Quantum and classical particle systems, 7.5hp

PhD Course at Chalmers/GU, spring term 2017. Held by J. Björnberg and J. Lamers.

Link to the official page for the course.

Lectures in MVL14 on Wednesdays at 10.00 - 11.45.

Plan for the lectures

Part I: percolation

Reading material: Probability on graphs by Geoffrey Grimmett.

  1. (25/1) Introduction to percolation (Grimmett Sections 3.1 and 3.3)
  2. (1/2) Mean-field percolation: the complete graph (Grimmett Sections 11.1 and 11.2)
  3. (8/2) Hexagonal lattice: RSW-estimates and Cardy's formula (Grimmett Sections 5.5 and 5.7)
  4. (15/2) Cardy's formula: continued (Grimmett Section 5.7)
  5. (22/2 - first half) Cardy's formula: finished

Part II: classical Ising and FK-models

  1. (22/2 - second half) Ising and FK-model (Grimmett Sections 8.1 and partly 8.2)
  2. (1/3) FK-models: phase-transition and stochastic comparisons (Grimmett Sections 8.2 - 8.3, partly 4.1)
  3. (8/3) Relation to loop- and six-vertex models

Part III: quantum models (Heisenberg, XXZ and TFIM)

  1. (15/3) Spin in quantum mechanics and spin operators
Note: no lecture on the 22/3!
  1. (29/3) Addition of spins and the mean-field QHF
  2. (5/4) Probabilistic representations: XXX, XXZ, TFIM, correlations
  3. (12/4) Large cycles in the mean-field QHF

Part IV: quantum integrability (Jules Lamers)

Note: no lecture on the 19/4!

Reading material: lecture notes (arXiv:1501.06805).

  1. (24/4) Brief overview of quantum integrability. XXZ spin chain: definition and symmetries (notes Sections 1 and 2.1)
  2. (3/5) Bethe's method for XXZ spin chain (notes Sections 2.2– 2.3)
  3. (10/5) Six-vertex model (notes Section 3)
  4. (17/5 - MVL15) Quantum inverse-scattering method (notes Section 4)

Homework sheets