Classical Spin Models: An Introduction

Keisuke Fujii

This is a short review on an interdisciplinary field of quantum information science and statistical mechanics. We first give a pedagogical introduction to the stabilizer formalism, which is an efficient way to describe an important class of quantum states, the so-called stabilizer states, and quantum operations on them. Furthermore, graph states, which are a class of stabilizer states associated with graphs, and their applications for measurement-based quantum computation are also mentioned. Based on the stabilizer formalism, we review two interdisciplinary topics. One is the relation between quantum error correction codes and spin glass models, which allows us to analyze the performances of quantum error correction codes by using the knowledge about phases in statistical models. The other is the relation between the stabilizer formalism and partition functions of classical spin models, which provides new quantum and classical algorithms to evaluate partition functions of classical spin models.

Mirjana Božić, Z. Marić

Marek Czachor

Bertrand Berche, Alexander López

A didactic description of the thermodynamic properties of classical spin systems is given in terms of their quantum counterpart in the Hamiltonian limit. Emphasis is on the construction of the relevant Hamiltonian, and the calculation of thermal averages is explicitly done in the case of small systems described, in Hamiltonian field theory, by small matrices.

John B. Parkinson, DAMIAN J. J. FARNELL

Wolfhard Janke E

This lecture gives an introduction to data correlations and error estimation in Monte Carlosimulations. The basic concepts are illustrated for studies of second-order phase transitions in classical spin systems such as the Ising model and for models exhibiting first-order phasetransitions such as the q-state Potts model. 1 Introduction The statistical mechanics of complex systems poses many hard problems which are oftendifficult to solve by analytical approaches. Numerical simulation techniques will therefore be indispensable tools on our way to a better understanding of such systems. Applicationsrange from physics and chemistry to biological systems and even sociology and economy. Examples include (spin) glasses, disordered magnets, or biologically motivated studies ofprotein folding, to mention only a few important problems in classical physics. The broad class of quantum statistical problems in condensed matter and elementary particles physicsas well as non-perturbative approaches to quantum gravity are further important examples.