Abstract:In this lecture I will introduce the basic Hamiltonians used to model quantum magnets and discuss quantum Monte Carlo methods for studying their finite-temperature and ground state properties. I will discuss how a path integral in d+1 dimensions is constructed for a d-dimensional quantum system and then show how an simpler formulation of quantum statistical mechanics is obtained by considering instead a Taylor expansion, which forms the basis of the stochastic series expansion (SSE) quantum Monte Carlo method for studies at temperatures T>0 (and ground state properties can be obtained by taking the low-temperature limit). I will also show how projector methods used to study the ground state can be cast in a form very similar to SSE by using the basis of valence bonds (singlet pairs).
Title: Stochastic series expansion and ground state projection for Heisenberg quantum antiferromagnets
Abstract: Following the previous lecture introducing the basics of classical Monte Carlo simulations and path integrals, in this lecture I will explain the essential principles and some technical details of the stochastic series expansion (SSE) quantum Monte Carlo method, using the 2D Heisenberg antiferromagnet as an example. Following the outline of the principles I will demonstrate the use of a program available online. I will also show how projector methods used to study the ground state can be cast in a form very similar to SSE by using the basis of valence bonds (singlet pairs).
Title: Criticality in classical and quantum magnets
Date: 2014.11.6[video:lecture 3]
Abstract: In this lecture, after a brief introduction to finite-size scaling methods for studying classical critical points, I will discuss similar techniques for analyzing quantum Monte Carlo data for quantum spin systems. I will discuss examples of how the standard antiferromagnetic (Neel) order in the S=1/2 Heisenberg model on the square lattice can be destroyed by quantum fluctuations induced by other interactions, giving way either to a quantum paramagnet, a spin liquid, or a valence-bond solid.
Title: New perspectives on deconfined quantum criticality
Date: 2016.6.7[video:20160607-Anders W. Sandvik-蒙特卡洛4]
Abstract: The concept of the deconfined quantum critical point (DQCP)  has been controversial and stimulated many numerical simulations of quantum spin models for more than 10 years. The theory of the DQCP predicts a continuous quantum phase transition between the standard Neel antiferromagnet (AFM) and a valenc-bond crystal (VBC) in two dimensions. As the transition is approached from the VBC, a second length scale, which diverges faster than the standard correlation length, governs an emergent U(1) symmetry and the deconfinement length scale of fractional S=1/2 excitations (spinons). I will review the history of the DQCP theory and numerical simulations of the AFM-VBS transition and present recent work  suggesting that the DQCP is even more unusual than previously anticipated. The longer length scale affects scaling preoperties more dramatically than in previously known systems with two divergent length scales, and this is the reason for the previously puzzling behaviors found in simulations.
 T. Senthil et al., Science 303, 1490 (2004).
 H. Shao, W. Guo, A. W. Sandvik, Science 352, 213 (2016).