The Lyapunov Spectrum of Quantum Thermalisation

The eigenstate thermalisation hypothesis resolves the paradox of emergent thermal or classical behaviour in a closed quantum system by focussing upon local observations. This permits the remainder of the system to act as a bath, thermalisation arising due to a process of de-phasing that gradually reveals the thermal nature of local observables measured in an eigenstate. This is very different from thermalisation in closed classical systems, which is driven by dynamical chaos. We show how quantum thermalisation in closed systems can be recast in a way that is directly related to classical thermalisation. Local observables can be accurately captured by projecting states onto a suitable variational manifold. Evolving on this manifold using the time-dependent variational principle projects the quantum dynamics onto a (semi-)classical Hamiltonian dynamics. Thermalisation in this setting is driven by dynamical chaos. We carry out this procedure for an infinite spin chain in two ways --- using the matrix product state ansatz for the wavefunction and for the thermofield double purification of the density matrix --- and extract the full Lyapunov spectrum of the resulting chaotic dynamics. This provides an alternative perspective upon eigenstate thermalisation, pre-thermalisation and integrability.