[Paper Review] Quantum many-body scars

Christopher J. Turner, Alexios A. Michailidis+3 more|arXiv (Cornell University)|Nov 9, 2017

Quantum many-body systemsPhysics and Astronomy52 references191 citations

TL;DR

The paper identifies a band of special, scarred eigenstates in a kinetically constrained Fibonacci chain, causing non-ergodic, long-lived dynamics despite overall thermalization.

ABSTRACT

Certain wave functions of non-interacting quantum chaotic systems can exhibit "scars" in the fabric of their real-space density profile. Quantum scarred wave functions concentrate in the vicinity of unstable periodic classical trajectories. We introduce the notion of many-body quantum scars which reflect the existence of a subset of special many-body eigenstates concentrated in certain parts of the Hilbert space. We demonstrate the existence of scars in the Fibonacci chain -- the one- dimensional model with a constrained local Hilbert space realized in the 51 Rydberg atom quantum simulator [H. Bernien et al., arXiv:1707.04344]. The quantum scarred eigenstates are embedded throughout the thermalizing many-body spectrum, but surprisingly lead to direct experimental signatures such as robust oscillations following a quench from a charge-density wave state found in experiment. We develop a model based on a single particle hopping on the Hilbert space graph, which quantitatively captures the scarred wave functions up to large systems of L = 32 atoms. Our results suggest that scarred many-body bands give rise to a new universality class of quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

Motivation & Objective

• Motivate exploration of weak ergodicity breaking in isolated quantum systems beyond integrable and MBL scenarios.
• Demonstrate the existence of a band of scarred many-body eigenstates embedded in the thermal spectrum of a constrained 1D model.
• Show that scars produce observable, robust dynamical signatures starting from experimentally accessible initial states.
• Provide an effective theoretical framework (FSA) to describe scarred states as a single-particle hopping problem on a Hilbert-space graph.

Proposed method

• Study a constrained 1D Fibonacci chain with H = sum_i P_i X_{i+1} P_{i+2} acting on a nontrivial constrained Hilbert space.
• Resolve translation, inversion, and parity symmetries to fully diagonalize systems up to L = 32.
• Identify a band of special eigenstates with large overlaps with a Z2 density-wave state and analyze their energy spacing (~1.33) and structure.
• Construct an effective forward-scattering (Lanczos) basis |n> at fixed Hamming distance from |Z2>, yielding a tridiagonal HFS A Hamiltonian H_FSA.
• Compare H_FSA eigenstates with exact special-band eigenstates to validate the approximation (err(n) ~ 0.2% for L=32).
• Use participation ratios to show special states are concentrated in subregions of Hilbert space and exhibit enhanced PR2 compared to average.

Experimental results

Research questions

• RQ1 Do constrained quantum systems host non-ergodic eigenstates coexisting with thermalizing ones?
• RQ2 Can a tight-binding (Lanczos/FSA) description capture many-body scarred states in a kinetically constrained model?
• RQ3 What are the dynamical signatures of scarred eigenstates starting from experimentally accessible initial states?
• RQ4 Is the observed non-ergodicity indicative of a new universality class beyond ETH, MBL, and integrability?

Key findings

• A band of special eigenstates (Z2-band) coexists with thermalizing states in the middle of the spectrum.
• The energy spacing within the band is approximately Omega ≈ 1.33, matching half the observed oscillation frequency.
• The number of special eigenstates scales linearly with system size, indicating a subextensive but robust scar presence.
• A forward-scattering (FSA) effective model on a Hilbert-space graph with L+1 sites captures the scarred states with high fidelity (overlaps with |n> basis).
• FSA predicts and explains the observed long-time oscillations when starting from Z2-type density-wave states, consistent with experiments.
• Zero modes exist with a Gaussian DOS spike at E=0, with counts tied to Fibonacci numbers depending on boundary conditions and symmetry sectors.