[Paper Review] Constraint-induced breaking and restoration of ergodicity in spin-1 PXP models

Bhaskar Mukherjee, Zi Cai+1 more|arXiv (Cornell University)|Apr 1, 2021

Quantum many-body systems64 references29 citations

TL;DR

This paper investigates how different hardcore constraints in one-dimensional spin-1 PXP models can induce or restore ergodicity, demonstrating that constraint engineering can lead to Hilbert space fragmentation (Model-I), emergent local conserved quantities (Model-II), and full ergodicity (Model-III). Using forward scattering approximation (FSA), the authors show that Z2 oscillations—indicative of non-ergodicity—vanish in Model-III, revealing that ergodicity can be restored by tuning constraints, not just broken.

ABSTRACT

Eigenstate Thermalization Hypothesis(ETH) has played a pivotal role in understanding ergodicity and its breaking in isolated quantum many-body systems. Recent experiment on 51-atom Rydberg quantum simulator and subsequent theoretical analysis have shown that hardcore kinetic constraint can lead to weak ergodicity breaking. In this work, we demonstrate, using 1d spin-1 PXP chains, that miscellaneous type of ergodicity can be realized by adjusting the hardcore constraints between different components of nearest neighbor spins. This includes ETH violation due to emergent shattering of Hilbert space into exponentially many subsectors of various sizes, a novel form of non-integrability with an extensive number of local conserved quantities and strong ergodicity. We analyze these different forms of ergodicity and study their impact on the non-equilibrium dynamics of a Z2 initial state. We use forward scattering approximation (FSA) to understand the amount of Z2-oscillation present in these models. Our work shows that not only ergodicity breaking but an appropriate choice of constraints can lead to restoration of ergodicity as well.

Motivation & Objective

• To explore how varying hardcore constraints in spin-1 PXP models alter ergodicity properties, including ETH violation and non-ergodic behavior.
• To investigate whether ergodicity can be restored in systems that initially break ergodicity via constraint engineering.
• To analyze the non-equilibrium dynamics of a Z2 initial state across three distinct constraint-based models using forward scattering approximation (FSA).
• To identify the role of emergent symmetries and conserved quantities in shaping the spectral and dynamical properties of constrained quantum systems.
• To provide a systematic classification of ergodicity types—fragmented, non-integrable with local conserved quantities, and strongly ergodic—based on constraint structure.

Proposed method

• Define three distinct spin-1 PXP models (I, II, III) by modifying the local projection operators Pi,i+1 to enforce different forbidden configurations: |++⟩ (Model-I), |+0⟩/|0+⟩ (Model-II), and both (Model-III), while preserving PXP form.
• Use exact diagonalization and symmetry projection (K=0, I=+1) to access large system sizes and analyze the Hilbert space structure, including connectivity and degeneracies.
• Apply forward scattering approximation (FSA) to compute the time evolution of the Z2 initial state, tracking the norm of the error vector δn to quantify deviation from ideal coherent oscillations.
• Identify emergent subsectors in Model-I via state connectivity analysis, showing exponential fragmentation into blocks of varying size, including isolated inert states.
• Construct local conserved quantities in Model-II by identifying a set of commuting operators that label disconnected patches of spin-1/2 PXP models.
• Use FSA to compute βn and δn at each step, where δn = ||H−|vn⟩ − βn|vn−1⟩||, to detect when the approximation fails and thus infer the presence of non-ergodic dynamics.

Experimental results

Research questions

• RQ1 How do different hardcore constraints in spin-1 PXP models affect the ergodicity of the many-body spectrum?
• RQ2 Can a system initially exhibiting weak ergodicity breaking (e.g., due to Hilbert space fragmentation) be restored to full ergodicity through constraint engineering?
• RQ3 What is the role of local conserved quantities in stabilizing non-ergodic behavior in constrained quantum systems?
• RQ4 How does the Z2 oscillation amplitude in the non-equilibrium dynamics of a Z2 initial state vary across models with different constraints?
• RQ5 To what extent can forward scattering approximation (FSA) accurately capture the dynamics and ergodicity properties of these constrained PXP models?

Key findings

• Model-I, which allows |++⟩ configurations on top of the traditional spin-1 PXP constraint, exhibits exponential Hilbert space fragmentation into subsectors of varying size, including isolated inert states with zero energy.
• Model-II, which further allows |+0⟩/|0+⟩ configurations, develops an extensive number of local conserved quantities, though the system remains non-integrable due to degeneracies in the conserved quantity spectrum.
• Model-III, which forbids both |00⟩ and |++⟩ configurations, restores strong ergodicity: scar states disappear, and the spectrum becomes fully thermalized.
• The FSA error δn first becomes non-zero at the 5th step in Model-I, indicating the onset of non-trivial dynamics and Z2-oscillation suppression due to fragmentation.
• In Model-II, the FSA error δII₃ is non-zero and scales as 50(2L−9)/[(2L−5)(6L²−45L+95)], confirming persistent non-ergodicity due to the structure of conserved quantities.
• In Model-III, the FSA error δIII₂ appears at the 2nd step, with δIII₂ = 1/[4(4L−11)], indicating rapid loss of Z2 oscillations and strong ergodicity, consistent with ETH-satisfying dynamics.