Professor Fabian Essler

Abstract:

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨𝝁⁢|𝒪|⁢𝝀⟩ in the same macrostate scale as exp⁡(−𝑐𝒪⁢𝐿⁢ln⁡(𝐿)−𝐿⁢𝑀𝒪 𝝁,𝝀), where the probability distribution function for 𝑀𝒪 𝝁,𝝀 is well described by Fréchet distributions and 𝑐𝒪 depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp⁡(−𝑑𝒪⁢𝐿2), where 𝑑𝒪 depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections.

Abstract:

The presence of long-lived oscillations in the expectation values of local observables after quantum quenches has recently attracted considerable attention in relation to weak ergodicity breaking. Here, we focus on an alternative mechanism that gives rise to such oscillations in a class of systems that support kinematically protected gapped excitations at zero temperature. An open question in this context is whether such oscillations will ultimately decay. We provide strong support for the decay hypothesis by considering spin models that can be mapped to systems of weakly interacting fermions, which in turn are amenable to an analysis by standard methods based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. We find that there is a time scale beyond which the oscillations start to decay that grows as the strength of the quench is made small.

Abstract:

We demonstrate that nonlinear response functions in many-body systems carry a sharp signature of interactions between gapped low-energy quasiparticles. Such interactions are challenging to deduce from linear response measurements. The signature takes the form of a divergent-in-time contribution to the response – linear in time in the case when quasiparticles propagate ballistically – that is absent for free bosonic excitations. We give a physically transparent semiclassical picture of this singular behaviour. While the semiclassical picture applies to a broad class of systems we benchmark it in two simple models: in the Ising chain using a form-factor expansion, and in a nonintegrable model — the spin-1 AKLT chain — using time-dependent density matrix renormalization group (tDMRG) simulations. We comment on extensions of these results to finite temperatures.