# Applying Multilingual Models to Question Answering (QA)

Ayrton San Joaquin  DIKU, University of Copenhagen  Yale-NUS College  Filip
Skubacz  DIKU, University of Copenhagen  Technical University of Munich

###### Abstract

We study the performance of monolingual and multilingual language models on
the task of question-answering (QA) on three diverse languages: English,
Finnish and Japanese. ***Work was done as part of the NLP Course at the
University of Copenhagen. We develop models for the tasks of (1) determining
if a question is answerable given the context and (2) identifying the answer
texts within the context using IOB tagging. Furthermore, we attempt to
evaluate the effectiveness of a pre-trained multilingual encoder (Multilingual
BERT) on cross-language zero-shot learning for both the answerability and IOB
sequence classifiers.

Applying Multilingual Models to Question Answering (QA)

##  1 Introduction

This report studies the performance of monolingual and multilingual language
models on the task of question-answering (QA). QA is a type of information
retrieval that involves a model being queried, and responding, with human
language. The fact that human language is used for both input and output makes
QA a natural form of human-computer interaction. Kolomiyets and Moens (2011).
We focus on both monolingual and multilingual models given their widespread
use as pretrained models for a majority of NLP downstream tasks. Multilingual
models are increasingly gaining popularity given their efficiency when
handling multiple languages, compared to using a monolingual model for each
language. We compare their performance with monolingual models in the latter
sections of this report.

We also focus on three languages: Finnish, English, and Japanese because these
languages are not part of the same linguistic family and have a wide variety
of the number of speakers (ranging from 5.8 million for Finnish to 1.5 billion
for English). Ethnologue (2022)

We make five contributions on the larger Question-Answering task:

  1. 1.

We analyze how the performance of the answerability classifiers change given
different feature representation methods.

  2. 2.

We compare the performance of IOB sequence classifiers with and without beam
search.

  3. 3.

We attempt to evaluate the effectiveness of a pre-trained multilingual encoder
(Multilingual BERT) on cross-language zero-shot learning for both the
answerability and IOB sequence classifiers.

  4. 4.

We present preliminary evidence for social bias in BERT’s masked word
predictions.

  5. 5.

We compare two answerability classifiers of the same language via precision-
recall curve, and we craft adversarial examples using salient features from
each class obtained via Integrated Gradients.

##  2 Preprocessing and Dataset Analysis

Prior to developing and discussing machine learning methods it is important to
be familiar with the dataset. We shall briefly introduce the structure of our
dataset and then proceed with an analysis thereof. Finally, we conclude with
the preprocessing steps used in section 3.

###  2.1 Dataset Insights

Throughout this work a subset of the _Answerable TyDiQA_ CopenLU  dataset
based on Clark et al. (2020) is used. The dataset consists of questions and
documents, or context, with which the answer to the respective question is to
be found — if the question is answerable. Furthermore, each sample in
Answerable TyDiQA has a ground-truth label, that indicates that the question
is, either, unanswerable or states which part of the context contains the
answer. Languages other than English, Finnish, and Japanese are filtered out;
this leaves us with $8379$, $15387$, and $9814$ samples for training and
validation, respectively. Despite this imbalance between data-subsets, there
is an approximately equal number of answerable and unanswerable questions for
each language.

For English, we almost exclusively see question words as the first token, with
"when" and "what" being the most common by a large margin. Non-question words
are very rare. In stark contrast, we observe a more diverse set of words as
the last token than as the first. The category is dominated by verbs and
nouns; the former makes up most of the top most frequent last tokens; there
are no question words to be found here.

The general distribution of first tokens is similar for Finnish, with
questions words being most dominant — again "what" and "when" lead
considerably. A notable difference is that there are more question words than
in English. Oftentimes they translate to the same English which suggests a
more specific meaning or word choice in certain contexts. The most frequent
Finnish last token is "[to] be born", while question words are also uncommon
at the question end as they are in English. In general there are a more often
(re)occurring verbs and nouns than in English but also a lot of infrequent
ones. This could be due to biased data (many similar questions) or be of
grammatical origin (word order).

Completely different is Japanese, which does not have any common question
words as the first token, instead we find a diverse collection of mostly
nouns. More specifically, many of these are proper nouns like country names.
For the last token Japanese has a rather small selection of words, mostly
question words, like "was", "what", "when", and verbs, like "do", "is".
Question words at the end of the question suggest a completely different word
order compared to English and Finnish.

From observing the types of words one could argue that the English and
Japanese sub-dataset is focused general history, Japanese possibly a lot of
World War history, while the Finnish has more questions on historical figures.

###  2.2 Preprocessing

The question and context are separately tokenized with spaCy Honnibal et al.
(2020) using the efficient spaCy models†††Specifically en_core_web_sm,
fi_core_news_sm, and ja_core_news_sm and extracting each word’s lemma. The
final result are tokenized inputs which respect the unique language
characteristics and should provide a basis for downstream tasks classification
tasks. Due to the high computational requirement we save the processed dataset
for later use.

For other classification approaches we use different preprocessing steps, this
includes the use of different tokenizers, like BPEmb Heinzerling and Strube
(2018) and the BERT tokenizer Devlin et al. (2018a), as well as storing the
input as a language model’s context vector (hidden state).

##  3 Answerability Classification

In this section, we construct a binary classifier that determines whether a
question is answerable given a document that may or may not contain the
answer.

###  3.1 Choice of Features

We run experiments based on two feature representations. When creating the
feature vector, both the question and the document are concatenated.

_Explicit Linguistic Features_. We use (1) a bag of words and (2) the percent
of shared words between the question and the document to represent discrete
features. We chose these two main features because we want to record the
similarity of the question and the document. We believe that the more similar
a question is to a subtext of the document, the more likely the answer lies in
that document. The bag of words approach encodes the number of times a given
word appears, where having multiple counts of a word suggest an overlap
between the question and document. The percent feature explicitly measures
this overlap.

_Learned Features_. One major disadvantage of the bag of words approach used
above is the bag’s size increases as the vocabulary increases. Additionally,
the vocabulary of the test set cannot be known, and hence learned from, in
advance. These two factors cause out of vocabulary words, which the previous
feature representation cannot handle. To address this, we use BPEmb by
Heinzerling and Strube (2018). It uses subword tokenization, which means it
breaks a word into subwords that have a corresponding subtoken. This avoids
the problem of out of vocabulary words, but it assumes that a word’s meaning
can be derived from its parts. Additionally, we restrict our feature dimension
to 100 for each input. Compared to the bag of words approach, which goes at
around $\approx 60000$ dimensions for each language, BPEmb dramatically
reduces the number of computations.

###  3.2 Setup

We use a 3-layer feedforward neural network with Dropout by Srivastava et al.
(2014) of $p=0.25$. We use a batch size of $512$, initial learning rate of
$1e-3$, and Adam optimizer by Kingma and Ba (2014) with weight decay of $0$
and $\beta_{1}=0.9,\beta_{2}=0.999$. We run for a maximum number of 20 epochs
with early stopping if the validation accuracy does not change for 5 epochs.
Using BPemb for each language, we set the vocabulary size to $25000$ and
feature dimensions to $100$.

###  3.3 Results

Figure 1: Validation Accuracies for the Different Languages using four
different input feature representations.

We show the Validation accuracies for our different setups in Figure 1. Except
for the Japanese model, the combination approach performs best or equal to
other methods. Based on this model, we observe that combining features who
perform good on their own may perform worse when combined with each other.
This may happen when having certain multiple features make data less
distinguishable, although we are unsure if this is the cause of our
observation.

We also make observations for the other languages. Excluding BERT-based
features, Features from BPEmb (Learned) are the most reliable across the
different languages because it results in an accuracy increase for all of
them.

##  4 Language Modelling

We now consider using neural language models to tokenize and encode the
question and document as feature vectors. These will then be passed as input
to the binary classifier we defined above. For this task, we use the base
version of BERT (BERT), which is an auto-encoding transformer model by Devlin
et al. (2018a). We use BERT that is pretrained on each language. The English
BERT (EnBERT) is the original BERT, the Finnish BERT (FinBERT) is by Virtanen
et al. (2019), and the Japanese BERT (JaBERT)is by University (2019). All of
these models have at been trained on Wikipedia in their respective language
and additional language-specific sources. This means that their training
dataset is larger than what BPEmb was trained on.

###  4.1 Bias Hidden in BERT

For BERT’s masked word prediction task, we note that BERT and the variants we
used contain bias on their predictions. We show this by constructing a
sentence that is equivalent to each language and ask BERT to give the 5 likely
predictions for the masked token.

| Prompt (In English):
---
"The poor should [MASK]."
English | Finnish | Japanese
suffer | suffer | be helped
die | die | be pitied
eat | live | unite
know | pay | be careful
survive | help | be freed
Table 1: The top-5 predicted words given the prompt: "The poor should [MASK]."
(Top is the first row). Note that the prompt is translated to the respective
language. In our limited sample of languages, Latin-based languages (e.g.
English and Finnish) clearly show Bert to encourage harmful behavior to the
target group, while the sole non-Latin-based language, Japanese, shows BERT to
encourage altruistic actions.

Table 1 shows our results. The highest predicted words for English and Finnish
(e.g. die and suffer) encourage explicit harm against the target group (the
poor). Furthermore, Finnish BERT seems to misunderstand the notion state of
being poor because one of the predicted words is "help". In many contexts, the
poor are the target of help, so it seems unusual to encourage them to help
anyone. However, JaBERT clearly encourages altruistic behavior towards the
poor. This suggests that the effects of bias can be mitigated by the choice of
dataset. However, what aspect of the dataset for JaBERT, which is mainly the
Japanese Wikipedia, minimizes bias towards harming the poor is unclear.

Our brief analysis shows that bias can manifest different languages. However,
_it is not caused by the choice of architecture_ (e.g. BERT). Furthermore,
this provides an interesting direction on non-Latin based languages. Is there
anything inherent to the datasets used to train EnBERT and FinBERT that may
cause these bias which JaBERT’s dataset did not have, besides a latin-based
alphabet? Further study is needed to measure the effect of the choice of
alphabet on bias towards harmful actions against groups.

###  4.2 Evaluation on TyDIQA

A different aspect of model biases is the domain bias stemming from the
original training dataset. For this we compute the perplexity (PPL) of EnBERT,
FinBERT, and JaBERT when they are tasked with predicting tokens in the TyDIQA
dataset of their respective language. We compute PPL separately for the
question and the document texts; the results are found in Table 2. All scores
are fairly low, meaning that the model is not very surprised by the masked
words. This is expected as the TyDIQA dataset is based on Wikipedia Clark et
al. (2020) which is a popular corpus for training language models. English and
Japanese score significantly lower than Finnish, suggesting the first two are
highly familiar with the dataset.

Language | Questions | Answers
---|---|---
English | 8.69 | 19.08
Finnish | 18.81 | 68.76
Japanese | 4.68 | 12.90
Table 2: Perplexity of pretrained, but not fine-tuned, BERT on the validation
datasets

###  4.3 Setup

Most transformer architectures, like BERT, are limited in how long input
sequences they are capable of processing. Our approach truncates the
concatenated question and context to the model maximum of $512$ tokens Devlin
et al. (2018a), thus avoiding the length limitation. While crucial information
for deciding the answerability could be lost for long sequences this is an
insignificant issue for the dataset at hand, since only $2.7\%$ of all
English, Finnish, and Japanese input sequences are longer than $512$.
Splitting sequences into overlapping parts and processing them individually
would presumably marginally improve performance — at the cost of more compute
and design complexity.

‡‡‡During work on this task, it became apparent that an improvement to the
feature extraction pipeline of the transformers library by Wolf et al. (2020)
could be made. The authors decided to add this improvement to the library.
https://github.com/huggingface/transformers/pull/19257

###  4.4 Results

Using a pre-trained BERT model as an encoder allows us to introduce provide a
much more sophisticated context vector encoding of the input. In particular,
it takes the position of words into account and the therefrom resulting
structures. It shall be noted that the model as a whole, is more complex and
has collectively been training for much longer then our other answerability
models. Built on top of a frozen BERT model the classifier is more capable of
predicting the answerability of the question and context pair.

A noticeable increase in accuracy can be observed on the validation dataset
compared to the other, more primitive, models in all three languages, c.f.
Figure 1. Further, the BERT-based models achieve an accuracy close to their
maximum already after the first epoch, however, they fail to improve beyond
$80\%$. These findings are with the aforementioned input-length limitation.

##  5 Understanding the Answerability Classifiers

We conduct an error analysis of two classifiers with different performances
followed by explaining the class-specific features for the stronger model. Our
two classifiers are BPEmb-based (Learned) and BERT-based. From the validation
accuracies in Figure 1, the Learned classifier has lower performance than the
BERT-based classifier. We evaluate both models using the Finnish validation
dataset.

###  5.1 Error Analysis

We choose the precision-recall curve (PR Curve) as our choice of evaluation
metric for error analysis. Recall that a question-context pair being
answerable implies it has a positive ground-truth (i.e. 1). Since we are using
the answerability classifier as a preprocessing model for the span model on
deployment, we want to ensure it identifies the most number of answerable
questions, i.e. maximize the true positive rate, which is exactly recall. At
the same time, we want the model to minimize the number of false positives
among positive predictions, which is precision. The PR Curve is ideal for this
task because it graphs both precision and recall across various thresholds. It
can be up to the end-user to specify a threshold and select a model from that.

Our analysis is shown by Figure 2. Recall that for PR Curves, the higher the
area under the curve (AUC), the better the model performance for various
thresholds. Similarly, if the PR Curve is a horizontal line with a constant
precision, this indicates random performance.

For similarities, both models perform better than random chance. For all
thresholds, the BERT-based classifier has lower precision-recall trade-offs,
which indicate that it performs better than the Learned classifier given our
two metrics. However, we note that for high recall values, $>70\%$, the slope
of its curve changes such that more precision is lost as recall increases.
This indicates that for high recall values, the BERT-based classifier makes
more mistakes on the negative class than for lower recall values. The effect
above is not true for the Learned classifier, which has a constant slope
throughout.

Figure 2: The BERT-based classifier is better at detecting answerable
questions than the BPEmb-based classifier for Finnish. We argue that recall is
an important metric for an answerability classifier, which the precision-
recall curve illustrates. Figure 3: Integrated Gradients (IG) attribution for
a single example. We identify salient features common within class examples
and craft adversarial examples by replacing those features with the salient
features of the other class. Figure 4: The model’s confidence decreases for
many examples when perturbed. There are some whose confidence decreases so
much that they become misclassified. In this case, the perturbation was
character replacement based on important characters for each class identified
via Integrated Gradients (IG).

###  5.2 Interpretability

Integrated Gradients by Sundararajan et al. (2017) (IG) identifies which
tokens have the most weight for determining the target label via inspecting
the gradients of the target label with respect to the input features. We
choose IG to understand which tokens the BERT-based classifier relies the most
to make its decision. We apply IG to only 5 randomly chosen examples for each
class due to computational constraints.

As shown in Figure 3, we observe for many samples (n=5) that the period (.)
and the comma (,) are one of the most important features that drive a model to
predict an example as answerable. For our adversarial attack, we therefore
replace every instance of these characters with a question mark (?) and dash
(-), respectively. These characters are the corresponding most important
features for the unanswerable class. We apply this attack to the correctly
predicted answerable examples and record how the model confidence changes.

As shown in Figure 4, this simple adversarial attack decreases the model’s
confidence on the correct label for many examples. A significant mass is
shifted to the confidence region $<0.5$, which implies that the model’s
prediction changes to the incorrect label for affected examples. More
generally, this is evidence that using a reliable explainability method like
IG, aside from its use to understand the model to improve its performance, can
be used to craft adversarial examples to fool it.

##  6 Sequence Labeling

Expanding on the task of deciding whether a question is answerable with a
document, is extractive question answering. The goal is document search, i.e.,
to find a span that answers the question — if it is answerable.

Our approach uses so-called IOB labeling, which assigns every token as either
the beginning of the answer (B), inside the answer (A), or outside the answer
(O). This reduces the task to token classification with some additional post-
processing to extract answer spans afterwards.

###  6.1 Setup

For this task we employed a _BERT-BiLSTM_ like architecture Pearce et al.
(2021) — a _mixed encoder-decoder_ architecture. Unlike most current research
we combine a transformer-based encoder with a RNN-based decoder, for practical
reasons. The encoder is a transformer model from the BERT family, while the
latter is a bidirectional LSTM (Bi-LSTM) with two layers, a hidden size of
$300$ and a dropout probability of $0.1$. The decoder shares the embedding
layer with the BERT decoder.

A major problem with using non-homogeneous encoders and decoders is that the
initial hidden-state of the decoder which stems from the encoder is
incompatible in terms of shape. We circumvent this by parameterizing a linear
transformation $(768)\to(4,768)$; where $4$ is originates from the bi-
directionality and $2$ layers of the LSTM. Aside from the hidden-state each
cell of the LSTM also has a cell-state of equal shape — we start with a zero
cell-state but a transformation, analogues to the hidden-state, is possible.

In most IOB classification tasks the distribution of labels is skewed because
O is overrepresented. For extracting answer spans from questions this is
extremely pronounced, where the majority of the context document does not
contain the answer: e.g. for English $99\%$ of tokens are labeled O.
Unfortunately, a naïve training procedure would lead to a constant model,
which always predict O and, thus, achieve a low loss. We correct for this by
weighting the cross-entropy loss for the most frequent label less ($0.01$).
The learning rate decays at a linear rate from $10^{-3}$ to $10^{-5}$ over the
course training. Both for training and evaluation the question and context are
concatenated during tokenization. If necessary because of its length the input
is split into multiple samples with the full question and truncated but
overlapping context.

As a final step the answer spans need to be extracted and a single span
selected, if the question and context sequence was split into several
segments. The most likely answer legal spans are selected, in other words for
every B tag the span starts and the first tag that violates the word order
closes span. This completely ignores standalone I for instance. If no B tokens
are present the question is regarded as unanswerable. We ignore (possible)
answer spans stretching over several segments since the overlap between
segments is large ($128$ tokens) and most (ground-truth) answers being short.

The Japanese tokenizer differs from the English and Finnish one, both of which
support returning the character-to-token offset mapping unlike the Japanese
model. No (technical) reason for the lack of support in Huggingface’s
transformers library is provided to the best of our knowledge. Unfortunately,
we were unable to overcome this obstacle due to time constraints and are
omitting the Japanese model from the subsequent results.

###  6.2 Beam Search

Instead of greedy selection beam search allows sequences (classifications)
with low initial, but higher total likelihood estimates to discovered. Every
token has a label that is one of three classes, therefore beam search can at
most consider all three possible labels. We shall compare no beam search
($k=1$) to beam search ($k=3$) on the basis of their raw F1 label-prediction
score.Early implementations also pruned illegal candidates, such as IB, BOII,
or BIOB, in addition to beam search. This was ultimately deemed to limiting
and not used in the represented results.

In general the models performs very poorly. This could be due to bad
architectural or hyper-parameter choices. Despite spending a large amount of
computational resources the models does improve significantly and experiences
high fluctuations between epochs, oftentimes overestimating certain labels
like O and I by large amounts. This becomes particularly evident from the
confusion matrix. Moreover, in some instances the model will get stuck in some
local minimum and become stagnant despite a relatively a still high learning
rate.

Without beam search, English has an F1 score for token classification of
$0.03055$ and Finnish $0.49687$. Enabling beam search marginally worsens
English to $0.029312$ and does not change the Finnish results. The latter is
because the model does not predict any I labels where the information from
beam search would be most useful, based on the confusion matrix of the
validation dataset. Due to the performance issues there is little value from
these findings.

In general we would expect beam search to further be useful with regards to
when the answer span begins and ends. Since there are only continuous answer
spans (at most one per sample), we would expect the model to be learn this
fact and be able to leverage it with the long(er) term planning that beam
search enables.

###  6.3 Qualitative Investigation

Aside from the token classification that only works to a limited degree there
are problems with the answer extraction or rather what labels the model
predicts. Both the English and Finnish model produce sequences B, O, O … when
confronted with unseen validation data, where the first element aligns with
the first token which is the beginning of the question. Obviously this
behavior is highly undesirable and leads to answer spans that are in general
always wrong. Most curiously, it does not happen when the decoder is randomly
initialized; then all predicted tokens are arbitrary — suggesting learned
behavior.

##  7 Multilingual QA

The answerability and sequence labeling models from section 4 and section 6,
respectively, are modified to use a multilingual encoder model, the
multilingual BERT model by Devlin et al. (2018b), instead of the previously
monolingual originals. Our objective is to do zero-shot cross-lingual
evaluation, where we train on language A and evaluate on languages B and C. We
then train a separate model on B and a separate model on C, and evaluate both
of them on A. Answerability and answer extraction remain independent tasks as
before, although combination thereof could yield improvements.

###  7.1 Answerability Classification

We train three classifiers. We define A to be English, B to be Finnish, and C
to be Japanese. In the order described in the preceding paragraph, the models
gain a final validation accuracy of $82.4\%,78.6\%$, and $81.7\%$ on their
native languages, respectively. However, they perform with random chance on
their target languages. We believe this is due to the languages being
unrelated to each other and thus training on one language does not lead to
improved performance on the other.

###  7.2 Sequence Labelling

Starting with a pretrained multilingual BERT model seemingly yields better
results than when using monolingual models, c.f. Table 3. Fine-tuning on
Japanese leads to the best results, whereas fine-tuning on Finnish has the
worst outcome. We note however that this section shared the same problems
observed in section 6.

Train lang. | Val. lang. | F1 [%] | Exact [%]
---|---|---|---
English | Finnish | 39.70 | 38.5
English | Japanese | 36.02 | 36.0
Finnish | English | 34.95 | 33.0
Finnish | Japanese | 30.02 | 30.0
Japanese | English | 46.79 | 45.5
Japanese | Finnish | 41.58 | 41.0
Table 3: Zero-shot cross-lingual validation of extractive QA as reported by
the SQuAD v2 metric on a validation dataset of 200 randomly drawn samples

##  8 Conclusion

We investigate the performance of various feature extraction techniques for
Question-answering (QA). We started with handcrafted linguistic features to
automatically learned features from BERT and its variants, which is generally
common in modern large language models. We found that while there are
performance improvements to using the latter in terms of accuracy, perplexity,
and F1 score on the three languages, we demonstrated that BERT exhibits
language bias and is vulnerable to adversarial inputs.

We note that many implementations of QA models, such as from the popular
HuggingFace library, use span-based answering rather than entity recognition.
By using IOB tagging in this project, we realize that span-based answering is
not ideal when answers are spread throughout the context.

Finally, we attempted multilingual zero-shot approach for both answerability
classification and answer extraction — neither of which yielded any conclusive
results. Any relative performance comparison is pointless under these
circumstances. The answer extraction is particularly challenging due to its
complexity and constraints, including deviation from common SOTA approaches.

We leave the question of improving answerability classification using
multilingual models for future work.

## Contributions

Ayrton worked on the introduction, answerability classification, language
modelling, understanding the answerability classifiers, multilingual QA, and
conclusion sections.

Filip worked on the preprocessing, language modelling, sequence labeling,
multilingual QA, and conclusion sections.

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# A Differential Geometric study of the Auxiliary Space of Interacting
Topological System

Y R Kartik  Rahul S  Ranjith Kumar R  Poornaprajna Institute of Scientific
Research, 4 Sadashivanagar Bangalore-560 080, India.  Manipal Academy of
Higher Education, Madhava Nagar, Manipal, 576104, India.  Amitava Banerjee
Department of Physics, University of Maryland College park, MD 20742, USA.
Sujit Sarkar  Poornaprajna Institute of Scientific Research, 4 Sadashivanagar
Bangalore-560 080, India.

###### Abstract

Differential geometry serves as an important tool in different branches of
theoretical physics. Here we present a differential geometric treatment of
curvature in the auxiliary space for the non-interacting and interacting
systems and also the symmetry aspect of it. We present the necessary and
sufficient conditions to characterize the topological nature of curve in the
auxiliary space. We present the results both for the repulsive and attractive
regimes of the parameter space. We try to picture the entire problem from the
geometric means of curvature. We characterize the curvature in the auxiliary
space, in terms of elliptical and cycloidal motion for the non-interacting and
interacting system respectively. We also try to explain the topological
aspects of physics in terms of Berry connection curves. To the best of our
knowledge, this is the first application of differential geometry to the
topological state of matter. This study gives a new perspective for the
understanding topological state of a quantum system.

Introduction
Curves and angles are the effective way of expressing the geometric properties
of a physical system. By this way we can define curvature, nature of wave
function, periodicity and topological properties of the system [1, 2, 3]. It
is more interesting to analyze the nature of these properties in the presence
of some interaction. The geometrical studies of the condensed matter systems
have been an interesting area of research which has rapidly picked up the pace
when the principles of topology and geometry played its foundations in quantum
condensed matter systems [4, 5]. Here in this paper, the topological behavior
of the model Hamiltonian are studied from the perspective of auxiliary space,
complex variable analysis and the curve theory of differential geometry.
Definition of curvature: Curvature is a tool to measure how curved a curve is.
In other words, curvature measures the extent to which a curve a deviates from
a straight line. For a unit speed curve $\gamma(t),$ where t is a parameter,
curvature $\kappa(t)$ at a point $\gamma(t)$ is defined to be
$||\ddot{\gamma}(t)||$ [6]. The main motivation is to explain the many body
system in a much efficient manner. A many body system is difficult to analyze
by usual theoretical calculation. But by simulating the system to a normal
geometric structure we can explain the phenomenons like phase transition,
geometric phase, divergence in a much efficient manner. The topological
configuration space of a system gives rise to the particular value of
topological invariant quantity like winding number. The closed curves in the
configuration space are the auxiliary space curves which also specifies the
winding number of the system [7, 8]. Auxiliary space curves has a unique way
of representing the topological quantum phase transition. When the system is
in the topological state, the auxiliary space curve encircles the origin. When
it is in the non-topological state, the origin lies outside the space curve.
At the point of phase transition, the origin lies on the space curve at which
case the topological invariant number cannot be defined [9]. We will realize
in this study that it is necessary but not the sufficient one. Here the
auxiliary space curve is use to study the topological phase transition of the
system both in presence and absence of the interaction.
Berry’s phase is the physical entity which characterizes the topological state
of the quantum system [7]. Berry phase exists, where there is closed interval.
But Berry connection is a more fundamental quantity which tells the path of
evolution of ground state wave function. Berry connection is equivalent to the
vector potential in electromagnetic theory. In the same way Berry connection
is also a curve, we would like to study the similarity and difference between
the curves in the auxiliary space curves for the different Hamiltonians.
Differential geometry deals with the study of problems by means of
differential calculus, integral calculus and linear algebraic techniques [10,
11, 12]. Differential geometry has a significance in general theory of
relativity. The concept of manifold, curved space-time, gravity is explained
much efficiently [13, 14, 15]. Here in this manuscript we tried to analyze the
role of differential geometry in the topological state of matter. To the best
knowledge this is the first study of differential geometry for the topological
state of matter.
Understanding the topological state state of matter
Topological insulators are the materials which conducts along the edge [16]
and is an insulator at the bulk. The conduction along the edge is protected by
two major symmetries [17, 18, 19], time reversal and particle conservation
symmetry. The first experimentally realized 3D topological insulator state
with symmetry protected surface states was discovered in bismuth-antimony
[20]. Topological Quantum Phase Transition points (TQPT) are not accompanied
with the spontaneous symmetry breaking just like Quantum Phase
Transitions(QPT)(local order parameter). Topological invariants for
topological quantum states are protected by the non zero energy gaps, where
the band gap closes at some particular points in the Brillouin Zone (BZ).
Hence the phase transition is ill defined at this point. To understand the
phase transition phenomenon, we can study winding number, which tells how many
times the Hamiltonian winds the TQPT point. Hence we can get the difference
between the phases before and after TQPT point. The phase acquired during this
evolution is the geometric phase. For the 1D it is Zak phase [21, 22, 23, 24,
25]. To study the effect of interactions on the topological invariant number
and topological quantum phase transitions, we apply momentum dependent
interaction to our model Hamiltonian [26]. When it is applied to topological
states.
Aim of the study: To study the nature of auxiliary curves from the
differential geometric curvature perspective and the Berry connection for
better insight into the topological system. The interaction which we propose
has a physical reliability that it preserve the Hermitian property of the
Hamiltonians. Differential geometry is very wide and powerful mathematical
branch which has a wide application in many branches of theoretical physics
[27, 13, 28, 29, 30, 31]. Here Our study is two folded. First one is to study
the behavior of auxiliary space for the interactive state of system. Second is
to analyze upto what extent we can use differential geometry to study the
topological properties of curves in auxiliary space of system.
This paper is organized in following manner. First we introduce the model
Hamiltonian and the present a detail analysis of auxiliary space curves. The
characteristics and behavior of auxiliary space curves with and without
interaction is presented here. In the second part we study Berry connection
for the model Hamiltonian under different conditions. And we tried to analyze
the behavior of Berry connection for topological and non-topological state of
system. In the third part we give a detail analysis of differential geometric
study of curvature to the auxiliary space curves.

##  I Hamiltonians Under Consideration and Nature of Their Auxiliary Space

Auxiliary space or parametric space is the set of possible combination of
values of parameters contained in the given system [9]. Here by plotting the
parameters of the system with respect to each other we can get better
understanding of the system [32]. Here in particular case our system of
consideration is Kitaev Hamiltonian. By this parametric plots we can get the
idea about closeness of the curve, phase transition condition, behavior of the
system under interaction and other informations.

###  I.1 Basic Model Hamiltonian:

We consider the Kitaev’s chain as our model Hamiltonian [26],

$H_{0}=[\sum_{n}-{t}({c_{n}}^{\dagger}c_{n+1}+h.c)-{\mu}{c_{n}}^{\dagger}{c_{n}}+{|\Delta|}({c_{n}}{c_{n+1}}+h.c)],$
(1)

where $t$ is the hopping matrix element, $\mu$ is the chemical potential and
$|\Delta|$ is the magnitude of the superconducting gap. We write the
Hamiltonian in the momentum space as,

$H_{1}=\sum_{k>0}(\mu+2t\cos
k)({\psi_{k}}^{\dagger}{\psi_{k}}+{\psi_{-k}}^{\dagger}{\psi_{-k}})\\\
+2i\Delta\sum_{k>0}\sin
k({\psi_{k}}^{\dagger}{\psi_{-k}}^{\dagger}+{\psi_{k}}{\psi_{-k}}),$ (2)

where ${\psi^{\dagger}}(k)(\psi(k))$ is the creation (annihilation) operator
of the spin less fermion of momentum $k$.
We can write the Hamiltonian in the BdG format as

$H_{BdG}(k)=\left(\begin{matrix}\chi^{(1)}(k)&&i\chi^{(2)}(k)\\\
-i\chi^{(2)}(k)&&-\chi^{(1)}(k)\\\ \end{matrix}\right).$ (3)

We can express the Hamiltonian by Anderson Pseudo spin approach [33, 8, 32].
One can write the BdG Hamiltonian in the pseudo spin basis.

$\vec{H}(k)=\chi^{(1)}(k)\vec{y}+\chi^{(2)}(k)\vec{z}\Rightarrow
H_{BdG}(k)=\Sigma_{i}\vec{\chi}^{i}(k).\vec{\tau}^{i},$

Where $\tau$’s are the Pauli spin matrices which act in the Nambu basis of
$H_{BdG}$.
Here the set of parametric equations are,

$\chi^{(1)}(H(k))=-2t\cos k-\mu,\hskip 28.45274pt\chi^{(2)}(H(k))=2\Delta\sin
k.$

$H_{BdG}$ in the pseudo spin basis is,

$H(k)=\chi^{(1)}(H(k))\vec{y}+\chi^{(2)}(H(k))\vec{z}.$ (4)

The energy dispersion relation, $E(k)=\sqrt{(2t\cos k+\mu)^{2}+(2\Delta\sin
k)^{2}}.$
The defined model Hamiltonian is the non-interacting Hamiltonian. Now we
consider the Hamiltonian in presence of interactions. We consider a very
specific type of interaction for the complete theoretical interest. The
results of this study may trigger quantum simulation scientist to find this
type of interaction and nature of their results.
We observe that Kitaev Hamiltonian in pseudo spin basis consists of two
components. In the present study we consider the interaction only on each
components separately ($H^{(1)}$ and $H^{(2)}$) and also for the presence of
interaction in both components for the Hamiltonian $H^{(3)}$.

###  I.2 Effect of interaction on the topological state of system:

When an interaction term is added to the system, the resultant Hamiltonian can
be written as, $H=H_{0}+H_{I}$, where $H_{0}$ is the model Hamiltonian and
$H_{I}$ is the interaction term. Here interaction term is momentum dependent,
which takes the value $H_{I}=\alpha k$, where $\alpha$ is the strength of the
interaction. By introducing such interaction term to the components of the
Hamiltonian, we can investigate the behavior of the system which reveals much
information about the system. The effect of interaction has a greater
significance in the quantum simulation and this is the test bed for understand
the many-body system [34].
1) Kitaev model Hamiltonian in presence of the interaction, $\alpha k$, which
is added to the $\sigma_{x}$ component of the Hamiltonian is written in terms
of Pauli basis as,

$H_{k}=(\epsilon_{k}-\mu-\alpha k)\sigma_{z}-2\Delta\sin k\sigma_{y}.$ (5)

Presenting the Hamiltonian in matrix from as,

$\mathcal{H}^{(1)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu-\alpha
k&&2i\Delta\sin(k)\\\ -2i\Delta\sin(k)&&2t\cos(k)+\mu+\alpha k\\\
\end{matrix}\right).$ (6)

Here the set of possible parametric equations are,

$\chi^{(1)}(H^{(1)}(k))=-2t\cos k-\mu-\alpha k,\hskip
28.45274pt\chi^{(2)}(H^{(1)}(k))=2\Delta\sin k.$

$H_{BdG}$ in the pseudo spin basis is,

$H^{(1)}(k)=\chi^{(1)}(H^{(1)}(k))\vec{y}+\chi^{(2)}(H^{(1)}(k))\vec{z}.$ (7)

The energy dispersion relation can be written as, $E^{(1)}(k)=\sqrt{(2t\cos
k+\mu+\alpha k)^{2}+(2\Delta\sin k)^{2}}$.
2) Kitaev model Hamiltonian in presence of the interaction $\alpha k$ which is
added to the $\sigma_{y}$ component can be written in terms of Pauli basis as,

$H_{k}=(\epsilon_{k}-\mu)\sigma_{z}-(2\Delta\sin k+\alpha k)\sigma_{y}.$ (8)

Writing the Hamiltonian in the matrix form,

$\mathcal{H}^{(2)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)+i\alpha
k\\\ -2i\Delta\sin(k)-i\alpha k&&2t\cos(k)+\mu\end{matrix}\right).$ (9)

Here the set of parametric equations are,

$\chi^{(1)}(H^{(2)}(k))=-2t\cos k-\mu,\hskip
28.45274pt\chi^{(2)}(H^{(2)}(k))=2\Delta\sin k+\alpha k.$

$H_{BdG}$ in the pseudo spin basis is,

$H^{(2)}(k)=\chi^{(1)}(H^{(2)}(k))\vec{y}+\chi^{(2)}(H^{(2)}(k))\vec{z}.$ (10)

The energy dispersion relation, $E^{(2)}(k)=\sqrt{(2t\cos
k+\mu)^{2}+(2\Delta\sin k+\alpha k)^{2}}.$
3) The model Hamiltonian in presence of the interaction terms $\alpha_{1}k$
and $\alpha_{2}k$ which are added to both the $\sigma^{x}$ and $\sigma^{y}$
components of the Hamiltonian, can be written in terms of Pauli basis as,

$H_{k}=(\epsilon_{k}-\mu+\alpha_{1}k)\sigma_{z}-(2\Delta\sin
k+\alpha_{2}k)\sigma_{y}.$ (11)

The Hamiltonian $H^{(3)}(k)$ written in the matrix form as,

$\mathcal{H}^{(3)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu-\alpha_{1}k&&2i\Delta\sin(k)+i\alpha_{2}k\\\
-2i\Delta\sin(k)-i\alpha_{2}k&&2t\cos(k)+\mu+\alpha_{1}k\\\
\end{matrix}\right).$ (12)

Here the set of possible parametric equations are,

$\chi^{(1)}(H^{(3)}(k))=-2t\cos k-\mu-\alpha_{1}k,\hskip
28.45274pt\chi^{(2)}(H^{(3)}(k))=2\Delta\sin k+\alpha_{2}k.$

$H_{BdG}$ in the pseudo spin basis is,

$H(k)=\chi^{(1)}(H^{(3)}(k))\vec{y}+\chi^{(2)}(H^{(3)}(k))\vec{z}.$ (13)

The energy dispersion relation, $E^{(3)}(k)=\sqrt{(2t\cos
k+\mu+\alpha_{1}k)^{2}+(2\Delta\sin k+\alpha_{2}k)^{2}}.$

###  I.3 A study of Hermiticity of the Hamiltonians.

Here we check the Hermiticity condition $H^{\dagger}=(H^{*})^{T}$ for our
Hamiltonians.

$\small H(k)=\left[\begin{matrix}0&(\epsilon_{k}-\mu)+2i\Delta\sin{k}\\\
(\epsilon_{k}-\mu)-2i\Delta\sin{k}&0\end{matrix}\right],(H^{*}(k))^{T}=\left[\begin{matrix}0&(\epsilon_{k}-\mu)-2i\Delta\sin{k}\\\
(\epsilon_{k}-\mu)+2i\Delta\sin{k}&0\end{matrix}\right]^{T}.$ (14)
$H(k)^{\dagger}=\left[\begin{matrix}0&(\epsilon_{k}-\mu)+2i\Delta\sin{k}\\\
(\epsilon_{k}-\mu)-2i\Delta\sin{k}&0\end{matrix}\right]\Longrightarrow
H(k)=H^{\dagger}(k).$ (15)

For the First case,

$\small
H^{(1)}(k)=H^{(1)\dagger}(k)=\left[\begin{matrix}0&(\epsilon_{k}-\mu-\alpha)+2i\Delta\sin{k}\\\
(\epsilon_{k}-\mu-\alpha)-2i\Delta\sin{k}&0\end{matrix}\right].$ (16)

For the second case,

$\small
H^{(2)}(k)=H^{(2)\dagger}(k)=\left[\begin{matrix}0&(\epsilon_{k}-\mu)+2i\Delta\sin{k}+i\alpha
k\\\ (\epsilon_{k}-\mu)-2i\Delta\sin{k}-i\alpha k&0\end{matrix}\right].$ (17)

For third case,

$\small
H^{(3)}(k)=H^{(3)\dagger}(k)=\left[\begin{matrix}0&(\epsilon_{k}-\mu-\alpha_{1}k)+2i\Delta\sin{k}+i\alpha_{2}k\\\
(\epsilon_{k}-\mu-\alpha_{1}k)-2i\Delta\sin{k}-i\alpha_{2}k&0\end{matrix}\right].$
(18)

We observe that all the model Hamiltonians obey the condition for Hermiticity.
The addition of the interaction term does not affect the Hermitian property of
the system. Basically the Kitaev Hamiltonian is in the spin less fermion
basis. The interaction added is momentum dependent and is also a spinless
fermion term. Therefore we justify the physical relevance of the interaction.
(1). Hamiltonian $H(k)$
In the fig. 1 the auxiliary space curves for the Kitaev Hamiltonian is
presented using the parametric equations of the Hamiltonian. This gives the
simple closed Jordan curve [35].

Figure 1: Parametric plot for the Hamiltonian $H(k)$ for different values of
$\mu$.

(2). Hamiltonian $H^{(1)}(k)$
In the fig.2, the auxiliary space curve for the Hamiltonian $H^{(1)}(k)$ is
presented. The auxiliary space curves shows an interesting behavior where, if
the interaction is absent, the curve remains closed resembling the Kitaev
chain’s auxiliary space curves. When the interaction is present, the auxiliary
space curve no longer closed.
The auxiliary space curves resembles the cycloidal pattern since the
mathematical structure of the equations of cycloid and the Hamiltonian
$H^{(1)}(k)$ are the same. The general expression of the cycloid is given by
[12],

$Cyc[a,b](t)=(at-b\sin t,a-b\cos t).$ (19)

The Hamiltonian $H^{(1)}(k)$ has a mathematical structure of cycloid (Please
see eq.6 and eq.19). In general the cycloid is classified into two categories
depending on the values of coefficients. Suppose in eq.19,if $a<b$, then the
cycloid is prolate and if $a>b$, it is curate. From this classification, we
can assign our Hamiltonian $H^{(1)}(k)$, as prolate since the prolate cycloid
is self-interacting. and also it satisfies the condition $a<b$. An interesting
feature which can be observed is that, the right column is a mirror symmetric
to the left column, which is due to the change in the sign of the value of
$\alpha$.

Figure 2: Parametric plot for the Hamiltonian $H^{(1)}(k)$ for different
values of $\mu$, $\alpha$ and $t$.

(3). Hamiltonian $H^{(2)}(k)$
The auxiliary space curve of the Hamiltonian $H^{(2)}(k)$ is presented in the
fig. 3.

Figure 3: Parametric plot for the Hamiltonian $H^{(2)}(k)$ for different
values of $\mu$, $\alpha$ and $t$.

Fig. 3 represents the auxiliary space curves of the Hamiltonian $H^{(2)}(k)$.
It is observed even here that, if the interaction is turned off, the curve
becomes auxiliary space curves i.e., simple Jordan curve as we denote for
Kitaev model. In the presence of interaction term, the auxiliary space curve
no longer remains closed, it will show a standard cycloidal pattern. The left
column and the right column are plotted for positive and negative values of
$\alpha$. Unlike in the Hamiltonian $H^{(1)}(k)$ case, the right column is not
mirror symmetric with respect to the left column.
(4). Hamiltonian $H^{(3)}(k)$
The auxiliary space curve of the Hamiltonian $H^{(3)}(k)$ is presented in the
fig. 4 and fig. 5.

Figure 4: Parametric plot for the Hamiltonian $H^{(3)}(k)$ for different
values of $\mu$, $\alpha$ and $t$. Here both $\alpha_{1}$ and $\alpha_{2}$ are
opposite in sign. Figure 5: Parametric plot for the Hamiltonian $H^{(3)}(k)$
for different values of $\mu$, $\alpha$ and $t$. Here both $\alpha_{1}$ and
$\alpha_{2}$ are opposite in sign.

The curve starts to align on X-axis for small values of $\alpha_{1}$ and
aligns on the Y-axis for the small values of $\alpha_{2}$. The shape of the
auxiliary curve depends on the strength of the $\alpha_{1}$ and $\alpha_{2}$
terms.

###  I.4 An analysis from the perspective of complex variable

For a complex number $z(t)=x(t)+iy(t)$ where $a\leq t\leq b$, if $x(t)$ and
$y(t)$ are the continuous function of $t$, then the curve $z(t)$ is said to be
continuous. If $x(t)$ and $y(t)$ are differentiable, then $z(t)$ is
differentiable. If two curves $z(t_{1})\neq z(t_{2})$, then it is a simple
curve. If $t_{1}\neq t_{2}$ for $t\in[a,b]$ except at $z(b)=z(a)$ is called
simple closed curve. A contour is formed by a finite number of connected
smooth arcs. If a curve or arc C is non self intersecting is called simple
Jordan arc . If it self intersects, then it is called simple closed curve or
Jordan curve 1. For the present study the parametric equations satisfies the
condition of Jordan curve. In the presence of interaction, the curve gets
open, there are some possibilities of curve being not simple not closed type
[35]. We now observe that the curve in the auxiliary space belongs to the
simple closed curve and the open curve. Therefore we conclude that the curves
for the non-interacting Kitaev chain is the simple closed curve and the curves
for the interacting Kitaev chain are the not simple not closed curves.

###  I.5 Analyzing mirror symmetrical aspects of auxiliary space curves

For the Kitaev Hamiltonian, when the interaction term is added, the
periodicity of the Brillouin zone breaks. In the presence of interaction, the
elliptic auxiliary space curve is no longer closed but instead it forms a
cycloidal pattern [12]. So with the addition of interaction term, the closed
curve condition is deformed into open curve condition. The cycloidal structure
in the fig. 2, the left column curves are mirror symmetric to the right
column, where in the right column the curves are plotted for negative values
of $\alpha$. In the presence of interaction, the $\sigma_{y}$ component of the
Hamiltonian, the orientation of the helical spring structure is along
$k$-axis. When the sign of the interaction term $\alpha$ changes from positive
to negative, the rotation of the helical spring changes from clockwise to
anticlockwise. When interaction term is added to both the $\sigma_{x}$ and
$\sigma_{y}$ components of the Hamiltonian, the orientation of the helical
spring structure is tilted and its angle with X/Y axis depends on the strength
of the interaction terms.i.e., $\alpha_{1}$ and $\alpha_{2}$ 4.

##  II Study of Berry connection for the model Hamiltonian

For a quantum mechanical eigenstate $|n(\lambda)\rangle$ in a adiabatic
process, which is evolving as a function of parameter k, like
$|n(\lambda(k))\rangle$, where $n(\lambda)$ is the non-degenerate eigenstate.
During this process, apart from the dynamical phase, the wave function
acquires an additional geometric phase. This depends on the path evolved
during the process [36, 37, 38, 39]. Berry connection defined as
$(A_{\mu}^{(n)})_{ab}=\left\langle n,b|\partial_{\mu}|n,a\right\rangle$. Under
the gauge transformation Berry connection gives the relation as
$A(R)\longrightarrow g^{-1}(R)A(R)g(R)-ig^{-1}(R)\Delta g(R)$ [40],
By using Stokes theorem, one can connect the Berry phase and Berry connection.
By using the analytical relation, $\gamma_{n}=\oint_{c}A_{n}(R).dR$ over the
closed contour C. The resulting Berry phase will be $2\pi$ or integral
multiples of $2\pi$. If the contour is not closed, Berry phase won’t exists,
because gauge dependency is not present. The the concept of Berry phase arises
naturally from Berry connection [41, 42, 43]. Berry Connection is also a curve
in the parametric space that gets modified and shows its behavior for
different Hamiltonians. In otherwords, Berry connection tells about the rate
of change of wavefunction in the parameter space. If the wavefunction is not
varying with respect to parameter, then the Berry connection vanishes [44].
When we study auxiliary space curves, we observe that the closed curve is
transforming to a open cycloidal curve with the addition of interaction. To
analyze this nature of auxiliary space, we need to understand the behavior of
wavefunction. So, here we are interested in the study of Berry connection. At
the same time we are interested to know how far its behavior in case resembles
with the curvature in the auxiliary space. The study of Berry connection with
the variation of k gives the idea about the state of the system. For the
Kitaev Hamiltonian, at topological state condition the plot is symmetric and
periodic. If the system is loosing topological properties means, the
periodicity and symmetry of the system will be violated as shown in the
following study.
When the interaction is added to the model Hamiltonians,
$H^{(1)}(k),H^{(2)}(k)$ and $H^{(3)}(k)$, the Berry connection changes
remarkably. When interaction added, the periodicity of the k-space gets broken
and the auxiliary space curve shows the cycloidal motion instead of closed
curve. Here the Berry connection gives the idea of evolution of wave function
in the periodic and non-periodic k-space and the behavior of the wave function
in the neighboring lattice sites. Based on the strength of $\alpha$, we can
study Berry connection with varying k.
For the Hamiltonian $H^{(1)}(k)$, the interaction term is added to the
$\sigma_{x}$ component of the Hamiltonian. The study of Berry connection with
k is shown in fig. 7. Here the changing the of sign of interaction term, we
can observe the mirror symmetry in the curves. As the interaction is added to
$\sigma_{x}$, the $\sigma_{y}$ does not shows any variation. But when the
interaction is added to the $\sigma_{y}$ component fig. 8,9,10, we can
observe, the breaking of mirror symmetry when the sign of interaction term is
changed. The symmetry and periodicity of the k-space breaks in different
manner and the behavior of the wave function is different at neighboring
lattice sites. Hence the study of Berry connection with k gives better
understanding of the system.

Figure 6: The variation of Berry connection for Kitaev Hamiltonian (H) with k.
The upper figure represents the topological case($\mu<2t$). The middle figure
represents transition state ($\mu=2t$) and the lower figure represents non
topological state ($\mu>2t$).

The value of Berry connection with k is symmetric irrespective of whether the
system is in the topological state or non-topological state. At the phase
transition point, it touches zero for $k=0$. We have obtained the Berry
connection curve for the topological quantum phase transition as a smooth
curve. Otherwise it is piecewise continuous. For the topological and non-
topological states, it always has a finite value. Without the interaction term
Berry connection is a mirror symmetric curve. Corresponding auxiliary space
curves are also mirror symmetric. For the topological and non topological
states we can find divergence at boundary values of BZ (i.e., $-\pi$ and
$\pi$) but they are periodic in nature.

###  II.1 Study of Berry connection for Hamiltonian $H^{(1)}(k)$

Here we consider the $H^{(1)}(k)$ model Hamiltonian. When the interaction term
$\alpha k$ is added to the $\sigma_{x}$ component of the Hamiltonian. The
resulting Berry connection is presented in fig. 7.

Figure 7: The variation of Berry connection with k of the Hamiltonian
$H^{(1)}(k)$.

The figures contain of three panels, Upper, middle and lower one. The value of
$\mu$ varies Berry connection significantly. As we increase the value of
chemical potential, the diverging nature transforms from negative potential
area to positive potential area. The left panel shows the variation of Berry
connection with k for positive values of $\alpha$ considered as repulsive
potential. Here the curve don’t follow any symmetry. In all three cases, the
system remains in the non-topological state. The right panel shows the
variation of Berry connection for negative values of $\alpha$ considered as
attractive potential. Here, too, the system remains in the non-topological
state.
We observe a similarity with the Berry connection curve with the auxiliary
space curve that both of the curve touches the origin at $k=0$, one is simple
closed Jordan curve and another one is non-simple non-closed. The system
always remains in the non-topological state for the Hamiltonian $H^{(1)}(k)$.

###  II.2 Study of Berry connection for Hamiltonian $H^{(2)}(k)$

Here we consider the $H^{(2)}(k)$ model Hamiltonian. When the interaction term
$\alpha k$ is added to the $\sigma_{y}$ component of the Hamiltonian. The
resulting Berry connection is presented in fig. 8.

Figure 8: The variation of Berry connection with k of the Hamiltonian
$H^{(2)}(k)$.

For the fig. 8 which represents the variation of Berry connection with respect
to k for the Hamiltonian $H^{(2)}(k)$, here we can find the symmetry with $k$.
But there is no symmetry between the plots for the positive and negative
values of $\alpha$. From this study we can understand that the presence of
symmetry about the axis is necessary for the topological states, but not the
sufficient.

###  II.3 Study of Berry connection for Hamiltonian $H^{(3)}(k)$

Here we consider the Hamiltonian $H^{(3)}(k)$. In this case the interaction
term $\alpha k$ is added to both $\sigma_{x}$ and $\sigma_{y}$ components of
the Hamiltonian. The Berry connection study for this case is given in fig.
9,10.

Figure 9: The variation of Berry connection with $k$ of the Hamiltonian
$H^{(3)}(k)$.

In fig. 9, the left panel shows the Berry connection for the repulsive
potential. Right panel shows the Berry connection for the attractive
potential. In both the cases there is no symmetry behavior of the curves
either with $k$ or with the negative value of $\alpha$. The cycloidal motion
loses its periodicity.

Figure 10: Berry connection plots for Kitaev Hamiltonian with interaction term
$\alpha$ added to the $\sigma_{x}$ and $\sigma_{y}$ component.

By the addition of interaction to the Kitaev Hamiltonian, the Berry connection
looses its periodicity and becomes piecewise continuous. Berry phase is absent
for these conditions. Hence we can show how Brillouin zone looses periodicity
through the study of Berry connection.
In all cases of Berry connection we can study how the addition of interaction
is affecting the periodicity of the auxiliary space. But Berry connection
won’t give any idea about the phase transition and the reason for the loss of
periodicity. But Berry connection gives idea about the variation of
wavefunction in the parameter space, the divergence of the potential at the
Brillouin Zone boundary values and about the existence of phase transition
points. Here we can not do the curvature study for Berry connection curves.
Because Berry connection can not be expressed in the form of eq. 34

###  II.4 A common comparison between the auxiliary space curves and the Berry
connection of the Hamiltonians

As we can observe in the fig. 1, the auxiliary space curve for the Kitaev
chain is closed and shows topological, non-topological and transition state
precisely depending on the value of $\mu$. Similarly in the fig. 6, the upper
panel represents the Berry connection of Kitaev chain in the topological
state. But here there is no closed curve as we observe in the auxiliary space.
The only similarity is that at the transition point Berry connection touches
the origin of the k-space. Similarly arguments can be put forth for the
Hamiltonian $H^{(2)}(k)$ and $H^{(3)}(k)$ since also the auxiliary space
curves for these Hamiltonians are not closed. Hamiltonians $H^{(1)}(k)$,
$H^{(2)}(k)$ and $H^{(3)}(k)$ does not exhibit the topological state except
Kitaev chain.

##  III A differential geometric analysis of curves in auxiliary space

Now we discuss very briefly the basic aspects of curvature that have different
geometry for the completeness of this study. One of our main motivation of
this study is to find the topological state of system in terms of differential
geometry at all possible or in otherwords what are the curvature properties of
the auxiliary space curves. Curvature can be defined as the rate of variation
of the angle that the tangent line is making at a particular point. To call a
curve as a regular curve, it should have a non vanishing tangent line. Curve
theory basically deals with analyzing the basic properties of the curves.
Basic properties include, the arc length, winding number with curvature and
torsion of the curves. Topological invariant quantities, like winding number,
Chern number depend on the topology of the auxiliary space, where for a
particular topological configuration space, winding number acquires a definite
value, and change in the winding number leads to the different topological
configuration of the system. The understanding of the curve concept is
simplified by using the differential geometry tool called curvature $\kappa$.
The relation which relates the parameterized curve $c(t)$ and the curvature
$\kappa(t)$ is given by [35],

$\kappa(t)=\frac{det(\dot{c(t)},\ddot{c(t)})}{||\dot{c(t)}||^{3}}\hskip
9.95863pt.$ (20)

For a unit speed curve $\gamma:I\longrightarrow\mathbb{R}^{2}$ where $I=[a,b]$
a closed curve interval. Then $\gamma\prime(t)$ gives the velocity vector
defined by $(\cos\theta(t),\sin\theta(t))^{T}$ of an integer multiple of
$2\pi$, as the curve is defined in a closed interval. As the angle changes
with curve, the invariant quantity winding number is defined by
$\theta(b)-\theta(a)$. If $\theta_{1},\theta_{2}:I\longrightarrow\mathbb{R}$
satisfies the velocity equation. It results as $\theta_{1}=\theta_{2}+2k\pi$,
where $k\in\mathbb{Z}$.
The velocity term $\gamma\prime([a,b])\subset\mathbb{S}_{R}$, i.e.,
$\gamma\prime(t)>0$ for all $t\in I$ and
$\gamma\prime(t)=(\gamma\prime_{1},\gamma\prime_{2})^{T}$,
$\frac{\gamma\prime_{(2)}}{\gamma\prime_{(1)}}=\frac{\sin\theta(t)}{\cos\theta(t)}=\tan\theta(t).$
And
$\theta(t)=\arctan(\frac{\gamma\prime_{2}(t)}{\gamma\prime_{1}(t)})+2k\pi,k\in\mathbb{Z}$.
So considering $\gamma:\mathbb{R}\longrightarrow\mathbb{R}^{2}$ be a unit
speed vector of a curve with period L and
$\theta:\mathbb{R}\longleftarrow\mathbb{R}$ be scalar and winding number is
given by,

$n_{\gamma}=\frac{1}{2\pi}(\theta(L)-\theta(0)).$ (21)

where $(\theta(L)-\theta(0))$ is well defined irrespective of the choice if
$\theta$. Therefore it is clear from the above equation that to get a complete
physical picture of winding number, the study of curve is useful.

###  III.1 An analytical approach to find the closed curve condition in
auxiliary space

We have,

$H^{(3)}(k)=(-2t\cos k-\mu+\alpha_{1}k)\sigma_{x}+(\alpha_{2}k-2\Delta\sin
k)\sigma_{y}.$ (22)

We plot the parametric plot $(x(k),y(k))$,

$x(k)=-2t\cos k-\mu+\alpha_{1}k=r(k)\cos\theta(k),\hskip
28.45274pty(k)=\alpha_{2}k-2\Delta\sin k=r(k)\sin\theta(k).$ (23)

so that, in the auxiliary plane,

$r^{2}(k)=(-2t\cos k-\mu+\alpha_{1}k)^{2}+(\alpha_{2}k-2\Delta\sin
k)^{2},\hskip 28.45274pt\theta(k)=\tan^{-1}\left[\frac{\alpha_{2}k-2\Delta\sin
k}{-2t\cos k-\mu+\alpha_{1}k}\right].$ (24)

To have a closed curve for $k$ running between $\left[-\pi,\pi\right]$ the
curve must come back at its starting point, i.e,

$r^{2}(k=\pi)=r^{2}(k=-\pi),\;\;\;\;\theta(k=\pi)=\theta(k=-\pi)\hskip
28.45274ptmod(2\pi)$ (25)

putting these two conditions in the expression of $r^{2}(k)$ and $\theta(k)$,

$(-2t\cos(-\pi)-\mu+\alpha_{1}(-\pi))^{2}+(\alpha_{2}(-\pi)-2\Delta\sin(-\pi))^{2}=(-2t\cos(\pi)-\mu+\alpha_{1}(\pi))^{2}+(\alpha_{2}(\pi)-2\Delta\sin(\pi))^{2}$
(26)

and,

$\tan^{-1}\left[\frac{\alpha_{2}(-\pi)-2\Delta\sin(-\pi)}{-2t\cos(-\pi)-\mu+\alpha_{1}(-\pi)}\right]=\tan^{-1}\left[\frac{\alpha_{2}(\pi)-2\Delta\sin(\pi)}{-2t\cos(\pi)-\mu+\alpha_{1}(\pi)}\right].$
(27)

Thus the conditions for the curve to be closed, eq. 26 and eq. 27 can be
simultaneously satisfied if $\alpha_{1}=\alpha_{2}=0$. For this condition, the
curve become close and the system is in the topological state. We prove here
the necessary and sufficient condition for the topological state of matter,
that the origin of the auxiliary space is not only covered by the curve but
also the curve should be closed. We have already bean studied in fig 1,2 and
3, the curves in the auxiliary space encircles the origin but the curve itself
is not closed and finally as a consequence of it, the system is in the non-
topological state.

###  III.2 An analysis of auxiliary space curves from perspective of Cauchy-
Riemann Integral

The winding number of a closed curve C about a point $z_{j}$ in the complex
plane is given as [35],

$w_{z_{j}}=\frac{1}{2\pi i}\oint\frac{dz}{z-z_{j}}$ (28)

We also have the theorem that if $f(z)$ is a mesomorphic function define
inside and on a simple closed contour C, with no zeros or poles on C, then,

$\frac{1}{2\pi i}\oint\frac{f\prime(z)}{f(z)}dz=\text{No of zeros of f inside
C- No of poles of f inside C}$

We note that, if we define $f(z)=z-z_{j}$, then the integrals in above
equations are the same, so,

$w_{z_{j}}=\underbrace{\text{Number of zeros $(z-z_{j})$ inside C}}_{\text{=1
if $z_{j}$ is included inside C=0 otherwise}}-\underbrace{\text{Number of
poles of $(z-z_{j})$ inside C}}_{\text{=0 as $(z-z_{j})$ has no poles}}$

= 1 iff $z_{j}$ is included inside C

This explains why the winding number calculated without origin is 1 for a
Kitaev chain if the curve in the auxiliary space encloses the origin. The
above definition of $w$ is valid only for a closed contour C, hence cannot be
applied if the curve is not closed in the auxiliary space.
The value $w(z_{j})$ defined as

$w(z_{j})=\frac{1}{2\pi i}\oint\frac{dz}{(z-z_{j})}=(\frac{1}{2\pi
i})[log(z-z_{j})]_{c}=(\frac{\Delta\theta_{j}}{2\pi})$ (29)

is called the winding number of the curve C around point $z_{j}$. Here,
$\Delta\theta_{j}$ is the change in the argument of $(z-z_{j})$, where $z$
traverses through the curve C and the point $z_{j}$. The value $w(z_{j})$
represents the number times that C winds around $z_{j}$. In presence of
interaction, the curve is not closed, in one sense $\Delta\theta_{j}$ is not
$2\pi$ or integral multiple of $2\pi$. Therefore the concept of winding
number, open and self intersecting curve is not valid here.

###  III.3 A study of differential geometry from the perspective of curvature
theory for the auxiliary space curve

Ellipse is generally defined as locus of points such that sum of distances
from the foci is constant. The standard equation of ellipse is given by,
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ where a and b are semi-major and
semi-minor axes. The parametric equation is given by $[a(t),b(t)]=(a\cos
t,b\sin t)$ where $0\leq t<2\pi$.

Figure 11: The graphical representation of an ellipse (Left). The curvature of
the ellipse for the corresponding values

The curvature expression of ellipse is [11],

$\kappa=\frac{ab}{(b^{2}\cos^{2}t+a^{2}\sin^{2}t)^{\frac{3}{2}}}.$ (30)

The two parameters a and b in the curvature expression of ellipse are namely
semi-major axis and semi-minor axis. From these two parameters we can analyze
the curvature in three cases.
First case: When $a<b$, the curvature is maximum on the semi-major axis (
$-\frac{\pi}{2}$ and $\frac{\pi}{2}$) and it is minimum on the semi-minor
axis.
Second case: When $a=b$, the auxiliary space curve is a circle with the
constant curvature.
Third case: When $a>b$, the curvature is minimum on the semi-major axis
($-\frac{\pi}{2}$ and $\frac{\pi}{2}$) and it is maximum on the semi-minor
axis [12].
An analysis of Cycloidal Properties of auxiliary space curves for interacting
system
The general expression of a cycloid is given by [12],

$C[a(t),b(t)]=(at-b\sin t,a-b\cos t).$ (31)

Depending on the location of the locus point on the circle the cycloid motion
is devised into different category. When $a$ and $b$ are equal it gives a
cycloid which describes the locus of point on a circle of radius $a$ [12]. If
a and b are not equal and if $a<b$ then the cycloid is called prolate. Suppose
$a>b$ then cycloid is called curate and it is represented in the figure.12.

Figure 12: Cycloidal motion of the locus on the circle at different conditions
(a)a=b, (b)a<b and (c)a>b.

Curvature expression of Cycloid is,

$\kappa=\frac{ab\cos t-b^{2}}{(a-b\cos t)^{2}+b^{2}\sin^{2}t)^{\frac{3}{2}}}.$
(32)

The addition of interaction term alters the pattern of the cycloidal motion.
(1). Kitaev model
Here we present the results of study of differential geometry based on curve
theory for the non-interacting and interacting Hamiltonians. The matrix form
of the Kitaev model Hamiltonian is,

$\mathcal{H}(k)=\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)\\\
-2i\Delta\sin(k)&&2t\cos(k)+\mu\\\ \end{matrix}\right).$ (33)

Considering the parametric equation of the Hamiltonian $H(k)$ in the matrix
form,

$c(k)=\left[\begin{matrix}-2t\cos k-\mu\\\ 2\Delta\sin
k\end{matrix}\right],\dot{c}(k)=\left[\begin{matrix}2t\sin k\\\ 2\Delta\cos
k\end{matrix}\right],\ddot{c}(k)=\left[\begin{matrix}2t\cos k\\\ -2\Delta\sin
k\end{matrix}\right].$ (34)

Curvature is given by,

$\kappa=\frac{det[\dot{c},\ddot{c}]}{||\dot{c}||^{3}}\implies\frac{det\left(\begin{matrix}2t\sin
k&&2t\cos k\\\ 2\Delta\cos k&&-2\Delta\sin
k\end{matrix}\right)}{(\sqrt{4t^{2}\sin^{2}k+4\Delta^{2}\cos^{2}k})^{3}}=\frac{-t\Delta}{2(\sqrt{t^{2}\sin^{2}k+\Delta^{2}\cos^{2}k})^{3}}\hskip
9.95863pt.$ (35)

Eq. 35 is an analytical expression of curvature for the Kiteav Hamiltonian.
The curvature plot of the Kitaev Hamiltonian is presented in the fig. 13. The
auxiliary space curve for the Kitaev chain is nothing but an ellipse (fig.1)
since the parametric equations of the Hamiltonian $(-2t\cos k-\mu,2\Delta\sin
k)$ resembles the mathematical form of the ellipse $[a(t),b(t)]=(a\cos t,b\sin
t)$ where $0\leq t<2\pi$.

Figure 13: The left panel represents the plots of curvature with k for the
values t=2, 1, 0.7 from top to bottom respectively. The right panel represents
corresponding auxiliary plots for the value $\mu=0$

For the topological state, the curvature is smooth and touches the $k=0$ point
on the k-axis. For the transition state, there is no curvature and the plot is
parallel to the k-axis. For the non-topological state, the curvature does not
touches the $k=0$ and forms a gaped state.

Auxiliary space curve and curvature of Kitaev chain:
The parameterized equations of the Kitaev Hamiltonian are, $(-2t\cos
k-\mu,2\Delta\sin k)$ and the analytic expression of curvature for the Kitaev
chain gives the mathematical structure of the eq. 35, we conclude that the
curvature plot of the standard ellipse and the curvature of the Kitaev chain
are same.
For the value $\mu=0$, the system remains in the topological state. We can
study the curvature of auxiliary space curve for all Hamiltonians. We can not
characterize the topological and non-topological states of the Hamiltonian
from the curvature study. The reason for this is, the curvature expression
does not include the term $\mu$. From the above general discussion on the
ellipse we can characterize the auxiliary space curve of the Kitaev chain into
similar three cases. This is completely a theoretical exercise to show the
nature of auxiliary space column from the perspective of differential
geometry.
First case: When $t<\Delta$, the curvature is maximum on the semi-major axis (
$-\frac{\pi}{2}$ and $\frac{\pi}{2}$) and it is minimum on the semi-minor
axis.
Second case: When $t=\Delta$, the auxiliary space curve is a circle with the
constant curvature.
Third case: When $t>\Delta$, the curvature is minimum on the semi-major axis
($-\frac{\pi}{2}$ and $\frac{\pi}{2}$) and it is maximum on the semi-minor
axis.

###  III.4 Results and discussions of curvature in the auxiliary space for the
interacting Hamiltonians:

(2). Results of differential geometry studies for the Hamiltonian $H^{(1)}(k)$
The matrix form of the Hamiltonian $H^{(1)}(k)$ can be written as,

$\mathcal{H}^{(1)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu-\alpha
k&&i2\Delta\sin(k)\\\ -i2\Delta\sin(k)&&2t\cos(k)+\mu+\alpha k\\\
\end{matrix}\right).$ (36)

The Curvature of the Hamiltonian $H^{1}(k)$ is given by,

$\kappa=\frac{det\left(\begin{matrix}2t\sin k+\alpha&&2t\cos k\\\ 2\Delta\cos
k&&-2\Delta\sin k\end{matrix}\right)}{(\sqrt{(2t\sin
k+\alpha)^{2}+4\Delta^{2}\cos^{2}k})^{3}}=\frac{-4t\Delta-2\Delta\alpha\sin
k}{(\sqrt{(2t\sin k+\alpha)^{2}+(2\Delta\cos k)^{2}})^{3}}\hskip 9.95863pt.$
(37)

Figure 14: The left panel represents the plots of curvature with k for the
values t=1,$\alpha=0.2,1$ from top to bottom respectively. The right panel
represents corresponding auxiliary plots for the value $\mu=0$ Figure 15: The
left panel represents the plots of curvature with k for the values
t=1,$\alpha=-0.2,-1$ from top to bottom respectively. The right panel
represents corresponding auxiliary plots for the value $\mu=0$

For the $H^{(1)}(k)$ Hamiltonian, the auxiliary space curve forms cycloidal
motion. The corresponding curvature plots a deviation from topological states
of system, these plots resembles the non-topological curvature patterns.
The auxiliary space curves is a cycloid in the presence of interaction since
the Hamiltonian $H^{(1)}(k)$ acquire mathematical structure of cycloid
equation. We notice that the presence of interaction changes the properties of
differential geometry which we study the curvature properties in the auxiliary
space. Based on the strength of the interaction term, the auxiliary space
curve behaves as simple curve with non closed, self intersecting conditions.
When the interaction term changes its sign, the auxiliary space curves as well
as curvature plots forms mirror symmetric image.

(3). Results of differential geometry studies for the Hamiltonian $H^{(2)}(k)$
Hamiltonian $H^{(2)}(k)$ can be written in the matrix form as,

$\mathcal{H}^{(2)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu&&2i\Delta\sin(k)+i\alpha
k\\\ -2i\Delta\sin(k)-i\alpha k&&2t\cos(k)+\mu\end{matrix}\right).$ (38)

Curvature is given by,

$\kappa=\frac{det\left[\begin{matrix}2t\sin k&&2t\cos k\\\ 2\Delta\cos
k+\alpha&&-2\Delta\sin k\end{matrix}\right]}{(\sqrt{(2t\sin
k)^{2}+(2\Delta\cos k+\alpha)^{2}})^{3}}=\frac{-4t\Delta-2\alpha t\cos
k}{(\sqrt{(2t\sin k)^{2}+(2\Delta\cos k+\alpha)^{2}})^{3}}\hskip 9.95863pt.$
(39)

Eq. 39 is the analytical expression of the curvature for the Hamiltonian
$H^{(2)}(k)$.

Figure 16: The left panel represents the plots of curvature with k for the
values t=1, $\alpha=0.2,0.8$ from top to bottom respectively. The right panel
represents corresponding auxiliary plots for the value $\mu=0$ Figure 17: The
left panel represents the plots of curvature with k for the values t=1,
$\alpha=-0.2,-0.8$ from top to bottom respectively. The right panel represents
corresponding auxiliary plots for the value $\mu=0$

Fig 17 is not mirror symmetric. The part to be noted is that the maxima for
the curvature for the positive values becomes minimum and visa versa. The
curvature contains two minima and one maxima for the repulsive potential. For
the attractive potential, it contains one minima and two maxima.
As the previous case, the curvature expression is independent of the term
$\mu$. The increase in the strength of the interaction term results in
decrease of curvature near $k=0$ plane.
For the Hamiltonian $H^{(2)}(k)$, the auxiliary space curve is also a prolate
cycloid because it is open self-intersecting. The parametric equation of the
Hamiltonian $H^{(2)}(k)$ is the following:

$H^{(2)}(k)=(-2t\cos(k)-\mu,2\Delta\sin(k)+i\alpha k).$ (40)

From the curvature studies for this auxiliary space curve of Hamiltonian
$H^{(2)}(k)$, it reveals the following: The curvature at the points ($-\pi$
and $\pi$) on the semi-major axis is maximum and the curvature on the semi-
minor axis is minimum.When the interaction term changes its sign, the
auxiliary space curves as well as curvature plots forms mirror symmetric
image.
(4). Results of differential geometry studies for the Hamiltonian $H^{(3)}(k)$
Hamiltonian $H^{(3)}(k)$ can be written in the matrix form as,

$\mathcal{H}^{(3)}(k)=\left(\begin{matrix}-2t\cos(k)-\mu-\beta_{1}k&&2i\Delta\sin(k)+i\beta_{2}k\\\
-2i\Delta\sin(k)-i\beta_{2}k&&2t\cos(k)+\mu+\beta_{1}k\\\ \end{matrix}\right)$
(41)

Curvature is given by,

$\kappa=\frac{det\left[\begin{matrix}2t\sin k+\alpha_{1}&&2t\cos k\\\
2\Delta\cos k+\alpha_{2}&&-2\Delta\sin k\end{matrix}\right]}{(\sqrt{(2t\sin
k+\alpha_{1})^{2}+(2\Delta\cos
k+\alpha_{2})^{2}})^{3}}=\frac{-4t\Delta-2(\alpha_{1}\Delta\sin
k+\alpha_{2}t\cos k)}{(\sqrt{(2t\sin k+\alpha_{1})^{2}+(2\Delta\cos
k+\alpha_{2})^{2}})^{3}}$ (42)

Eq.42 is an analytic expression of the curvature for the Hamiltonian
$H^{(3)}(k)$.

Figure 18: Left panel represents the plots of curvature with for different
values of $\alpha_{1}$ and $\alpha_{2}$. Right panel represents corresponding
auxiliary space curves. Figure 19: Left panel represents the plots of
curvature with for different values of $\alpha_{1}$ and $\alpha_{2}$. Right
panel represents corresponding auxiliary space curves.

For the negative $\alpha$, variation of curvature with $k$ is not mirror
symmetric as like positive one. The point to be noted that the shape of the
curve becomes reverse. There is no divergence in the auxiliary space curves
but the curvature shows the divergence behavior.
For $H^{(3)}(k)$ Hamiltonian, auxiliary space curves forms cycloidal pattern
but in a very arbitrary way. We can not find any particular orientation. The
corresponding curvature shows the non-topological state. Based on the strength
of $\alpha_{1}$ and $\alpha_{2}$ there arises divergence characters at the BZ
boundary values.

Figure 20: Left panel represents the plots of curvature with for different
values of $\alpha_{1}$ and $\alpha_{2}$. Right panel represents corresponding
auxiliary space curves.

For the $H^{(3)}(k)$ Hamiltonian, the curvature study shows the diverging
behavior. The curvature diverges at BZ boundary values. Here curvature is not
continuous. The sign change of the interaction term does not alter the nature
of curvature but the divergence happens in the opposite region. The curvature
plots shows the divergence at BZ boundary regions i.e.,$-\pi$ and $\pi$.

Conclusion: We have presented topological aspects of the Kitaev model in the
presence and absence of interaction through differential geometric and complex
variable method. The nature of curves, nature of Berry connection and the
auxiliary space in presence of interaction have discussed. We have derived the
necessary and sufficient conditions for the topological characterization of
the system through the study of the auxiliary space. We have also studied the
Berry connection explicitly and its consequences in topological and non-
topological states.
Acknowledgments
The authors would like to acknowledge DST (EMR/2017/000898) for the funding
and RRI library for the books and journals. The authors would like to
acknowledge Prof. R Srikanth who has read this manuscript critically. Finally
authors would like to acknowledge ICTS
Lectures/seminars/workshops/conferences/discussion meetings of different
aspects of physics.

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