# Spin glass

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## Contents

### Introduction

A spin glass is a certain class of magnetic systems. Especially, spin glasses are highly frustrated disordered materials. Disorder is given either by a disordered structure or by a disordered magnetic doping. Frustration comes into play as the interactions between the spins may be in conflict, which means that there may exist spin coupling such that no spin configuration satisfying all interactions can be found. Exact calculations of ground states of Ising spin glasses and the ground state calculations of 2D planar Ising spin glasses are examples of a part of our research.

### Ising models

Ising spin glass models are very important in statistical physics. An Ising spin glass model can be formulated as a graph as follows. Consider a graph $G=(V,E)$ with vertex set $V$, representing lattice sites (spins), and edge set $E$, representing near-neighbour interactions. Each edge $e=(i,j) \in E$ is assigned a real weight $J_{ij}$ representing the interaction energy or coupling constant. Every vertex $i \in V$ has a magnetic spin variable $S_i$ associated with it. $S_i$ can take two values $S_i = \pm 1$. Here we interpret +1 as spin up and -1 as spin down. A state or a spin configuration $S$ is an assignment of $\pm 1$ to every spin variable $S_i, \forall i \in V$. The energy of a spin configuration $S$ (without an external magnetic field) is given by the Hamiltonian $H=-\sum_{e=(i,j)\in E}J{_ij}S_iS_j.$ Thus a spin configuration $S$ minimizes the energy of two adjacent spins if they point in the same direction if the interaction energy connecting them is positive or point in different directions if the interaction energy is negative. Furthermore, a spin configuration of minimum energy $H$ is called a ground state.

### Example

In the figure we depicted a $\pm J$ Ising spin glass with three spins. The interaction energy between spins $S_1$ and $S_2$ and between $S_1$ and $S_3$ is positive and negative between $S_2$ and $S_3$.

As mentioned a spin configuration $S$ satisfies an interaction if the two spin system attains minimal energy. In this example there exists no spin configuration $S$ minimizing the energy on all couplings.

### Ongoing research

Spin glasses are of high interest with an ongoing and rich history of research. Not only to physicists but e.g. as the computation of ground states for spin glasses can be modeled as a combinatorial optimization problem which is NP-hard in general, spin glasses are also interesting from the optimization point of view.