Mathematics Colloquia and Seminars

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In this talk, I will consider the family of AKLT models defined on decorated versions of simple, connected graphs. The decoration consists of replacing each edge with a chain of n sites, for a fixed n called the decoration parameter.

I will show that, when the decoration parameter is larger than a linear function of the maximal degree of the graph, the model has a nonvanishing spectral gap above the ground state energy. As part of the proof, I will introduce a family of operators called matchings operators, show that they form a basis of the algebra generated by the Casimir operator of k spin-1/2s, and obtain the coefficients of the projection on the symmetric subspace in that basis.