Download Modern Topics in Quantum Integrable Systems

Modern Topics in Quantum Integrable Systems
SMSTC Advanced Course - Semester 2
Organisers/Lecturers:
Panagiota-Maria Adamopoulou, Anastasia Doikou, Christian Korff and Robert Weston
Quantum integrable systems appear in many branches of mathematical physics (e.g. statistical
mechanics, quantum mechanics, high-energy physics and string theory) but also have overlap with
areas in pure mathematics in particular algebra and geometry. This short course will review some
recent developments of `modern usage’ of quantum integrable systems in different areas whose
selection is highly biased by the interests of the organisers. This short course is a first attempt to
gauge interest and might be followed in the future by more in depth courses.
Aims
To give several short lectures on various active research topics in quantum integrable systems which
go beyond a typical research seminar by spending more time to introduce ideas and notions and
presenting through tutorials an introduction to some of the techniques involved. We are not aiming
at being comprehensive but simply trying to showcase some recent developments in slightly more
depth than is possible in research seminars.
Prerequisites
Starting point for each of the lecture series will be only a basic mathematical background and we will
attempt to explain all notions from scratch. A liking of algebra (in particular representation theory),
combinatorics and mathematical physics would be of an advantage but is not a must.
Contents
There will be several lecture series within this course each one focussing on one topic.
1. Braid groups, quantum algebras & Bethe ansatz (AD)
Schedule: 14:00-15:30 on 1, 3 and 8 June 2016
This is basically an introduction to quantum integrable systems & the Quantum Inverse Scattering
Method (QISM). The Yang-Baxter equation (YBE) and its solution, the so called R-matrix are
introduced. YBE is the fundamental equation within the QISM context. We introduce the equation
and also provide systematic means for solving it via its structural similarity with the braid group.
The braid group and certain quotients, such as the Hecke and Temperley-Lieb algebras are also
discussed. The quantum Lax operator is then introduced, as well as the fundamental algebraic
equation governing the underlying quantum algebras (Yangians and q-deformed Lie algebras). We
construct tensor representations of the underlying algebras, and eventually build the closed
(periodic) transfer matrix of a spin chain-like system. The integrability of the system is shown, and
the corresponding local momentum and Hamiltonian are extracted. Finally, we present
representations of the Uq(sl(2)) algebra and discuss in detail the algebraic Bethe ansatz technique
for diagonalizing the generalized XXZ spin chain.
1.1. Lecture 1: Yang-Baxter equation & the R-matrix; braid group & Hecke algebras
1.2. Lecture 2: The quantum Lax operator & quantum groups
1.3. Lecture 3: The algebraic Bethe ansatz method for the XXZ spin chain
2. Quantum Integrability & Quantum Schubert Calculus (CK)
Tentative Schedule: 14:00-15:30 on 10, 15 and 17 June 2016
An introduction to the combinatorial description of quantum Schubert calculus using
exactly solvable models from statistical mechanics. We start out with some elementary
combinatorics (exact matchings on the honeycomb lattice, plane partitions, nonintersecting paths) and consider their weighted counting which leads to the notion of
partition functions in statistical mechanics. We use the latter to define so-called YangBaxter algebras and identify within these algebras commutative subalgebras which are
isomorphic to the quantum cohomology and quantum K-theory rings of Grassmannians.
The spectrum of these rings can be computed via the Bethe ansatz which leads to simple
combinatorial models and explicit algorithms to compute (K-theoretic) Gromov-Witten
invariants. The lectures will be supplemented by tutorials after each lecture where (with
audience participation) sample computations will be presented which make the abstract
ideas presented in the lectures concrete.
2.1. Lecture 1 – Dimers and free fermions
2.2. Lecture 2 – Quantum cohomology and Bethe’s ansatz
2.3. Lecture 3 – Interacting dimers and quantum K-theory
3. The qKZ Equation and Solvable Lattice Models (RW)
Schedule: 14:00-15:30 on 22, 24 and 29 June 2016
The quantum Knizhnik-Zamolodchikov (qKZ) equation is a system of linear difference equations.
In the context of representation theory (of quantum affine algebras) it was discovered by Frenkel
and Reshetikhin in the early 1990s. The equation can be viewed as a finite-difference version of
the Knizhnik-Zamolodchikov equation which is in turn a system of PDEs satisfied by correlation
functions of primary fields in certain conformal field theories.
The qKZ equation's main physical applications are in 2D integrable quantum field theory and 2D
solvable lattice models of statistical mechanics. In these lectures I shall describe the formulation,
solution and use of the qKZ equation in solvable lattice models: in brief, solutions give exact results
for correlation functions of the 2D lattice models and for the eigenvectors of the associated 1D
quantum integrable systems.
Thus the qKZ equation can be viewed as an alternative to the Bethe Ansatz approach.
3.1. Lecture 1: The formulation and origin of the qKZ equation.
3.1.1. The equation
3.1.2. The representation theory of quantum affine algebras
3.1.3. Matrix elements of vertex operators and the qKZ equation
3.2. Lecture 2: Applications of the qKZ equation in solvable lattice models
3.2.1. Ground states of quantum transfer matrices
3.2.2. Correlation functions
3.3. Lecture 3: Solutions of the qKZ equation.
3.3.1. q-Calculus
3.3.2. Various Solutions
4. The PDE-ODE/Integrable Models Correspondence (PMA)
Schedule: 14:00-15:30 on 1 and 6 July 2016
The aim is to give an introduction to a correspondence between certain linear ODEs defined in the
complex plane and quantum integrable models.
First, we will discuss a particular eigenvalue problem concerning the Schrödinger equation and
derive a functional relation satisfied by associated spectral determinants. The spectrum of the
ODE will be encoded in Bethe Ansatz-type equations.
Then we will show how the above correspondence can be generalised to certain nonlinear
integrable PDEs. To showcase this generalisation, we will consider the sinh-Gordon model and
discuss all the necessary ingredients for the connection: symmetry reductions and asymptotic
solutions to Painleve equations, Lax pair, reduction groups.
There will be 1-2 lectures on this topic, details will be announced during the course.
Delivery
Lectures will normally take place on Wednesdays and Fridays. Each lecture will be followed
by a tutorial or Q&A session. The format might slightly change in each lecture series but we
roughly aim to have each lecture + follow-on tutorial lasting about 1.5 hours in total.