11institutetext: FH Aachen - University of Applied Sciences, 52428 Jülich,
Germany 22institutetext: University of Kassel, 34127 Kassel, Germany 22email:
<EMAIL_ADDRESS>
22email<EMAIL_ADDRESS>
# STAMP 4 NLP – An Agile Framework for Rapid Quality-Driven NLP Applications
Development

Philipp Kohl 11 Oliver Schmidts 11 Lars Klöser 11 Henri Werth 11 Bodo Kraft 11
Albert Zündorf 22

###### Abstract

The progress in natural language processing (NLP) research over the last
years, offers novel business opportunities for companies, as automated user
interaction or improved data analysis. Building sophisticated NLP applications
requires dealing with modern machine learning (ML) technologies, which impedes
enterprises from establishing successful NLP projects. Our experience in
applied NLP research projects shows that the continuous integration of
research prototypes in production-like environments with quality assurance
builds trust in the software and shows convenience and usefulness regarding
the business goal. We introduce STAMP 4 NLP as an iterative and incremental
process model for developing NLP applications. With STAMP 4 NLP, we merge
software engineering principles with best practices from data science.
Instantiating our process model allows efficiently creating prototypes by
utilizing templates, conventions, and implementations, enabling developers and
data scientists to focus on the business goals. Due to our iterative-
incremental approach, businesses can deploy an enhanced version of the
prototype to their software environment after every iteration, maximizing
potential business value and trust early and avoiding the cost of successful
yet never deployed experiments.

###### Keywords:

Natural Language Processing Process Model Machine Learning Best Practices
Avoiding Pitfalls Quality Assurance.

##  1 Introduction

The field of artificial intelligence in general and natural language
processing as one of its sub-fields offers tremendous novel business
opportunities in a steadily growing market [13]. Recent progress in NLP
research shows the potential for business applications, leading to a demand
for more advanced NLP applications [7].

The state-of-the-art in NLP differs from research to industrial domains.
Besides the progress in research, the application of ML-based NLP in many
enterprises is severely limited [6]. The black-box behavior of ML models,
missing know-how, complex technological landscape, and the decision on an
appropriate tool stack discourage enterprises from implementing NLP approaches
[1, 10]. They discard promising projects due to the combination of high and
uncertain effort estimation [23].

Many ML projects fail because of exceeding budgets, deadlines or they do not
meet the business requirements [11]. The late integration of several projects
can lead to a services shutdown [27, 17]. We minimize the risk of these
situations with agile methodology to handle the uncertainties and generate
business value and feedback on the application as early as possible. This
increases the quality and trust in the software for all involved stakeholders
[5].

We propose a new process model adjusted for developing NLP applications:
Standardized Modeling Process for Natural Language Processing (STAMP 4 NLP).

With STAMP 4 NLP, we merge software engineering principles with best practices
from data science to improve and accelerate the development cycle and
integrate prototypes with every iteration into a test or production
environment. STAMP 4 NLP provides a transparent development process, including
roles, tasks, artifacts, and best practices.

Our main contributions111https://github.com/philipp-kohl/stamp4nlp are:

  * •

A novel process model for developing NLP applications, with formally specified
roles, activities, and artifacts focusing on quality, and early business
value.

  * •

Usage of predefined environment and software templates based on prior
experiences for accelerating the development start.

##  2 Related Work

Knowledge Discovery in Databases (KDD) [9] represents one of the first process
models for data mining. It offers a generic guided process of the technical
tasks to reveal patterns in data and building knowledge.

Cross Industry Standard Process for Data Mining (CRISP-DM) [4] also considers
business requirements and models the application development process in an
applied context in contrast to KDD. Modern ML process models originate from
CRISP-DM. It consists of six stages: Business Understanding, Data
Understanding, Data Preparation, Modeling, Evaluation, and Deployment.
Depending on the stage’s results, it allows transitions to previous stages.

CRISP-ML(Q) [24] extends CRISP-DM for machine learning and explicitly
considers the differences between data mining (revealing patterns in data) and
machine learning (training and inference). Studer et al. focus on quality
assurance on every specific task. CRISP-ML(Q) merges Business Understanding
and Data Understanding into a single stage and adds the Monitoring and
Maintenance stage, addressing particular challenges of machine learning
applications not considered by CRISP-DM.

While CRISP-DM and CRISP-ML(Q) mainly focus on application creation, Weber et
al. [28] introduced an approach with defined transitions between model
development and model operation. Thus, they cover the whole model lifecycle
from planning over production until retirement. Their process does not
explicitly consider business requirements.

Similar to KDD, Amershi et al. [2] focuses on the modern, mainly technical
process for developing a machine learning model but also incorporates
fundamental operational analytics with transitions to previous stages.

In contrast to the mentioned process models, we focus on NLP. We treat manual
data annotations and annotation guidelines as central project artifacts. Our
approach is especially suitable for supervised NLP tasks as information
extraction. Further, our focus is on strong quality assurance with different
levels of applied tests. We leverage approaches and best practices from
software engineering, combining them with machine learning approaches, such as
supporting versioning of code, data, models, and tracking experiment results
[16].

Agile software development [5] uses iterations and increments instead of
transitions between process stages. This leads to a stronger focus on running
software during the development process. The incremental aspect allows
isolated investigation of the experimental effects and the usage of continuous
integration and delivery (CI/CD) [8]. Developers receive feedback, and
stakeholders gain value and trust in the application with every deployed
increment. In comparison to [24] with task-agnostic quality assurance, we
measure technical and business-oriented metrics on a higher level after every
iteration [18, 19, 22].

Inspired by Spring [14], maven archetypes [25] and NLPf [20], we deliver STAMP
4 NLP with a framework, which supports the developer with a predefined
development environment, code for well-known standard tasks [12, 26] for
creating a rapid prototype as basline or proof of concept, which the developer
enhances over iterations and increments. STAMP 4 NLP decreases the risk of not
deployed valuable experiments and failing projects.

##  3 Process Model

STAMP 4 NLP is an instantiable, iterative, and incremental process model for
developing natural language processing applications with a focus on quality,
business value, and simplified prototyping.

STAMP 4 NLP uses agile methodology [5] by establishing software in increments
developers enhance in various iterations (c.f. Figure 1).

Figure 1: STAMP 4 NLP consists of five subprocesses arraned in two nested
loops: Development Loop (green + solid) for developing the NLP application and
model. Evolution Loop (blue + dashed) surrounds the Development Loop and
covers mainly the interaction with the customer: creation and refinement of
the project goals with associated requirements and integration and monitoring
of the application. Note: The solid and dashed connectors have the same
meaning of sequence flow but help distinguish the loops in black and white
print.

The Evolution Loop works in sprints (e.g., 2-4 weeks) to create a new release
candidate that the developers integrate into the customer’s test or production
environment to gain trust in the software and receive feedback for further
improvements to fit the customer’s needs [8]. The workflow is as follows: the
developers start with the Evolution Loop to define or refine the requirements,
followed by multiple iterations in the Development Loop (hours to days) to
create a model that satisfies the specified requirements. Deployment and
testing are also a part of the inner loop. Once the software fulfills them,
the developers leave the inner loop and proceed to the outer loop to integrate
and monitor the release candidate. Depending on the monitoring results, they
trigger a new iteration.

The process model provides particular levels of transparency. At first, every
stakeholder can follow the complex process, even if they are unfamiliar with
NLP. They can also track the order of tasks by consulting the documentation.
Thereby the responsibilities are clearly defined, and every involved party
knows their tasks. Clearly defined responsibilities are helpful to prevent
conflicts and create rational processes that the users can monitor and
optimize. We use theMEDIATION [19] approach to provide the application state
transparent via a dashboard with project specific business metrics for all
stakeholders.

The development environment equips the user with a standard set of tools for
rapidly building a proof of concept while maintaining the flexibility to let
the user choose other preferred libraries. Creating a project instance
provides folder structure, development environment, documentation, and
boilerplate code (e.g., REST-Service with predefined endpoints, etc.). The
standard folder facilitates the automatic loading and storing of data, models,
and results to their destination by convention. Furthermore, no settling-in
period for already STAMP-involved members; they know where to find code,
documentation, experiments, and data.

ML metrics serve as a common benchmark for different models. High or low
metrics show the averaged performance but do not allow making conclusions
which use-cases a model cannot handle appropriately. Model interpretability is
a current research subject [1]. To minimize the interpretability gap in
machine learning and to receive more feedback on a low application level, we
incorporated CheckList [18] for creating behavioral tests aiming for specific
capabilities the NLP application should cover. Besides CheckList, we use
MEDIATION [19] for testing the NLP application on a business level: the
developers and stakeholders define test cases strongly related to their
intended use cases in the form of annotated documents. Thus, this additional
test set serves as an indicator of the business readiness of the NLP
application. In combination with the CI/CD approach, we receive this feedback
for every iteration and increment. Involving testers or real user groups into
the increment testing generates feedback for business and practical usage.

STAMP 4 NLP supports the user to keep the reproducibility of experiments and
models as high as possible. We incorporated parts of Pineau’s reproducibility
list [16] into the process by documentation and tools supporting versioning of
code, data, models, and tracking experiment’s results.

In the following, we give a short description of each subprocess with its
primary artifacts. We show exemplarily a detailed BPMN diagram of the Domain
Adoption and Customization in Figure 2. Our GitHub repository provides the
other subprocess diagrams and the detailed description of each task and
artifact.

###  3.1 Goal Specification

Description: The Goal Specification aims to establish a common business
understanding. The stakeholders define and refine the business goals, and
their associated technical, machine learning, business, and MEDIATION
requirements and update the documentation accordingly. It includes an
evaluation of all data sources and the data provision for the data scientists.
This stage involves all currently relevant stakeholders to minimize the bias
and possibly wrong model assumptions.

Artifacts: The primary artifacts are the refined and reviewed requirements,
test cases, and access to all mandatory data sources.

###  3.2 Domain Discovery and Data Selection

Description: Data scientists and domain experts prepare the annotation process
(c.f. Figure 2). They identify NLP tasks and corresponding annotation schemas
helpful to fulfill the business goals. Additionally, data scientists include
domain knowledge from experts to steadily improve annotation guidelines and
collect and evaluate data samples for the annotation process and necessary
metadata.

Artifacts: The primary artifacts are the annotation guidelines, the new corpus
versions prepared for annotation, and documentation about licenses, data
protection, and data security.

Figure 2: Showing the task order of the subprocess Domain Adoption and
Customization as BPMN Diagram with the involved roles displayed as swimlanes.
Additional information is provided by Documentation and Data lane. Create new
corpus version, Training, and Apply Data Augmentation are automatic tasks,
which run without human interaction. Training requires the execution of one
command as trigger.

###  3.3 Domain Adoption and Customization

Description: This subprocess (c.f. Figure 2) includes data annotation and
model training. The annotation process setup involves planning and, if
necessary a domain training for annotators. The annotated texts build a new
corpus version, which is used for training a new model. To minimize the
annotation effort, we want to stop further annotating when noticing the
resulting model’s metrics stagnate. We incorporated the continuous integration
and delivery approach of Schreiber et al. [21]. Thus, annotators receive
feedback after each annotation session. The feedback can motivate to continue
or stop annotating because the further annotations do not impact the model’s
performance remarkably.

Artifacts: The primary artifacts are the new corpus version, the new model,
and metric reports about the model’s performance.

###  3.4 Application Engineering

Description: Software developers package the model and all necessary
dependencies. A CI/CD pipeline deploys the application in a test or
production-like environment and runs evaluations to ensure the quality gates
(software quality, machine learning metrics MEDIATION and behavioral tests via
CheckList [18]) defined in the goal specification. The software package is
versioned to provide transparency, whether the made modifications improved the
previous version and for fallback solutions. Depending on the evaluation
results, the application stays in the Development Loop or transitions into the
Evolution Loop for integrating the software into the customer’s application
landscape (c.f. Figure 1).

Artifacts: The primary artifacts are the software package and a quality
assurance report.

###  3.5 Customer Integration and Evaluation

Description: Software developers integrate the packaged NLP application in the
customer’s application landscape. On the customer side, a monitoring service
checks the model’s performance. The resulting reports build the basis for a
refinement of the quality gates comparable to [22, 19] ensuring the
fulfillment of the business requirements during the production phase.

Artifacts: An operation manual documents the deployment, and an integration
plan explains the integration in the customer application. The performance
reports support recommendations and business decisions.

##  4 Project Template

STAMP 4 NLP facilitates focusing on the project-specific challenges such as
analyzing the data and the domain, annotating, experimenting with different
concepts, and deep learning architectures. To decrease the overhead data
scientists face while starting a new project, we offer a template with
development environment, folder structure222similar to
https://drivendata.github.io/cookiecutter-data-science/, tools, code and
process documentation. Furthermore, the framework can generate customizable
implementations for specific NLP tasks (e.g., named entity recognition (NER)
[12], or text classification [26]) into the project, helping the developers
implement a first baseline or proof of concept.

The template serves the paradigm convention over configuration [3]. Therefore
the template comes with, but is not limited to a standard set of tools and
libraries. If the user stays with the standard, no additional configuration is
needed. But the user has the opportunity to use additional libraries or tools,
resulting in extra configuration. The basic configuration provides, for
example, the library spacy333https://spacy.io/ to create prototypes quickly.
Depending on the business goal, it is necessary to preserve more control over
used architecture and training routines. Therefore the developer can exchange
the conventional added spacy module with PyTorch444https://pytorch.org/ or
similar frameworks. The same applies to the folder structure, environment, and
infrastructure. We recommend to start with the standard configuration and
specialize on demand.

##  5 Example

This section demonstrates a simplified STAMP 4 NLP usage over a few iterations
to show the intuition behind the process model. We focus on a real-world
project we performed with our business partners: The profile extraction from
social media messages of an advertising group conversation. We use named
entity recognition (NER) as a standard NLP task, for which we can use existing
approaches. NER describes the task of finding domain-relevant terms in
documents: e.g., persons, brands, products, and their prices. On top of that,
we implement a business layer to aggregate the named entities to a profile.

First iteration – Requirements Analysis and Dry Run: The first iteration
focuses on the requirements analysis and the infrastructure test run (also
called dry run). Instantiating the process model provides a development
environment including a prototypical web-service, a pre-configured CI/CD
pipeline, and prepared documentation. We define and document the NER as the
applied NLP task. Based on the documentation, we invoke the framework for
generating a reference implementation for NER as a first baseline. Among
others, we define the corresponding machine learning metrics we want to
achieve with the NLP application. We skip the most tasks of all other
subprocesses for this iteration since its the iteration’s goal to ensure the
infrastructure: training a model, embed the model into a software package,
deploy into a test environment, model evaluation, publish results via a
dashboard. The scores do not matter at this stage.

Figure 3: An excerpt of a Grafana Dashboard visualizes the F1, precision, and
recall score for the NER component over several iterations of annotating and
architecture decision. Precision measures the correctness of positive
classifications (here for a named entity) by penalizing false positives.
Recall shows if our model has found all named entities by penalizing false
negatives. F1 is the harmonic mean of precision and recall. It is commonly
used for model evaluation and comparison.

Second iteration – Baseline: The second iteration focuses on creating a first
baseline. This iteration covers the Development Loop exclusively. We prepare
the annotation process by defining the annotation guidelines, deciding on a
suitable corpus format, and transferring the data to the corpus format in
Domain Discovery and Data Selection. Domain Adoption and Customization mainly
focuses on annotating a subset of the data in this iteration to create the
first baseline with the standard implementation. We embed the model into our
software architecture and add a business layer to combine the named entities
to a profile. Our CI/CD approach packages and deploys the software into the
test environment, where the framework performs detailed quality assurance. The
results are published on a dashboard (c.f. Figure 3).

Additional iterations – Beat the baseline: Further iterations focus isolated
on specific aspects to enhance the baseline. We incorporate the aspects as
isolated as possible to ensure the cause-effect relationships:

Annotations

Annotating new data points, improve existing annotations, decide to
incorporate new labels, enhance annotation guideline based on gained
experience.

Architectural Decisions

Use different common neural network components or pretrained contextualizes
embedding layers [15, 7].

If the application fulfills the requirements, we have a possible release
candidate and exit the Development Loop and continue the Evolution Loop. If
the results do not show the expected behavior or do not fulfill the business
need, we begin a new iteration to investigate the cause and start new
experiments improving the application’s quality.

##  6 Limitations and Drawbacks

Our proposed process model has a strong focus on supervised NLP. We have many
subprocesses with corresponding roles and artifacts for problems that include
the manual annotation of corpora. They may be unbeneficial for unsupervised
tasks. We further define clear responsibilities and processes but assume that
a practical application involves loose compliance to those in some cases. The
process needs to be adapted to each project individually. A significant
benefit of the instantiable framework is a low application barrier. The
resulting standardized configuration and black box code can lead to laborious
error detection or adaptions when moving too far from these conventions.

##  7 Summary

We introduced STAMP 4 NLP a novel instantiable and iterative-incremental
process model to develop NLP applications. It supports developers to create
valuable and deployable increments rapidly, results in earlier feedback, and
improves quality and trust in the application for all stakeholders.

Our approach equips the user with templates, development environment, and
documentation to reduce the starting and integration overhead. That minimizes
implementation barriers, avoids common pitfalls, and sets the focus on the
business goal. Thus, STAMP 4 NLP reduces the risk of failing projects.

In the future, we plan to create a benchmark project for different groups to
work on: some groups work with the STAMP 4 NLP and others work from scratch.
Thus, we want to measure various project milestones, key performance
indicators and observe the challenges the different teams face and pitfalls
they could avoid. Furthermore, we want to use it for educational purposes to
set the focus appropriately with incremental depth increase. We want to
improve the development of generic NLP tasks, including unsupervised problem
settings.

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11footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author.33footnotetext:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Converting Lattices into Networks: The Heisenberg Model and Its
Generalizations with Long-Range Interactions

Chi-Chun Zhou  Yao Shen  Yu-Zhu Chen  and Wu-sheng Dai

###### Abstract

In this paper, we convert the lattice configurations into networks with
different modes of links and consider models on networks with arbitrary
numbers of interacting particle-pairs. We solve the Heisenberg model by
revealing the relation between the Casimir operator of the unitary group and
the conjugacy-class operator of the permutation group. We generalize the
Heisenberg model by this relation and give a series of exactly solvable
models. Moreover, by numerically calculating the eigenvalue of Heisenberg
models and random walks on network with different numbers of links, we show
that a system on lattice configurations with interactions between more
particle-pairs have higher degeneracy of eigenstates. The highest degeneracy
of eigenstates of a lattice model is discussed.

##  1 Introduction

Lattice models are important because they can be not only treated
theoretically, for example the Heisenberg models can be solved exactly at
lower dimensions [1, 2, 3], but also implemented experimentally, such as
constraining ultracold atoms in optical lattices [4, 5]. Among many lattice
models, the Heisenberg model which describes the magnetism of a system [1, 2,
3] is of significance. To solve the Heisenberg model, methods such as the
Bethe Ansatz and the Jordan Wigner transformations are proposed [1, 6]. The
solution of the Heisenberg model at different dimensions [7, 8, 9] or under
certain boundary conditions [10, 11, 12] is discussed.

In this paper, we solve the Heisenberg model by using a group theory method.
In the group theory method suggested in the present paper, we reveal the
relation between the Casimir operator of the unitary group and the conjugacy-
class operator of the permutation group. We propose a generalization of the
Heisenberg model which can be solved exactly.

(1) Converting lattice configurations into networks. We consider a model on
lattice configurations with arbitrary numbers of interacting particle-pairs by
converting the lattice configuration into networks. Usually, lattice models
with certain number of neighbours or interacting particle-pairs are taken into
account. For example, $3$, $4$, $6$, etc. It is because only countable lattice
configurations exist under the constraint of symmetries [13, 14]. For example,
there are $17$ kinds of two-dimensional lattice models and there are $230$
kinds of three-dimensional lattice models. In this paper, we map the lattice
configuration to networks with different modes of links, as shown in Fig. (1).
We show that the number of nearest particles is characterized by the number of
links in the network. The advantage of doing this is that we are free to
consider a systems with arbitrary number of interacting particle-pairs which
in lattice configurations is restricted by symmetries. By studying the
relation between degeneracy of eigenstates and the number of the interacting
particle-pairs, we show that models on lattice configuration with more
interacting particle-pairs have higher degeneracy of eigenstates. Thus,
lattice models with interactions between all particle-pairs should have a
highest degeneracy of eigenstates.

(2) The Long-Range Interactions. The lattice model with long-range interaction
between inter-particles is considered. Usually, short-range interactions are
considered in lattice models, as a result, only the interaction between
nearest, second nearest, third nearest, etc, particle-pairs need to be
considered. However, if long-range interactions, such as the inverse square
potential and the harmonic oscillator potential, exist, interactions between
more particle-pairs need to be considered. For example, one-dimensional models
with long-range interactions is considered in [15]. Ising models on the
hypercube lattices with long-range interaction is considered in [16]. In this
paper, the model on lattice configuration with interaction between all
particle-pairs is with long-range interactions.

Figure 1: Converting lattice configurations into networks. By holes or
impurities, we mean that they can not interact with other particles. The
periodic boundary condition is applied.

This paper is organized as follows: In Sec. 2, we solve the Heisenberg model
on lattice configurations with interactions between all particle-pairs by
revealing the relation between the Casimir operator of the unitary group and
the conjugate class operator of the permutation group. In Sec. 3, we give a
series of exactly solvable models by generalizing the Heisenberg model from
this relation. In Sec. 4, we investigate the relation between degeneracy of
eigenstates and the number of links in a network by numerically calculating
the eigenvalues of Heisenberg models and random walks on a network. We discuss
the highest degeneracy of eigenstates for a interacting many-body system.
Conclusions and discussions are given in Sec. 5.

##  2 The Heisenberg model on lattice configurations with interactions between
all particle-pairs

In this section, we consider the Heisenberg model on lattice configurations
with interactions between all particle-pairs.

A special lattice configuration: lattice configurations with interactions
between all particle-pairs. As shown in Fig. (1), for various kinds of lattice
configurations the difference varies in the mode of links between particles
other than the position of particles. The network with links between all
particle-pairs considers the largest number of particle-pairs interactions, as
shown in Fig. (2). In the following, we will make no distinguish between
lattice configurations and networks.

Figure 2: The lattice configuration with interactions between all particle-
pairs and the one-dimensional lattice configuration are two special lattice
configurations: one considers the largest number of links and the other
considers the smallest number of links.

The Hamiltionian of the Heisenberg model with interactions between all
particle-pairs. Here, we show that the Hamiltonian of the Heisenberg model on
the lattice configuration with interactions between all particle-pairs can be
expressed in terms of the Casimir operators of $U\left(2\right)$ group. We
give the eigenvalue spectrum of the system. For the sake of clarity, in this
section, we directly give the result. The detail of calculations will be given
in Sec. 3. For the Heisenberg model on lattice configurations with
interactions between all particle-pairs, the Hamiltonian reads

$H=\sum_{\left(i,j\right)}\frac{1}{2}\mathbf{S}_{i}\cdot\mathbf{S}_{j},$ (2.1)

where $\mathbf{S}_{i}$ is the spin of $ith$ particle and
$\sum_{\left(i,j\right)}$ runs all pairs of particles. With simple
transformations, the Hamiltonian, Eq. (2.1), can be written as

$H=\sum_{\left(i,j\right)}\tau_{ij}-\sum_{\left(i,j\right)}\mathbf{e}\cdot\mathbf{e},$
(2.2)

where $\tau_{ij}$ if the action of permutation of particles with index $i$ and
$j$. and $\mathbf{e}$ is the identity matrix. By using the relation between
the Casimir operator of unitary group and the conjugacy-class operator of
permutation group, which, for the sake of clarity, will be provided in Sec.
3.1, one can rewrite Eq. (2.2) in the form

$H\mathbf{=}P_{\left(1^{N-2},2\right)}=\frac{1}{2}C_{2}-C_{1},$ (2.3)

where $C_{2}$ and $C_{1}$ are the Casimir operator of the unitary group
$U\left(2\right)$ with order $2$ and $1$, respectively.
$P_{\left(1^{N-2},2\right)}$ is the conjugacy-class operator of the
symmetrical group.

The eigenvalue spectrum. The Hamiltionain, Eq. (2.3), shows that the
eigenvalue of the Hamiltionain can be obtained once the eigenvalues of the
Casimir operator of the unitary group $U\left(2\right)$ are given. For a
system consisting of $N$ particles, the irreducible representation of
$U\left(2\right)$ is indexed by a single parameter $s$, an integer ranges from
$0$ to the largest integer smaller than $N/2$ [17]. By using the eigenvalue of
the Casimir operator of $U\left(2\right)$ in a given representation [17], we
give the eigenvalue of the Hamiltionain, Eq. (2.3):

$E_{s}=2s^{2}-2Ns-2s+\frac{1}{2}N^{2}-\frac{1}{2}N.$ (2.4)

The degeneracy of energy $E_{s}$ is $\omega_{s}$

$\omega_{s}=\frac{\left(1+N-2s\right)^{2}N!}{\left(1+N-s\right)!s!}.$ (2.5)

##  3 A generalization of the Heisenberg model

In this section, we construct a series of exactly solvable models. These
models are the generalization of the Heisenberg model using the relation
between the Casimir operator of the unitary group and the conjugacy-class
operator of permutation group. This method provides us an easier way to give
the spectrum of the many-body spin system. As shown in Sec. 2, setting $m=2$
recovers the Heisenberg model. For the generalized model, bosonic and
fermionic realizations are discussed below.

###  3.1 The Casimir operator of the unitary group and the conjugacy-class
operator of the permutation group: a brief review

The Casimir operator of the unitary group $U\left(m\right)$. The unitary group
$U\left(m\right)$ has $m$ linear-independent Casimir operators. The Casimir
operator of order $l$, denoted by $C_{l}$, is [17]

$C_{l}=\sum_{k_{1},k_{2},...,k_{l-1}}E_{k_{1}k_{2}}E_{k_{2}k_{3}}...E_{k_{l-2}k_{l-1}}E_{k_{l-1}k1}$
(3.1)

where $E_{kl}$ is the generator of $U\left(m\right)$ .

The irreducible representation of $U\left(m\right)$ is indexed by $m$
nonnegative integers denoted by
$\left(a\right)=\left(a_{1},a_{2},\ldots,a_{m}\right)$ with $a_{1}\geqslant
a_{2}\geqslant...\geqslant a_{m}\geqslant 0$ [17]. The eigenvalue of the
Casimir operator $C_{l}$, denoted by $\left\langle C_{1}\right\rangle$, under
the representation indexed by $\left(a\right)$ is [17]

$\displaystyle\left\langle C_{1}\right\rangle$ $\displaystyle=S_{1}$ (3.2)
$\displaystyle\left\langle C_{2}\right\rangle$
$\displaystyle=S_{2}-\left(m-1\right)S_{1}$ (3.3) $\displaystyle\left\langle
C_{3}\right\rangle$
$\displaystyle=S_{3}-\left(m-\frac{3}{2}\right)S_{2}-\frac{1}{2}S_{1}^{2}-\left(m-1\right)S_{1}$
(3.4) $\displaystyle\ldots,$ (3.5)

where

$S_{k}=\sum_{i=1}^{m}\left[\left(a_{i}+m-i\right)^{k}-\left(m-i\right)^{k}\right].$
(3.6)

The conjugacy-class operator of the permutation group $S_{N}$. The sum of all
the group elements which belong to the same conjugacy class gives a conjugacy-
class operator [18]. The conjugacy-class operator commutes with all the group
elements [18]. For the permutation group of order $N$, each integer partition
of $N$ (a representation of $N$ in terms of the sum of other integers) gives a
corresponding conjugacy-class operator [18]. In this paper, we focus on the
conjugacy-class operator corresponding to the integer partition
$\left(\lambda\right)=\left(1^{N-2},2\right)$ with superscript $N-2$ standing
for $1$ appearing $N-2$ times. By definition, the conjugacy-class operator
reads

$P_{\left(1^{N-2},2\right)}=\sum_{1=i<j}^{N}\tau_{ij},$ (3.7)

where $\tau_{ij}$ is the exchange of the $ith$ and the $jth$ particle.

###  3.2 A relation between the Casimir operator of the unitary group and the
conjugate class operator of the permutation group

In this section, we introduce a relation between the Casimir operator of the
unitary group and the conjugacy-class operator of the permutation group. This
relation makes it easier to calculate the eigenvalue spectrum of the
Heisenberg model .

The relation between $C_{l}$ and $P_{\left(1^{N-2},2\right)}$. The conjugacy-
class operator $P_{\left(1^{N-2},2\right)}$ of the permutation group satisfies

$P_{\left(1^{N-1},2\right)}=\frac{1}{2}C_{2}-\frac{m}{2}C_{1},$ (3.8)

where $C_{1}$ and $C_{2}$ are the Casimir operators of order $1$ and $2$ of
the unitary group.

Proof. Let $a_{k}^{\dagger i}$ with superscript ranging from $1$ to $N$ and
subscript ranging from $1$ to $m$ represents creating the $ith$ particle in
the state $k$. $a_{k}^{i}$ is the annihilation operator, which represents the
conjugate of $a_{l}^{\dagger j}$. The commutation relation between
$a_{k}^{\dagger i}$ and $a_{l}^{j}$ is

$\displaystyle\left[a_{k}^{i},a_{l}^{\dagger j}\right]$
$\displaystyle=\delta_{ij}\delta_{kl},$ (3.9)
$\displaystyle\left[a_{k}^{i},a_{l}^{j}\right]$
$\displaystyle=\left[a_{k}^{\dagger i},a_{l}^{\dagger j}\right]=0.$ (3.10)

The generator $\tau_{ij}$ of $S_{N}$ can be expressed in terms of $a_{l}^{j}$
and $a_{l}^{\dagger j}$ [18]:

$\tau_{ij}=\sum_{k,l=1}^{m}a_{k}^{\dagger i}a_{l}^{\dagger
j}a_{l}^{i}a_{k}^{j}.$ (3.11)

The generator of $U\left(m\right)$ can also be expressed as:

$E_{kl}=\sum_{i=1}^{N}a_{k}^{\dagger i}a_{l}^{i}.$ (3.12)

Substituting Eq. (3.11) with $\tau_{ij}$ into Eq. (3.7) gives the conjugacy-
class operator:

$P_{\left(1^{N-2},2\right)}=\sum_{1=i<j}^{N}\sum_{k,l=1}^{m}a_{k}^{\dagger
i}a_{l}^{\dagger j}a_{l}^{i}a_{k}^{j}.$ (3.13)

Substituting Eq. (3.12) with $E_{kl}$ into Eq. (3.1) gives the Casimir
operator:

$\displaystyle C_{2}$
$\displaystyle=\sum_{i,j}^{N}\sum_{k,l=1}^{m}a_{k}^{\dagger i}a_{l}^{\dagger
j}a_{l}^{i}a_{k}^{j}C,$ (3.14) $\displaystyle C_{1}$
$\displaystyle=\sum_{i=1}^{N}\sum_{l=1}^{m}a_{l}^{\dagger i}a_{l}^{i}.$ (3.15)

So the operator relation between $P_{\left(1^{N-2},2\right)}$ and $C_{l}$
becomes

$\frac{1}{2}C_{2}-\frac{m}{2}C_{1}=P_{\left(1^{N-1},2\right)}+\frac{1}{2}\sum_{k,l=1}^{m}\sum_{i=1}^{N}a_{k}^{\dagger
i}a_{l}^{\dagger i}a_{l}^{i}a_{k}^{i},$ (3.16)

where Eq. (3.13) is used. The last term of Eq. (3.16) can be written as

$\sum_{k,l=1}^{m}\sum_{i=1}^{N}a_{k}^{\dagger i}a_{l}^{\dagger
i}a_{l}^{i}a_{k}^{i}=\sum_{i=1}^{N}\left[\sum_{l=1}^{m}n_{l}^{i}-\left(\sum_{l=1}^{m}n_{l}^{i}\right)\left(\sum_{l=1}^{m}n_{l}^{i}\right)\right],$
(3.17)

by introducing $N_{i}\equiv\sum_{l=1}^{m}n_{l}^{i}$ with $n_{l}^{i}$ equaling
to $a_{l}^{\dagger i}a_{l}^{i}$. $N_{i}$ represents the number of $ith$
particle, that is $1$. Therefore, the equation

$\sum_{k,l=1}^{m}\sum_{i=1}^{N}a_{k}^{\dagger i}a_{l}^{\dagger
i}a_{l}^{i}a_{k}^{i}=0$ (3.18)

holds. Eq. (3.8) is proved.

###  3.3 The generalization

In this section, by using the relation between the Casimir operator of the
unitary group and the conjugacy-class operator of the permutation group, Eq.
(3.8), we construct a series of exactly solvable models which are
generalizations of the Heisenberg model. They are models on lattice
configurations with interactions between all particle-pairs.

The Hamiltionian. The Hamiltonian of the model is

$H=\frac{1}{2}C_{2}-\frac{m}{2}C_{1}$ (3.19)

with $C_{1}$ and $C_{2}$ the Casimir operators of $U\left(m\right)$. Such
systems consist of $N$ $m$-states particles. The Hamiltonian $H$ in Eq. (3.19)
exchanges all particle-pairs in the system, that is

$H\left(v_{1}\otimes v_{2}\otimes v_{3}\otimes...\otimes
v_{N}\right)=v_{2}\otimes v_{1}\otimes v_{3}\otimes...\otimes
v_{N}+v_{1}\otimes v_{3}\otimes v_{2}\otimes...\otimes v_{N}+\ldots.$ (3.20)

Therefore, the generalized models are on lattice configurations with
interactions between all particle-pairs.

The eigenvalue spectrum. Eq. (3.19) shows that, the eigenvalue of the
Hamiltionain can be obtained provided the eigenvalues of the Casimir operator
of the unitary group $U\left(m\right)$ are given. The eigenvalues of the
Casimir operator of the unitary group $U\left(m\right)$ are indexed by $m$
nonnegative integers denoted by
$\left(a\right)=\left(a_{1},a_{2},...,a_{m}\right)$ with $a_{1}\geqslant
a_{2}\geqslant...\geqslant a_{m}\geqslant 0$ [17]. The summation of
$\left(a\right)$ is $N$, i.e., $\sum_{i=1}^{m}a_{i}=N$. By using the
eigenvalue of the Casimir operator given in Eqs. (3.2)-(3.3), we give the
eigenvalue of the Hamiltionain, Eq. (3.19),

$E_{\left(a\right)}=\frac{1}{2}\sum_{i=1}^{m}a_{i}^{2}-\sum_{i=1}^{m}a_{i}i+\frac{1}{2}N.$
(3.21)

The degeneracy of $E_{\left(a\right)}$ is

$\omega_{\left(a\right)}=N!{\displaystyle\prod\limits_{i<j}^{m}}\frac{\left(a_{i}-a_{j}-i+j\right)}{\left(j-i\right)}\frac{{\displaystyle\prod\limits_{i^{\prime}<j^{\prime}}^{l}}\left(a_{i^{\prime}}-a_{j^{\prime}}-i^{\prime}+j^{\prime}\right)}{{\displaystyle\prod\limits_{i^{\prime\prime}=1}^{l}}\left(l+a_{i^{\prime\prime}}-i^{\prime\prime}\right)!},$
(3.22)

where $l$ is the number of positive integers in $\left(a\right)$.

A bosonic realization: an example.  By expressing the generators of
$U\left(m\right)$ in terms of the Boson operators $a_{i}^{\dagger}$ and
$a_{j}$ with commutation relation

$\displaystyle\left[a_{i},a_{j}^{\dagger}\right]$ $\displaystyle=\delta_{ij},$
(3.23) $\displaystyle\left[a_{k},a_{j}\right]$
$\displaystyle=\left[a_{k}^{\dagger},a_{l}^{\dagger}\right]=0,$ (3.24)

the Hamiltonian of the system, Eq. (3.19), can be written as

$H=\frac{1}{2}\sum_{i,j=1}^{m}a_{i}^{\dagger}a_{j}a_{j}^{\dagger}a_{i}-\frac{m}{2}\sum_{i=1}^{m}a_{i}^{\dagger}a_{I}.$
(3.25)

By using Eqs. (3.21) and (3.22) with $a_{1}=N$ and $a_{i}=0$ for $i>1$, we can
obtain the eigenvalue and degeneracy of the system:

$E_{n}=\frac{1}{2}n^{2}-\frac{1}{2}n,$ (3.26)
$\omega_{n}=\frac{n!{\displaystyle\prod\limits_{l=1}^{m-1}}\left(l+n\right)^{2}}{\left(m-1\right)!\left(n+m-1\right)!}.$
(3.27)

A fermionic realization: an example. By expressing the generators of
$U\left(m\right)$ in terms of the Fermi operators $b_{i}^{\dagger}$ and
$b_{j}$ with commutation relation

$\displaystyle\left\\{b_{i},b_{j}^{\dagger}\right\\}$
$\displaystyle=\delta_{ij},$ (3.28)
$\displaystyle\left\\{b_{i}^{\dagger},b_{j}^{\dagger}\right\\}$
$\displaystyle=\left\\{b_{i},b_{j}\right\\}=0,$ (3.29)

the Hamiltonian of the system, Eq. (3.19), can be written as

$H=\frac{1}{2}\sum_{i,j=1}^{m}b_{i}^{\dagger}b_{j}b_{j}^{\dagger}b_{i}-\frac{m}{2}\sum_{i=1}^{m}b_{i}^{\dagger}b_{i}$
(3.30)

The eigenvalue and degeneracy, by using Eqs. (3.21) and (3.22) with
$a_{i}=a_{j}=1$, are expressed as

$E_{n}=-\frac{n^{2}}{2}+\frac{1}{2}n,$ (3.31)
$\omega_{n}=\frac{m!}{\left(m-n\right)!n!}.$ (3.32)

Notice that, in Bose and Fermi cases, such system have no-fixed particle
numbers.

##  4 The degeneracy of eigenstates of a lattice model

In a lattice model, usually the nearest number of particle is a finite
numbers, such as 3, 4, 6, and so on, due to the constraint of symmetries.
Therefore, one only considers interactions between nearest particle-pairs,
second nearest particle-pairs, and so on. In this paper, by converting the
lattice configurations into networks, we can consider models with arbitrary
number of interacting particle-pairs. That is, we can consider models on
networks with different modes of links. In this section, we investigate the
relationship between degeneracy of eigenstates and the number of links in
networks, by numerically calculating the Heisenberg models and random walks on
network with different number of links. It shows that models on networks with
more links tend to have higher degeneracy of eigenstates.

###  4.1 Heisenberg models and random walks on networks with different numbers
of links

Various models can be considered on a network or lattice configurations. In
this section, we consider Heisenberg models and random walks on networks with
different modes of links.

Heisenberg models. In this section, we study the isotropic Heisenberg models
with Hamiltionain $H$ reads

$H=\sum\frac{1}{2}\mathbf{S}_{i}\cdot\mathbf{S}_{j},$ (4.1)

where $\mathbf{S}_{i}$ is the spin of $ith$ particle and $\sum$ represents
summation over particle-pairs with links. The eigenvalue spectrum of
Heisenberg models, Eq. (4.1), in different dimensions and on networks with
random links are given in Figs (3) and (4).

Figure 3: The eigenvalue of Heisenberg model consisting of $9$ particles in
different dimensions. as shown, $1$D represents a line configuration, $2$D
represents a plain configuration. Figure 4: The eigenvalue of Heisenberg model
consisting of $9$ particles on networks with random links. In a network, for
any two particles, there is a probability to form a link between them.

Fig. (5) shows the number of distinct eigenvalues of Heisenberg models
consisting of $N$ particles on networks with different number of links. It
shows the tendency that degeneracy of eigenstates increases with the increase
of the number of links when the number of links is relatively large.

To be notice that the number of distinct eigenvalues is not strictly monotonic
decreasing as the number of links in the networks, see Fig. (5), because the
number of links and how the links are formed decide the eigenvalues
simultaneously. Here, we only consider the number of links in the networks.

Figure 5: degeneracy of eigenstates of Heisenberg models consisting of
different numbers of particles on lattice with different numbers of links.
degeneracy here is revealed on the number of distinct eigenvalues.

The random walk. Here, we construct random walks on networks with different
modes of links. For example, the matrix of a random walk on a one-dimensional
lattice configuration consists of $5$ positions reads

$\left(\begin{array}[c]{ccccc}0&\frac{1}{2}&0&0&\frac{1}{2}\\\
\frac{1}{2}&0&\frac{1}{2}&0&0\\\ 0&\frac{1}{2}&0&\frac{1}{2}&0\\\
0&0&\frac{1}{2}&0&\frac{1}{2}\\\
\frac{1}{2}&0&0&\frac{1}{2}&0\end{array}\right),$ (4.2)

where periodic boundary conditions are applied. On a one-dimensional lattice
configuration, one can only go left or right with equal probability $1/2$. The
random walk on a one-dimensional lattice configuration with a hole at position
$1$ reads

$\left(\begin{array}[c]{ccccc}0&0&0&0&0\\\ 0&0&1&0&0\\\
0&\frac{1}{2}&0&\frac{1}{2}&0\\\ 0&0&\frac{1}{2}&0&\frac{1}{2}\\\
0&0&0&1&0\end{array}\right),$ (4.3)

where a hole means that any links to the hole is forbidden. Under such
assumptions, one can not reach a hole. Thus, the probability to reach a hole
is $0$. as shown in Table. (1) and Fig. (6), especially in three-dimension
spaces, degeneracy of eigenstates increases with the number of links in the
networks. The number of links is adjusted by changing the number of holes in
the network.

Table 1: The number of distinct eigenvalues of random walks on different lattice configurations. Shape | Position of holes | Number of links | Number of distinct eigenvalues
---|---|---|---
$1\times 25$ | $-$ | $25$ | $13$
$1\times 25$ | $4$ | $23$ | $25$
$1\times 25$ |  $2$ and $9$ | $21$ | $21$
$1\times 25$ |  $3$, $8$, and $13$ | $19$ | $17$
$5\times 5$ | $-$ | $50$ | $6$
$5\times 5$ | $(1,4)$ | $46$ | $17$
$5\times 5$ |  $(1,2)$ and $(2,4)$ | $42$ | $23$
$5\times 5$ |  $(1,3)$, $(2,3)$, and $(3,3)$ | $40$ | $21$
$3\times 3\times 3$ | $-$ | $81$ | $4$
$3\times 3\times 3$ | $(2,1,1)$ | $75$ | $8$
$3\times 3\times 3$ |  $(1,2,1)$ and $(3,3,1)$ | $69$ | $15$
$3\times 3\times 3$ |  $(2,2,1)$, $(3,3,1)$, $(3,3,2)$, and $(2,3,3)$ | $59$ | $16$
Figure 6: degeneracy of eigenstates of random walks on lattice configurations
with different numbers of links. degeneracy here is revealed on the number of
distinct eigenvalues.

###  4.2 The highest degeneracy of eigenstates of a lattice model

In Sec. 4.1, it shows that degeneracy of eigenvalues increases with the number
of links in a network. That is, models on configurations with larger number of
interacting particle-pairs have higher degeneracy of eigenstates. In this
section, we propose the assumption that the lattice model with interactions
between all particle-pairs, such as the generalized model proposed in Sec.
3.2, should have the highest degeneracy of eigenstates.

The highest degeneracy of eigenstates of the generalized Heisenberg model. For
the generalized Heisenberg model consisting of $N$ particles with the
dimension of single-particle-Hilbert space $m$, by using Eq. (3.21), the
number of distinct eigenvalues is $P(N,m)$, where $P(N,m)$ is the restrict
integer partition number [19] that is the number of ways to express $N$ as sum
of other integers with the number of summands no larger than $m$. It is
because one irreducible representation gives a distinct eigenvalue of Casimir
operator and thus gives a distinct eigenvalue of the system. The irreducible
representation is labeled by a set of number
$\left(a\right)=\left(a_{1},a_{2},...,a_{m}\right)$ with $a_{1}\geqslant
a_{2}\geqslant...\geqslant a_{m}\geqslant 0$ [17]. The summation of
$\left(a\right)$ is $N$, thus, the number of $\left(a\right)$ equals the
number of ways to represent $N$ in terms of other integers the number of
summands smaller than $m$, that is $P(N,m)$. as shown in Fig. (7).

Figure 7: The smallest number of distinct eigenvalues of the generalized
Heisenberg model.

For these cases, the average degeneracy of an energy level reads
$m^{N}/P(N,m)$. For example, setting $m=3$ and $N=6$, the smallest number of
distinct eigenvalues is $P(N,m)=7$, the average degeneracy is $104.14$. as
shown in Table. (2), the average degeneracy of an energy level in creases fast
with the number of particles or the dimension of single-particle Hilbert
space.

Table 2: The average degeneracy of an energy level at the upper limit of energy level degeneracy. The dimension of single-particle Hilbert space | Number of particles | The average degeneracy
---|---|---
$2$ | $5$ | $10.7$
$2$ | $6$ | $16.0$
$2$ | $7$ | $32.0$
$2$ | $8$ | $51.2$
$2$ | $9$ | $102.4$
$2$ | $10$ | $170.7$
$3$ | $5$ | $48.6$
$3$ | $6$ | $104.1$
$3$ | $7$ | $273.4$
$3$ | $8$ | $656.1$
$3$ | $9$ | $1640.3$
$3$ | $10$ | $4217.8$

##  5 Conclusions and discussions

The difficulty of solving the interacting quantum many-body system is due to
the properties of the space configuration [20, 21, 22] and the interaction
mode between inter-particles including the classical interaction and the
quantum exchange interaction [23, 24, 25, 26]. Few models of quantum
interacting many-body systems could be solved exactly in reality. Common
practices are simplifying the models or using approximation methods instead of
solving the system directly, we conclude that to investigate the system from
the perspective of properties of the space and symmetries of the system is
useful and effective.

In this paper, we consider models on lattice configurations with arbitrary
numbers of interacting particle-pairs by converting the lattice configurations
into networks with different modes of links. There are three main purposes of
our work: (1) we introduce a group theory method to solve the Heisenberg model
on lattice configurations with interactions between all particle-pairs by
revealing the relation between the Casimir operator of the unitary group and
the conjugacy-class operator of the permutation group. (2) We generalize the
Heisenberg model by this proposed relation and thus give a series of exactly
solvable models. (3) We show that degeneracy of the eigenvalue increases with
the number of interacting particle-pairs. Thus, there is a highest degeneracy
of eigenstates of a lattice model. The generalized Heisenberg model is a model
on lattice configurations with interactions between all particle-pairs, and
thus have the highest degeneracy of the eigenvalue among lattice models. The
smallest number of distinct eigenvalues for the generalized Heisenberg model
is a restrict integer partition function.

##  6 Acknowledgments

We are very indebted to Dr Yong Xie and Dr Guanwen Fang for their
encouragements. This work is supported by the Fundamental Research Funds For
the Central Universities No.2020JKF306.

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