Classical and Quantum Monte Carlo Methods Or: Why we know as little as we do about interacting fermions

Classical and Quantum Monte Carlo Methods

• Or: Why we know as little as we do about interacting fermions

Outline

• Introduction to MC

• Quantum and classical statistical mechanics

• Classical Monte Carlo algorithm for the Ising model

• Quantum Monte Carlo algorithm for the Hubbard model

• “Sign problems”

Introduction: Monte Carlo

Introduction: Monte Carlo

Statistical mechanics

Statistical mechanics

Stochastic summation

Stochastic summation (cont.)

Importance Sampling

Sampling Technique

The Metropolis Algorithm

Outline

• Introduction to MC

• Quantum and classical statistical mechanics

• Classical Monte Carlo algorithm for the Ising model

• Quantum Monte Carlo algorithm for the Hubbard model

• “Sign problems”

The Hubbard Model

Determinantal MC

Determinantal MC (2)

Determinantal MC (3)

Determinantal MC (4)

Determinantal MC (5)

Monte Carlo Evaluation

Sign Problem

Sign Problem (2)

Sign Problem (3)

Summary

Summary

References