A proposal for solving the high-Tc superconductivity conundrum via artificial

intelligence

A research proposal by Yoon Tiem Leong

School of Physics, Universiti Sains Malaysia, Penang

March 2019

Key words: Artificial intelligence; Machine learning; Many-body physics problem; High-Tc

superconductivity; Extended Hubbard Model; Phase classification

Executive Summary of Research Proposal

The EHM [6], a USM-grown theoretical model, is a potential resolution to the ‘Holy Grail’

problem of HTSC. Latest AI-inspired algorithms provide a promising tool for overcoming the

major technical bottleneck of applying EHM to solve the HTSC conundrum which involves

the classification of the phases inherently embedded in the model Hamiltonian. AI-inspired

algorithms will be used to solve for the ground states, classify the quantum phases and the

dependence of the transition temperatures in the EHM. This is a computational condensed

matter physics project requiring heavy HPC facilities. The technical details and procedures of

calculations in this project are to be based heavily on, but not exclusively, that published in

[24], [32], [33], [36], [37]. The generic machine learning platform TensorFlow [46], will be

used for performing ML calculations, in conjunction with a list of open-source QMC codes

[38, 39, 40, 43] for generating configurations for the 2D Hubbard and EHM models to

sample data for classifying the phases. Other than deploying existing software packages, we

shall also need to develop our own codes for handling part of the calculations. The project is

strategically segregated into stages with progressive level of complexity. Classical results

and quantum results of selected prototypical many-body systems using AI-inspired

algorithms will first be reproduced, followed by the reproduction of the known results for

the prototype 2D Hubbard model. Finally, the EHM modelled will be dealt with based on the

ML techniques and computational experience picked up in the preceding stages by

numerically investigating the superconducting and the pseudo-gap sectors, and the

interplay between them. Apart from contributing to the resolution of the HTSC conundrum

[47], the know-how gained in this highly theoretical project can be translated into other

productivity applications which the industries are desperately needing.

Problem Statement

High-temperature (high-Tc) superconductivity (HTSC) in cuprates [1] is one of the most

profound physics problems since 1986. The strong electronic correlations, which is

intrinsically a quantum many-body effect, responsible for the superconducting, pseudo-gap

[45] and other measured phases in unconventional high-Tc superconducting cuprates

remain an unresolved problem in condensed matter physics to date. This is famously known

as the ‘HTSC conundrum’ [47]. Arriving at a working theoretical model for resolving the

conundrum in HTSC constitutes one of the ‘Holy Grails’ in condensed matter physics. Apart

from arriving at the right theoretical model, the other major challenge concerns with the

difficult task of solving these many-body physics models which are, generically,

computationally expensive if not practically formidable. Specifically, classification of

quantum phases in a generic quantum many-body model Hamiltonian with existing,

conventional computational approaches is known to be particularly so. We propose a

resolution to the ‘Holy Grail’ problem of HTSC based on a model, so called the Extended

Hubbard Model (EMH), published by B. S. Lee [6], B. S. Lee, T. L. Yoon and R. Abd-Shukor [7]

in 2010 and 2017 respectively (T. L. Yoon is the leader of this proposal). One of the technical

bottlenecks of this research proposal is to classify the phases inherently embedded the EMH

without having any prior knowledge of the solutions embodied in the exponentially large

state space of the many-body system. Latest AI-inspired algorithms which are well suited

with the ability to classify, identify, or interpret exponentially large data sets provide a

promising tool for overcoming this major bottleneck.

Hypothesis

The many-body physics responsible for the unconventional high-Tc superconducting and

pseudo-gap phases in cuprates are correctly captured in the renown Hubbard model and

Hubbard-type models. In principle, given the underlying Hamiltonian of a many-body

system, it is possible to derive all we need to know about the physical and macroscopic

details of the system via a spectrum of conventional computational techniques (such as

quantum Monte Carlo [2,3,9], density matrix renormalization group [4], and tensor

networks [5]). We distinguish two distinct categories of computational technique to solve

many-body physics, namely, the conventional (i.e., non-artificial intelligence) and artificial

intelligence (AI) approaches. Those AI-inspired computational approaches, which rise rapidly

only in the very recent years, could deliver a much feasible, cheap and relatively promising

way for solving quantum many-body problems that are otherwise proven extremely

expensive and often intractable via non-AI approaches. Specifically, AI-inspired algorithms

are expected to provide a powerful, computationally cheap, convenient, and accurate tool

for quantum phase identification in quantum many-body problems over existing,

conventional approaches.

Research Questions

1. How to technically classify, via the novel AI approaches, the quantum phases as well

as the parameters driving these phases as embedded in the EHM?

2. What are the undiscovered physics embedded in the EHM relevant to the pseudo-

gap and superconducting phases in the HTCS cuprates?

3. Could there be any other unexpected or serendipitous new physics embedded in the

EHM?

4. Could we finally nail down the ‘Holy Grail’ of unconventional HTCS in the EHM with

the new arsenal of AI-inspired computational approaches?

Literature Reviews

One of the main tasks in theoretical condensed matter physics is to infer the macroscopic

properties of a physical system from their microscopic description, i.e., its underlying

Hamiltonian. The most drastic changes in the macroscopic properties of a physical system

occur at phase transition. Different phases can be identified by an order parameter that is

zero in the disordered phase and nonzero in the ordered phase. Whereas in many known

models the order parameter can be determined by symmetry considerations of the

underlying Hamiltonian, there are states of matter where such a parameter can only be

defined in a complicated nonlocal way [8], such as the models for unconventional high Tc-

superconductivity [1] in cuprates, which is the intended system we wish to investigate in

this proposal. Theoretical condensed matter physicists wish to predict the details of phase

transition, such as the critical temperature for the onset of superconducting [1] or pseudo-

gap [45] phases, and the physics driving the transitions among the various phases

embedded in a many-body quantum model. However, the theories for describing phase

transition [10, 11, 12] generally involve non-local, long-range order effects which are often

computationally formidable.

Serious models for unconventional high-Tc superconductivity generically involve

quantum many-body effects. Quantum many-body (QMB) problems are known to be

computationally daunting due to the nontrivial correlations encoded in the exponential

complexity of the many-body wave function Ψ. An exponential amount of information is

needed to fully encode a generic many-body quantum state, rendering reliable numerical

solutions for the ground state technically difficult to come by. Conventionally, many-body

calculations are performed through highly sophisticated computational methods with some

extent of approximations, such as quantum Monte Carlo (QMC) methods [2,3,9], density

matrix renormalization group [4], tensor networks [5], matrix product states (MPS) [13, 14,

15] or general tensor networks [16, 17, 18]. However, there are many instances where these

conventional approaches fail, due to, e.g., the sign problem [19] or the inefficiency in

handling the exponentially huge degree of freedom inherent in these systems.

Fundamentally, the quandary in QMB lies in the failure of finding a general strategy to

reduce the exponential complexity of the full many-body wave function down to its most

essential features (this is known as the ‘curse of dimensionality’ [32]).

As an independent development in condensed matter and quantum statistical

physics, since 2017 or so we are witnessing a rapid rise of AI, a. k. a. machine learning (ML)

technique in the present context, as a powerful tool for solving many-body problems,

classical as well as quantum ones, that are very expensive to calculate via non-machine

learning approaches. Machine learning provides a complementary paradigm to the

conventional computational approach. The ability of modern machine learning techniques

to classify, identify, or interpret massive data sets such as images provides physicists with a

new and suitable tool to deal with the exponentially large data sets embodied in the state

space of condensed-matter systems. The fundamental benefit of applying machine learning

in physical problems is that this approach can extend beyond the limits of conventional

approaches by obtaining solutions based on partial or even no prior physical knowledge,

and thereby extrapolate them to unexplored data. This benefit has been demonstrated in

many papers as recent as 2016 [23], including identifying phase transition by unsupervised

learning approaches in classical systems [20, 21, 22, 23]. In [23], Wang applied principle

component analysis (PCA) to classify the two phases in the classic Ising model. Later, van

Nieuwenburg et al. [22] proposed the so-called confusion scheme to obtain critical points

successfully for several Ising-like models. The application of ML, as demonstrated in these

publications, represents a landmark in the study of phase transitions.

In the year 2017, another pioneering paper appearing in Science by Carleo and

Troyer [24] has created a spark of frenzy among computational condensed matter physicists

as it successfully shown a new direction by using artificial neural network (ANN), a subset of

artificial intelligence algorithm, to achieve correct phase classification and transitions in

certain prototypical quantum many-body systems, without knowing a-priori the boundary of

the phases. ANN is in general inspired by the human brain neural network. It uses the model

of neurons which are connected with each other by synapses. The neurons are divided into

the input, hidden and output layers connected with weighted synapses as seen in Fig. 1.

The key ingredient of the success of Carleo and Troyer’s [24] approach is that the ANN can

effectively and efficiently compress the essential information of a many-body wave function

in high-dimensional systems in terms of ‘weights’, a set of internal parameters representing

the neutral network. Neural network representation of the many-body ground state Ψ

tremendously reduces the dimensionality to represent it to a level much lower than the

maximum Hilbert space. The parameters of the neural network are then optimized either by

static variational Monte Carlo (VMC) sampling [25] or time-dependent VMC [26, 27]. The

artificial neural network architecture used by Carleo and Troyer [24] is known as restricted

Boltzmann machine (RBM). Such an AI-inspired strategy to represent the ground state of a

quantum many-body system could potentially reduce the computationally cost by many

Figure 1. Neurons layers and synapses

orders of magnitude, apart from the ability to access parametric regions that are otherwise

not possible using conventional numerical approaches such as the popular QMC.

Since the pioneering work by Carleo and Troyer, many works using neural network to

address QMB problems have been published. To name a few, the Bose-Hubbard model has

been studied by using a similar method in [28, 29], while the feedforward neural network

has been applied to the Fermi-Hubbard model [30]. Another method used to study QMB

problem, deep neural network DNN, is reported in [31] where a large class of many-body

lattice Hamiltonians is studied using deep Boltzmann machine. Similar approach employing

convolutional neural network (CNN) has also been recently reported by [23, 32], where CNN

is shown to efficiently encode phases of matter and discriminate phase transitions in

classical correlated many-body systems. In a follow-up of an earlier work [32], Carrasquilla’s

group has reported in [33, 42] an auxiliary field QMC (AFQMC) technique to sample

statistical instances of the wavefunction of a fermionic system. A CNN was then deployed to

discriminate between two fermionic phases of a fermionic Hubbard-type quantum lattice

models. The authors showed that their machine learning approach, coupled with auxiliary

field QMC, can locate the phase transition point in selected systems which are known to be

plagued by severe sign problem. Based on the literature findings since 2017 pertaining to

applying machine learning in quantum many-body problems, it seems that the formidable

fermion sign problem that haunts generations of condensed matter theorists seems to have

a new weapon, i.e., machine-learning, to tame the beast. At this point of writing, the most

up-to-date and technical summary of how machine learning rises to overcome the curse of

dimensionality in quantum many-body can be found in a March 2019 article in Physics

Today [48]. The essential ideas of solving quantum many-body problem with various ML

algorithms, which would be used in this proposal, are clearly explained in [48].

However, our literature search reviews that only prototypical many-body models,

e.g., Ising-like and Heisenberg spin-like models are explored via ML approaches. There has

not been much comprehensive work of using AI-inspired approaches to study the full-

fledged 2D Hubbard model [34, 35], which is arguably the most studied QMB model since its

first proposal in 1963. Specifically, there has not been any comprehensive works of using AI-

inspired approaches to explore the Hubbard model to explain the interplay of

superconductivity, pseudo-gap and other observed phases in HTSC, nor how the transition

temperatures associated with these phases are predicted as a function of the free

parameters of the model. It is conventionally believed that 2D Hubbard model provides a

working model that captures some if not all essential features in copper-oxide

superconductors. It remains a daunting model to be completely solved despite relentless

computational efforts spent for so many years to abstract the physics out of it due to the

inherent sign problem. To illustrate the significance of the Hubbard model, as recent as

2017, we still see heroic effort such as that by two back-to-back papers in Science (Zheng et

al. and Huang et al., [36]) reporting their consorted, yet independent effort for obtaining the

ground state of the prototype 2D Hubbard model using state-of-the-art computational

methods , including Auxiliary field quantum Monte Carlo (AFQMC), density matrix

renormalization group (DMRG), hybrid DMRG, density matrix embedding (DMET) and

infinite projected entangled pair states (iPEPS). These are all ‘conventional’, non-machine

learning methods. On the other hand, the physics of superconductivity, pseudo-gap, and

their dynamical interplay in the 2D Hubbard model have been explored numerically using

another non-machine-learning approach known as cluster dynamical cluster approximation

(DCA) in a paper by Gull in 2013 [37]. The paper uses the DCA technique to map out the

phases (including the pseudo-gap, superconducting, Fermi liquid and Mott phases) along

with other physical insights, e.g., the transition temperatures as a function of the dopant

concentration, in the 2D Hubbard model. It is well known that these non-AI inspired

approaches are highly expensive and expert knowledge-demanding.

Independently, there are also alternative views on the physical origin of the pseudo-

gap and superconductivity in high-Tc cuprates, such as that proposed by B. S Lee [6] and the

following-up paper by B. S. Lee, T. L. Yoon and R. Abd-Shukor [7]. B. S. Lee was a well-

recognized theoretical physics professor retired from USM, while Yoon is the leader of this

research proposal. [6] proposes a physically-motivated model, known as the Extended

Hubbard Model (EHM) for explaining the physical origin of the pseudo-gap phase observed

in HTSC cuprates, whereas [7] further explores the physical signatures and consequences in

the superconducting sector in the EHM via a hybrid approach of theoretical mean field

theory and conventional numerical procedures. EHM offers a physically motivated

mechanism for modelling the novel phases in HTSC, which are traced to a common origin,

namely a Jahn-Teller type interaction induced by an electron interacting with a nonlinear 𝑄2

mode of the oxygen clusters in the CuO2 plane of the HTSC cuprates. The EHM is an

extended version of the prototype 2D Hubbard model and admits additional terms for the

lattice-electron interactions induced by the 𝑄2 mode. Due to its theoretically attractive

features, the EHM is a fully qualified contending many-body model for solving the

unconventional high-Tc superconductivity conundrum, the ‘Holy Grail’ of condensed matter

physics. This proposal makes this highly original, Malaysian-grown model the main subject

of research. It serves a rare chance where our very own home-grown theoretical model

could strike a truly profound achievement in the international frontier of condensed matter

physics.

The use of AI has a long history and its applications are already causing continuous

yet profound impact to the world we are living in. However, based on the literature review

as illustrated above, the application of machine learning in the area of QMB is a very recent

one. This means that there is still a lot of unexplored avenues to pursue. We, a group of

aspiring physicists and computer scientists, are jumping at the opportunity of such

promising research frontier to apply AI-inspired approaches to solve a ‘Holy Grail’ problem

in theoretical condensed matter physics.

Relevance to Government Policy, if any

Our proposed project is advancing one of the prioritised research areas as stated in the

Malaysian Ministry of Education research grant policy, namely, Artificial Intelligence and

Internet of Everything, which is an integral part in any research policy aimed for long term

sustainability. The scope and scale of our proposal, which is computational-cum-theoretical

in nature, offers a natural advantage against the pragmatic constraints in the context of

Malaysian research environment.

Objective(s) of the Research

1. To numerically solve for the ground states, classify and map out the quantum

phases in the prototype 2D Hubbard model via the AI-inspired machine learning

algorithms as originally proposed by the pioneering papers [24], [32], [33].

2. To work out the dependence of the superconducting transition temperature in

the EHM

3. To elucidate the physics and interplay of pseudo-gap, Mott, superconducting and

other observed phases in HTSC cuprates within the framework of EHM.

4. To establish a successful showcase of applying AI techniques in solving highly

non-trivial computational problems in the pure academic frontier of theoretical

condensed matter physics.

Description of Methodology

a. This is a computational condensed matter physics project. All calculations are to

be implemented in high-performance computing (HPC) facilities where open-

source AI platforms and Quantum Monte Carlo packages are to be deployed. To

this end, availability of HPC hardware, specifically, large amount of RAM, hard

disks, GPUs and powerful parallel machines are highly desirable.

b. The technical details and procedures of calculations in this project are to be

based heavily on, but not exclusively, that published in [24], [32], [33], [36], [37].

c. Python programming will be learned and used as it is the ‘default’ programming

language in most AI platforms. We shall also develop our own programming

codes to batch the implementation of ML calculations and perform some

numerical tasks. Mathematica, Fortran and shell script are the candidate

programming language of choice, depending on circumstances and necessity.

d. The generic machine learning platform developed by Google, which is popularly

used by the condensed physics research community, TensorFlow [46], will be

used for performing our machine learning calculations, in conjunction with a list

of open-source codes, as listed below:

(i) NetKet [38]. It is listed as one of open-source computing software

developed in the Center for Computational Quantum Physics (CCQ)

(https://www.simonsfoundation.org/flatiron/center-for-computational-

quantum-physics/) under the Simon Foundations

(https://www.simonsfoundation.org/about). NetKet comes with a

complete set of well-documented manual plus tutorials. It is the machine

learning package used for the publication of the pioneering paper [24].

(ii) QUEST (QUantum Electron Simulation Toolbox) [39]. The theoretical

background and other technical details of QUEST are described in the

documentation within the QUEST portal [39]. QUEST is a determinant

QMC (DQMC) implementation that is particularly suitable for generating

data from a generic 2D Hubbard model.

(iii) ALF package [40] which implements the auxiliary field QMC (AFQMC).

(iv) An AFQMC library, which is still under development at the time of this

proposal, by CCQ (see

https://www.simonsfoundation.org/flatiron/center-for-computational-

quantum-physics/software/auxiliary field-quantum-monte-carlo-2/).

(v) CPMC-Lab [43]. This is a Matlab package with a graphical interface made

readily available by the project leader of the AFQMC in CCQ, Shiwei

Zhang. It is a constrained-path and phaseless auxiliary field Monte Carlo

code for the Hubbard model.

We shall make use of the above QMC codes for generating 2D Hubbard model

configurations as sample data for training, testing and classifying for phases in

selected Hamiltonians. Our TensorFlow platform will be installed in a GPU-

enabled HPC. All software packages required for this research project are open-

source and require no cost for the licenses. We shall also maintain these

packages ourselves as our research group has the necessary expertise to do so.

e. Stage 1: To begin with, the numerical results of the classical Ising model as

reported in [24] shall be reproduced using NetKet and TensorFlow as a warming-

up training to familiarize with the computational handling of the package. The

numerical results to recognize the phases in classical spin systems using fully

connected and convolutional neural networks in the prototype Ising model as

reported in the 2017 Nature Physics paper [32] shall also be reproduced as part

of the Stage 1 exercise.

f. Stage 2: Once the classical results of phase classification are derived and verified,

we shall proceed to reproduce the quantum results as reported in [33] (which is

a follow-up of [32]), where the fermionic sign problem was reported to have

been circumvented. To this end, we shall make use of the NetKet and TensorFlow

packages which were used by the authors of [32] and [33] to obtain these

published results.

g. The proposed tasks in Stage 1 and Stage 2 are meant as warming-up exercises for

this research project. It is expected to last for a few months to a year, during

which the know-how to deploy machine learning algorithms in NetKet, use

Python programming and run computations on a GPU-enabled TensorFlow

platform to solve the QMB problems would have been picked up. Specifically, the

know-how to deploy Restricted Boltzmann machine (RBM) and Deep Boltzmann

machine (DBM) on prototype spin models, including transverse-field Ising model,

antiferromagnetic Heisenberg model, Bose-Hubbard model (these models are

well documented as part of the tutorials exercise in the NetKet code) would

become familiarized.

h. Stage 3: The next stage in the planned progress line is to apply the relevant

machine learning algorithms as reported in [24], [32], [33] on the 2D Hubbard

model based on the experience gained from the warming-up stage. The results

for the phases classified via various ML algorithm should be compared and

verified against that obtained via the non-AI approaches as reported in [36] and

[37].

i. Stage 4: The next stage would be to confront the EHM directly. EHM is a

generalisation of the prototype 2D Hubbard model which have never been

subjected to any ML investigation before. Armed with the knowledge and

experience picked up until this stage, we shall proceed to classify the phases

inherently embedded in the model Hamiltonian using ML algorithms.

j. Stage 5: We will then attempt to numerically investigate the two sectors

embedded in the EHM, i.e., the superconducting and the pseudo-gap sectors,

and the interplay between them. The numerical investigation at this stage does

not necessarily involve AI algorithms but relies on conventional computational

methodology. The dependence of the critical temperature with doping

concentration and other free parameters in the EHM model can be calculated by

following the theoretical method as detailed in [41]. The computation approach

in [41] requires the generation of QMC data from the model Hamiltonian to

determine two independent quantities, i.e., the helicity modulus and the pairing

correlation function. Critical temperature can be deduced once these two

quantities are known. To carry out the calculations as prescribed in [41], we

would have to develop our own numerical code to implement the computation

procedure.

(e) Expected Results/Benefit

Novel theories/New findings/Knowledge

1. The knowledge of applying various machine learning tools for the purpose of

solving quantum-many body problem, which can be easily translated into other

general, real-life applications, such as productivity optimization, pattern

recognition, identification of signals buried under noisy background, etc.

2. A practical know-how for using AI-spired techniques for predicting phases of

generic quantum or classical many body systems.

3. A resolution for the unconventional high-Tc superconductivity conundrum.

Specific or Potential Applications

The knowledge of applying various machine learning tools for the purpose of solving

quantum-many body problem, which can be easily translated into other general, real-life

applications, such as productivity optimization, pattern recognition, identification of signals

buried under noisy background, etc.

Impact on Society, Economy and Nation

This research proposal basically has two aspects which are in principle distinct from each

other, namely the AI aspect and the physics of HTSC. The former is a very, very powerful and

versatile tool while the latter concerns academic researchers committed to understand the

fundamentals of how nature work.

As for the latter aspect, in the context of a developing country like Malaysia who is

stuck at the bottle neck of a middle-income range, the disinterest to advocate pure

fundamental research is hampering the nation in the long run. We need to establish a

society worthy of protecting, one which embraces intellectual appreciation for endeavour

that do not appear to provide immediate, tangible gain. Supporting pure, fundamental

research (which this research proposal is one), has a long lasting, though gradual, impact on

the overall intellectual level of the nation.

On the other hand, the benefit and potential significance of the AI component in this

research proposal is immediately apparent. The highly non-trivial experience gained from

practically meddling with the AI tools can immediately be translated into practical

applications which the nation and industries are desperately needing. Being a very versatile

tool, the experience gained from applying AI in solving theoretical many-body problems,

which are highly mathematical and non-trivial in nature, can be easily translated into

applications for optimizing productivity in almost all aspect of economic and research

activities. Our nation is yearning for capable human capital in this AI age. This research

proposal, which is AI intensively by nature, rightly fits the bill for a nation which is already

far lagging behind in the face of the AI era.

Intellectual Property (IP)

This is a theoretical cum computational research project on fundamental physics that

produces universal knowledge that cannot be filed for IP.

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Flow Chart of Research Activities

Verify that these classical results are successfully reproduced and comparable to that published in the relevant papers listed in the references.

Plan, prepare, design and test-run ML calcultions of selected prototype classical systems, e.g., ferromagnetic Ising model in [32,34] Ising. The purpose is to learn how to to use these reported ML algorithm s to reproduce the phases in these classical systems.

Purchase, upgrade and install high-performing computing (HPC) facilities purchased with grant money. Recruit a graduate student.

Results justified?

Troubleshoot

Generate the phase classification and transition of selected classical systems that have never published before using the novel ML technique (e.g., unsupervised machine learning techniques of [44]), such as that of water or other selected Lennard-Jones systems.

Publication of the ML derived results on the phase classification of these selected classical systems.

Stage 1 completes

No

Yes

Troubleshoot

Plan, prepare, design and test-run ML calcultions of selected prototype quantum systems, e.g., transverse-field Ising (TFI) model and antiferromagnetic Heisenberg (AFH) model of [24], the Hubbard-like fermion model of [33] and Fermi-Hubbard Hamiltonian of [42]. ML algoriths to use: Restricted Boltzmann machine (RBM) and Deep Boltzmann machine (DBM).

Results justified?

Verify that these quantum results are successfully reproduced and comparable to that published in the relevant papers listed in the references.

Stage 2 completes

Stage 2 begins

Stage 3 begins

Apply the relevant machine learning algorithms as reported in [24], [32], [33] to perform a full-fledged classification and transition of phases in the prototype 2D Hubbard model.

No

Yes

Stage 1 begins

Troubleshoot Results justified?

Explore the prototype 2D Hubbard model to explain the interplay of superconductivity, pseudo-gap and other observed phases in HTSC, and how the transition temperatures associated with these phases are predicted as a function of the free parameters of the model.

Compare and verify the results for the prototype 2D Hubbard model phases classified via selected ML algorithms against that obtained via the non-AI approaches as reported in [36] and [37].

Stage 3 completes

No

Yes

Numerically evaluate the dependence of the critical temperature with doping

concentration and other free parameters in the EHM model by following the

theoretical method as detailed in [41]. This will involve application of AFQMC

on the EMH to generate sample data from the model Hamiltonian to

determine the helicity modulus and the pairing correlation function.

Stage 4 completes

Stage 4 begins

Begin stage 5

Numerically investigate the interplay of the two sectors embedded in the EHM, i.e., the superconducting and the pseudo-gap sectors. Propose resolution of the HTSC conundrum within the EHM.

Publish of obtained results

Stage 5 completes

Classify the phases inherently embedded in the model Hamiltonian of EHM

using ML algorithms and experience picked up during the previous stage of

investigating the prototype 2D Hubbard model.