Abstract
In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coherent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypothesis (ETH) in theories with a Schwarzian sector. We also elucidate the role of FZZT-type boundary conditions in the Schwarzian theory, shedding light on the physics of microstates associated with ZZ branes and FZZT branes in low dimensional holography.
Highlights
Situations the appearance of the Schwarzian is intimately related to conformal (Virasoro) symmetry and its breaking [1, 3, 4, 8, 23]: an action of the type (1.1) arises as the universal description of such effects in all instances mentioned
In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states
In order to understand the thermalization of unitary closed quantum systems, this approach proposes to study the properties of eigenstates or typical pure states of the associated Hamiltonian and the degree to which operator expectation values in these states approximate those in thermal ensembles
Summary
Situations the appearance of the Schwarzian is intimately related to conformal (Virasoro) symmetry and its breaking [1, 3, 4, 8, 23]: an action of the type (1.1) arises as the universal description of such effects in all instances mentioned. Which behavior ensues depends on the parameters of the theory as well as the state, but is universally classified by elliptic, parabolic and hyperbolic monodromy, (or equivalently, which coadjoint Virasoro orbit the state lies on) This allows us to establish an extended version of the eigenstate thermalization hypothesis (ETH) [16, 21] which includes the usual notion of ETH for simple operators, and extends it to more complex operators, in particular including out-of-time-order correlation functions (OTOCs).. A crucial novel ingredient in our story is the inclusion of effects of FZZT branes [32, 33] in the Liouville description, which descend to certain coherent states in the Schwarzian, as we shall explain Effects of this type have recently been pointed out to arise in the study of the late-time behavior of a certain matrix model designed as a topological completion of JT gravity [34].
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