Thermalization/Relaxation in integrable and free field theories: an Operator Thermalization Hypothesis

Abstract

Free or integrable theories are usually considered to be too constrained to thermalize. For example, the retarded two-point function of a free field, even in a thermal state, does not decay to zero at long times. On the other hand, the magnetic susceptibility of the critical transverse field Ising is known to thermalize, even though that theory can be mapped by a Jordan-Wigner transformation to that of free fermions. We reconcile these two statements by clarifying under which conditions conserved charges can prevent relaxation at the level of linear response and how such obstruction can be overcome. In particular, we give a necessary condition for the decay of retarded Green's functions. We give explicit examples of composite operators in free theories that nevertheless satisfy that condition and therefore do thermalize. We call this phenomenon the Operator Thermalization Hypothesis as a converse to the Eigenstate Thermalization Hypothesis.

Operator thermalisation in $d>2$: Huygens or resurgence

“…First, in singlet models [10], the calculated correlation functions were observed to display exponential decay in even dimensions d > 2. This decay is directly related to the thermalisation later extracted in [13] by arguments which however do not resolve a difference between even and odd dimensions. Since the odd-dimensional singlet model correlation functions do not decay exponentially, the results appear to be in tension with each other.…”

“…Sabella-Garnier et al formulated the operator thermalisation hypothesis in [13] and considered the thermalisation properties of operator correlation functions in a fixed background rather than operator expectation values in the presence of of assumptions on the energy spectrum, as done by the eigenstate thermalisation hypothesis, ETH [1,2]. They discuss thermalisation in terms of an exponentially fast return to equilibrium of operator expectation values in response to a perturbation by the operator in question.…”

“…On the other hand, to which extent it is also a sufficient condition is less understood. In recent work [13,14], a simple criterion has been formulated that guarantees the absence of thermalisation of a given operator, characterised by a lack of exponentially decaying contributions to its linear response function. In addition, it was conjectured that a converse statement can be made and any operator that fails this non-thermalisation condition in fact thermalises.…”

Quantum Chaos in Topologically Massive Gravity

“…In two dimensions, conformal symmetry is powerful enough to allows us to calculate the retarded two-point function for T 00 operator. This calculation of retarded correlator is closly adapted from [32,39].…”

From spin chains to real-time thermal field theory using tensor networks

“…To make this point apparent, let us note that in (1+1)dimensional CFT the retarded two-point function on a line at finite β is fixed by the conformal symmetry, see ref. [17] for a recent overview. For an operator of scaling dimension ∆ and at p = 0, the retarded correlator has 6) in the case M h = 0.…”