Finite-size effects in 2D QFT

Prerequisite

Basic knowledge of Quantum Mechanics, Quantum Field Theory and Statistical Mechanics is needed, as well as reasonable knowledge of lineal algebra and complex analysis.

Introduction

The aim of this course is to spell out the methods for the evaluation of the finite volume/temperature effects in two-dimensional relativistic quantum field theories (CFT). I will focus on three CFT with one local degree of freedom: the free massive boson, the Ising Field Theory and the sinh-Gordon model. The last one is an interacting CFT with factorised scattering. The finite-volume dynamics of such theories is solved by the Thermodynamical Bethe ansatz (TBA). Here I will present a refined version of the TBA known as exact cluster expansion.

Keywords: Partition function, Path integral, Modular invariance, Casimir energy, Factorised scattering, Gaudin norm, Thermodynamical Bethe Ansatz.

Syllabus

I. Quantum field Theory (QFT) of a free relativistic massive boson with doubly periodic boundary conditions.

Three equivalent representations of the partition sum: 1) as a functional integral over the bosonic field, 2) as a thermal trace of the evolution operator, 3) as a sum over relativistic particles winding around the two cycles of the periodic rectangle. Computation of the free energy in the three representations. Zero-point energy, modular invariance, Casimir energy. Massless limit.

II. Ising field theory on infinite and doubly periodic lattices.

Vdovichenko map onto a gas of loops. Equivalence with the QFT of a free Majorana fermion. Splitting the partition function into four computable blocks. Modular invariance, Casimir energy, Massless limit.

III. Interacting QFTs with factorised scattering.

Properties of the simplest two-particle scattering matrix and general solution in case of diagonal scattering. Scattering matrix for the Sinh-Gordon model. Bethe ansatz and Bethe-Yang equations.

IV. Evaluation of the partition function in the limit of one large and one small cycle

1. Evaluation of the thermal trace by a resolution of the identity. Gaudin measure for the integral over multi-particle states. Graph expansion of the Gaudin determinant. Kirchhoff’s Matrix-Tree theorem. Exact cluster expansion and derivation of the TBA equation.

2. Reformulation of the QFT as a gas of loops winding the two periodic directions and phase factors associated with their intersections. Evaluation of the sum over winding loops by a Hubbard-Stratonovich transformation. TBA equation as a Schwinger-Dyson equation for the Hubbard-Stratonovich fields.

This two-month course will cover about half of the material. The other half will be covered by another two-month course during the next year.

Record

Yes

Audience

Graduate, Postdoc

Language

English

Lecturer Intro

Ivan Kostov obtained his PhD in 1982 from the Moscow State University, with scientific advisers Vladimir Feinberg and Alexander Migdal. Then he worked in the group of Ivan Todorov at the INRNE Sofia, and since 1990 as a CNRS researcher at the IPhT, CEA-Saclay, France. Currently he is emeritus DR CNRS at IPhT and a visiting professor at UFES, Vitoria, Brazil.