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We study the matrix elements of few-body observables, focusing on theoff-diagonal ones, in the eigenstates of the two-dimensional transverse fieldIsing model. By resolving all symmetries, we relate the onset of quantum chaosto the structure of the matrix elements. In particular, we show that a generalresult of the theory of random matrices, namely, the value 2 of the ratio ofvariances (diagonal to off-diagonal) of the matrix elements of Hermitianoperators, occurs in the quantum chaotic regime. Furthermore, we explore thebehavior of the off-diagonal matrix elements of observables as a function ofthe eigenstate energy differences, and show that it is in accordance with theeigenstate thermalization hypothesis ansatz.

Eigenstate thermalization in the two-dimensional transverse field Ising model: II. Off-diagonal matrix elements of observables