Ergodicity Breaking in Quantum Discrete Time Models

Abstract

In recent years, we have witnessed astonishing technological progress in the capabilities of digital quantum simulators, which hold the potential to solve certain problems that are classically intractable. A particularly promising application is the simulation of the dynamics of quantum many-body systems. Although these devices inherently realise discrete-time models, they can also be programmed to approximate continuous-time evolution. Moreover, in certain circuit models, such as dual-unitary or random unitary circuits, one can analytically compute various quantities of interest. Therefore, there are two aspects to studying discrete-time models. On the one hand, they are inherently realised in digital quantum processors, and on the other hand, they can enable exact calculations.

In this thesis, we focus on studying quantum discrete-time models which display ergodicity breaking mechanisms. The first major result is the demonstration of ergodicity breaking in dual-unitary circuits (DUCs) via quantum many-body scars (QMBS). We present the main properties of DUCs and key results from the literature, followed by a systematic approach to obtain DUCs that host QMBS states, and we highlight the consequences using numerical evidence.

We next consider a different toy model that hosts so-called asymptotic quantum many-body scars (AQMBS), which are exact eigenstates only in the thermodynamic limit. However, their effects can also be observed for finite system sizes, in which they lead to anomalous thermalisation. The considered system hosts both exact QMBS and AQMBS states. The latter are shown to arise from the remnants of gapless excitations that participate in a ground-state localisation transition. The repercussions of these states are demonstrated using classical tensor network simulations and experimental results from a trapped-ion quantum computer, which constitute the first experimental detection of AQMBS.

In the final results chapter we present Complete Hilbert Space Ergodicity (CHSE), a notion for ergodicity in a unitarily evolving quantum system, closely aligned with the classical concept. CHSE can be achieved by breaking time translational invariance in discrete-time models, moving beyond the Floquet paradigm by constructing aperiodic drives. We investigate how conserved quantities, both local and non-local, affect CHSE. We find that in the presence of conserved quantities, whether from QMBS, Hilbert space fragmentation, or symmetries, a system can still exhibit CHSE within dynamically decoupled subspaces.

In conclusion, we have studied several discrete-time models exhibiting ergodicity breaking. Exploiting known properties of DU circuits, we demonstrated the presence of QMBS in provably chaotic and thermalising systems. We also presented the first experimental demonstration of AQMBS states, on quantum computers, underscoring their relevance for investigating new physical phenomena.