Course on "Quantum Spin Simulations", April 5-23, 2010.

Instructor: Anders Sandvik Tutor Denis Dalidovich

This course introduces quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Exact diagonalization and quantum Monte Carlo algorithms and their computer implementations are discussed in detail (including the use of lattice symmetries in complete and Lanczos diagonalization studies, and quantum Monte Carlo methods based on the stochastic series expansion as well as ground-state projection in the valence-bond basis). Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S=1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states in two dimensions.

PART 1: Introduction to classical and quantum spin systems and quantum magnetism
[April 05] Classical and quantum spin systems and their significance; origin of quantum antiferromagnetism
[April 06] Spin-wave theory of the Neel state and its quantum fluctuations
[April 07] Non-magnetic states, frustrated interactions, examples of quantum phase transitions
[April 08] Quantum and classical phase transitions, criticality and finite-size scaling behavior
[April 09] Classical Monte Carlo simulations, symmetry-breaking, autocorrelation functions
Ising Program used in tutorials April 12-13: [Ising.f90]
PART 2: Exact diagonalization methods
[April 12] Bit representation of spins, constructing the hamiltonian matrix, fixed-magnetization blocks
[April 13] Translational symmetry and momentum states, reflection symmetry (parity)
[April 14] Semi-momentum states, spin-inversion symmetry, thermodynamics, the Lanczos method
[April 15] Computer implementation of the Lanczos method, convergence properties
[April 16] Properties of spin chains, standard Heisenberg model, frustrated chain, long-range interactions
[April 19] Momentum states in 2D, quantum rotor states of the Heisenberg model
Programs used in tutorial and homework April 15,19,20: [program list and instructions]
PART 3: Quantum Monte Carlo methods
[April 19] Path-integral formulation of quantum statistical mechanics
[April 20] Stochastic series expansion (SSE), implementation for the S=1/2 Heisenberg model
[April 21] More on SSE implementation, tests of correctness, properties of long Heisenberg chains
[April 22] Properties of Heisenberg ladders, 2D systems, spin stiffness, quantum criticality
[April 23] Calculations with the valence-bond basis, Neel-VBS transition in "J-Q" models
Programs used in tutorial and homework April 23: [SSE code and instructions]