Dynamics and correlations in quantum many-body systems

Thesis

This thesis is a collection of three exact results on correlation and response functions in integrable systems.

In the first part we study transport (two-point functions) in the 1D Hubbard model. First, we analyze the limit of large on-site repulsion and characterize spin transport as a function of temperature. Then, we consider the case where the model displays non-abelian symmetries, e.g., spin $SU(2)$, and argue that in this case transport is anomalous with dynamical scaling exponent $z=3/2$ and follows \acrshort{kpz} scaling.

In the next two parts we focus instead on (perturbative) nonlinear response functions, which can be expressed as $n$-point correlators with $n>2$. Given the lack of a systematic understanding of the information which can be extracted from these, we compute them in integrable 1D systems. We therefore focus on two distinct scenarios. In the first scenario, we consider a finite temperature system perturbed by some inhomogeneous field coupling to local charges --- thus the perturbations can only accelerate quasiparticles, but not create/annihilate them. In this context, we develop a diagrammatic framework based on generalized hydrodynamics that allows a systematic calculation of nonlinear response functions. We show that, in the hydrodynamic limit, nonlinear response functions qualitatively distinguish between non-interacting and interacting systems. In the second scenario, we instead consider a zero temperature system whose ground state coincides with the quasiparticle vacuum. In this simpler setting we consider more general perturbations that can also create and annihilate quasiparticles. Focusing on the Ising chain as a paradigmatic model we show that the long-time limit of four-point functions (third-order response) grows linearly in time. We interpret these divergences in terms of semiclassical processes where quasiparticles propagate ballistically and scatter when their trajectories intersect.

Authors: Fava, M

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Condensed matter theory