Spin glass

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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.


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.

simple frustrated system


Ongoing research
