression classifier. The results show an AUC of 80%.

Another study used an ensemble of 3 different networks for MFCCs, Mel-
spectrograms, and extra features (binary label for the presence of respiratory
diseases and fever/myalgia) respectively to predict coronavirus infection with
an AUC of 77.1% [11]. The model also generalized to Latin American and South
Asian audio samples. A relatively recent work [12] achieved a sensitivity and
specificity of 85% and 99.2% respectively, along with an AUC of 0.99 for
detecting COVID-19 positive cough samples using a ResNet-50 based multi-branch
ensemble learning architecture trained on MFCCs, Mel-frequency spectrograms,
and clinical features.

##  III Proposed Method

###  III-A Feature Extraction

The information of the rate of change in spectral bands of a signal is given
by its cepstrum. It conveys the different values that construct the formants
and timbre of a sound. The cepstral coefficients can be extracted using
Equation 1.

$C(x(t))=F^{-1}(log(F[x(t)]))$ (1)

Figure 1: Extracting Cepstral Coefficients from an audio signal

Peaks are observed at periodic elements of the original signal while computing
the log of the magnitude of the Fourier transform of the audio signal followed
by taking its spectrum by a cosine transformation as shown in Fig 1. The
resulting spectrum lies in the quefrency domain [13]. Humans perceive
amplitude logarithmically, hence a conversion to the Log-Amplitude Spectrum is
perceptually relevant, as shown in Figure 1. Mel scaling is performed on it by
using Equation 2 on frequencies measured in Hz.

$Mel(f)=2595*log(1+f/700)$ (2)

Chaotic dynamics have indicated that nonlinear phenomena exist in vocal
signals. Mel frequency cepstral coefficients (MFCCs) [14] are known to be able
to extract critical information from the non-linear and low-frequency regions
of sound. Mel-spectrograms can capture small changes in the lower frequency
regions since the Mel scale contains unequal spacing in the frequency bands,
unlike a typical frequency spectrogram having equally spaced frequency bands
[15]. Cough sounds contain more energy in the lower frequencies [16],
consequently MFCCs are an apt representation for the cough sound recordings.
It is also widely used in audio detection tasks.

The cough sound recordings of the dataset have audio samples with duration
ranging from approximately 0.79 seconds to 14.73 seconds, as shown in the
histogram plot in Fig. 2. Additionally, the input dimensions must be constant
across all the training examples to be able to feed into any neural network,
thus implying that the MFCC matrices of all the examples need to have a fixed
dimensional size. To achieve this, each of the examples must compulsorily have
a constant duration resulting in a fixed number of samples when sampled with a
constant sampling rate. Out of the 1280 recordings, it was observed that 804,
996, 1130 recordings have duration lesser than 5,6 and 7 seconds respectively.
Choosing the right duration for all the recordings is crucial. Choosing a
small duration will trim out important information from the sound. On the
contrary, choosing the maximum duration i.e 14.73 seconds will result in a
tremendous amount of sparse values in the inputs. Considering this trade-off,
we chose 154350 samples for each example, which is equivalent to 7 seconds
when sampled at 22050 samples/second, as the constant number of samples for
all the examples because a credible majority of the recordings have a duration
lesser than 7 seconds. Losing valuable information is a graver concern than
having too many sparse values since the dataset is relatively small. Moreover,
it can be observed from Figure Fig. 2 that only a few recordings have a
duration above 7 seconds. Only 150 examples had to be trimmed down while the
others had to be padded with zeros to make all 1280 recordings have a constant
number of samples of 154350.

Figure 2: Distribution of duration of audio samples in the dataset

After the required trimming and padding of the examples, 15 MFCC coefficients
were chosen per frame of each example since the lower end of the frequency
axis of the cepstrum contains the most relevant information to our particular
task viz. formants, spectral envelope, etc. Moreover, choosing a higher number
of cepstral coefficients would proportionally increase the complexity of the
model. This may result in a higher variance problem since the dataset is not
very large. 15 MFCC coefficients were extracted, using the Librosa [17] python
library, per frame of each recording. A single frame contained 2048 samples. A
hop length of 512 frames was used for the framing window. A duration of 7
seconds was chosen for each cough sound recording. The 7 second sound samples
resulted in MFCC matrices having dimensions of 15x302. MFCC features of the
sound recordings as its input and produces the probability score of the
classification as output.

###  III-B Data Augmentation

Initially, a simple neural network was trained using the provided dataset as
described in Section IV. However, due to high class imbalance, present in the
ratio of approximately 1:12 with respect to COVID-19 positive-to-negative
sounds, the performance was expectedly poor. To overcome this, upsampling of
the COVID-19 positive sound recordings was performed with data augmentation
techniques using the Python
Audiomentations222https://pypi.org/project/audiomentations/0.6.0/ package. The
classes were used with different values of probability parameters to produce a
wide range of audio recordings varied with respect to various aspects of the
sound signal according to the probability value. And to address them a range
of different augmentation functions were used. The following functions were
exploited to augment the data:

  * •

TimeStretch- The signal is time-stretched without changing the pitch.

  * •

PitchShift- The pitch of the sound is varied without changing the tempo.

  * •

Shift- The samples are shifted forwards or backwards, with or without
rollover.

  * •

Trim- The leading and trailing silence of a sound signal is removed.

  * •

Gain- The audio is multiplied by a random amplitude factor to increase or
reduce the volume.

The augmentation functions were selected and used with full consideration of
the changes that they might bring to the original sound signal. While
augmenting the data, care was taken that the essential features of a sound
signal that is the major contributing factor to be diagnosed as COVID-19
positive remain more or less the same, otherwise this upsampling would have
rather decreased the performance of the model. Pitch shifts aim to look at how
a COVID-19 cough may differ among individuals since pitch is a major component
that defines an individual’s voice. Trimming the silent parts of the sound
helps reduce sparse and needless components. Varying the volume would not
affect the contributing features that lead to the diagnosis. Thus, the applied
transformations result in physically realizable coughs.

The data that was finally used for training after the augmentation had 1280
recordings of cough sound: 965 COVID-19 negative and 315 COVID-19 positive
subjects. The class ratio was improved to approximately 1:3, as shown in Fig.
3.

Figure 3: Class distribution of the augmented dataset

###  III-C Model Architecture

The sound recordings converted to MFCC matrices were used as input features
separately into a multilayer perceptron (MLP) network [18], an LSTM Recurrent
Neural Network [19], and our CNN. Those literature architectures are shown in
Section II. The evaluation results obtained from the described architectures,
including our proposed model and the baseline model, are shown in Table I. The
baseline model of the DiCOVA challenge used a Random Forest classifier trained
with fifty trees. The participants were instructed to submit the output
probabilities corresponding to each sound recording name.

Figure 4: Proposed CNN architecture

The blind test set was provided in the DiCOVA 2021 challenge, along with the
data. It consisted of 233 new cough sound recordings on which the models were
to perform a classification between the COVID-19 positive and negative
examples. The MFCC feature matrices are visually comprehensible and the recent
results of CNN image representations show considerable performance gains for
the visual recognition tasks. Therefore we propose ConvNets architecture as
illustrated in Fig. 4 and elucidated below:

  1. 1.

Convolution layer with 64 filters, kernel size of 3x3, stride of 1x1, valid
padding followed by ReLU activation function, accepting an input shape of
302x15x1.

  2. 2.

Max pooling layer with a pool size of 2x2.

  3. 3.

Another Convolution layer with 32 filters, kernel size of 2x2, stride of 1x1,
valid padding followed by ReLU activation function.

  4. 4.

Batch normalization layer

  5. 5.

The resultant shape was then flattened for the subsequent fully connected
layers.

  6. 6.

Fully connected layer having 256 units with kernel, bias, and activity
regularizers, followed by ReLU activation function.

  7. 7.

A dropout layer with a rate of 0.5.

  8. 8.

Another fully connected layer having 128 units with kernel, bias, and activity
regularizers, followed by ReLU.

  9. 9.

Another dropout layer with a rate of 0.3.

  10. 10.

Output layer having 1 neuron with Sigmoid activation function.

##  IV Experiment SettingsSetup

The dataset provided for Track 1 of the DiCOVA challenge is a subset of the
Project Coswara database[20] and contains a total of approximately 1.36 hours
of cough sound recordings from 75 COVID-19 positive subjects and 965 COVID-19
negative subjects. In addition, the organizers of this challenge adopt a blind
test set containing 233 samples, so the results obtained in this paper on the
test set are given by the organizers of the challenge.

For evaluating the proposed model performance, the metric have been used is
the AUC. To adhere to the rules of the DiCOVA competition, the evaluation
parameters have been calculated based on the mean confusion charts from the
5-fold cross-validation.

The model was trained and evaluated using the stratified k-fold cross-
validation technique [21] with 5 folds, similar to the number of folds
provided in the challenge. The Adam optimizer [22] was used with an initial
learning rate of 0.0001 while training on the examples which were further
divided into mini-batches of size 32, to implement mini-batch gradient descent
with respect to the binary cross-entropy loss function. The training took
place over 200 epochs per fold. In the randomness experiment, we only change
the random seeds at the beginning of the pipeline. For better training, the
order of the training samples is shuffled randomly after every epoch. All the
model training and evaluation tasks in this work have been executed in a
system with Google Colab, 16 GB RAM, and NVIDIA Tesla V100 GPU.

##  V Results

TABLE I: Model AUC scores on the blind test set Model Architecture | Test AUC
---|---
Random Forest (baseline) | 70.69
MLP | 56.57
LSTM | 70.67
ConvNets | 72.33
ConvNets and Data Augmentation | 87.07

As shown in Table I, among the four models, our proposal architecture gave the
best performance on the cough sounds. Base on the class imbalance problem, we
proposed our ConvNet combine Data Augmentation. This helped to improve the AUC
score percentage from 72.33% to 87.07%, outperformed the baseline model by
23%.

Fig. 5 depicts the receiver operating characteristic (ROC) [23] curves
obtained from the model evaluation on each fold. The validation accuracy
averaged over all five folds resulted in 94.61% with a standard deviation of
2.62%. In a similar manner, a mean ROC-AUC score percentage of 97.36 was
achieved over the folds.

Figure 5: ROC curves depicting model performance on each fold

Confusion matrices for each fold were evaluated corresponding to the decision
threshold giving 80% sensitivity. Averaging these confusion matrices over the
folds for each model, the approximate confusion matrix that was obtained as
shown in Fig. 6.

Figure 6: Averaged model decisions computed at 80% sensitivity

Lastly, the proposed ConvNet model with Data Augmentation achieved a Test AUC
score of 87.07 on the blind test set, hence securing the top rank of the
DiCOVA 2021 challenge leaderboard 333Challenge Leaderboard:
https://competitions.codalab.org/competitions/29640#results. This can be
claimed as a truly unbiased evaluation of the proposed model.

##  VI Conclusion and Future Work

With just the limited amount of data that was provided, and some data
augmentation techniques, the proposed ConvNet model gave a commendable
performance on previously unseen data. It achieved a Test AUC score of 87.07
on the blind test set which outperformed the baseline model by 23%. Our model
is comparatively lightweight since no transfer learning techniques involving a
large number of pre-trained were used.

Possible further developments in this area may be the mixture of different
sound representations [24] such as Chroma Feature, Spectral Contrast, Tonnetz,
etc. Additional features from breathing sound may be extracted and interlaced.
Such features may be able to formulate distinguishing patterns in the coughing
characteristics among patients infected with respiratory diseases like asthma.
Using more features may tremendously improve the model performance provided it
is trained on a larger dataset with a more balanced class distribution to
avoid variance. The proposed model architecture may be trained on a database
that is constantly being updated and integrated with web-based applications to
boost the process of diagnosing COVID-19, even amongst asymptomatic
individuals.

## Acknowledgment

We would like to thank the organizers of the DiCOVA 2021 Challenge for
conducting this impactful competition and providing the dataset. We are also
grateful to NIT Silchar for supporting us to carry out the work.

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# Spin-1/2 Ising-Heisenberg Cairo pentagonal model in the presence of an
external magnetic field: Effect of Landé g-factors

Hamid Arian Zad<EMAIL_ADDRESS>A. I. Alikhanyan National Science
Laboratory, Alikhanian Br. 2, 0036 Yerevan, Armenia  ICTP, Strada Costiera 11,
I-34151 Trieste, Italy  Andrea Trombettoni<EMAIL_ADDRESS>Department of
Physics, University of Trieste, Strada Costiera 11, I-34151 Trieste, Italy
CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via Bonomea 265, I-34136
Trieste, Italy  Nerses Ananikian<EMAIL_ADDRESS>A. I. Alikhanyan National
Science Laboratory, Alikhanian Br. 2, 0036 Yerevan, Armenia  CANDLE
Synchrotron Research Institute, Acharyan 31, 0040 Yerevan, Armenia  BLTP,
Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia

###### Abstract

In the present paper, a study of the magnetic properties of a spin-1/2 Ising-
Heisenberg Cairo pentagonal structure is presented. The model has been
investigated in Ref. Rodrigues  in the absence of external magnetic field.
Here, we consider the effects of an external tunable magnetic field. By using
the transfer matrix approach, we investigate the magnetic ground-state phase
transition, the low-temperature magnetization process, and how the magnetic
field influences the various thermodynamic parameters such as entropy,
internal energy and specific heat. It is shown that the model exhibits
intermediate magnetization plateaux accompanied by a double-peak in the
specific heat curve versus temperature. The position of each magnetization
jump is in accordance with the merging and/or separation of the two peaks in
the specific heat curve. Considering different g-factors for the nodal Ising
spins and spin dimers also results in arising different intermediate plateaux
and to remarkable alterations of the thermodynamic properties of the model.

##  I Introduction

Obtaining a deep understanding of low-dimensional magnetic materials and
quantum spin systems with competing interactions or geometrical frustration
have become an intriguing subject of research in in condensed matter physics,
material science and inorganic chemistry. Many investigations focused on
quantum ferrimagnetic chains have been carried out, due to that they present a
relevant combination of ferromagnetic and antiferromagnetic phases. As a
result of zero and finite temperature phase transitions, these materials
present various ground states and thermal properties Kitaev ; Gu ;
Dillenschneider ; Ivanov ; Werlang ; Sachdev ; Rojas2 ; RojasM ; Strecka .
Spin ladders have been also exensively studied, such as the ones consisting of
square-shaped topological units along the ladder, which allows one to solve
them using various methods Strecka ; Koga ; Okamoto ; Muller ; Vuletic ;
Notbohm ; Buttner ; Blundell ; Bacq ; Arian1 ; Feng2007 ; Giuseppe2020 .
Analogously, 2-D lattices like spin-1/2 Heisenberg antiferromagnets on the
kagome lattice have been also widely examined within the Lanczos
diagonalization both for the ground state properties up to $48$ spins
Moessner2019  and at finite temperature up to $42$ spins Schnack2018  and
using DMRG White2011 ; Kolley2015 .

During the past two decades it has become possible to synthesize a large
variety of compounds such as
$\mathrm{A}_{3}\mathrm{Cu}_{3}(\mathrm{PO}_{4})_{4}$ with
$\mathrm{A}=\mathrm{Ca},\mathrm{Sr}$ Drillon1 ,
$\mathrm{Cu}_{3}\mathrm{Cl}_{6}(\mathrm{H}_{2}\mathrm{O})_{2}\cdot
2\mathrm{H}_{8}\mathrm{C}_{4}\mathrm{SO}_{2}$ Okamoto1999 ; Okamoto2003 , the
ferromagnetic diamond chains in polymeric coordination compound
$\mathrm{Cu}_{3}(\mathrm{TeO}_{3})_{2}\mathrm{Br}_{2}$ Uematsu  and the
natural mineral azurite
[$\mathrm{Cu}_{3}(\mathrm{CO}_{3})_{2}(\mathrm{OH})_{2}$] Kikuchi1 ; Kikuchi2
, which can be properly studied in terms of Heisenberg spin models. Recently,
a mixed $3d/4f$ cyclic coordination cluster with a ground-state spin of $S=60$
was studied experimentally in Baniodeh , where the authors synthesised the
$\mathrm{Gd}$-containing isotropic member of a new series of cyclic
coordination clusters, forming a nano-torus with alternating gadolinium and
iron ions with a nearest neighbour $\mathrm{Fe}–\mathrm{Gd}$ coupling and a
frustrating next-nearest neighbour $\mathrm{Fe}–\mathrm{Fe}$ coupling. This
arrangement corresponds to a cyclic delta or saw-tooth chain Baniodeh .
Motivated by the compound $\mathrm{Bi}_{2}\mathrm{Fe}_{4}\mathrm{O}_{9}$, in
Ref. Rojas2012  it was given a general solution for the frustrated Ising model
on the so-called 2-D Cairo pentagonal lattice, i.e. a planar lattice where the
tiling is achieved with nonregular pentagons. In Ref. Rousochatzakis2012  the
antiferromagnetic Heisenberg model on the Cairo pentagonal lattice was
studied. Afterwards, F. C. Rodrigues et al. designed an interesting spin model
for a stripe of the Cairo pentagonal Ising-Heisenberg lattice, and
investigated in detail zero-temperature phase transition and some
thermodynamic parameters for such model in Rodrigues . The advantage of the
Cairo pentagonal Ising-Heisenberg stripe geometry is to make possible
analytical calculations, and also the possibility of considering the effect of
the interactions of classical spins interacting with quantum ones. Another
advantage, exploited in the following, is that it allows for an evaluation of
the effect of external magnetic fields.

Phase transitions in spin models with competing interactions has been one of
the most interesting topics of condensed matter physics and statistical
mechanics during the last decades Arian1 ; Feng2007 ; Saadatmand ; Valverde ;
Ananikian2012 ; Abgaryan1 ; Strecka1 ; Arian2 ; Campa20 . Further studies to
investigate quantum spin models have provided exact outcomes for the ground-
state phase transition in the presence of an external magnetic field, which
can be induced through the exchange couplings Sahoo1 ; Sahoo2 ; Giri ;
Hovhannisyan . The ground-state and thermodynamics of the spin ladders
constituted by higher spins have been also examined Ivanov ; Giri .

The magnetization curves of low-dimensional quantum ferromagnets and
antiferromagnets are also a topical issue of current research interest, and
they often exhibit intriguing features such as magnetization plateaux. Exactly
solvable quantum spin models for which magnetization varies smoothly as a
function of the magnetic field until reaches its saturation magnetization
include spin-1/2 quantum chains in a transverse magnetic field Hida ;
Verkholyak , spin-1/2 quantum spin ladders Arian1 ; Cabra and spin-1/2 Ising-
Heisenberg diamond chains in a transverse magnetic field Gu ; Rojas2 ; RojasM
; Ananikian2012 ; Abgaryan1 ; Strecka1 . The specific heat of magnetic
materials has attracted as well much attention, since it may exhibit an
anomalous thermal behavior when the parameters of the Hamiltonian such as
coupling constants, spin exchange anisotropy and magnetic field are changed.
Such a function can be typically approximated under a certain thermodynamic
condition by the Schottky theory Strecka ; Karlova2016 . The associated round
maximum of the specific heat, the so-called Schottky peak, has been
experimentally detected in various magnetic compounds Bernu ; Misguich ;
Helton .

When practicable, the possibility of using an analytical approach to describe
the ground-state diagram and magnetic and thermodynamic properties of quantum
spin systems such as magnetization, entropy, internal energy and specific
heat, is certainly unvaluable. A very useful method is the transfer-matrix
formalism which has widely been applied to a number of strongly correlated
systems at zero-temperature and low temperature for studying the ground- and
low-lying state properties of spin models. In the following of the analytical
discission in Ref. Rodrigues , we here investigate the magnetic and
thermodynamic properties of the spin-1/2 Ising-Heisenberg Cairo pentagonal
model in the presence of an external magnetic field using the same transfer
matrix technique. We also consider the possibility to have in this model
different Landé g-factors Vadim2015 ; Vadim2020 , showing that the diffence
among them plays an important role in the magnetic and thermodynamic behaviors
of the model.

Figure 1: Schematic structure of the spin-1/2 Ising-Heisenberg Cairo
pentagonal model. Yellow circles represent Heisenberg dimer spins and green
circles between them, localized on the of external vertices of each pentagon,
represent the Ising nodal spins. The dashed rectangle represents an unit
block.

The paper is organized as follows. In Sec. II we describe the model and
present its thermodynamic solution with in the transfer-matrix formalism. In
Sec. III, we numerically discuss the magnetization process and the
thermodynamic behavior of the model in the presence of an external homogeneous
magnetic field. Finally, the most significant results will be summarized
together in Sec. IV.

##  II Model and exact solution within the transfear matrix formalism

The Hamiltonian of the spin-1/2 Ising-Heisenberg Cairo pentagonal model shown
in Fig. 1 can be written as

$\begin{array}[]{lcl}H=\sum\limits_{i=1}^{N}\big{[}\mathcal{H}_{i-1,i}^{ab}+\mathcal{H}_{i,i}^{cd}\big{]}+{H_{z}},\end{array}$
(1)

where the Hamiltonians of two sub-units block cell-1
($\mathcal{H}_{i-1,i}^{ab}$) and cell-2 ($\mathcal{H}_{i,i}^{cd}$) in each
block together with the Zeeman term are given by

$\begin{array}[]{lcl}\mathcal{H}_{i-1,i}^{ab}=-J({\bm{\sigma}}_{a,i}\cdot{\bm{\sigma}}_{b,i})_{\Delta}-J_{0}\big{(}s_{1,i-1}+s_{4,i}\big{)}{\sigma}_{a,i}^{z}\\\
\quad\quad\quad\quad-J_{0}\big{(}s_{2,i-1}+s_{3,i})\sigma_{b,i}^{z},\\\
\mathcal{H}_{i,i}^{cd}=-J^{\prime}({\bm{\sigma}}_{c,i}\cdot{\bm{\sigma}}_{d,i})_{\Delta^{\prime}}-J_{0}\big{(}s_{3,i}+s_{4,i}\big{)}{\sigma}_{c,i}^{z}\\\
\quad\quad\quad-J_{0}\big{(}s_{1,i}+s_{2,i})\sigma_{d,i}^{z},\\\
H_{z}=-\mu_{B}B_{z}\sum\limits_{i=1}^{N}{\big{[}}g_{1}{\big{(}}\sigma_{a,i}^{z}+\sigma_{b,i}^{z}+\sigma_{c,i}^{z}+\sigma_{d,i}^{z}{\big{)}}\\\
\quad\quad\quad+\dfrac{1}{2}g_{2}{\big{(}}s_{1,i}+s_{2,i}+s_{3,i}+s_{4,i}{\big{)}\big{]}},\end{array}$
(2)

where $N$ is the number of unit blocks. Two different Landé g-factors $g_{1}$
and $g_{2}$ are considered. The XXZ interaction between pair spins of
$ab$-dimer can be given by

$\begin{array}[]{lcl}J({\bm{\sigma}}_{a,i}\cdot{\bm{\sigma}}_{b,i})_{\Delta}\equiv
J\big{(}\sigma_{a,i}^{x}\sigma_{b,i}^{x}+\sigma_{a,i}^{y}\sigma_{b,i}^{y}\big{)}+\Delta\sigma_{a,i}^{z}\sigma_{b,i}^{z}.\end{array}$
(3)

Analogously, for the $cd$-dimer we have following definition

$\begin{array}[]{lcl}J^{\prime}({\bm{\sigma}}_{c,i}\cdot{\bm{\sigma}}_{d,i})_{\Delta^{\prime}}\equiv
J^{\prime}\big{(}\sigma_{c,i}^{x}\sigma_{d,i}^{x}+\sigma_{c,i}^{y}\sigma_{d,i}^{y}\big{)}+\Delta^{\prime}\sigma_{c,i}^{z}\sigma_{d,i}^{z}.\end{array}$
(4)

The nodal spins $s^{\alpha,i}$ ($\alpha=\\{1,2,3,4\\}$) localized on the wings
of each pentagon are Pauli operators, taking values $(-1,\;1)$. The pure
Ising-type exchange coupling $J_{0}$ represents interaction between nodal
spins $s^{\alpha,i}$ and Heisenberg dimer spins.
$\bm{\sigma}=\\{{\sigma}^{x},{\sigma}^{y},{\sigma}^{z}\\}$ are Pauli operators
of Heisenberg dimers (with $\hbar=1$) and are given by

$\sigma^{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma^{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma^{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$

The final part of the Hamiltonian (1) accounts for the Zeeman,s energy of
magnetic moments in the external magnetic field $B={B}_{z}$. We will write
$H_{z}=\sum_{i=1}^{N}\mathcal{H}^{i}_{z}$, with
$\mathcal{H}^{i}_{z}=(\mathcal{H}_{i-1,i}^{ab})^{z}+(\mathcal{H}_{i,i}^{cd})^{z}$
representing the Zeeman,s energy of each two cells in the block Hamiltonian,
and $H=\sum_{i=1}^{N}\mathcal{H}_{i}$, with
$\mathcal{H}_{i}=\mathcal{H}_{i-1,i}^{ab}+\mathcal{H}_{i,i}^{cd}+\mathcal{H}^{i}_{z}$.
From now on, we consider $J_{0}$ as energy unit for all other parameters with
$B/J_{0}$, $J/J_{0}$, $\Delta/J_{0}$ and $T/J_{0}$ being dimensionless
parameters, restoring $J_{0}$ in equations and plots when needed for clarity.
Moreover, we will set $\mu_{B}=1$.

The commutation relation between each of two different block Hamiltonians
$[\mathcal{H}_{i},\mathcal{H}_{j}]=0$ enables us to extract the partition
function of the model under consideration from the following formula

${Z}=Tr\Big{[}\displaystyle\prod_{i=1}^{{N}}\exp\big{(}-\beta{\mathcal{H}_{i}}\big{)}\Big{]},$
(5)

where $\beta=1/k_{B}T$ with $k_{B}$ the Boltzmann constant and $T$ the
temperature (for simplicity we set $k_{B}=1$). Hence, we can write the
$4\times 4$ transfer matrix $\mathcal{W}$ as it follows:

$\begin{array}[]{lcl}\mathcal{W}=\mathcal{T}_{ab}\mathcal{T}_{cd}=\sum\limits_{k=1}^{4}\exp\big{[}-\beta\mathcal{E}_{k}(s_{1,i-1}\;s_{2,i-1}\mid
s_{4,i}\;s_{3,i})\big{]}\times\\\
\quad\quad\quad\quad\quad\sum\limits_{k=1}^{4}\exp\big{[}-\beta\bar{\mathcal{E}}_{k}(s_{1,i}\;s_{4,i}\mid
s_{2,i}\;s_{3,i})\big{]}.\end{array}$ (6)

$\mathcal{T}_{ab}$ represents the transfer matrix of the cell-1 and
$\mathcal{T}_{cd}$ represents the transfer matrix of the cell-2. Symbol $\mid$
distinguishes the interplay between rows (pair nodal spins
$\\{s_{1,i-1},\;s_{2,i-1}\\}$) and the columns (nodal spins
$\\{s_{4,i},\;s_{3,i}\\}$) of the transfer matrix $\mathcal{T}_{ab}$, and
between rows (pair nodal spins $\\{s_{4,i},\;s_{3,i}\\}$) and the columns
(nodal spins $\\{s_{2,i},\;s_{3,i}\\}$) of the transfer matrix
$\mathcal{T}_{cd}$. Moreover, $\mathcal{E}_{k}$ denote the eigenvalues of the
Hamiltonian $\mathcal{H}_{i-1,i}^{ab}+(\mathcal{H}_{i-1,i}^{ab})^{z}$ and
depend on the spins $s_{\alpha,i}$ (with $\alpha=1,2,3,4$). One obtains

$\begin{array}[]{lcl}\mathcal{E}_{1,2}=-\frac{\Delta}{4}-\frac{g_{2}B}{4}(s_{1,i-1}+s_{2,i-1}+s_{3,i}+s_{4,i})\\\
\quad\quad\quad\pm\frac{1}{4}\left(2g_{1}B+J(s_{1,i-1}+s_{2,i-1}+s_{3,i}+s_{4,i})\right),\\\
\mathcal{E}_{3,4}=\frac{\Delta}{4}-\frac{g_{2}B}{4}(s_{1,i-1}+s_{2,i-1}+s_{3,i}+s_{4,i})\\\
\quad\quad\quad\pm\frac{1}{2}\sqrt{J_{0}^{2}\left(s_{1,i-1}+s_{4,i}-s_{2,i-1}-s_{3,i}\right)^{2}+J^{2}}.\end{array}$
(7)

Analogously, the corresponding eigenenergies for $cd$-dimer are given by

$\begin{array}[]{lcl}\bar{\mathcal{E}}_{1,2}=-\frac{\Delta^{\prime}}{4}-\frac{g_{2}B}{4}(s_{1,i}+s_{2,i}+s_{3,i}+s_{4,i})\\\
\quad\quad\quad\pm\frac{1}{4}\left(2g_{1}B+J^{\prime}(s_{1,i}+s_{2,i}+s_{3,i}+s_{4,i})\right),\\\
\bar{\mathcal{E}}_{3,4}=\frac{\Delta^{\prime}}{4}-\frac{g_{2}B}{4}(s_{1,i}+s_{2,i}+s_{3,i}+s_{4,i})\\\
\quad\quad\quad\pm\frac{1}{2}\sqrt{J_{0}^{2}\left(s_{1,i}+s_{2,i}-s_{4,i}-s_{3,i}\right)^{2}+J^{\prime
2}}.\end{array}$ (8)

Hereafter, for simplicity, we consider the case $J^{\prime}=J$ and
$\Delta^{\prime}=\Delta$. Since we look for eigenvalues of the transfer matrix
$\mathcal{W}$ in the thermodynamic limit $N\rightarrow\infty$, as usual the
largest eigenvalue $\Lambda_{max}$ is the only one determining the
thermodynamic properties of the system Yeomans . Hence, the free energy per
block can be obtained from the largest eigenvalue of the transfer matrix (6)
as

$\begin{array}[]{lcl}f=-\frac{1}{\beta}\lim\limits_{N\rightarrow\infty}\ln\frac{1}{N}{Z}=-\frac{1}{\beta}\ln\Lambda_{max}.\end{array}$
(9)

The specific heat, entropy, and magnetization per block can be defined as

$\begin{array}[]{lcl}{M}=-\Big{(}\frac{\partial f}{\partial
B}\Big{)}_{T}\;,\;{S}=-\Big{(}\frac{\partial f}{\partial
T}\Big{)}_{B}\;,\quad{C}=-T\Big{(}\frac{\partial^{2}f}{\partial
T^{2}}\Big{)}_{B}.\end{array}$ (10)

##  III Results and discussion

This section introduces the results obtained from the study of the possible
ground-state phase transitions, magnetization process, entropy, internal
energy and specific heat behavior of the spin-1/2 Ising-Heisenberg Cairo
pentagonal model in different cases.

###  III.1 Magnetization and ground-state phase diagram

We start by considering the case $g_{1}=g_{2}$ for the Landé g-factors.
Various possible magnetic ground-states of a unit-block of the spin-1/2 Ising-
Heisenberg Cairo pentagonal model can be identified in terms of six different
phases with the corresponding magnetization plateax at zero, one-eighth, one-
fourth, three-eighth and one-half of saturation magnetization and fully
polarized state (FPS). The magnetic ground-state phase diagrams in the
($B/J_{0}-\Delta/J_{0}$) plane are displayed in Figs. 2(a) and (b) for fixed
values of, respectively, $J=0.5J_{0}$ and $J=1.5J_{0}$ by supposing the same
g-factors for the nodal Ising spins and Heisenberg dimers, i.e., $g_{1}=1$ and
$g_{2}=1$. In Fig. 2(a) we see that at anisotropy range $\Delta>-J_{0}$ there
is a single magnetization jump from the zero plateau to the FPS saturation
magnetization. When the anisotropy decreases, the magnetization curve reveals
intermediate magnetization plateau at one-eighth of the saturation
magnetization at $0.05J_{0}\lesssim B\lesssim 0.1J_{0}$. Meanwhile,
intermediate plateaux at three-eighth and one-forth of the saturation value
are present in the magnetic field range $0.1J_{0}\lesssim B\lesssim 0.7J_{0}$
when the anisotropy changes in the interval $-3J_{0}\lesssim\Delta\lesssim-
J_{0}$.

With a further decrease of the anisotropy $\Delta\lesssim-3J_{0}$ another
intermediate plateau at one-half of saturation value is observed for the
magnetic field range $B\gtrsim 0.7J_{0}$. It is quite visible from Fig. 2(b)
that the increment of $J/J_{0}$ leads to broaden the area of one-half plateau.

Figure 2: The ground-state magnetic phase diagram of the model in the
($B/J_{0}-\Delta/J_{0}$) plane by considering fixed values: (a) $J=0.5J_{0}$
and $g_{1}=g_{2}=1$; (b) $J=1.5J_{0}$ and $g_{1}=g_{2}=1$; (c) $J=0.5J_{0}$,
$g_{1}=1$ and $g_{2}=2$; (d) $J=1.5J_{0}$, $g_{1}=1$ and $g_{2}=2$.

Let now study the case $g_{1}\neq g_{2}$, considering the ratio $g_{2}/g_{1}$
being an integer. For $g_{1}=1$ and $g_{2}=2$, as in Figs. 2(c) and 2(d), the
model has ground-states with completely different magnetization intermediate
plateaux at zero, one-sixth and one-third normalized with respect to its
saturation value.

Fig. 3 illustrates the magnetic field dependences of the magnetization per
block in the unit of its saturation value for various fixed values of the
exchange anisotropy parameter $\Delta/J_{0}$. Panel 3(a) demonstrates the
magnetization per block against the magnetic field at low temperature
($T=0.02J_{0}$) and fixed value of the isotropic coupling constant
$J=0.5J_{0}$ and $g_{1}=g_{2}=1$, where several values of the anisotropy
$\Delta/J_{0}$ have been considered. On the other hand, panel 3(b) displays
this quantity versus the magnetic field at a larger coupling constant,
$J=1.5J_{0}$, for the same set of other parameters of the panel 3 (a). As
mentioned before, the magnetization curve shows intermediate plateaux at zero,
one-eighth, one-fourth, three-eighth and one-half of saturation magnetization
for anisotropy range $\Delta/J_{0}<0$. It is shown in these figures the
magnetization jumps accompanied with the first-order ground-state phase
transition between different magnetization plateaux. Critical magnetic fields
at which magnetization jumps occur are observable as well. These points
strongly depend on both parameters $J/J_{0}$ and $\Delta/J_{0}$.

Figure 3: (a) Magnetization in units of the saturation value $M/M_{s}$ as a
function of external magnetic field $B/J_{0}$ at temperature $T=0.02J_{0}$ for
$J=0.5J_{0}$, $g_{1}=g_{2}=1$ and several exchange anisotropy ratios
$\Delta/J_{0}$. (b) $J=1.5J_{0}$, where other parameters are taken as in panel
(a). Magnetization for the cases (c) $J=0.5J_{0}$, $g_{1}=1$, $g_{2}=2$; (d)
$J=1.5J_{0}$, $g_{1}=1$, $g_{2}=2$; (e) $J=0.5J_{0}$, $g_{1}=1$, $g_{2}=3$;
and (f) $J=1.5J_{0}$, $g_{1}=1$, $g_{2}=3$.

It is evident from Fig. 3(b) that by tuning $J/J_{0}$ the magnetization jump
occurs at different critical magnetic fields. From the discontinuous ground-
state phase transition perspective, when the coupling constant $J/J_{0}$
increases, the transition from the state with magnetization $M/M_{s}=1/4$ to
that of with $M/M_{s}=1/2$ occurs at lower magnetic field, while the
transition from ground-state with the related magnetization $M/M_{s}=1/2$ to
the FPS occurs at stronger magnetic fields compared to the case $J=0.5J_{0}$.

Figure 4: Temperature dependence of the magnetization $M/M_{s}$ for several
selected magnetic fields and fixed $\Delta=-1.5J_{0}$, assuming (a)
$J=0.5J_{0}$ and $g_{1}=g_{2}=1$; (b) $J=1.5J_{0}$ and $g_{1}=g_{2}=1$; (c)
$J=0.5J_{0}$, $g_{1}=1$ and $g_{2}=2$; and (d) $J=1.5J_{0}$, $g_{1}=1$ and
$g_{2}=2$.

Again assuming different g-factors $g_{1}=1$ and $g_{2}=2$, we observe
significant changes in the magnetization curve of the model. For instance, as
shown in Fig. 3(c), when fixed values $J=0.5J_{0}$, $g_{1}=1$ and $g_{2}=2$
are considered, intermediate plateaux at zero, one-sixth and one-third of
saturation magnetization appear in the magnetization curve. By increasing the
coupling constant $J/J_{0}$, see Fig. 3(d), the width of one-sixth plateau
decreases, while the one-third plateau gets wider so that the magnetization
jump between these plateaux occurs at a lower magnetic field.

Fig. 4(a) displays the temperature dependence of the magnetization for several
fixed values of the magnetic field for $J=0.5J_{0}$, $g_{1}=g_{2}=1$ and at a
fixed $\Delta=-1.5J_{0}$. The main effect of the temperature on the
magnetization is to aggregate the magnetization curves which are strictly
discontinuous at finite, low temperatures. The magnetization gaps in this
figure denote the magnetization jump from one plateau to another. Numbers in
parentheses indicate the magnetization plateaux normalized with respect to its
saturation value. For example, $(0)\longrightarrow(1/4)$ means jumping from
the zero plateau to the one-fourth plateau in the magnetization curve. The
magnetization gaps corresponding to the jumps from one-fourth to one-half and
from one-half to the saturation magnetization are quite evident in Fig. 4(b).

The results illustrated in Figs. 4(c) and 4(d) highlight that the
magnetization behavior against temperature for the case $g_{1}=1$ and
$g_{2}=2$ is substantially different from the case $g_{1}=g_{2}=1$.

For the more general case $g_{1}=1$ and $g_{2}=ng_{1}$ with
$n=1,\;2,\;3,\;\cdots$, each magnetization plateau can be successfully
identified in terms of the considered Landé g-factors. By inspection we find

$\begin{array}[]{lcl}\begin{cases}\text{$g_{1}=1$}\\\ \text{$g_{2}=1$}\\\
\end{cases}\Rightarrow M/M_{s}=\begin{cases}\text{$\frac{2}{8}$}\\\
\text{$\frac{3}{8}$}\\\ \text{$\frac{4}{8}$}\\\
\text{$\frac{1}{8}$}\end{cases}\equiv\begin{cases}\text{$\dfrac{\frac{\alpha}{2}g_{1}}{\alpha(g_{1}+g_{2})}$}\\\
\text{$\dfrac{\frac{\alpha}{2}g_{1}+g_{2}}{\alpha(g_{1}+g_{2})}$}\\\
\text{$\dfrac{2(g_{1}+g_{2})}{\alpha(g_{1}+g_{2})}$}\\\
\text{$\dfrac{g_{2}}{\alpha(g_{1}+g_{2})}$}\end{cases},\\\ \\\
\begin{cases}\text{$g_{1}=1$}\\\ \text{$g_{2}=2$}\\\ \end{cases}\Rightarrow
M/M_{s}=\begin{cases}\text{$\frac{1}{6}$}\\\ \text{$\frac{2}{6}$}\\\
\end{cases}\equiv\begin{cases}\text{$\dfrac{\frac{\alpha}{2}g_{1}}{\alpha(g_{1}+g_{2})}$}\\\
\text{$\dfrac{g_{2}}{\alpha(g_{1}+g_{2})}$}\\\ \end{cases},\\\ \\\
\begin{cases}\text{$g_{1}=1$}\\\ \text{$g_{2}=3$}\\\ \end{cases}\Rightarrow
M/M_{s}=\begin{cases}\text{$\frac{1}{8}$}\\\ \text{$\frac{5}{16}$}\\\
\text{$\frac{3}{16}$}\end{cases}\equiv\begin{cases}\text{$\dfrac{\frac{\alpha}{2}g_{1}}{\alpha(g_{1}+g_{2})}$}\\\
\text{$\dfrac{\frac{\alpha}{2}g_{1}+g_{2}}{\alpha(g_{1}+g_{2})}$}\\\
\text{$\dfrac{g_{2}}{\alpha(g_{1}+g_{2})}$}\end{cases};\end{array}$ (11)

similar expressions could be carried out by increasing $n$. In Eqs. (11) it is

$\alpha=2+[1+(-1)^{g_{1}+g_{2}}].$

Figs. 3(e) and 3(f) illustrate the validity of the results (11). E.g., in Fig.
3(e) it is shown the magnetization per saturation against the magnetic field
for the situation $g_{1}=1$ and $g_{2}=3$, where $J=0.5J_{0}$. It is clear
that there are magnetization intermediate plateuax at $M/M_{s}=1/8$,
$M/M_{s}=5/16$ and $M/M_{s}=3/16$, in agreement with the corresponding
relationships noted in Eq. (11).

###  III.2 Entropy and internal energy

The magnetic field variations of the magnetization near the ground-state phase
boundaries may manifest themselves also in unusual behaviors of basic
thermodynamic quantities, so we hereby explore in the following the
temperature dependence of the entropy and internal energy for different
magnetic fields.

Figure 5: (a) Entropy of the spin-1/2 Ising-Heisenberg Cairo pentagonal model
as a function of the temperature for several fixed values of the magnetic
field $B/J_{0}$ at fixed coupling constant $J=0.5J_{0}$. (b) The same as in
panel (a), but with $J=1.5J_{0}$. Panels (c) and (d) illustrate the
temperature dependence of the internal energy for the same selected magnetic
fields and fixed values of other parameters as in panels (a) and (b),
respectively. In all panels it is assumed fixed anisotropy $\Delta=-1.5J_{0}$
and $g_{1}=g_{2}=1$.

Figs. 5(a) and 5(b) show thermal variations of the entropy at different
magnetic fields on a logarithmic scale for $\Delta=-1.5J_{0}$, assuming
$J=0.5J_{0}$ and $J=1.5J_{0}$, respectively. As argued in Ref. Rodrigues  in
the absence of the magnetic field, the residual entropy at $J=0.5J_{0}$ is
given by $S\longrightarrow\mathrm{ln}(3)$ when $T\longrightarrow 0$. By
applying a weak magnetic field ($0<B\lesssim 0.2J_{0}$) the residual entropy
at $J=0.5J_{0}$ decreases such that $S<\mathrm{ln}(3)$ when $T\longrightarrow
0$. When the magnetic field becomes larger than $\approx 0.2J_{0}$, the system
is highly influenced by magnetization one-fourth and one-half plateaux with
residual entropy $S\longrightarrow 0$ when $T\longrightarrow 0$. For the case
$J=1.5J_{0}$, even if a weak magnetic field is applied, the system is
dominated by ground-state phases associated to the one-fourth and one-half
plateaux with residual entropy $S\longrightarrow 0$ when $T\longrightarrow 0$.
Figs. 5 (c) and 5 (d) display the internal energy $U=f+TS$, for the same set
of parameters considered in Figs. 5 (a) and 5 (b), respectively. Obviously, by
applying an external magnetic field the internal energy remarkably decreases
at low temperatures.

Figure 6: (a) Entropy of the spin-1/2 Ising-Heisenberg Cairo pentagonal model
as a function of the the ratio $T/J_{0}$ for several fixed magnetic fields and
$J=0.5J_{0}$. (b) The same as in panel (a), but with $J=1.5J_{0}$. In (c) and
(d) we plot the temperature dependence of the internal energy of the model for
the same selected magnetic fields and the same fixed values of other
parameters as in panels (a) and (b), respectively. In all panels ii is
$\Delta=-1.5J_{0}$, and $g_{1}=1$ and $g_{2}=2$.

Let consider now $g_{1}=1$ and $g_{2}=2$. Figs. 6(a) and 6(b) show the entropy
as a function of the temperature on the logarithmic scale for the case when
these different g-factors are considered. By inspecting Fig. 6(a) and
comparing it with Fig. 5(a), one can observe a non-trivial difference in the
entropy behavior versus temperature when an external magnetic field is
applied. For more clarity, if we focus on the three selected magnetic fields
$B=\\{0.2J_{0},\;0.4J_{0},\;0.6J_{0}\\}$ we realize that, independently form
the ratio $J/J_{0}$, the entropy behaves in a remarkably different way from
the case $g_{1}=g_{2}=1$. As depicted in Figs. 6(c) and 6(d), the internal
energy has been increased at low temperatures.

###  III.3 Specific heat

Let us now examine the effects of exchange coupling $J/J_{0}$ and the magnetic
field $B/J_{0}$ on the temperature dependence of the specific heat for
anisotropy $\Delta=-1.5J_{0}$. We display in Fig. 7(a) the specific heat of
the model under consideration as a function of the temperature for several
fixed values of the magnetic field by supposing $J=0.5J_{0}$ and
$g_{1}=g_{2}=1$. The blue solid line marked with hexagons represents the
specific heat curve for $J=0.5J_{0}$ and $\Delta=-1.5J_{0}$ when $B=0$. In
Fig. 7 (a) one can see that the specific heat exhibits a double-peak when
$B=0$. When the magnetic field increases, two maxima of the double-peak
gradually merge together and make a single peak at higher temperature, see for
instance the red dashed line marked with cycles corresponding to $B=0.2J_{0}$.
With a further increase of the magnetic field ($B\gtrsim 0.2J_{0}$), an
anomalous Schottky type maximum arises at higher temperatures (green dashed
line marked with hexagons). Comparison between Figs. 7 (a) and 3 (a) shows
that the existence of the double-peak in the specific heat curve denotes the
magnetization zero plateau in magnetization curve. Therefore, one can conclude
that the spin-1/2 Ising-Heisenberg Cairo pentagonal model is in the AFM state
when a double-peak is observed in the specific heat curve. The appearence of a
single Schottky-peak nearby the critical magnetic field $B\approx 0.3J_{0}$
(when $M/M_{s}=1$) indicates that the system is in the FPS. Remarkably, a
small change in the height of the first peak at low temperature, $T\approx
0.1J_{0}$, is accompanied with the magnetization jump from the zero plateau to
the one-fourth plateau, see the evolution of specific heat from the red dashed
line marked with cycles to the gold dotted line marked with diamonds
illustrated in Fig. 7(a). In fact, the height of the first peak gradually
increases as the magnetic field increases, then slightly decreases for the
magnetic field interval $0.2J_{0}\lesssim B\lesssim 0.3J_{0}$.

Figure 7: The specific heat of the spin-1/2 Ising-Heisenberg Cairo pentagonal
model as a function of the temperature for several values of the magnetic
field $B/J_{0}$: (a)$J=0.5J_{0}$, $\Delta=-1.5J_{0}$ and $g_{1}=g_{2}=1$; (b)
$J=1.5J_{0}$, $\Delta=-1.5J_{0}$ and $g_{1}=g_{2}=1$; (c) $J=0.5J_{0}$,
$\Delta=-1.5J_{0}$, $g_{1}=1$ and $g_{2}=2$; and (d) $J=1.5J_{0}$,
$\Delta=-1.5J_{0}$, $g_{1}=1$ and $g_{2}=2$.

To gain further insight into the effects of the coupling constant $J/J_{0}$ on
the specific heat, let us also examine the specific heat behavior of the
spin-1/2 Ising-Heisenberg Cairo pentagonal model for different values of
$J/J_{0}$, which can be of particularly interest especially when the model is
in the presence of a tunable magnetic field. We have depicted in Fig. 7(b) the
typical dependences of the specific heat on the temperature or several
magnetic fields, assuming $J=1.5J_{0}$, $\Delta=-1.5J_{0}$ and
$g_{1}=g_{2}=1$. The interesting point from this figure is that the specific
heat does not show the single Schottky-peak for range $B\lesssim 0.7J_{0}$.
The particular double-peak temperature dependence is observed whose peaks
discontinuously change in height upon increasing the magnetic field. The rise
and fall of the height of first peak appeared at lower temperatures is
accompanied with the presence of magnetization plateaux.

Finally, the specific heat has a Schottky-type maximum at higher temperature
and for large magnetic fields $B\gtrsim 0.7J_{0}$, revealing the model is in a
FPS. Another interesting thing is that the Schottky-type maximum appears for
magnetic fields quite larger than the case $J=0.5J_{0}$. This phenomenon
indicates the existence of a magnetization intermediate plateau at one-half of
saturation magnetization for the case $J=1.5J_{0}$. Consequently, when the
coupling constant $J/J_{0}$ take larger values, the single Schottky peak
occurs for magnetic fields larger than the ones for $J=0.5J_{0}$. This
scenario is in agreement with the width alterations of intermediate
$1/2-$plateau, as shown in Fig. 2(b).

For completeness, Fig. 7(c) shows the specific heat versus temperature for a
number of selected magnetic fields, assuming $\Delta=-1.5J_{0}$, $J=0.5J_{0}$,
$g_{1}=1$ and $g_{2}=2$. It is clear than the Schottky maximum appears at
lower magnetic field and higher temperature intervals. Increasing $J/J_{0}$,
as reported in Fig. 7(d), the distance between the peaks of the double peak
structure increases and the single maximum arises at considerably lower
magnetic fields ($B\gtrsim 0.4J_{0}$) with respect to the case when
$g_{1}=g_{2}=1$.

##  IV Conclusions

In this paper, we have investigated the thermodynamic and magnetic properties
of the spin-1/2 Ising-Heisenberg Cairo pentagonal model in the presence of an
external tunable magnetic field. The magnetization process, entropy, internal
energy and the specific heat of this model have been studied by means of the
solution obtained within the transfer-matrix formalism. The possibility of
having different Landé g-factors was considered.

We showed that the magnetization exhbits intermediate plateaux at zero, one-
fourth and one-half of the saturation magnetization when the same Landé
g-factors are assumed for the nodal Ising spins and dimer spins. The width of
these plateaux depends on the both the isotropic and anisotropic Heisenberg
exchange couplings considered for the dimers. By decreasing the anisotropic
exchange coupling, the magnetization intermediate one-fourth and one-half
plateaux gradually appear. Increasing the isotropic coupling constant results
in widening the width of one-half plateau and decreasing the width of one-
fourth one. The magnetic field variations considerably affect the entropy and
the internal energy when the temperature goes to zero. When different
g-factors are considered, different intermediate plateaux are obtained. We
also observed a substantial change in entropy behavior and internal energy of
the model. We focused on the case in which the ratio between the Landé
g-factors is an integer, and it would be interesting to consider in the future
both the more general case in which the ratio is a rational number and the
situation in which the g-factors are incommensurable.

It has been already shown that in the absence of magnetic field the specific
heat curve of the model manifests a visible double-peak structure Rodrigues .
Here, we have shown that turning on and varying the magnetic field and the
isotropic coupling constant can remarkably modify the shape and the
temperature position of this double-peak. One of he most interesting result we
found is provided by the magnetic field dependence of the specific heat, for
which the rises and falls of the first peak are well in agreement with the
entity of magnetization plateaux. We concluded that one can recognize the
lowest-energy eigenstate of the model by investigating the evolution of first
peak of the double-peak. With a further increase of the magnetic field, the
first peak gradually disappears and a single Schottky peak is created at
higher temperatures. This single Schottky peak indicates that the model is in
the fully polarized phase. As a result, varying the exchange interactions
between dimer spins and selecting different Landé g-factors have notable
effects on the specific heat behavior versus temperature.

Finally, we mention that it would be interesting to consider more general
configurations in which spin-1/2 Ising-Heisenberg Cairo pentagonal chains are
connected, starting from the case in which they are merged as a $Y$-junction
wiht a variable leg, a geometry which has been considered for XX and Ising
quantum spin models Crampe ; Tsvelik ; Giuliano .

## Acknowledgments

The authors are grateful to O. Rojas and S. Ruffo for useful discussions.
H.A.Z. and N.A. acknowledge the receipt of the grant from the Abdus Salam
International Centre for Theoretical Physics (ICTP), Trieste, Italy, and the
CS MES RA in the frame of the research project No. SCS 18T-1C155, and the
CSMES RA in the frame of the research project No. SCS 19IT-008. N.A. and A.T.
acknowledge support from the CNR/MESRA project ”Statistical Physics of
Classical and Quantum NonLocal Hamiltonians: Phase Diagrams and
Renormalization Group”.

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