Dynamics and equilibration in integrable and solvable interacting models

Abstract

This thesis studies the dynamics and equilibration of interacting quantum systems in which integrability, or the structure of conserved charges, shapes the long-time behavior. The In- troduction (Chapter 1) sets up machinery that the later chapters draw on: linear response and the fluctuation-dissipation theorem, large deviations and cumulant-generating functions, quantum state ensembles, and spectral diagnostics of chaos. With these in place, the four research chapters trace a single arc. They move from closed-system transport, to full counting statistics in boundary-driven steady states, to projected ensembles in systems with extensively many conserved charges, and finally to chaos diagnostics suited to near-term quantum hardware. The common technical thread is matrix product state methods (introduced in Chapter 2), combined with analytic input from (quasi-)local conserved quantities, to access regimes where integrability or solvability produces behavior outside the reach of generic thermalization arguments. Chapter 3 applies time-dependent DMRG to thermal transport in a chiral Z3 clock chain tuned along an integrable, time-reversal symmetric line. The thermal current’s finite overlap with a local conserved charge Q(2) obtained from the transfer matrix yields a Mazur bound that saturates the Drude weight at finite temperature, validated against a sum rule. Chapter 4 studies the boundary-driven XXZ chain, whose non-equilibrium steady state under maximally polarising Lindblad operators admits an exact matrix product solution. In the XX limit we obtain closed-form spin correlators, entropy per site, and SCGFs of local observables. For ∆ > 0 a cosine of a rational multiple of π, the MPS yields numerically exact SCGFs and rate functions for the local z-magnetisation, with finite-size corrections decaying exponentially at a rate discontinuous in ∆. A double-peak structure in the x-magnetisation density for ∆ ≲ 1 is suggestive of short-range ferromagnetic ordering. Chapter 5 uses the projected ensemble—subsystem states conditioned on measurements of the complement, to probe deep thermalisation in systems with extensively many (quasi-)local conserved charges. In a Floquet spin chain deep in its MBL regime and the ℓ-bit model as a 1-local archetype, the late-time projected ensemble converges to a Scrooge ensemble except when the measurement operator aligns with the conserved charges, a behaviour we link to the emergence of Porter–Thomas bitstring statistics and justify with semi-analytical arguments. Chapter 6 carries these diagnostics onto probes scalable on current quantum hardware, studying the mixed-field quantum Ising model on Erdős–Rényi graphs whose connectivity interpolates the dynamics between localized, chaotic, and permutation-symmetric integrable regimes–a crossover with implications for the trainability of variational algorithms such as QAOA. The regimes are characterized through convergence of the projected ensemble to Haar, a partial spectral form factor as a resource-efficient SFF proxy, and operator Krylov complexity, which is maximized in the chaotic regime. Taken together, the four chapters trace a path in which the role of conservation laws on quantum dynamics is investigated with progressively sharper probes.