This is the webpage for the Trinity 2022 lecture series on "Quantum Matter II" within the Oxford Masters Course in Mathematical and Theoretical Physics (MMathPhys). This series of 20 lectures is a follow-on to Quantum Matter I, taught in Hilary term, and focuses on field theoretic and non-perturbative approaches to strongly correlated systems. Since in various ways it explores departures from Landau's two hugely successful theories - namely, of "ordinary" metals and the classification of phases of matter and transitions between them - this course is informally titled "Life after Landau", and focuses on three "big ideas": Renormalization, Bosonization, and Competing Orders.

Course Summary and Topics Covered

Modern condensed matter physics is increasingly focused on understanding the properties of strongly interacting systems. Traditional techniques that rely on diagrammatic perturbation theory about the independent electron approximation are often insufficient to provide an adequate description of the rich phenomena possible in this setting. Instead, their study requires a variety of ideas often also invoked in the study of quantum field theories in the non-perturbative regime. This course will cover some of the major themes in the study of correlated systems, including applications of the renormalization group to interacting Fermi systems, bosonization, superfluid-Mott insulator transitions, and competing orders.

Topics to be covered:

• Renormalization group for Interacting Fermions: momentum-shell RG for φ4 theory; RG and the Fermi surface; BCS and CDW as competing instabilities; RG in D=1 and emergence of Luttinger liquids [~ 8 hrs of lectures]

• Bosonization: Fermion-boson dictionary; application to spinless fermions; sine-Gordon and Kosterlitz-Thouless flow; emergence of insulators from commensuration [~6 hours of lectures]

• Competing Orders: Bose-Hubbard model and Superfluid-Mott transition; duality; commensurability and “Lieb-Schulz-Mattis” constraints; Landau-forbidden phases and transitions in the Bose-Hubbard model. [~6 hours of lectures]

Although the lectures will be self-contained, this course is designed as a follow-on to Quantum Matter I. Familiarity with ideas introduced in Advanced Quantum Theory and Renormalization Group will be useful but not essential.

Assessment: Homework Completion.

Course Book: No single textbook, but useful books include Quantum Field Theory in Condensed Matter by R. Shankar (Cambridge University Press, 2017); Quantum Phase Transitions by S. Sachdev (Cambridge University Press, 2011); and Quantum Physics in One Dimension by T. Giamarchi (Oxford University Press, 2003). Will be supplemented by bespoke lecture notes.

Lectures

Change in lecture location: Lectures will be held from 2-4pm on Tuesday and Thursday. In Weeks 1 and 5 and Thursday of Week 3, they will be in the Fisher Room of the Denys Wilkinson Building; in Weeks 2 and 4 and Tuesday of Week 3, they will be in the Lindemann Lecture Theatre in the Clarendon Complex.

The lectures will be held in the Fisher Room of the Denys Wilkinson Building, from 2-4pm on Tuesdays and Thursdays in Weeks 1-5 of term. Since a 2 hour block is quite long, there will be a 3' break in the middle to allow a chance to stretch your legs, caffeinate, etc. The lectures will be live, but recordings will be available. Students are strongly encouraged to attend wherever possible, since the material is taught at quite a high level, and thus interaction is quite important to getting something out of the lectures.

Problem Classes

The tutor for the course will be Dr. Abhishodh Prakash, who will lead 3 problem classes this term. Each class is run in conjunction with a problem sheet, which you must complete and submit (e.g. by scanning handwritten pages or by working on a tablet) electronically to your tutors, who will then provide feedback to students in the homework completion option or for those who need credits for a DPhil requirement. For full homework completion credit, work must be submitted by the deadline assigned. Please be considerate of Abhi's time, as he is also busy with lots of research responsibilities!

The class times and locations are:

Class 1 Monday 16 May 2:00 - 4:00 pm Lindemann Lecture Theatre (due Monday 9 May)

Class 2 Monday 23 May 4:00 - 6:00 pm Martin Wood Lecture Theatre (due Monday 16 May)

Class 3 Wednesday 01 June 4:00 - 6:00 pm Lindemann Lecture Theatre (due Thursday 26 May)

If you are an MMathPhys student taking this course and require Homework Completion credit, please follow the usual instructions to upload your work.

If you are a DPhil student taking this course as part of a requirement for your course and need Homework credit, your work can either be emailed directly to Abhishodh or else handed in at the Beecroft reception addressed to him.

Lecture Plan

This is a very rough plan of the lectures, and liable to change during the course. Bold entries represent how of topics were actually covered during the respective lecture.

Week 1

• Tuesday, April 26 (Fisher): The RG ethos; Momentum-shell RG for complex scalar field theory

• Thursday, April 28 (Fisher): Complete scalar field theory RG. Fermionic path integrals. Start fermion RG (in d=2)

Week 2

• Tuesday, May 3 (Lindemann) : Complete fermion RG. Emergence of Fermi liquid theory. Kohn -Luttinger and generalization to d=3

• Thursday, May 5 (Lindemann) : Nesting and the CDW instability; breakdown of Fermi liquids in d=1. Motivating bosonization.

Week 3

• Tuesday, May 10 (Lindemann) : Bosonization I: massless Dirac fermions & free bosons; preview the bosonization dictionary

• Thursday, May 12 (Fisher): Bosonization II: boson duality & bosonization dictionary; bosonizing spinless lattice fermions to sine Gordon

Week 4

• Tuesday, May 17 (Lindemann) : Bosonization III: sine-Gordon RG flow

• Thursday, May 19 (Lindemann) : Superfluid-Mott transition in mean-field and beyond.

Week 5

• Tuesday, May 24 (Fisher): Competing Orders and Lieb-Schulz Mattis

• Thursday, May 26 (Fisher): Boson-Vortex Duality; Bose-Hubbard example of a transition beyond Landau theory.

Lecture Notes and Problems:

Comprehensive lecture notes (~3MB, 105pp) are available on the Canvas Page (requires Oxford SSO; updated and corrected during the course). The final version will be uploaded here at the conclusion of the course.

Problem sets are available on this page, and are also embedded in the Canvas lecture notes at the end of each Part.

Problems for Part I, due by Monday of week 3

Problems for Part II, due by Monday of week 4

Problems for Part III, due by Thursday of week 5

Note: COVID-19 has made the standard email torrent into a deluge. Some simple guidelines to ensure you receive an answer quickly:

• Please first read this page and the MMathPhys handbook to see if your question is answered there

• If your question pertains to content or organisation of these lectures, then I am your point of contact.

• If your question pertains to organisation of your year overall, examination logistics, etc., please contact Lizzy Griffiths.

Due to the sheer volume of correspondence, emails that are misdirected or for which the resolution is clearly available in course resources may go unanswered.