# Carrot and Stick: Inducing Self-Motivation with Positive & Negative Feedback

Jimin Sohn1, Jeihee Cho2, Junyong Lee2, Songmu Heo3, Ji-Eun Han4, David R.
Mortensen5,
1 GIST, South Korea 2 Yonsei University, South Korea
3 Korea University, South Korea, 4 KT, South Korea, 5 Carnegie Mellon
University, USA
<EMAIL_ADDRESS>
###### Abstract

Positive thinking is thought to be an important component of self-motivation
in various practical fields such as education and the workplace. Previous
work, including sentiment transfer and positive reframing, has focused on the
positive side of language. However, self-motivation that drives people to
reach their goals has not yet been studied from a computational perspective.
Moreover, negative feedback has not yet been explored, even though positive
and negative feedback are both necessary to grow self-motivation. To
facilitate self-motivation, we propose CArrot and STICk (CASTIC) dataset,
consisting of $12,590$ sentences with 5 different strategies for enhancing
self-motivation. Our data and code are publicly available at here.

Carrot and Stick: Inducing Self-Motivation with Positive & Negative Feedback

Jimin Sohn1, Jeihee Cho2, Junyong Lee2, Songmu Heo3, Ji-Eun Han4, David R.
Mortensen5, 1 GIST, South Korea 2 Yonsei University, South Korea 3 Korea
University, South Korea, 4 KT, South Korea, 5 Carnegie Mellon University, USA
<EMAIL_ADDRESS>

##  1 Introduction

Interest in positive psychological aspects of language has growing in the
field of NLP. Ziems et al. (2022), Sharma et al. (2023), and Maddela et al.
(2023) introduce the task of Positive Reframing, aiming to shift the negative
perspective of a statement into a positive one without altering the original
content. Njoo et al. (2023) proposed a new benchmark analyzing how empowerment
is conveyed in language.

Previous research has only focused on reframing negative thoughts into
positive ones, ignoring the value of non-positive language. In this work, we
propose a new approach to appropriately utilize negative (and positive)
language as feedback, so as to induce self-motivation via stimulation.

Figure 1: An example of positive reframing Ziems et al. (2022) and feedback
generated using our CASTIC framework.

Self-motivation is an internal drive that leads a person to take action
towards a goal, which is significant in various real-world domains such as
education and business. One popular theoretical approach to motivation is
Maslow (1958), proposing that motivation is derived from five basic needs:
physiological, safety, belongingness & love, esteem, and self-actualization,
which are hierarchically organized.

Researchers have attempted to enhance the motivation of people by giving
feedback relevant to their situations. In Kim and Lee (2019), it was found
that when students received negative feedback, they achieved more accurate
self-assessment of skills compared to positive feedback, while positive
feedback enhanced students’ self-efficacy and boosted confidence in their
ability to achieve goals. Hence, the findings from Kim and Lee (2019) suggest
that a balanced use of positive and negative feedback is necessary to optimize
self-motivation. Wisniewski et al. (2020) also concluded that positive
feedback was effective in enhancing confidence and motivation while negative
feedback helped to clearly identify areas of deficiency and motivate
improvement. Although negative feedback might seem demotivating, it helped in
identifying areas that need improvement, guiding future efforts, and avoiding
past mistakes. The analysis of the results for each type of feedback in
Wisniewski et al. (2020) also aligns with Kim and Lee (2019), indicating that
for optimal self-motivation, both positive and negative feedback should be
used in a balanced manner.

Figure 2: Overall procedure of generating CASTIC dataset

In this work, we introduce a new benchmark named CArrot and STICk (CASTIC)
meant measure induction of self-motivation. We address the task by providing
both positive and negative feedback. As far as we can determine, dealing with
negative aspects of language in the context of motivation is methodologically
novel within NLP. The proposed dataset is generated with a three step
procedure and evaluated with both quantitative and qualitative approaches.

##  2 CASTIC Dataset

Obstacle Types | Definition & Example | Severity score
---|---|---
Relationships | Conflict situations with nearby people | not serious
_fight with friends, mother’s nagging_
Health | Physical or mental illness | serious
_migraine, stomachache, burn out_
Fear, overwhelmed | Anxiety about what will happen in the future or previous failings | serious
_Anxiety about past failures_
Lack of resource | A lack of supplies needed for work | serious
_lack of internet connection, lost laptop_
Annoyance | Irritation by trivial, annoying situations | not serious
_noisy circumstance_
Rest, Entertainment | Lack of motivation due to desire for simple entertainment | not serious
_game, movie, dating_
etc | Any situation other than the above | serious
_Internet/banking system error, bad weather_

Table 1:  The seven types of obstacles blocking someone from reaching their
goal and the corresponding severity score. The definition is indicated at the
top and a corresponding example is given in italics.

Large Language Models tend to generate sentences with a positive bias Chen et
al. (2023). However, from the perspective of motivation, unconditional
positive support is not always what is needed. Stimulating feedback relevant
to the person’s circumstance is more effective in achieving goals. Therefore,
instead of always giving positive feedback, the model should provide either
negative or positive feedback depending on how seriously the obstacle
interferes with the task to be done. To give language models this ability, we
propose CASTIC dataset that provides appropriate feedback for given sentences.
In this section, we present our overall procedure for data generation and
provide a taxonomy of the types of obstacles and strategies for giving
feedback.

###  2.1 Data Collection

The overall procedure of generating CASTIC is provided in Fig. 2. The prompt
for extracting TODO, Obstacle, and generating the final feedback sentence are
provided in Appendix A.3.

##### Input sentence

We use input sentences from Positive Psychology Frames (POSREF, Ziems et al.
(2022) collected from Twitter with simple keyword #stressed. We use only the
original text column from the dataset.

##### Obstacle and TODO Extraction Module

We use Orion-14B-Chat Chen et al. (2024) with an Apache-2.0 License to
generate datasets as it is an open-source large language model (LLM) with
outstanding performance in comprehensive evaluations and supporting even
extremely long texts Chen et al. (2024). The model first extracts TODO and
Obstacle from the input sentence. TODO is the goal that people aim to achieve
and Obstacle is the challenge or obstacle that hinder people from achieving
TODO. The model gets to respond "None" when there is no specific TODO in a
given sentence, and the feedback is generated considering only TODO.

##### Obstacle Type and Severity Score

We annotate which of the seven categories the extracted Obstacle belongs to.
It is worth noting that a sentence can have multiple obstacles and therefore
can have multiple types. The Severity Score is assigned corresponding to the
obstacle type in Table 1. Severity Score means how seriously the Obstacle
blocks the person from TODO. If the sentence has multiple obstacles, the
sentence is considered “not serious” only when all the obstacle types are
considered “not serious”. Categories and corresponding severity scores were
determined according to the criteria by which we manually checked and
classified all input data.

##### Feedback Generation

To generate feedback inducing self-motivation, we use five motivation
strategies referenced from the widely known Motivation Theory’s “five needs”
Maslow (1958). A detailed explanation of each need is provided in Appendix
A.2. Feedback was created with LLM (Orion-14B-Chat) using the TODO, Obstacle,
and Severity Score from the previous step and each of the five needs. The
severity score determines whether the feedback is positive or negative, and
each of five needs determines which aspects to emphasize to motivate the
person. We reviewed each feedback generated by the model.

###  2.2 Data Distribution

We evaluate the statistics of seven obstacle types in our CASTIC dataset in
Table 2. As one sample can have multiple obstacle types, the total number does
not indicate the number of distinct samples. The statistics of frequently
appearing words in the dataset is provided in Appendix A.6.

Obstacle Types | Train # | Validation #
---|---|---
Relationships | 110 | 80
Health | 1,000 | 305
Fear | 5,165 | 1,205
Lack of resource | 235 | 80
Annoyance | 2,015 | 785
Rest | 20 | 50
etc | 1,455 | 135
serious | 7,855 | 1,725
not serious | 2,145 | 915
Total | 10,000 | 2,640
Table 2:  Summary statistics of each obstacle type in the CASTIC dataset.

##  3 Self-Motivation Framework

Fine-tune | Model | Param. | R-1 | R-2 | R-L | BLEU | BScore | Sim | PPL
---|---|---|---|---|---|---|---|---|---
w/o Fine-tune | GPT | 116M | 11.79 | 0.47 | 8.35 | 0.08 | 82.20 | 0.121 | -
M2M-100 | 483M | 2.10 | 0.16 | 1.89 | 3.20 | 75.66 | 0.088 | 176.36
T5 | 60M | 0.49 | 0.00 | 0.49 | 1.66 | 84.52 | 0.452 | 295.87
Falcon | 7B | 10.59 | 0.76 | 7.49 | 0.19 | 82.07 | 0.19 | 106.02
Mistral | 7B | 12.47 | 1.04 | 8.76 | 0.29 | 82.77 | 0.16 | 202.30
BART-L | 406M | 18.10 | 3.76 | 13.00 | 1.81 | 84.94 | 0.458 | 188.65
w/ Fine-tune | GPT | 116M | 27.68 | 6.62 | 20.11 | 3.77 | 88.11 | 0.429 | 69.52
M2M-100 | 483M | 30.12 | 8.76 | 22.44 | 5.99 | 88.74 | 0.437 | 32.84
T5 | 60M | 30.28 | 10.04 | 23.74 | 5.5 | 88.76 | 0.480 | 25.40
Falcon | 7B | 29.63 | 8.59 | 20.84 | 4.59 | 88.30 | 0.522 | 33.49
Mistral | 7B | 27.96 | 13.19 | 23.66 | 2.64 | 88.63 | 0.496 | 27.61
BART-L | 406M | 33.93 | 10.04 | 23.59 | 7.00 | 88.98 | 0.498 | 24.11
Table 3:  Self-Motivation results Performance of models with and without fine-
tuning on CASTIC dataset on ROUGE-1 (R-1), ROUGE-1 (R-2), ROUGE-L (R-L), BLEU,
and BERTScore (BScore). Param., Sim, PPL indicates the number of parameters of
each model, cosine similarity and perplexity, respectively.

###  3.1 Task Formulation

To generate feedback, we extract the Obstacle $O_{i}$ and TODO $T_{i}$ from
the input sentence. Then, the Severity Score $SS_{i}$ is assigned based on the
Obstacle Type $OT_{i}$ of the $O_{i}$. Then, Feedback Type $FT_{i}$ is labeled
as either positive or negative according to the severity score.

$FT_{i}=\begin{cases}\text{Positive if}\;SS_{i}=\text{serious},\\\
\text{Negative if}\;SS_{i}=\text{not serious}\end{cases}$

Based on Motivation Strategy $M_{i}$, the final Feedback $F_{i}$ to induce
self-motivation is generated.

$F_{i}=\\{FT_{i},M_{i},O_{i},T_{i}\\}$

###  3.2 Evaluation

####  3.2.1 Experimental Setup

Dataset The CASTIC dataset contains 9,990 samples in the train split and 2,600
samples in the validation split. All the data are in English.
Model We use BART-L Lewis et al. (2019), GPT-2 Radford et al. (2019), M2M-100
Fan et al. (2021), T5 Raffel et al. (2020) model to test the dataset. The
number of parameters per model is described in Table 3.

####  3.2.2 Evaluation Metric

Quantitative metric In various studies, the BLEU Papineni et al. (2002) and
ROUGE Lin (2004) metrics are utilized to evaluate the semantic similarity with
the ground truth. In this paper, we use BLEU, ROUGE-1, ROUGE-2, ROUGE-L, and
BERTScore Zhang et al. (2019) for qualitative results following previous work
Ziems et al. (2022). Cosine similarity between generated output and ground
truth is measured using the sentence transformer all-MiniLM-L6-v2 Reimers and
Gurevych (2019). Perplexity Bengio et al. (2000) is measured using GPT-2
Radford et al. (2019) from Hugging Face. It is worth noting that we discard
the empty generation samples when measuring the scores.
Qualitative metric Following Chiang and Lee (2023), we use GPT-3.5 Brown et
al. (2020) to evaluate the effect of our dataset on inducing self-motivation.
The prompt is illustrated in Appendix A.4. From the model’s generated
feedback, we randomly sample 100 sentences and ask GPT-3.5 how motivating
(Motivation) and how fluent (Fluency) the feedback is. The rating scale is
from 1-5 with 1 being the lowest.

###  3.3 Experimental Result

##### Overall Result

In Table 3, we evaluate the result of the experiment. The models can learn
each motivation strategy and generate feedback well. We illustrate the example
of generated feedback in Table 7 of Appendix A.7. Overall, BART-L shows the
best performance both in zero-shot and fine-tuning experiments.

Strategy | GPT | GPT-2 | M2M-100 | T5 | BART-L
---|---|---|---|---|---
Physiological Needs | 62.49 | 76.17 | 74.9 | 77.51 | 74.45
Safety Needs | 72.23 | 75.26 | 73.31 | 74.64 | 71.94
Love and Belonging | 72.83 | 76.97 | 76.2 | 76.67 | 65.95
Self-actualization | 67 | 75.32 | 75.87 | 74.95 | 72.32
Esteem Needs | 78.21 | 80.41 | 79.01 | 80.8 | 68.88
AVG. | 70.55 | 76.83 | 75.86 | 76.91 | 70.71
STD. | 6.01 | 2.12 | 2.09 | 2.48 | 3.32

Table 4:  F1 score (%) of motivation strategies classification.

##### LLM Evaluation Result

We compare the score for models that are fine-tuned on the CASTIC dataset and
pre-trained models without fine-tuning. In Table 5, we illustrate the self-
motivation feedback resulting from GPT-3.5. The models fine-tuned with our
dataset show better performance compared to the others. Specifically, the
average motivation score for the fine-tuned models is 2.7, whereas models
without fine-tuning achieve an average motivation score of 1.66. Additionally,
in terms of fluency, the fine-tuned models attain an average score of 4,
outperforming the zero-shot result. The result indicates that the model fine-
tuned on our dataset generates fluent outputs.

Model | w/ Fine-tune | w/o Fine-tune
---|---|---
Motivation | Fluency | Motivation | Fluency
GPT | 2.3 | 3.57 | 1.17 | 2.03
M2M-100 | 2.32 | 3.85 | 1.00 | 1.00
BART-L | 2.54 | 4.34 | 1.82 | 3.06
T5 | 3.03 | 4.18 | 2.12 | 2.47
Mistral | 3.05 | 3.82 | 1.82 | 2.67
Falcon | 2.99 | 4.35 | 2.03 | 2.10

Table 5:  LLM Evaluation results The average rate of generated feedbacks on
with and without fine-tuned models in the terms of how motivating and fluent
the feedback is.

##### Motivation Strategy Classification

In Table 4, we evaluate F1 score per 5 motivation strategies. We add one
additional linear layer that outputs 5 classes corresponding to each
motivation type. The result shows that each model can learn and distinguish
the characteristics between motivation strategies with over 70% F1 score for
every strategy.

## Limitation

We acknowledge that the severity score, which is determined based on the
severity of the obstacle, can be subjective. However, just as people
previously judged the negative and positive levels of words and annotated them
to obtain negative scores in the field of Sentimental analysis, we present the
standards by creating our own dataset and annotating it.

More significantly, we did not test the output of the trained models on human
participants to determine, empirically, whether they induced greater levels of
self-motivation.

## Ethics Statement

In this work, we used POSREF Ziems et al. (2022) which is a publicly available
dataset. The creators of POSREF have already considered ethical issues when
creating the dataset, but we additionally have manually checked every input
sentence and filtered out inappropriate ones. We didn’t find any obvious
ethical concerns, such as violent or offensive content. We used the dataset
consistent with the intended use. We used LLM in the process of creating and
validating the dataset and we performed the verification of the output
ourselves, meaning there were no issues with human annotators.

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##  Appendix A Appendix

###  A.1 Implementation Detail

Motivation Result We train BART-L and M2M-100 with learning rate 1e-4, and
batch size 32 for 5 epochs with a maximum sequence length of 128. For GPT and
T5, we use learning rate 3e-5, and a weight decay of 0.01 for 5 epochs. For
training and inference, we use a NVIDIA H100 80GB GPU. We run the experiment
once for all the models.
Motivation Strategy Classification We train each model with learning rate 1e-4
and batch size 32 for 2 epochs with output class number as 5. For training and
inference, we use a NVIDIA H100 80GB GPU. We run the experiment once for all
the models.

###  A.2 Maslow’s Motivation Theory

In this section, we provide detailed explanation of Maslow’s Motivation Theory
Maslow (1958)’s five needs.
Physiological Needs are desires to maintain a constant, normal state such as
with respect to homeostasis, hormones, and nutrients. We applied this strategy
for desiring rest, sleep, food, water, air and homeostasis.
Safety Needs are the desires for a safe environment such as financial and
health security, stability in employment, protection from accidents, and a
safe environment. The need derives from human’s nature or preferring a safe,
orderly, predictable, organized world and disfavoring unexpected,
unmanageable, or other dangerous things.
Love & Belonging Needs are desires for love and affection from friends, lovers
or spouses. Humans strive to have affectionate relations with people.
Esteem Needs are desires for stable, firmly based, high evaluation of
themselves. Humans want to feel real respect from themselves and others for
their capacities and achievements.
Self-actualization refers to the desire for self-fulfillment in actualizing a
person’s full potential. Humans want to become everything that one is capable
of becoming.

###  A.3 Prompt for Dataset Generation

####  A.3.1 Obstacle and TODO Extraction

We extract Obstacle and TODO of the negative input sentence with
Orion-14B-Chat using the prompt in Figure 3.

Figure 3: Prompt for Obstacle and TODO extraction

####  A.3.2 Generating Feedback

We generate feedback inducing self-motivation with Orion-14B-Chat using prompt
in Figure 3. The type of feedback, positive/negative determined by severity
score was applied in the feedback.

Figure 4: Prompt for Generating Feedback

###  A.4 Prompt for LLM Evaluation

Motivation We evaluate how motivating the feedback generated by model trained
in our dataset is with Chat-GPT using prompt in Figure 5 following Chiang and
Lee (2023).
Fluency We evaluated how fluent the generated feedback by model trained in our
dataset is using prompt in Figure 6.

Figure 5: Prompt for evaluating motivation with LLM Figure 6: Prompt for
evaluating fluency with LLM

###  A.5 Samples of CASTIC dataset

In Table 6, we illustrate the example of dataset.

(a) Input

(b) Feedback

Figure 7: Word frequency analysis for train of CASTIC

(a) Input

(b) Feedback

Figure 8: Word frequency analysis for validaiton of CASTIC

###  A.6 Word frequency analysis

The word frequency figure illustrates the distribution of terms within our
CASTIC dataset, providing key insights into the linguistic patterns and
content characteristics of the corpus. As depicted in Figure 7(a), 7(b), 8(a),
8(b), the x-axis represents individual words ordered by their frequency, while
the y-axis shows the occurrence count without prompt words. The figure reveals
several prominent trends: notably, common words such as ‘goal’, ‘love’, and
‘focus’ dominate the corpus, reflecting the dataset’s focus on emotional
motivation. These findings underscore the dataset’s thematic concentration and
provide a foundation for understanding its linguistic structure and contextual
relevance within the field of natural language processing. Moreover,
comparative analyses with benchmark datasets reveal distinctive word usage
patterns unique to our dataset, highlighting its potential contributions to
advancing research in automated language understanding and generation.

Input | Obstacle | TODO | Obstacle Type | Severity score | Motivation strategy | Feedback
---|---|---|---|---|---|---
Still home from school sick gonna pay big time when i get back should be doing homework all weekend on the plane :$($ | Sick |  Take some rest and do homework all weekend on the plane. | Health | serious | physiological need |  | Take care of yourself
---
by resting and doing
your homework on the plane.
Your body will thank you for it!
safety needs |  | Take care of yourself, prioritize
---
your health and well-being. Rest
and study on the plane will help
you feel recharged and ready
to tackle any obstacles that come your way.
love and belonging |  | Take care of yourself and focus
---
on your studies. Your well-being is
important for your future relationships.
self-actualization |  | Take care of yourself and prioritize
---
your well-being. Rest and study will
help you achieve your goals
and grow as a person.
esteem needs |  | Rest and study will boost
---
your health and increase your
confidence, giving you the chance
to show your true potential and achieve success.

Table 6: Examples of feedback in the dataset.

###  A.7 Qualitative result

In table, we examine one original text for each of the motivation strategies
in our dataset along with the BART model.

Input | Motivation strategy | Feedback
---|---|---
I already hate using computers, so having to apply for a job online is not for me. | physiological need |  | You have the capability and determination to succeed
---
in applying for a job online. Take it one step at
a time and focus on what you need to accomplish.
_Think about how much easier it will be to apply_
_for a job online without having to leave your_
_house! You can take care of business and still_
_maintain a constant, normal state._
safety needs |  | You have the capability and determination to succeed
---
in applying for a job online. Take it one step at
a time and focus on what you can control. You can do it!
_Don’t let fear of computers hold you back. Apply_
_for that job online and take control of your future._
love and belonging |  | Feeling lonely? Apply for a job online and make new
---
connections. You’ll find love and belonging, which
are important for your overall well-being.
_Feeling lonely? Finding a job online could connect_
_you with new friends and a sense of community._
self-actualization |  | You have the potential and the desire to succeed.
---
Let’s focus on applying for a job online
and fulfilling your potential.
_Don’t let fear of computers hold you back._
_Take control and apply for that job_
_online. You have the potential to succeed_
_and this is your chance to show it._
esteem needs |  | You have the capability and determination to succeed.
---
Let’s focus on applying for the job
online and show your skills.
_Feeling capable and respected is important, but_
_hating computers won’t get you far. It’s time to_
_overcome your fear and apply for that job online._
Table 7:  Examples of generated feedback of BART in a controlled setting.
Prediction feedback corresponds to each motivation strategy is at the first
row and the ground truth follows in italic.


# Proof of a Stable Fixed Point for Strongly Correlated Electron Matter

Jinchao Zhao1, Gabriele La Nave2, and Philip W. Phillips1 1Department of
Physics and Institute for Condensed Matter Theory, University of Illinois 1110
W. Green Street, Urbana, IL 61801, U.S.A.  2Department of Mathematics,
University of Illinois, Urbana, Il. 61801

(March 2023)

###### Abstract

We establish the Hatsugai-Kohmoto model as a stable quartic fixed point
(distinct from Wilson-Fisher) by computing the $\beta-$function in the
presence of perturbing local interactions. In vicinity of the half-filled
doped Mott state, the $\beta-$function vanishes for all local interactions
regardless of their sign. The only flow away from the HK model is through the
superconducting channel which lifts the spin degeneracy as does any ordering
tendency. The superconducting instability is identical to that established
previouslyPhillips et al. (2020). A corollary of this work is that Hubbard
repulsive interactions flow into the HK stable fixed point in the vicinity of
half-filling. Consequently, although the HK model has all-to-all interactions,
nothing local destroys it. The consilience with Hubbard arises because both
models break the $Z_{2}$ symmetry on a Fermi surface, the HK model being the
simplest to do so. Indeed, the simplicity of the HK model belies its
robustness and generality.

##  I Introduction

Proving Mott’s claim, in any dimension we care about ($d\geq 2$), that strong
electron correlation opens a gap in a half-filled band without changing the
size of the Brillouin zone still stands as a key unsolved problem in
theoretical physics. Demonstrating that the spectral function contains no
states in momentum space that cross the chemical potential would suffice as
proof of Mott’s claim. However, the inherent problem is that the model
employed in this context, namely the Hubbard model, contains strong on-site
repulsion in real space, thereby preventing any exact statement about the
corresponding spectral function in momentum space. There is of course an easy
way around this problem: add to the non-interacting band model an energy
penalty, $U$,

$H=\sum_{{{\mathbf{p}}}\sigma}\xi_{{{\mathbf{p}}}\sigma}n_{{{\mathbf{p}}}\sigma}+U\sum_{{{\mathbf{p}}}}n_{{{\mathbf{p}}}\uparrow}n_{{{\mathbf{p}}}\downarrow},$
(1)

whenever two electrons doubly occupy the same momentum state. In the above
$\xi_{{{\mathbf{p}}}\sigma}$ is the non-interacting band dispersion and
$n_{{{\mathbf{p}}}\sigma}$ is the occupancy. Since such an interaction does
not mix the momenta, a gap must open in the spectrum should $U$ exceed the
bandwidth as illustrated in Fig. 1. This is precisely the Hatsugai-Kohmoto
modelHatsugai and Kohmoto (1992) which was introduced in 1992 but attracted
essentially no attention untilLi et al. (2022); Zhong (2022); Yang (2021);
Setty (2021a); Zhu et al. (2021); Leeb and Knolle (2023); Zhu and Han (2021);
Setty (2021b); Tzeng et al. (2023); Zhong (2023); Zhao et al. (2023) our
demonstrationPhillips et al. (2020); Huang et al. (2022) that this model
offers an exact way of treating the Cooper instability in a doped Mott
insulator. Despite this utility, the HK model faces an uphill battle to
replace the ingrained Hubbard model as the knee-jerk response to the strong-
correlation problem with local-in space interactions. In bridging the gap from
the HK to the Hubbard model, three questions arise. 1) Does the HK model
remain stable to local-in-space interactions? If it does, then a
correspondence with the Hubbard model can be established. 2) Does the
resilience to local-in-space interactions give rise to a stable fixed point?
3) What about the obvious spin degeneracy that arises from the singly occupied
states in the spectrum? Are there ordering instabilities at $T=0$ that lift
the degeneracy?

Figure 1: The spectral functions of the HK model at different fillings and
interaction strength. (a) Mott insulating state, with $U>W$ and
$\braket{n}=1$. (b) Strongly repulsive Mott metal, with $U>W$ and
$\braket{n}<1$. (c) Weakly repulsive Mott metal, with $U<W$ and
$\braket{n}=1$.

It is these three questions that we answer in this paper. Briefly, we
construct the $\beta-$function explicitly and demonstrate that the answer to
all three leading questions is yes. The degeneracy is lifted spontaneously by
superconductivity as in a Fermi liquid. Central to our construction is the
leading lesson from Fermi liquid theory. As Fermi liquids admit a purely local
description in momentum space, their eigenstates are indexed simply by
momentum and spin. Consequently, in real space, Fermi liquids exhibit long-
range entanglement. It is precisely this long- range entanglement that makes
them impervious to any local interaction, that is, short-range Coulomb
interactions. The renormalization group analysis of a Fermi liquid shows
distinctlyPolchinski (1992); Shankar (1994) that should the sign of the
interaction be reversed and the electrons reside on opposites of the Fermi
surface, a superconducting instability arises. The added stipulation of the
electrons residing on opposite sides of the Fermi surface changes the scaling
dimension of the generic 4-fermion interaction from being irrelevant to
marginal. Also of note is that in terms of the pair annihilation operator,
$b_{k}=c_{k\uparrow}c_{k\downarrow}$, the Cooper pairing interaction

$V_{\rm
pair}=-g\sum_{{{\mathbf{k}}},{{\mathbf{k}}}^{\prime}}b_{{{\mathbf{k}}}}^{\dagger}b_{{{\mathbf{k}}}^{\prime}},$
(2)

is completely non-local in real space as it involves an interaction between
all pairs of electrons regardless of their separation. Such non-locality does
not make the BCS model unphysical because this interaction is the only
relevant perturbation that destroys a Fermi liquid as indicated by the 1-loop
flow equation,

$\frac{dg}{dt}=\frac{g^{2}}{4\pi},$ (3)

which exhibits a breakdown at a finite energy scale (the transition
temperature) signaling the onset of pairing and the eventual non-perturbative
physics of the BCS superconducting state. All of this is summarized in the
flow diagram in Fig. (2): short-range interactions ($V_{\rm local}$)
regardless of their sign flow to the Fermi liquid state while the Cooper term,
($V_{\rm pair}$), inherently non-local in real space, flows to strong
coupling. Once again, it is the real-space long-range entanglement of a Fermi
liquid that accounts for its resilience to any short-range interaction.

Figure 2: Perturbative low diagram for interactions in a Fermi liquid. Short-
range interactions regardless of their sign do nothing. Pairing leads to a
flow to strong coupling and the ultimate destruction of the Fermi liquid and
the onset of a superconducting state. The nature of the superconducting state
cannot be established based on perturbative arguments but requires BCS theory.

As it is ultimately the conservation of the charge currents in momentum space,
$n_{{{\mathbf{p}}}\sigma}$, that is at play here, it is natural to rewrite the
HK model as $\sum_{p}h_{p}$ with $h_{p}$ implicitly defined from Eq. (1). From
this, it is clear that the HK model has a momentum-diagonal structure of a
Fermi liquid. However, unlike a Fermi liquid, this model describes a Mott
insulatorPhillips et al. (2020); Huang et al. (2022); Hatsugai and Kohmoto
(1992) as shown in Fig. (1). While it is natural to criticize this model as
being unphysical because of the non-local in real space interactions, non-
locality by itself does not dismiss a model from being physically relevant as
$V_{\rm pair}$ in a Fermi liquid is non-local but nonetheless is the only
relevant pairing term that leads to the BCS instability. The real question is:
Do local interactions lead to flow away from the Hatsugai-Kohmoto model? That
is, does the Mott insulator that HK describes represent a fixed point in
direct analogy with the stability of a Fermi liquid to local interactions? We
show here that the answer to this question is a resounding yes; HK is a stable
fixed point. Even adding local terms of the Hubbard kind does nothing.

Analogies with Fermi liquid aside, a more fundamental reason is operative for
the resilience of the HK model to local perturbations. Haldane and
AndersonAnderson and Haldane (2001) argued that Fermi liquids contain an added
$Z_{2}$ symmetry on the Fermi surface that amounts to a particle-hole
interchange for one of the spin species. Operationally, if only doubly
occupied or empty sectors are present, then adding or removing an electron of
either spin is kinematically equivalent. However, removing an electron from a
singly occupied k-state can only be done in one way, thereby breaking the
$Z_{2}$ symmetry. Note even in the Hubbard model, the on-site repulsion
creates single occupancy in real space which will give rise to single
occupancy also in momentum space. HK suffices as it is the simplest term that
does soHuang et al. (2022). As long as this symmetry is already broken, adding
new interactions which also break this symmetry yields no relevant physics as
far as Wilsonian renormalization is concerned.

We carry out the full renormalization group analysis for the HK model and show
that as in a Fermi liquid, no local perturbation destroys the HK model,
thereby defining a stable fixed point. The only perturbation that destroys HK
physics is $V_{\rm pair}$ as in a Fermi liquid. We conclude then that the HK
model is more than a toy model. Rather it represents a stable fixed point for
a model non-Fermi liquid that describes a Mott insulator. It is a new fixed
point in quantum matter that describes a doped Mott insulator. The
superconducting state that ensuesPhillips et al. (2020) is the BCS analogue
for a doped Mott insulator.

##  II RG approach for $d\geq 2$ HK model: Tree level analysis

Our goal is to establish the stability of the HK model to weak local
interactions. What is central to the HK model are the two filling surfaces,
the lower filling surface (L-surface) separates singly- and empty states, and
the upper filling surface (U-surface) separates the doubly- and singly-
occupancy. The key argument of the RG process is that at weak coupling, only
modes around these filling surfaces participate. At different filling and
interaction strengths, the number of filling surfaces varies from zero (Mott
insulating state, Fig.1(a)), to one (strongly repulsive Mott metal, Fig.1(b))
or two (weakly repulsive Mott metal, Fig.1(c)). For simplicity, we work with
$d=2$ and rotationally invariant filling surfaces. The higher dimensional
result can always be achieved by adding rotational degrees of freedom and the
relevance of interactions that define the fixed point is not changed.

###  II.1 L-surface

We start with the analysis of the setup with only one spherical filling
surface of which the particle occupancy is singly- or empty on either side
(L-surface). This analysis follows the method WeinbergWeinberg (1996) and
ShankarShankar (1994) though we use the notation of PolchinskiPolchinski
(1992). We linearize the dispersion around the L-surface with radius $K_{L}$
to

$E\left({{\mathbf{K}}}={{\mathbf{n}}}(K_{L}+k)\right)=v_{L}k,$ (4)

where ${{\mathbf{n}}}=\frac{{{\mathbf{K}}}}{|{{\mathbf{K}}}|}$ is the unit
vector in the direction of ${{\mathbf{K}}}$ and $v_{L}$ is the isotropic
L-surface velocity. Note linearization restricts our analysis to the vicinity
of metallic state where Mott physics is relevant. Near the bottom of the band,
our analysis fails as the band dispersion is inherently quadratic, and weak
coupling physics obtains. We write the zero temperature partition function,

$\displaystyle Z=\int\mathcal{D}[c,\bar{c}]e^{-S_{0}},$ (5)
$\displaystyle\begin{split}&S_{0}=\int_{\Lambda}\frac{d^{d}{{\mathbf{K}}}}{(2\pi)^{d}}\int_{-\infty}^{\infty}d\omega\\\
&\left[\sum_{\sigma}\bar{c}_{{{\mathbf{K}}}\sigma}(i\omega-
v_{L}k)c_{{{\mathbf{K}}}\sigma}+U\bar{c}_{{{\mathbf{K}}}\uparrow}\bar{c}_{{{\mathbf{K}}}\downarrow}{c}_{{{\mathbf{K}}}\downarrow}{c}_{{{\mathbf{K}}}\uparrow}\right].\end{split}$
(6)

The integral over momentum is confined in a thin shell around the filling
surface, with distance $\Lambda$ as the cut-off. The partition function
factorizes at each momentum $K$, in exactly the same ways as a Fermi liquid,
however, mixing the up and down spins at the same momentum. After integrating
out the fast modes living within $s\Lambda<|k|<\Lambda$, we perform the
rescaling of the variables and fields:

$\displaystyle k^{\prime}$ $\displaystyle=s^{-1}k,$ (7)
$\displaystyle\omega^{\prime}$ $\displaystyle=s^{-1}\omega,$ (8)
$\displaystyle c^{\prime}_{{{\mathbf{K}}}^{\prime}\sigma}$
$\displaystyle=s^{3/2}c_{{{\mathbf{K}}}\sigma},$ (9) $\displaystyle
U^{\prime}$ $\displaystyle=s^{-2}U,$ (10)

to make the partition function invariant up to a constant. It is worth noting
that the HK repulsion has a scaling dimension of $-2$ and hence is strongly
relevant. This relevant term suppresses the contribution of spectral weight
from the other band that is $U$ away from the filling surface.

Now we consider the effect of perturbations on this fixed point. First,
consider perturbations that are quadratic in the fields,

$\delta
S^{(2)}=\int_{\Lambda}\frac{d^{d}{{\mathbf{K}}}}{(2\pi)^{d}}\int_{-\infty}^{\infty}d\omega\sum_{\sigma}\mu({{\mathbf{K}}})\bar{c}_{{{\mathbf{K}}}\sigma}c_{{{\mathbf{K}}}\sigma}.$
(11)

This action separates into slow and fast pieces, and the effect of integrating
out the fast modes produces a constant. After rescaling the momenta and the
fields, we have

$\mu^{\prime}(k^{\prime})=s^{-1}\mu(k).$ (12)

This relevant term is the chemical potential, which should be included in the
action kinetic term. As a result, the location of the fixed point definitely
depends on the filling of the system. We shall adjust the position of the
filling surface according to the chemical potential so as to make the system
truly fixed.

Next, we consider the quartic interaction in the most general form,

$\begin{split}\delta
S_{4}&=\int_{K}\int_{\omega}\bar{c}_{4}(\tau)\bar{c}_{3}(\tau)c_{2}(\tau)c_{1}(\tau)u(4,3,2,1),\\\
\bar{c}_{i}&\equiv\bar{c}_{{{\mathbf{K}}}_{i}\sigma_{i}},\\\ c_{i}&\equiv
c_{{{\mathbf{K}}}_{i}\sigma_{i}},\\\
u&(4,3,2,1)=u({{\mathbf{K}}}_{4}\sigma_{4},{{\mathbf{K}}}_{3}\sigma_{3},{{\mathbf{K}}}_{2}\sigma_{2},{{\mathbf{K}}}_{1}\sigma_{1}),\\\
\int_{K}&\equiv\prod_{i=1}^{4}\int_{\Lambda}d{{\mathbf{K}}}_{i}\int
d\Omega_{i}\delta({{\mathbf{K}}}_{1}+{{\mathbf{K}}}_{2}-{{\mathbf{K}}}_{3}-{{\mathbf{K}}}_{4}),\\\
\int_{\omega}&\equiv\prod_{i=1}^{4}\int_{-\infty}^{\infty}d\omega_{i}\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4}).\end{split}$
(13)

The delta functions put constraints on the integral region of momentum and
frequency. The delta function on frequency could be easily rescaled as
$\delta(\omega_{1}^{\prime}+\omega_{2}^{\prime}-\omega_{3}^{\prime}-\omega_{4}^{\prime})=s\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})$
since the integral over frequency extends to infinity. The delta function on
momentum, however, has a different scaling behavior as pointed out by
PolchinskiPolchinski (1992). The distance from the filling surface only gives
a contribution proportional to the cutoff $\Lambda$ which is a negligible
contribution compared with the filling momentum
$K_{L}:\delta({{\mathbf{K}}}_{1}+{{\mathbf{K}}}_{2}-{{\mathbf{K}}}_{3}-{{\mathbf{K}}}_{4})\approx\delta(K_{L}({{\mathbf{n}_{1}}}+{{\mathbf{n}}}_{2}-{{\mathbf{n}}}_{3}-{{\mathbf{n}_{4}}})+O(\Lambda))$.
When the momenta all point in different directions, the first term in the
delta function dominates and the delta function does not scale upon RG.
Following the same argument by PolchinskiPolchinski (1992), this quartic
operator is thus irrelevant at the tree level,

$u^{\prime}(4^{\prime},3^{\prime},2^{\prime},1^{\prime})=su(4,3,2,1).$ (14)

It is then easy to see that any further interactions are even more irrelevant.
This power-counting tree-level analysis already rules out the general short-
range interactions from being relevant. Upon decreasing the energy scale, we
find that the coupling becomes weaker and weaker while the HK repulsion gets
stronger and stronger. Consequently, Mott physics captured by the filling
surface is stable under local interactions.

There is an important subtlety in the kinematics and as a result, our
treatment of the delta function is not always valid. The first-term
contribution to the delta function could be set exactly to zero by putting an
additional constraint on the momentum directions. The second term which is
proportional to $\Lambda$ will be renormalized and generates a factor of $s$.
The rescaling of these direction-constrained interactions are

$u^{\prime}(4^{\prime},3^{\prime},2^{\prime},1^{\prime})=s^{0}u(4,3,2,1).$
(15)

Performing a Taylor expansion in $k$ and $\omega$ and comparing coefficients
of separate powers, we conclude that the leading term, with no dependence on
either variable, is marginal, while all the rest are irrelevant. Because there
are only two degrees of freedom in momentum, both of them are non-local by
definition. Consequently, to reach a definitive conclusion, we must proceed to
the 1-loop level to determine whether they are marginally irrelevant or
marginally relevant, or truly marginal. This calculation will be performed in
the next section.

###  II.2 U-surface

In the case that the filling surface is the occupancy boundary of the doubly
and singly occupied regions, we have to perform an extra particle-hole
transform before we write down the partition function in the path integral
language so as to obtain the correct linearized dispersion around the
U-surface with radius $K_{U}$,

$E\left({{\mathbf{K}}}={{\mathbf{n}}}(K_{U}+k)\right)=v_{U}k.$ (16)

This particle-hole asymmetry reflects the broken $Z_{2}$ symmetry of Mott
physics. The zero temperature partition function then becomes

$\displaystyle Z=\int\mathcal{D}[c,\bar{c}]e^{-S_{0}},$ (17)
$\displaystyle\begin{split}&S_{0}=\int_{\Lambda}\frac{d^{d}{{\mathbf{K}}}}{(2\pi)^{d}}\int_{-\infty}^{\infty}d\omega\\\
&\left[\sum_{\sigma}\bar{c}_{{{\mathbf{K}}}\sigma}(i\omega-
v_{U}k)c_{{{\mathbf{K}}}\sigma}+U\bar{c}_{{{\mathbf{K}}}\uparrow}\bar{c}_{{{\mathbf{K}}}\downarrow}{c}_{{{\mathbf{K}}}\downarrow}{c}_{{{\mathbf{K}}}\uparrow}\right].\end{split}$
(18)

choosing the same setup with cut-off $\Lambda$ results in the same scaling
rule for the variables and fields as in Eq.(10). The identical analysis on
quadratic and quartic perturbations thus reoccurs and we have the same
irrelevant tree-level behavior around this U-surface fixed point.

###  II.3 2 Filling Surfaces

In the weakly repulsive case, the 2 occupancy boundaries(L-surface and
U-surface) coexist. In order to discuss low-energy physics, we have to include
modes around both filling surfaces. Due to the factorizability of the
partition function in momentum space, we can safely achieve the bare partition
function by the product of Eq.(5) and Eq.(17). By setting the energy scale
$\Lambda$ around both filling surfaces to be the same, we arrive at the same
scaling of the variables and fields as in Eq.(10). In conclusion, regardless
of the number of filling surfaces, local interactions are always irrelevant at
the tree level and hence do not modify the fixed point defined by the filling
surfaces. We will move on to see the effect of marginal quartic interactions
on these fixed points.

##  III The perturbative expansion for 1-loop level corrections

With the tree-level analysis in hand, we have already ruled out the local part
of any perturbations. It is interesting to see how we can obtain a collective
effect such as superconductivity in a low-energy theory.

The RG process is carried out by integrating the fast modes and rescaling the
slow modes and the associated variables to keep the partition function
unchanged. Besides the terms that only contain slow modes(tree-level result),
we also need to include the terms that have both slow modes and fast modes and
add their contribution to the scaling equations. This process is
mathematically equivalent to calculating multipoint correlation functions with
interactions. In the HK model, the correlation functions could be calculated
using perturbative expansion for weak coupling as demonstrated in the
Appendix.

Figure 3: The one-loop graphs for $\beta(u)$ for quartic interactions. The
loop momenta lie in the shell of width $d\Lambda$ being eliminated. The
external frequencies being all zero, the loop frequencies are (a) equal for
ZS, (b) equal for ZS’, and (c) equal and opposite for the BCS graph.

The increment in $u(4,3,2,1)$

$\begin{split}du(4,3,2,1)=&\int
u(6,3,5,1)u(4,5,2,6)G(5)G(6)\delta(3+6-1-5)d5d6\\\ &-\int
u(6,4,5,1)u(3,5,2,6)G(5)G(6)\delta(6+4-1-5)d5d6\\\ &-\frac{1}{2}\int
u(6,5,2,1)u(4,3,6,5)G(5)G(6)\delta(5+6-1-2)d5d6.\end{split}$ (19)

is given by 3 diagrams as mentioned by Shankar, and we will follow the
nomenclature to call them ZS, ZS’ and BCS diagrams respectively.

We examine first the tree-level marginal interactions. We define the momentum
components on the filling surface of $K_{i}$ as $K_{i}^{F}$. Then a common
property of these marginal interactions is that the sum of the incoming and
outgoing $K_{i}^{F}$s is zero
${{\mathbf{K}}}_{1}^{F}+{{\mathbf{K}}}_{2}^{F}-{{\mathbf{K}}}_{3}^{F}-{{\mathbf{K}}}_{4}^{F}=0$.
The delta function on momentum thus scales as $s^{-1}$ and gives the marginal
power counting. This property reduces the marginal interactions into two
families: 1) forward scattering and 2) superconductivity or the Cooper
channel.

##  IV Forward scatterings at 1-loop

The forward scatterings are defined by a non-vanishing
${{\mathbf{P}}}={{\mathbf{K}}}_{1}^{F}+{{\mathbf{K}}}_{2}^{F}$. This
nomenclature comes from the $d=2$ 1-filling surface case, where there are only
two solutions:
${{\mathbf{K}}}_{1}={{\mathbf{K}}}_{3},{{\mathbf{K}}}_{2}={{\mathbf{K}}}_{4}$
or
${{\mathbf{K}}}_{1}={{\mathbf{K}}}_{4},{{\mathbf{K}}}_{2}={{\mathbf{K}}}_{3}$.
These two setups are equivalent to one another up to changing the Fermion
order.

###  IV.1 1 Filling surface

When there is only a single filling surface, the forward scattering is
determined only by the solution to
${{\mathbf{K}}}_{1}={{\mathbf{K}}}_{3},{{\mathbf{K}}}_{2}={{\mathbf{K}}}_{4}$.
Including spins, there are explicitly 3 choices:

$\displaystyle
F^{1}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{2})=u({{\mathbf{K}}}_{2}\sigma,{{\mathbf{K}}}_{1}\sigma,{{\mathbf{K}}}_{2}\sigma,{{\mathbf{K}}}_{1}\sigma),$
(20) $\displaystyle
F^{2}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{2})=u({{\mathbf{K}}}_{2}\bar{\sigma},{{\mathbf{K}}}_{1}\sigma,{{\mathbf{K}}}_{2}\bar{\sigma},{{\mathbf{K}}}_{1}\sigma),$
(21) $\displaystyle
F^{3}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{2})=u({{\mathbf{K}}}_{2}\sigma,{{\mathbf{K}}}_{1}\bar{\sigma},{{\mathbf{K}}}_{2}\bar{\sigma},{{\mathbf{K}}}_{1}\sigma).$
(22)

Due to the fact that any higher order terms in the Taylor expansion on
$\omega$ and $k$ are irrelevant, we can freely choose the frequency and
momentum deviations. For simplicity, we set all external legs to zero
frequency and almost on the filling surface ($\omega=0,k=\epsilon\ll
d\Lambda$). The tiny value of $\epsilon$ will be set equal to zero at the last
step. We need to keep it during the calculation to make the running momentum
$K$ and $K+Q$ different according to the requirement of the weak coupling
expansion.

First, we consider the ZS diagram in Fig.3 given by the first integral in
Eq.(19). Since $Q=\epsilon\ll d\Lambda$, both $K$ and $K+Q$ lie on the same
side of the filling surface for all eligible choices of $K$. As a result, the
directional integral of $K$ is over the full range,

$\begin{split}dF({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{2})=&\int\frac{d{{\mathbf{n}}}}{(2\pi)^{d-1}}F({{\mathbf{n}}}_{1},{{\mathbf{n}}})F({{\mathbf{n}}},{{\mathbf{n}}}_{2})\\\
&\int_{d\Lambda}\frac{dk}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}G(\omega,k)G(\omega,k+\epsilon).\end{split}$
(23)

Here $F({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})$ represents the appropriate
choice of $F$ that satisfies the momentum and spin conservation at each
vertex. The integral over $dk$ lies inside the thin shells to be integrated
out. However, there are two such shells. One of the shells lies inside the
filling surface while the other is outside the filling surface. For the outer
shell corresponding to $k\in[\Lambda-d\Lambda,\Lambda]>0$, the states belong
to the 0-occupancy region, which means we can replace the Green function by

$G(\omega,k)=\frac{1}{i\omega-v_{L}k}.$ (24)

For the inner shell, corresponding to $k\in[-\Lambda,-\Lambda+d\Lambda]<0$,
the states belong to the single-occupancy region, which means we can replace
the Green function by

$G(\omega,k)=\frac{1/2}{i\omega-v_{L}k}+\frac{1/2}{i\omega-v_{L}k-U}.$ (25)

The poles in the $\omega$ plane do not contribute if they lie on the same side
of the real axis. The only surviving contribution from ZS is thus

$\begin{split}dF(&{{\mathbf{n}}}_{1},{{\mathbf{n}}}_{2})=2\int\frac{d{{\mathbf{n}}}}{(2\pi)^{d-1}}F({{\mathbf{n}}}_{1},{{\mathbf{n}}})F({{\mathbf{n}}},{{\mathbf{n}}}_{2})\\\
&\int_{-\Lambda}^{-\Lambda+d\Lambda}\frac{dk}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{1/2}{i\omega-
v_{L}k}\cdot\frac{1/2}{i\omega-v_{L}k-U}.\end{split}$ (26)

The integral over $\omega$ and $k$ gives

$\int_{d\Lambda}\frac{dk}{2\pi}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{1/2}{i\omega-
v_{L}k}\cdot\frac{1/2}{i\omega-v_{L}k-U}=\frac{d\Lambda}{8\pi U}.$ (27)

As $U$ is strongly relevant, that is, $U^{\prime}=s^{-2}U$, this contribution
goes to zero much faster than $d\Lambda/\Lambda$. Thus the ZS diagram does not
yield any contribution under the RG analysis.

Now consider the ZS’ diagram. Due to the momentum transfer $Q$ of order
$K_{L}$ at the left vertex, not only is the magnitude of the loop momentum
restricted to lie within the shell being eliminated but also its angle is
restricted to a range of order $d\Lambda/K_{L}$. This suppression in ZS’
diagrams contributes to $d\Lambda^{2}/\Lambda K_{L}$. The $\beta$ function
thus vanishes as we take the limit $d\Lambda/\Lambda\rightarrow 0$.

Finally, the same kinematic reasons used to establish that ZS’ vanishes can be
adopted to show that the BCS diagram also does not renormalize $F$ at one
loop. Hence, the coupling constants for the forward scattering do not flow in
this order because $\beta(F)=0$.

###  IV.2 2 Filling surfaces

When there are 2 filling surfaces, there are at most 4 independent solutions
to ${{\mathbf{P}}}={{\mathbf{K}}}_{1}+{{\mathbf{K}}}_{2}$ as shown in Fig.4.

Figure 4: The solutions to
${{\mathbf{P}}}={{\mathbf{K}}}_{1}+{{\mathbf{K}}}_{2}$. The L-surface and
U-surface are plotted in dotted circles. The 4 solutions are marked by the
surfaces where the 2 momentums that sum up to ${{\mathbf{P}}}$ are.

The combined equation ${{\mathbf{P}}}={{\mathbf{K}}}_{1}+{{\mathbf{K}}}_{2}$
and ${{\mathbf{P}}}={{\mathbf{K}}}_{3}+{{\mathbf{K}}}_{4}$ thus have $4\times
4=16$ independent solutions. There are still explicitly 3 spin configurations
for each momentum solution. Thus the total number of marginal forward
scattering is $16\times 3=48$. We will not enumerate them here since they
remain un- changed in the same way as we discussed in the case of a single-
filling surface. The 1-loop correction contributes a term proportional to
either $d\Lambda/U$ from the ZS diagrams or $d\Lambda/K_{L}$ for the ZS’ and
BCS diagrams. Once again, forward scatterings remain fixed under RG; that is,
they do not flow under RG.

##  V Superconducting pairings at 1-loop

The incoming and outgoing momenta each summing to zero
${{\mathbf{K}}}_{1}=-{{\mathbf{K}}}_{2},{{\mathbf{K}}}_{3}=-{{\mathbf{K}}}_{4}$
constitute superconducting pairings. The 1-loop evolution of $V$s has a non-
vanishing contribution even for a Fermi liquid and hence we expect an
analogous contribution at the HK fixed point should one exist. We will analyze
the single and two filling surfaces separately.

###  V.1 1 Filling surface

${{\mathbf{K}}}_{1}$ and ${{\mathbf{K}}}_{3}$ can lie freely on the filling
surface. For simplicity, we set all external legs to zero frequency and on the
filling surface ($\omega=k=0$). Including the spin-singlet and spin-triplet
pairings, we have 2 choices,

$\displaystyle
V^{1}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})=u(-{{\mathbf{K}}}_{3}\sigma,{{\mathbf{K}}}_{3}\sigma,-{{\mathbf{K}}}_{1}\sigma,{{\mathbf{K}}}_{1}\sigma),$
(28) $\displaystyle
V^{2}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})=u(-{{\mathbf{K}}}_{3}\bar{\sigma},{{\mathbf{K}}}_{3}\sigma,-{{\mathbf{K}}}_{1}\bar{\sigma},{{\mathbf{K}}}_{1}\sigma)..$
(29)

These two spin configurations satisfy the antisymmetry for Fermions regardless
of the angular structure of the pairing interaction. The incoming momenta are
equal and opposite on the filling surface. The ZS and ZS’ diagrams are
suppressed by $d\Lambda^{2}/\Lambda K_{L}$ and hence do not contribute for the
same reason that ZS’ and BCS diagrams did not contribute to the flow of
forward scatterings. The BCS diagram now has a contribution since the running
momentum in the loop can now freely point in any direction, regardless of the
value of $K$. We rewrite Eq.(19) as

$\begin{split}dV^{1,2}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})=&-\frac{1}{2}\int\frac{d{{\mathbf{n}}}}{(2\pi)^{d-1}}V^{1,2}({{\mathbf{n}}}_{1},{{\mathbf{n}}})V^{1,2}({{\mathbf{n}}},{{\mathbf{n}}}_{3})\\\
&\int_{d\Lambda}\frac{dk}{2\pi}\int_{-\infty}^{\infty}d\omega
G(\omega,k)G(-\omega,k).\end{split}$ (30)

The integral over $dk$ also lies inside the two thin shells to be integrated
out. These two thin shells yield contribute differently. For the outer shell,
corresponding to $k\in[\Lambda-d\Lambda,\Lambda]>0$, the second line of
Eq.(30) yields a finite value $\frac{d\Lambda}{4\pi\Lambda}$. For the inner
shell, corresponding to $k\in[-\Lambda,-\Lambda+d\Lambda]<0$, the integral
region thus gives a value that is reduced by a factor of a quarter,
$\frac{d\Lambda}{16\pi\Lambda}$. In all, The renormalization group equation
for $V^{1,2}$ is

$\begin{split}\frac{dV({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})}{dt}=&-\frac{5}{32\pi
v_{L}}\int\frac{d{{\mathbf{n}}}}{(2\pi)^{d-1}}V({{\mathbf{n}}}_{1},{{\mathbf{n}}})V({{\mathbf{n}}},{{\mathbf{n}}}_{3})\\\
\equiv&-\frac{5}{32\pi
v_{L}}(V*V)({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3}),\end{split}$ (31)

where $dt=|d\Lambda|/\Lambda$ is the RG transform step size, and $*$ defines
the generalized convolution in $d$-dimensions. This is the Cooper instability
in the HK model. For the case of $d=2$, we can simplify this by going to
momentum eigenfunctions

$V_{l}=\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{il\theta}V(\theta),$ (32)

where $V(\theta)=V({{\mathbf{n}}}_{0},R_{\theta}{{\mathbf{n}}}_{0})$ and
$R_{\theta}$ is the rotation by degree $\theta$. We can obtain the $\beta$
function for each angular momentum $l$

$\frac{dV_{l}}{dt}=-\frac{5}{32\pi v_{L}}V_{l}^{2}.$ (33)

The flow tells us that the couplings $V_{l}$ are marginally relevant if
negative, and marginally irrelevant if positive. By integrating the flow, we
obtain

$V_{l}(t)=\frac{V_{l}(0)}{1+5tV_{l}(0)/32\pi v_{L}},$ (34)

which implies an instability at the energy scale
$\Lambda_{c}=\Lambda_{0}e^{32\pi v_{L}/5V_{0}}$. The energy scale in a thermal
system can be proportional to the temperature; thus we propose that the
approximate transition temperature of this metallic state scales as
$\Lambda_{c}$.

###  V.2 2 Filling surfaces

The two-filling surface case remains to be analyzed. The electrons around
different filling surfaces now have different contributions and can be grouped
into 3 different categories (intra-L, intra-U, and inter-LU)

$\displaystyle
V^{1}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})=u(-{{\mathbf{K}}}_{3}^{L}\bar{\sigma},{{\mathbf{K}}}_{3}^{L}\sigma,-{{\mathbf{K}}}_{1}^{L}\bar{\sigma},{{\mathbf{K}}}_{1}^{L}\sigma),$
(35) $\displaystyle
V^{2}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})=u(-{{\mathbf{K}}}_{3}^{U}\bar{\sigma},{{\mathbf{K}}}_{3}^{U}\sigma,-{{\mathbf{K}}}_{1}^{U}\bar{\sigma},{{\mathbf{K}}}_{1}^{U}\sigma),$
(36) $\displaystyle
V^{3}({{\mathbf{n}}}_{1},{{\mathbf{n}}}_{3})=u(-{{\mathbf{K}}}_{3}^{U}\bar{\sigma},{{\mathbf{K}}}_{3}^{U}\sigma,-{{\mathbf{K}}}_{1}^{L}\bar{\sigma},{{\mathbf{K}}}_{1}^{L}\sigma),$
(37)

for the spin-singlet configuration. Here the superscripts represent the
surface around which the momenta are located. The spin-triplet processes have
exactly the same RG structure and follow the same RG equations. We omit their
definition and work with only the spin-singlet processes.

The RG flow of the BCS couplings still follows Eq.(30). The running momentum
can be chosen freely on both filling surfaces. The integral of the Green
function around each filling surface follows the same process as Eq.(31). The
renormalization group equations are

$\displaystyle\frac{dV^{1}}{dt}=$
$\displaystyle-\frac{5}{32\pi}(\frac{V^{1}*V^{1}}{v_{L}}+\frac{V^{3\dagger}*V^{3}}{v_{U}}),$
(38) $\displaystyle\frac{dV^{2}}{dt}=$
$\displaystyle-\frac{5}{32\pi}(\frac{V^{3}*V^{3\dagger}}{v_{L}}+\frac{V^{2}*V^{2}}{v_{U}}),$
(39) $\displaystyle\frac{dV^{3}}{dt}=$
$\displaystyle-\frac{5}{32\pi}(\frac{V^{1}*V^{3}}{v_{L}}+\frac{V^{3}*V^{2}}{v_{U}}).$
(40)

With the simplification, $V^{1}=V^{2}=V^{3}=V$, we simplify the $\beta$
function to a single equation,

$\frac{dV}{dt}=-\frac{5}{32\pi}(\frac{1}{v_{L}}+\frac{1}{v_{U}})V*V.$ (41)

For the case of $d=2$, the instability for attractive $V$ obtains at the
energy scale

$\Lambda_{c}=\Lambda_{0}\exp\left({\frac{32\pi
v_{L}v_{U}}{5(v_{L}+v_{U})V_{0}}}\right).$ (42)

The energy scale in a thermal system can be proportional to the temperature.
This increase in the critical energy scale is consistent with the exactly
calculated pair susceptibilityZhao et al. (2022), which diverges at a higher
temperature for a small value of $U$ compared with the Fermi liquid result.
The comparison between different $T_{c}$ was shown in Figure(5)

Figure 5: The dependence of $T_{c}$ on superconducting pairing strength
$g=-V$. The curves from top to bottom are: $U<W$ (2 filling surfaces), $U=0$
(Fermi liquid), and $U>W$ (1 filling surface). The insert plot shows the
linear dependence of $\log(T_{c})$ on $1/g$ and the slope of each curve is
$0.8:1:1.6$. These $T_{c}$ exponents are consistent with the $\Lambda_{c}$
estimate.

Due to the fact that RG analysis only deals with perturbative interactions
around the fixed point at each step, the first order transition into a
superconductor at finite $U$Zhao et al. (2022) is absent.

##  VI Final Remarks

We have shown that the analogies with Fermi liqiud theory noted in the
introduction are borne out by a thorough RG analysis of the interactions that
can contribute to the HK model. For short-range repulsions, nothing flows away
from HK, thereby establishing it as a stable fixed point depicted in Fig. (6).
While it is surprising that a new quartic fixed point exists, it is ultimately
not surprising given that the momentum structure of the HK model is identical
to that of Fermi liquid theory. As a consequence, all of the interactions
which flow into Fermi liquid theory also flow into this quartic fixed point.
Once again as in Fermi liquid theory, superconductivity leads to flow away
from the HK fixed point (see Fig.6). Since Hubbard interactions are also local
in real space, they also cannot perturb away from the HK fixed point in the
metallic state. Consequently, the simplicity of the HK model belies its true
robustness and generality.

Figure 6: Perturbative flow diagram for interactions in a doped HK Mott
system. Short-range interactions regardless of their sign do nothing. Only
pairing leads to flow to strong coupling and the ultimate destruction of the
non-Fermi liquid HK metallic state and the onset of a superconducting state
distinct from that of a BCS superconductor. The nature of the superconducting
state cannot be established based on perturbative arguments but requires the
PYH theoryPhillips et al. (2020); Zhao et al. (2022).

While the similarity of the momentum structure of the theory with that of
Fermi liquid theory plays a key role in the stability of the fixed point, the
relation to Hubbard is ultimately driven by the $Z_{2}$ symmetry breaking of
the interaction in Eq. (1). As long as the interaction is repulsive, breaking
the $Z_{2}$ symmetry noted by Anderson and HaldaneAnderson and Haldane (2001)
leads to single occupancy. The phase diagram for the evolution of HK physics
from a simple Fermi liquid can then be plotted as a function of the singly
occupied region, $\Omega_{1}$, as shown in Fig. (7). Fermi liquids live only
in the region where $\Omega_{1}$ vanishes. The transition line for
superconductivity from a FL state is given by the green line. The entire
region in the $\Omega_{1}-g$ plane represents non-BCS superconductivity and is
governed by the quartic HK fixed point delineated here. In addition to
mediating non-Fermi liquid behvaiour, single occupancy in real or momentum
space leads to degeneracy. Degeneracy is a tell-tale sign of competing
ordering tendencies that are well known to obtain in strongly correlated
systems. For example, the pure HK model has a divergent ferromagnetic
susceptibility (See Appendix). What the HK model does is separate the
bifurcation of the spectral function into low and high energy branches per
momentum state, the inherent physics of a Mott insulator, from ordering
tendencies. Such ordering is ultimately needed to free the HK model of the
spurious ground-state degeneracy without generating flow away from the stable
fixed point. Hence, what the HK model ultimately does is offer a way of
treating Mott’s original conception of the gapped half-filled band. Mottness
sets the gap and ordering is secondary as borne out experimentally in all Mott
systems ranging from the vanadatesQazilbash et al. (2008) to the
cupratesCooper et al. (1990); Chen et al. (1991); Uchida et al. (1991).

Figure 7: The general phase diagram of the superconducting instability in the
HK fixed point. As long as the $\Omega_{1}$ region is non-zero, the system
breaks the Fermi Liquid $Z_{2}$ symmetry and flows into HK.

Acknowledgements We thank the NSF DMR-2111379 for partial funding this work.

## Appendix

###  VI.1 Detailed derivation of the Green function

Here we briefly review the exact partition and Green function of the HK model
at any filling. We start with the HK Hamiltonian,

$H_{HK}=\sum_{k}\left[\xi_{k}(n_{k\uparrow}+n_{k\downarrow})+Un_{k\uparrow}n_{k\downarrow}\right],$
(43)

where $\xi_{k}=\epsilon_{k}-\mu$. For each momentum sector, the HK Hamiltonian
is built out of commuting parts. Thus, the partition function factorizes, and
for each momentum sector we have

$Z_{k}=1+2e^{-\beta\xi_{k}}+e^{-\beta(2\xi_{k}+U)}.$ (44)

The Heisenberg equations in imaginary time for Fermion $c_{k\sigma}$
annihilation and the number operator
$n_{k\sigma}=c^{\dagger}_{k\sigma}c_{k\sigma}$ are

$\displaystyle\dot{c}_{k\sigma}$
$\displaystyle=\left[H,c_{k,\sigma}\right]=-(\xi_{k}+Un_{k\bar{\sigma}})c_{k\sigma},$
(45) $\displaystyle\dot{n}_{k\sigma}$
$\displaystyle=\left[H,n_{k,\sigma}\right]=0.$ (46)

Thus we have the time evolution of the fermi operator,

$c_{k\sigma}(\tau)=e^{-(\xi_{k}+Un_{k\bar{\sigma}})\tau}c_{k\sigma}(0).$ (47)

The average particle number is

$\braket{n_{k\sigma}}=\frac{e^{-\beta\xi_{k}}+e^{-\beta(2\xi_{k}+U)}}{1+2e^{-\beta\xi_{k}}+e^{-\beta(2\xi_{k}+U)}}.$
(48)

The imaginary time Green function is

$\begin{split}G_{k\sigma}(\tau)&=-\braket{c_{k\sigma}(\tau)c^{\dagger}_{k\sigma}(0)}\\\
&=-\mathrm{Tr}\left[e^{-\beta(H-F)}e^{-(\xi_{k}+Un_{k\bar{\sigma}})\tau}c_{k\sigma}(0)c^{\dagger}_{k\sigma}(0)\right]\\\
&=-\mathrm{Tr}\left[e^{\beta F}e^{-(\xi_{k}+Un_{k\bar{\sigma}})\tau}e^{-\beta
H}(1-n_{k\sigma})\right]\\\ &=-e^{\beta
F}\left[e^{-\xi_{k}\tau}+e^{-\beta\xi_{k}}e^{-(\xi_{k}+U)\tau}\right].\end{split}$
(49)

Performing the Fourier transform with the anti-periodic boundary condition
$G(\tau+\beta)=-G(\tau)$ leads to

$\begin{split}G_{k\sigma}(i\omega_{n})&=\int_{0}^{\beta}G_{k\sigma}(\tau)e^{i\omega_{n}\tau}\\\
&=-e^{\beta
F}\left[\frac{-e^{-\beta\xi_{k}}-1}{i\omega_{n}-\xi_{k}}+e^{-\beta\xi_{k}}\frac{-e^{-\beta(\xi_{k}+U)}-1}{i\omega_{n}-\xi_{k}-U}\right]\\\
&=\frac{1-\braket{n_{k\bar{\sigma}}}}{i\omega_{n}-\xi_{k}}+\frac{\braket{n_{k\bar{\sigma}}}}{i\omega_{n}-\xi_{k}-U}.\end{split}$
(50)

This result is exact for any value of $\mu$ and $U$.

###  VI.2 Weak coupling Expansion and Wick’s Theorem

According to the Gell-Mann-Low formula, the $2n$-point function under
interaction $V$ is given by

$\begin{split}&\braket{Tc_{k_{1}}^{\dagger}(\tau_{1})\cdots
c_{k_{n}}^{\dagger}(\tau_{n})c_{k^{\prime}_{1}}(\tau^{\prime}_{1})\cdots
c_{k^{\prime}_{n}}(\tau^{\prime}_{n})}_{I}\\\
&\quad=\frac{\braket{T\bar{c}_{k_{1}}(\tau_{1})\cdots\bar{c}_{k_{n}}(\tau_{n})c_{k^{\prime}_{1}}(\tau^{\prime}_{1})\cdots
c_{k^{\prime}_{n}}(\tau^{\prime}_{n})\exp\left(-\int_{0}^{\beta}d\tau
V\left[\bar{c},c\right]\right)}_{HK}}{\braket{T\exp\left(-\int_{0}^{\beta}d\tau
V\left[\bar{c},c\right]\right)}_{HK}},\end{split}$ (51)

where $\braket{}_{I}$ is the average using the full Hamiltonian $H_{HK}+H_{I}$
and $\braket{}_{HK}$ is the average under the HK Hamiltonian $H_{HK}$ only.

Since the HK path integral is decomposed into a series of products in the
momentum space, the creation and annihilation operators from different
momentum sectors can be safely factored out. For example, suppose
$k_{1},k_{2},\cdots,k_{n}$ are different from each other, and each
corresponding annihilation and creation operator appears
$l_{1},l_{2},\cdots,l_{n}$ times, then

$\begin{split}&\left\langle
T\prod_{j=1}^{l_{1}}c_{k_{1}\sigma_{1}^{j}}^{\dagger}(\tau_{1}^{j})\cdots\prod_{j=1}^{l_{n}}c_{k_{n}\sigma_{n}^{j}}^{\dagger}(\tau_{n}^{j})\prod_{j=1}^{l_{1}}c_{k_{1}\sigma_{1}^{j}}(\tau^{\prime}_{1}{}^{j})\cdots\prod_{j=1}^{l_{n}}c_{k_{n}\sigma_{n}^{j}}(\tau^{\prime}_{n}{}^{j})\right\rangle_{HK}\\\
&=(-1)^{P}\left\langle
T\prod_{j=1}^{l_{1}}c_{k_{1}\sigma_{1}^{j}}^{\dagger}(\tau_{1}^{j})\prod_{j=1}^{l_{1}}c_{k_{1}\sigma_{1}^{j}}(\tau^{\prime}_{1}{}^{j})\right\rangle_{HK}\cdots\left\langle
T\prod_{j=1}^{l_{n}}c_{k_{n}\sigma_{n}^{j}}^{\dagger}(\tau_{n}^{j})\prod_{j=1}^{l_{n}}c_{k_{n}\sigma_{n}^{j}}(\tau^{\prime}_{n}{}^{j})\right\rangle_{HK},\end{split}$
(52)

where $P$ is the times of permutation performed so as to separate the fermion
operators.

The deviation from Wick’s theorem could be observed on certain multi-point
correlation functions(e.g. 4-point function) for which all the momenta of the
fermion operators are identical. This contribution, however, is
thermodynamically suppressed in our calculation. Thus we employ Feynman
diagram rules and use the exact Green function to calculate the correlation
functions.

##  VII Magnetic instability

The singly occupied region of the HK model at finite $U$ in the HK model leads
to a singly occupied region which introduces spin degeneracy for each momentum
sector. This ground state degeneracy is extensive with the system size and
thus considered to be unphysical although a singly occupied region generally
exists in Mott systems with the traditional Hubbard model. The solution to the
degeneracy in the HK fixed point is ordering such as superconductivity,
ferromagnetism or a spin-density wave. The superconducting instability has
been discussed in the main body and we provide here an analysis of a possible
magnetic ordering of the HK model.

With an external magnetic field $B$, the partition function now reads,

$Z(B)=\sum_{k}\left(1+e^{-\beta(\xi_{k}-\mu B)}+e^{-\beta(\xi_{k}+\mu
B)}+e^{-\beta(2\xi_{k}+U)}\right).$ (53)

The magnetic susceptibility is achieved by a double derivative of the
partition function,

$\begin{split}\chi&=\left.-\frac{1}{\beta Z(B)}\frac{\partial^{2}Z}{\partial
B^{2}}\right|_{B=0}\\\
&=\left.\frac{\mu^{2}\beta}{Z(B)}\sum_{k}\left(e^{-\beta(\xi_{k}-\mu
B)}+e^{-\beta(\xi_{k}+\mu B)}\right)\right|_{B=0}\\\
&=\mu^{2}\Omega_{1}\beta,\end{split}$ (54)

where $\Omega_{1}$ is the extent of singly occupied region. The susceptibility
diverges as temperature goes to zero, signaling a ferromagnetic phase
transition at $T=0$, thus offering an avenue for resolving the ground state
degeneracy. As pointed out in the text, superconductivity also lifts the
degeneracy and leads to flow away from the HK fixed point.

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