Monotonicity of eigenstate thermalization hypothesis in two-dimensional systems

We study numerically the enveloping $f$-function of the fluctuation term in eigenstate thermalization hypothesis (ETH) statement. We concentrate on the energy (or entropy) dependence of this function in two-dimensional systems. Our numerical results show that it is, in general, a monotonically increasing function of the entropy. This is in agreement with the general expectation that fluctuations increase with increasing entropy. We show that the $f$-function locally flattens with increasing system-size. The flattening rate is directly proportional to the system size. We also show that the flattening rate is directly proportional to the particle number for systems of same spatial size. This variation of the $f$-function is important for physics at subleading order of the system-size. So, it is relevant for intermediate-size systems (upto a few hundred qubits) which are experimentally accessible. One exception we found is that the $f$-function of the order parameter of a thermal phase transition defy the monotonic behaviour.