GENERAL OVERVIEW:

This is a second course on the methods of mathematically rigorous statistical physics. It may be considered as a sequel to Thomas Bothner's TCC module on discrete models held in the Spring-2021 term. However, in a technical sense it is independent and self-contained.

I will assume known and I will mention only tangentially (for reference and comparison only) the basic facts about phase transitions in discrete spin models (like e.g. the Ising model). My main focus will be on lattice spin models with continuous spins and continuous internal symmetries. These systems exhibit very different large scale behaviour and their investigation requires very different mathematical methods from those of the discrete spin models. For example, combinatorial contour methods and Peierls-type arguments simply do not work here. As a physical consequence, Long Range Order (LRO) at low but still positive temperatures occurs only in 3 (and more) dimensions. (In contrast to the discrete contour models, where LRO may - and does - occur in 2-dimensions.) The methods developed and used are more analytic and probabilistic, but combinatorial elements are still at hand.

In the first half of the module I will discuss in full mathematical detail the Classical Heisenberg (a.k.a. O(N)) Models and some related variants. The two most important results being:

(C1) No LRO at positive temperatures in d=2 (Mermin-Wagner Theorem).

(C2) Existence of phase transition and LRO at low temperatures in d>=3 (Fröhlich-Simon-Spencer Theorem, and its consequences).

In the second half of the module I will turn to quantum spin systems, primarily to the Quantum Heisenberg Models. Due to non-commutativity of the observables new difficulties and challenges come up. These problems are also strongly linked to Bose-Einstein condensation in some particular lattice boson gases. The most important issues are:

(Q1) The quantum version of the Mermin-Wagner Theorem excludes long range order at positive temperatures in 2d.

(Q2) However, in the quantum setting the Néel LRO in the ground state of antiferromagnetic models is of major importance, even in 2d.

(Q3) Phase transition and Long Range Order at low temperatures in d>=3 in a restricted range of parameters (antiferromagnetic coupling and no external magnetic field) is proved in the celebrated Dyson-Lieb-Simon Theorem.

Finally, time permitting, I will present some elements of stochastic graphical representations relating some classes of quantum spin systems to probabilistic problems like random interchange dynamics and lattice loop models.

Plenty of beautiful (and relevant!) mathematics and plenty of hard open problems for future generations of mathematicians/mathematical physicists.

DETAILS:

The following topics are intended to be covered during the course. The main theorems will be presented in full detail.

1. Quick survey of the Ising model – for reference and comparison.

2. The Classical Heisenberg (a.k.a. O(N)) model and some variations.

3. The Classical Mermin-Wagner Theorem: No continuous symmetry breaking at positive temperatures in d=2.

4. Reflection positivity, Gaussian domination, infrared bounds. Fröhlich-Simon-Spencer Theorem: Continuous symmetry breaking at positive temperatures, in d>=3.

5. The Quantum Heisenberg (XXZ) models and their internal symmetries.

6. Bogoliubov's inequality and the Quantum Mermin-Wagner Theorem. No LRO at positive temperatures in d=2.

7. Other quantum correlation inequalities (Bogoliubov, Röpstorff, Falk-Bruch)

8. Reflection positivity - the quantum case. Dyson-Lieb-Simon Theorem: Néel order in antiferromagnetic systems at positive temperatures, in d>=3.

9. Bose lattice gas and the problem of Bose Einstein Condensation.

10. Stochastic/graphical representations of the Bose lattice gas and of quantum spin systems - their use and their limitations.

WHEN?

LECTURES HELD ON WEDNESDAYS 10:00-12:00 (UK TIME)

FIRST LECTURE: WEDNESDAY, 20 OCTOBER 2021

LAST LECTURE: WEDNESDAY, 1 DECEMBER 2021

LECTURES OF 24 NOVEMBER AND 1 DECEMBER WILLBE THREE HOURS LONG, FROM 10AM TO 1PM

WHERE?

THE LECTURES WILL BE HELD ON MICROSOFT TEAMS PLATFORM WITH ACCESS PROVIDED TO REGISTERED ATTENDANTS.

LECTURE NOTES AND/OR SLIDES:

1. Introduction, warming up, quick survey of main results about the Ising model (for comparison) [ver 27 Oct 2021]

2. The classical Heisenberg model (aka the O(\nu) model): LRO in three (and higher) dimension [ver 1 Nov 2021]

3. The quantum Heisenberg (XXZ) model: generalities, internal symmetries, etc. [ver 10 Nov]

4. Quantum correlation inequalities (Bogoliubov, Röpstorff, Falk-Bruch) [ver 24 Nov]

5. The quantum Heisenberg model ctd: therems and proofs [ver 24 Nov]

6. The Bose lattice gas and stochastic representations [ver 28 Nov]

ASSESSMENT FOR CREDIT:

Those PhD students who take this unit for credit will be assessed in the following way: I will provide some extra material for reading. This will be either a research paper or some theoretical part not fully covered in class. The student will be asked to read and understand the material and to write an essay of cca 15 pages about it, in style and format of a research paper.

Timing: The reading material will be provided towards the end of the TCC term (by end of November or early December). Deadline for handing in the essay sometime in the winter break (mid- or end of January).

Please let me know whether you take this unit for credit or for audit only.

BIBLIOGRAPHY:

RESEARCH PAPERS:

• to be uploaded soon

BOOKS, MONOGRAPHS, SURVEYS:

• David Ruelle: Statistical Mechanics: Rigorous Results. W.A. Benjamin, N.Y. 1969.

• Robert Griffiths: Rigorous results and theorems. In: Phase Transitions and Critical Phenomena. eds.: C. Domb and M.S. Green Academic Press, London 1972.

• Sacha Friedli and Yvan Velenik: Statistical Mechanics of Lattice Systems - a Concrete Mathematical Introduction. Cambridge: Cambridge University Press, 2017.