theoretical physics phd cambridge

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As a postgraduate student at the Cavendish laboratory you would be joining an institution with an illustrious history of innovation and discovery and a current programme that builds on that tradition.

Postgraduate Students in the Department study for one of the following qualifications:

• Master of Advanced Studies (MASt in Physics) , 9 Months
• MPhil in Data Intensive Science ,10 Months
• MPhil in Planetary Sciences and Life in the Universe (PSLU) , 10-month
• MPhil in Physics (by research) , 1 year
• MPhil in Scientific Computing (taught/research), 1 year
• PhD in Physics (by research), 3+ years
• Interdisciplinary PhD in Nanoscience and Nanotechnology (NanoDTC)   (initial common training period + research), 3.5+ years
• PhD in Computational Methods for Materials Science - 4 year

Applications for admissions

If you decide to make an application, you are advised to do so as early as possible. This will increase your chances not only of acceptance but also of being considered for funding. 

All candidates should consult the course directory for funding deadlines

We hope that the above information is helpful as we wish to encourage good applicants. Should you choose to apply, we will try to reach a decision in the Department without undue delay before passing on your papers for further consideration by the Colleges, the Degree Committee, and the Postgraduate Admissions Office. You should not take any steps to come to Cambridge before you receive an admission letter from the Postgraduate Admissions Office stating that you have satisfied all conditions.

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Virtual Postgraduate Open Days takes place every year. During the Open Day you can find out more about what it is like to be a Cavendish postgraduate, various research areas, the application process, student life and more. You will also get an opportunity to attend a Q&A session with the current staff.

This year the Postgraduate Open Day will take place between 4 th and 15 th November 2024.  You can find more information here.

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PhD in Applied Mathematics and Theoretical Physics

University of cambridge, different course options.

• Key information

Course Summary

Tuition fees, entry requirements, similar courses at different universities, key information data source : idp connect, qualification type.

PhD/DPhil - Doctor of Philosophy

Subject areas

Applied Mathematics Theoretical Physics Applied Physics

Course type

This is a three to four-year research programme culminating in submission and examination of a thesis containing substantial original work. PhD students carry out their research under the guidance of a supervisor, and research projects are available from the wide range of subjects studied within the Department. Students admitted for a PhD will normally have completed preparatory study at a level comparable to the Cambridge Part III (MMath/MASt) course. A significant number of our PhD students secure post-doctoral positions at institutions around the world and become leading researchers in their fields.

Assessment for the PhD is by submission of a thesis and oral examination only. There is no standard format for the thesis in mathematics (ie no prescribed word limit). Candidates should discuss the format appropriate to their topic with their supervisor.

The Mathematics Degree Committee oversees the examinations process and is responsible for approving the research title of the thesis, appointing examiners and scrutinising the reports of those examiners before making a decision on the outcome.

UK fees Course fees for UK students

For this course (per year)

International fees Course fees for EU and international students

Applicants for this course should have achieved a UK First class Honours Degree. The usual minimum entry requirement is a first-class honours degree, awarded after a four-year course in physics, mathematics or engineering, or a three-year degree together with a one-year postgraduate course on advanced mathematics and theoretical physics. Part III (MMath/MASt) of the Mathematical Tripos provides such a course. Note, however, that entry is competitive and a higher level of preparation may be required for research in some subject areas.

MSc Applied Mathematical Sciences

Heriot-watt university, phd applied mathematics, university of essex, applied mathematics phd, university of birmingham, applied mathematics mres, mphil applied mathematics.

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DAMTP PhD Opportunities

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• Postgraduate Study
• Research Programmes

Applications are welcome in all areas.

Research Topics

Our research page lists the various research groups and their interests. Note that in some cases there are overlaps. You need to determine which group or groups cover your range of interests, and where appropriate identify a potential supervisor as described below .

The following is a list of some,  but by no means all , of the current PhD opportunities in DAMTP:

• Astrophysics
• Atmosphere-Ocean Dynamics
• Biological Physics and Mechanics
• G K Batchelor Laboratory
• Institute of Theoretical Geophysics
• Soft Matter
• Solid Mechanics
• CDT in Data Intensive Science
• van der Schaar Lab
• Mathematics of Information
• Quantum Information and Foundations

Qualifications

The usual minimum entry requirement is a first class honours degree, awarded after a four-year course in mathematics, physics or engineering, or a three-year degree together with a one-year postgraduate course on advanced applied mathematics and theoretical physics. Please note that a very large majority of the successful applicants for PhD studentships with the High Energy Physics (HEP), General Relativity & Cosmology (GR ) , and Quantum Information (QI) groups will have taken Part III of the Mathematical Tripos .

Finding a supervisor  

The topic of the research thesis may be chosen from the wide range of subjects studied within the Department. It is expected that applicants to the PhD course will investigate the research interests and expertise of academic staff within DAMTP prior to making a formal application. This should be done by consulting the  dedicated page on finding a supervisor , as well  individual profiles of our academic staff .

Applicants are encouraged to make informal contact with potential supervisors prior to making an application. Applicants should clearly state in the 'Proposed supervisor' field of the application form the name(s) of those member(s) of academic staff with whom they wish to work, and provide a clear indication of the areas or topics in which they intend to undertake research in the 'Research Summary' field. We do not currently require submission of a separate detailed research proposal.

Applicants with Home fee status will be considered by the Department for a full Research Council studentship (including 3.5 years of maintenance).  Overseas applicants may be considered for partial support.  Receipt of this funding is not guaranteed and all applicants, irrespective of fee status, are expected to apply to other funding schemes for which they are eligible.  Applicants are advised to investigate potential sources of funding as early as possible.

Other funding opportunities for students studying in DAMTP include:

• NERC Doctoral Training Programme (includes Atmosphere-Ocean Dynamics, Theoretical Geophysics, and certain areas of Fluid and Continuum Mechanics)
• Several fully funded PhD studentships are available in Machine Learning and Artificial Intelligence
• Centre for Doctoral Training in Data Intensive Science
• Newnham Scholarship for Women in Theoretical Physics  (Deadline: 15 December 2023)

There are other awards available from Cambridge sources outside DAMTP. An up-to-date guide can be found here.

Applications must be made via the University Application Portal . Applicants are advised to consult the University Postgraduate Admissions website for details of the admissions process and to review the Course Directory entry for the PhD at DAMTP prior to making an application.

Applicants are encouraged to make informal contact with potential supervisors prior to making an application. Applicants should clearly state in the relevant field of the application form the subject area in which they intend to undertake research as determined under 'Research Topics' above.  If possible they should mention which DAMTP research group(s) they would like to consider their application. We do not expect applicants to submit a detailed research proposal. 

All applicants are required to submit two references, for MASt students this should be an updated reference which covers your performance in the final year at your previous institution. MMath students must include one from their College Director of Studies.

The DAMTP PhD course code is: MAAM21.

The University values diversity and is committed to equality of opportunity. The Department would particularly welcome applications from women, since women are, and have historically been, underrepresented in our student cohorts.

Deadlines and Offers

Those wishing to do a PhD in DAMTP are strongly encouraged to apply by 4 January 2024  for admission in October. However, later applications will still be considered where possible (up until the general University deadline). Applicants wishing to explore the possibility of a later start date (January or April) should contact DAMTP prior to submitting an application. Applicants will only be considered for admission in January or April in exceptional circumstances.

Earlier application deadlines apply for:

• High Energy Physics and General Relativity & Cosmology groups (15 December 2023 for full consideration)
• University Postgraduate Funding Competition (Gates US: 11 October 2023; All other funding: 4 January 2024)
• Cambridge Mathematics of Information (4 January 2024)
• NERC Doctoral Training Programme (4 January 2024)

Click here for further information about the Postgraduate Open Day.

If you have queries please contact the Faculty Postgraduate Office

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MASt in Mathematics (Theoretical Physics)

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Course closed:

Mathematics (Theoretical Physics) is no longer accepting new applications.

This course is an application stream for the Master of Advanced Study (MASt) in Mathematics; students should apply to only one of the application streams for this course.

This course, commonly referred to as Part III, is a nine-month taught masters course in mathematics. It is excellent preparation for mathematical research and it is also a valuable course in mathematics and its applications for those who want further training before taking posts in industry, teaching, or research establishments.

Students admitted from outside Cambridge to Part III study towards the Master of Advanced Study (MASt). Students continuing from the Cambridge Mathematical Tripos for a fourth-year study towards the Master of Mathematics (MMath). The requirements and course structure for Part III are the same for all students irrespective of whether they are studying for the MASt or MMath degree, or whether they applied through the Applied Mathematics (MASA), Pure Mathematics (MASP), Mathematical Statistics (MASS), or Theoretical Physics (MASTH) application stream.

There are around 280 Part III (MASt and MMath) students each year; almost all are in their fourth or fifth year of university studies. Each year the Faculty offers up to 80 lecture courses in Part III, covering an extensive range of pure mathematics, probability, statistics, applied mathematics and theoretical physics. They are designed to cover those advanced parts of the subjects that are not normally covered in a first-degree course, but which are an indispensable preliminary to independent study and research. Students have a wide choice of the combination of courses they take, though naturally, they tend to select groups of cognate courses. Examples classes and associated marking of (unassessed) example sheets are provided as complementary support to lectures.

As a taught masters course, the main emphasis is on lecture courses, and assessment is almost entirely based on written exams, which are normally taken at the end of the academic year starting in the last week of May, alongside a mathematical essay, normally due in early May. The standard graduation dates for successful candidates are usually in June and July.

Learning Outcomes

After completing Part III, students will be expected to have:

• studied advanced material in the mathematical sciences to a level not normally covered in a first degree;
• further developed the capacity for independent study of mathematics and problem-solving at a higher level; and
• undertaken an extended essay normally chosen from a list covering a wide range of topics.

Students are also expected to have acquired general transferable skills relevant to mathematics as outlined in the Faculty Transferable Skills Statement.

MASt students wishing to apply for a PhD at Cambridge must apply via the Postgraduate Admissions webpage for readmission by the relevant deadline. Details of entry requirements can be found in the relevant course listings on this site.

Applications to study in either of the Mathematics Departments will be considered on a case-by-case basis and offer of a place will usually include an academic condition based on Part III results.

The Postgraduate Virtual Open Day usually takes place at the end of October. It’s a great opportunity to ask questions to admissions staff and academics, explore the Colleges virtually, and to find out more about courses, the application process and funding opportunities. Visit the Postgraduate Open Day page for more details.

Details of activities hosted by the Faculty of Mathematics can be found on the Faculty website .

Departments

This course is advertised in the following departments:

• Faculty of Mathematics
• Department of Applied Mathematics and Theoretical Physics

Key Information

9 months full-time, study mode : taught, master of advanced study, department of applied mathematics and theoretical physics this course is advertised in multiple departments. please see the overview tab for more details., course - related enquiries, application - related enquiries, course on department website, dates and deadlines:, michaelmas 2024 (closed).

Some courses can close early. See the Deadlines page for guidance on when to apply.

Funding Deadlines

These deadlines apply to applications for courses starting in Michaelmas 2024, Lent 2025 and Easter 2025.

Similar Courses

• Mathematics (Applied Mathematics) MASt
• Mathematics (Pure Mathematics) MASt
• Mathematics (Mathematical Statistics) MASt
• Mathematics MPhil
• Pure Mathematics and Mathematical Statistics PhD

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Professor Stephen Hawking 1942-2018

Professor stephen hawking 1942 - 2018, friends and colleagues from the university of cambridge have paid tribute to professor stephen hawking, who died on 14 march 2018 at the age of 76.  .

Widely regarded as one of the world’s most brilliant minds, he was known throughout the world for his contributions to science, his books, his television appearances, his lectures and through biographical films. He leaves three children and three grandchildren.

Professor Hawking broke new ground on the basic laws which govern the universe, including the revelation that black holes have a temperature and produce radiation, now known as Hawking radiation. At the same time, he also sought to explain many of these complex scientific ideas to a wider audience through popular books, most notably his bestseller A Brief History of Time .

He was awarded the CBE in 1982, was made a Companion of Honour in 1989, and was awarded the US Presidential Medal of Freedom in 2009. He was the recipient of numerous awards, medals and prizes, including the Copley Medal of the Royal Society, the Albert Einstein Award, the Gold Medal of the Royal Astronomical Society, the Fundamental Physics Prize, and the BBVA Foundation Frontiers of Knowledge Award for Basic Sciences. He was a Fellow of The Royal Society, a Member of the Pontifical Academy of Sciences, and a Member of the US National Academy of Sciences.

He achieved all this despite a decades-long battle motor neurone disease, with which he was diagnosed while a student, and eventually led to him being confined to a wheelchair and to communicating via his instantly recognisable computerised voice. His determination in battling with his condition made him a champion for those with a disability around the world.

Professor Hawking came to Cambridge in 1962 as a PhD student and rose to become the Lucasian Professor of Mathematics, a position once held by Isaac Newton, in 1979. In 2009, he retired from this position and was the Dennis Stanton Avery and Sally Tsui Wong-Avery Director of Research in the Department of Applied Mathematics and Theoretical Physics until his death - he was also a member of the University's Centre for Theoretical Cosmology , which he founded in 2007. He was active scientifically and in the media until the end of his life.

Professor Stephen Toope, Vice-Chancellor of the University of Cambridge, paid tribute, saying, “Professor Hawking was a unique individual who will be remembered with warmth and affection not only in Cambridge but all over the world. His exceptional contributions to scientific knowledge and the popularisation of science and mathematics have left an indelible legacy. His character was an inspiration to millions. He will be much missed.”

Stephen William Hawking was born on January 8, 1942 in Oxford although his family was living in north London at the time. In 1959, the family moved to St Albans where he attended St Albans School. Despite the fact that he was always ranked at the lower end of his class by teachers, his school friends nicknamed him ‘Einstein’ and seemed to have encouraged his interest in science. In his own words, “physics and astronomy offered the hope of understanding where we came from and why we are here. I wanted to fathom the depths of the Universe.”

His ambition brought him a scholarship to University College Oxford to read Natural Science.There he studied physics and graduated with a first class honours degree.

He then moved to Trinity Hall Cambridge and was supervised by Dennis Sciama at the Department of Applied Mathematics and Theoretical Physics for his PhD; his thesis was titled ‘Properties of Expanding Universes.’ In 2017, he made his PhD thesis freely available online via the University of Cambridge’s Open Access repository. There have been over a million attempts to download the thesis, demonstrating the enduring popularity of Professor Hawking and his academic legacy.

On completion of his PhD, he became a research fellow at Gonville and Caius College where he remained a fellow for the rest of his life. During his early years at Cambridge, he was influenced by Roger Penrose and developed the singularity theorems which show that the Universe began with the Big Bang.

An interest in singularities naturally led to an interest in black holes and his subsequent work in this area laid the foundations for the modern understanding of black holes. He proved that when black holes merge, the surface area of the final black hole must exceed the sum of the areas of the initial black holes, and he showed that this places limits on the amount of energy that can be carried away by gravitational waves in such a merger. He found that there were parallels to be drawn between the laws of thermodynamics and the behaviour of black holes. This eventually led, in 1974, to the revelation that black holes have a temperature and produce radiation, now known as Hawking radiation, a discovery which revolutionised theoretical physics. 

He also realised that black holes must have an entropy – often described as a measure of how much disorder is present in a given system – equal to one quarter of the area of their event horizon: – the ‘point of no return’, where the gravitational pull of a black hole becomes so strong that escape is impossible. Some forty-odd years later, the precise nature of this entropy is still a puzzle. However, these discoveries led to Hawking formulating the ‘information paradox’ which illustrates a fundamental conflict between quantum mechanics and our understanding of gravitational physics. This is probably the greatest mystery facing theoretical physicists today.

To understand black holes and cosmology requires one to develop a theory of quantum gravity. Quantum gravity is an unfinished project which is attempting to unify general relativity, the theory of gravitation and of space and time with the ideas of quantum mechanics. Hawking’s work on black holes started a new chapter in this quest and most of his subsequent achievements centred on these ideas. Hawking recognised that quantum mechanical effects in the very early universe might provide the primordial gravitational seeds around which galaxies and other large-scale structures could later form. This theory of inflationary fluctuations, developed along with others in the early 1980’s, is now supported by strong experimental evidence from the COBE, WMAP and Planck satellite observations of the cosmic microwave sky. Another influential idea was Hawking’s ‘no boundary’ proposal which resulted from the application of quantum mechanics to the entire universe. This idea allows one to explain the creation of the universe in a way that is compatible with laws of physics as we currently understand them. 

Professor Hawking’s influential books included The Large Scale Structure of Spacetime, with G F R Ellis; General Relativity: an Einstein centenary survey, with W Israel; Superspace and Supergravity, with M Rocek (1981); The Very Early Universe, with G Gibbons and S Siklos, and 300 Years of Gravitation, with W Israel.

However, it was his popular science books which took Professor Hawking beyond the academic world and made him a household name. The first of these, A Brief History of Time, was published in 1988 and became a surprise bestseller, remaining on the Sunday Times best-seller list for a record-breaking 237 weeks. Later popular books included Black Holes and Baby Universes, The Universe in a Nutshell, A Briefer History of Time, and My Brief History. He also collaborated with his daughter Lucy on a series of books for children about a character named George who has adventures in space.

In 2014, a film of his life, The Theory of Everything, was released. Based on the book by his first wife Jane, the film follows the story of their life together, from first meeting in Cambridge in 1964, with his subsequent academic successes and his increasing disability. The film was met with worldwide acclaim and Eddie Redmayne, who played Stephen Hawking, won the Academy Award for Best Actor at the 2015 ceremony.

Travel was one of Professor Hawking’s pastimes. One of his first adventures was to be caught up in the 7.1 magnitude Bou-in-Zahra earthquake in Iran in 1962. In 1997 he visited the Antarctic. He has plumbed the depths in a submarine and in 2007 he experienced weightlessness during a zero-gravity flight, routine training for astronauts. On his return, he quipped “Space, here I come.”

Writing years later on his website, Professor Hawking said: “I have had motor neurone disease for practically all my adult life. Yet it has not prevented me from having a very attractive family and being successful in my work. I have been lucky that my condition has progressed more slowly than is often the case. But it shows that one need not lose hope.”

At a conference In Cambridge held in celebration of his 75th birthday in 2017, Professor Hawking said “It has been a glorious time to be alive and doing research into theoretical physics. Our picture of the Universe has changed a great deal in the last 50 years, and I’m happy if I’ve made a small contribution.”

And he said he wanted others to feel the passion he has for understanding the universal laws that govern us all. “I want to share my excitement and enthusiasm about this quest. So remember to look up at the stars and not down at your feet. Try to make sense of what you see and wonder about what makes the universe exist. Be curious, and however difficult life may seem, there is always something you can do, and succeed at. It matters that you don’t just give up.”

Stephen Hawking pictured with Newton's own annotated copy Principia Mathematica as part of Cambridge University Library's 600th anniversary.

Credit: Graham CopeKoga/Cambridge University Library

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Quantum Field Theoretical Study of Correlated Quantum Ising Model with Next-Nearest-Neighbour Interaction

• Published: 23 August 2024
• Volume 54 , article number  206 , ( 2024 )

Cite this article

• Ranjith R. Kumar 1 , 2 &
• Sujit Sarkar 1  

We present the results of quantum Ising and longer range quantum Ising model for strongly correlated emergent Hamiltonian. We show explicitly that the ordered ferromagnetic (FM) phase to the disorder quantum paramagnet (dqpI) quantum phase transition occurs only in the strongly correlated regime for the quantum Ising model otherwise only dqpI phase appears for a non-interacting and attractive regime. We show that short-range FM coupling and longer range coupling are competing with each other and also the effect of strong correlation in this competition. We also show the most interesting feature that the transverse field opposes the FM coupling of quantum Ising model but it favours the longer range coupling. We find the evidence of another disorder quantum paramagnetic (dqpII) phase due to the relevance of longer-range coupling. We also present the existence of another quantum phase the transition from dqpII phase to the FM phase. We also observe three different kinds of coexistence phases depending on the correlated regime. The fixed point analysis of the three sets of RG equations is carried out.

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No datasets were generated or analysed during the current study.

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Acknowledgements

The authors acknowledge Prof. Chandan Dasgupta and Prof. Prabir Mukherji for critically reviewing the manuscript. The author would like to acknowledge AMEF, DST (CRG/2021/000996) and RRI library for books and journals.

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Theoretical Sciences Division, Poornaprajna Institute of Scientific Research, Bidalur, Bengaluru, 562164, India

Ranjith R. Kumar & Sujit Sarkar

Graduate Studies, Manipal Academy of Higher Education, Madhava Nagar, Manipal, 576104, India

Ranjith R. Kumar

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S.S. formulated the problem and wrote the manuscript. R.R.K. performed the calculation and generated the figures.

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Correspondence to Sujit Sarkar .

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(A). Derivation of Bosonized Hamiltonian

We consider the Hamiltonian,

where \(\lambda _{1}\) and \(\lambda _{2}\) are the nearest and next nearest neighbor interactions. We may also write the Hamiltonian as,

The field, \(\phi \) , corresponds to the spin fluctuations and \(\theta \) is the dual to the field \(\phi \)  [ 5 , 8 , 25 ]. These fields are related by the following relations \({\phi }_R = \theta - \phi \) and \({\phi }_L = \theta + \phi \) . \({\phi }_R \) and \({\phi }_L \) are respectively the right and left component of the field [ 8 , 25 ].

The above two Hamiltonians are free from K . Therefore, now our main task is to find the analytical expression for spin-1/2 operators in terms of in terms of bosonized fields \(\phi \) and \(\theta \) and that also show how K appears in the quantum simulated model Hamiltonian.

We present spin operators interms of \(\phi \) , \(\theta \) , and K  [ 8 , 25 ], detail nature of the \(\phi \) and \(\theta \) fields and the relation between them have already mentioned in the main text (after ( 2 ) ). \(\phi \) and \(\theta \) represent the bosonic fluctuation of the system.

Here K is the Luttinger liquid parameter. We have already presented detail physical explanation of K (after ( 2 )) in the main text.

Neglecting oscillatory terms (i.e, \((-1)^n\) ) and higher order cosine terms (first term in the above equation), we have,

We write the final form of Bosonized Hamiltonian as, Now, we use bosonized form of ( 15 ), ( 16 ), and ( 17 ) in ( 11 ) to get the final version of the bosonized Hamiltonian.

where \(H_0 = \frac{v}{2} \int [(\partial _r \sqrt{K} \phi (r) )^2] dr \) with \(v=\lambda _{1}\) .

(B). Stability matrix for the interacting RG ( 3 )

\(\frac{d A_4 }{dl} = B_4 A_4 \) , \(A_4 = {( \delta K, \delta \mu , \delta \lambda _1 )}^{T} \)

where, \(\chi _{5} = -2 {{\lambda }^*}^2 {K}^{*} \) , \(\chi _{6} = 0 \) , \(\chi _{7} = + 2 {{K}^*}^2 {\lambda }^* \) .

\(\chi _{8} = - {\mu }^{*} {\lambda }^{*} \) , \(\chi _{9} = (2 - K^* - 1/4K^* ) - {\lambda }^* K^* \) , \(\chi _{10} = {\mu }^* {K^*} \) .

\(\chi _{11} = + 4 {\lambda }^* + { ({\mu }^* )}^2 /8 (2 + (1/2) (1/{K^*}^2 ) \) , \(\chi _{12} = {{\mu }^*}/4 (2 K^* - 1/(2*K) \) , \(\chi _{13} = (2 - 4 K^* ) \) .

(C). Stability matrix for the interacting RG ( 4 )

\(\frac{d A_5 }{dl} = B_5 A_5 \) , \(A_5 = {( \delta K, \delta \mu , \delta \lambda _1, \delta \lambda _2 )}^{T} \)

where, \(\chi _{14} = -2 {{\lambda _1}^*}^2 {K}^{*} \) , \(\chi _{15} = 0 \) , \(\chi _{16} = - 2 {{K}^*}^2 {\lambda }^* \) .

\(\chi _{17} =0 \) , \(\chi _{18} = {\lambda _1}^{*} {K}^{*} + ( 0.25/{K^*}^2 -1 ) \)

\(\chi _{19} = (2 - K^* - 1/4K^* ) - {\lambda _1}^* K^* \) , \(\chi _{20} = {\mu }^* {K^*} \) .

\(\chi _{21} = 0 \) .

\(\chi _{22} = - 4 {\lambda _1}^* + { ({\mu }^* )}^2 /8 (2 + (1/2) (1/{K^*}^2 )) \) , \(\chi _{23} = {{\mu }^*}/4 (2 K^* - 1/(2*K)) \) , \(\chi _{24} = (2 - 4*K) \) , \(\chi _{25} =0 \) . \(\chi _{26} = {\lambda _1 }^{*} {\lambda _2}^{*}/{\pi } + {\lambda _2}^{*} (-1 + 1/{4*K*K} ) \) .

\(\chi _{27} =0 \) , \(\chi _{28} = {\lambda _2}^{*} {K^*}/{\pi }\) , \(\chi _{29} = (2 -K^* - 0.25/{K^*} ) + {\lambda _1}^{*} {K^*}/{\pi } \) .

(D). Stability matrix for the interacting RG ( 5 )

\(\frac{d A_6 }{dl} = B_6 A_6 \) , \(A_6 = {( \delta K, \delta \lambda _1, \delta \lambda _2 )}^{T} \)

where, \(\chi _{33} = -4 \lambda _1 \) , \(\chi _{34} = (2 - 4*K) \) , \(\chi _{35} =0 \) . \(\chi _{36} = {\lambda _1 }^{*} {\lambda _2}^{*}/{\pi } + {\lambda _2}^{*} (-4 + 1/{4*K*K} ) \) .

\(\chi _{37} = {\lambda _2}^{*} {K^*}/{\pi }\) , \(\chi _{38} = (2 - 4*K^* - 0.25/{K^*} ) + {\lambda _1}^{*} {K^*}/{\pi } \) .

We have used these, three, sets of stability matrices analysis for the relevant, irrelevant and marginal coupling analysis. These all analysis have presented in a tabular (Tables 1 to 7) form in the main text.

(D). Derivation of Renormalization Group Equation

We write the partition function \(\mathcal {Z}\) in terms of fields as,

where \(S_E\) is the Euclidean action which can be written as \(S_E= -\int dr \mathcal {L} = -\int dr (\mathcal {L}_0 + \mathcal {L}_{int})\) , where \(r=(\tau ,x)\) . Thus, the partition function is given by

The first and second terms of the exponent of the above equation are \(\mathcal {L}_0 \) and \(\mathcal {L}_{int}\) respectively.

Now, we divide the fields into slow and fast modes and integrate out the fast modes. The field \(\phi \) is \(\phi (r)=\phi _s(r)+\phi _f(r)\) similarly the \(\theta \) is \(\theta (r)=\theta _s(r)+\theta _f(r)\) , where

Here \(\Lambda \) is the cut-off to start with and b is a factor greater than one. It is clear from the above definitions of faster and slower mode. One can make the average over the fast mode in order to get an effective action for the slower mode. Thus, \(\mathcal {Z}\) is

Using the relation \(\left\langle A \right\rangle _f = \int \mathcal {D}\phi _f e^{-S_f(\phi _f,\theta _f)} A \) , one can write

We write the effective action as

Taking \(\ln \) on both sides give, \( S_{eff}(\phi _s,\theta _s)= S_s(\phi _s,\theta _s) - \ln \left\langle e^{-S_{int}(\phi ,\theta )} \right\rangle _f. \) By writing the cumulant expansion up to the third order, we have

Now, we calculate the second order terms,

Following the previous calculations, we write,

Thus, all the correlation functions vanish and \(\lambda _2^2 \) term goes to zero. Now, we calculate \(\lambda _1\lambda _2\) term.

Here correlation function,

Similarly, we have

Thus, we have,

Finally, we obtain our RG equations for the Hamiltonian, H ( 18 ), using the above all equations (from ( 26 ) to ( 43 ), specially ( 40 ) to ( 43 ), we also follow ref. 15):

Similarly one can find the RG equation only for the coupling \(\lambda _1 \) .

Similarly one can find the RG equation in absence of \(\mu (=0)\) .

E. Spinless fermion representation of model Hamiltonian

Performing Jordan-Wigner transformation \(\sigma _{i}^x=1-2c_i^{\dagger }c_i\) and \(\sigma _{i}^z= - \prod _{j<i} (1-2c_{j}^{\dagger }c_{j}) (c_{i} + c_{i}^{\dagger })\) , the model Hamiltonian can be written in spinless fermionic form as

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Kumar, R.R., Sarkar, S. Quantum Field Theoretical Study of Correlated Quantum Ising Model with Next-Nearest-Neighbour Interaction. Braz J Phys 54 , 206 (2024). https://doi.org/10.1007/s13538-024-01584-x

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Received : 29 April 2024

Accepted : 13 August 2024

Published : 23 August 2024

DOI : https://doi.org/10.1007/s13538-024-01584-x

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