Operator thermalization vs eigenstate thermalization

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We study the characteristics of thermalizing and non-thermalizing operators in integrable theories as we turn on a non-integrable deformation.

• Specifically, we show that $\sigma^z$, an operator that thermalizes in the integrable transverse field Ising model, has mean matrix elements that resemble ETH, but with fluctuations around the mean that are sharply suppressed.
• This suppression rapidly dwindles as the Ising model becomes non-integrable by the turning on of a longitudinal field.
• We also construct a non-thermalizing operator in the integrable regime, which slowly approaches the ETH form as the theory becomes non-integrable.