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arXiv:2503.16327

Many exact area-law scar eigenstates in the nonintegrable PXP and related models

Andrew N. Ivanov, Olexei I. Motrunich

Published Mar 20, 2025• 1 citation
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Abstract

In this work, we present new, highly non-trivial area-law exact zero-energy eigenstates of the one-dimensional (1D) PXP and related models. We formulate sufficient conditions for a matrix product state to represent an exact zero-energy eigenstate of a given 1D kinetically constrained model and use them to prove our new states. We also demonstrate that all previously known exact eigenstates of PXP-type models satisfy these conditions, and, in fact, can be directly deduced from them. We discuss and demonstrate a remarkably effective general numerical technique for discovering finite-bond-dimension eigenstates residing in degenerate subspaces of a broad class of Hamiltonians. Our results highlight a previously unrecognized structure characteristic of the exponentially large nullspaces in kinetically constrained models, suggesting the possibly of extensively many increasingly complex area-law zero-energy eigenstates in the thermodynamic limit. The important implications of these emergent exact eigenstates for the general thermalization phenomenology are exemplified by one of the states introduced in this work, which we propose is a member of the primary $\mathbb{Z}_2$ quantum many-body scar tower responsible for long-lived revivals in the Rydberg atom chain experiment.

DOI
10.48550/arxiv.2503.16327

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In this work, we present new, highly non-trivial area-law exact zero-energy eigenstates of the one-dimensional (1D) PXP and related models. We formulate sufficient conditions for a matrix product state to represent an exact zero-energy eigenstate of a given 1D kinetically constrained model and use them to prove our new states. We also demonstrate that all previously known exact eigenstates of PXP-type models satisfy these conditions, and, in fact, can be directly deduced from them. We discuss and demonstrate a remarkably effective general numerical technique for discovering finite-bond-dimension eigenstates residing in degenerate subspaces of a broad class of Hamiltonians. Our results highlight a previously unrecognized structure characteristic of the exponentially large nullspaces in kinetically constrained models, suggesting the possibly of extensively many increasingly complex area-law zero-energy eigenstates in the thermodynamic limit. The important implications of these emergent exact eigenstates for the general thermalization phenomenology are exemplified by one of the states introduced in this work, which we propose is a member of the primary Z2\mathbb{Z}_2Z2​ quantum many-body scar tower responsible for long-lived revivals in the Rydberg atom chain experiment.
A recent experiment in the Rydberg atom chain observed unusual oscillatory quench dynamics with a charge density wave initial state, and theoretical works identified a set of many-body "scar states" showing nonthermal behavior in the Hamiltonian as potentially responsible for the atypical dynamics. In the same nonintegrable Hamiltonian, we discover several eigenstates at an infinite temperature that can be represented exactly as matrix product states with a finite bond dimension, for both periodic boundary conditions (two exact E=0 states) and open boundary conditions (two E=0 states and one each E=±sqrt[2]). This discovery explicitly demonstrates the violation of the strong eigenstate thermalization hypothesis in this model and uncovers exact quantum many-body scar states. These states show signatures of translational symmetry breaking with a period-2 bond-centered pattern, despite being in one dimension at an infinite temperature. We show that the nearby many-body scar states can be well approximated as "quasiparticle excitations" on top of our exact E=0 scar states and propose a quasiparticle explanation of the strong oscillations observed in experiments.
Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realised by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie algebra. We demonstrate why a careful usage of the forward scattering approximation (FSA), instead of any other method, is required to extract the most appropriate set of Lanczos coefficients in this case as the consequence of this approximate symmetry. Only such a method leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely|Z2⟩,|Z3⟩and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries.
Kinetically constrained systems, such as those representing Rydberg atom arrays in the blockade regime, have gathered considerable attention due to the presence of atypical eigenstates in their spectrum. The latter manifests itself through the presence of quantum many-body scars as well as unusually large zero-mode (ZM) subspaces which contain analytically tractable eigenstates with various entanglement scalings. In particular, some of the latter are excitable zero-modes (EZMs), meaning that they can be promoted to a non-zero energy for open boundary conditions. In this work, I present an efficient protocol for finding analytical expressions for translation-invariant EZMs in constrained systems, based on the eigendecomposition of the local unconstrained Hamiltonian. I demonstrate the power of this method on a decorated Rydberg chain. In that model, my protocol directly produces a continuous matrix-product-state manifold located entirely in the zero-energy eigenspace for periodic boundary conditions. The span of that manifold grows exponentially with the system, and for odd system sizes it covers the entire zero-momentum eigenspace with zero energy. I then show how the physical structure of the manifold, which is tied to the local eigenbasis, also allows one to derive analytical expressions for ZMs at non-zero momentum and for a polynomial number of exact scars at E=±3E=\pm\sqrt{3}E=±3​.
Quantum many-body scarred systems host nonthermal excited eigenstates immersed in a sea of thermal ones. In cases where exact expressions for these special eigenstates are not known, it is computationally demanding to distinguish them from their exponentially many thermal neighbors. We propose a matrix-product-state (MPS) algorithm, dubbed DMRG-S, to extract such states at system sizes far beyond the scope of exact diagonalization. Using this technique, we obtain scarred eigenstates in Rydberg-blockaded chains of up to 80 sites and perform a finite-size scaling study to address the lingering question of the stability for the Néel state revivals in the thermodynamic limit. Our method also provides a systematic way to obtain exact MPS representations for scarred eigenstates near the target energy without a priori knowledge. In particular, we find several new scarred eigenstates with exact MPS representations in kinetically constrained spin and clock models. The combination of numerical and analytical investigations in our work provides a new methodology for future studies of quantum many-body scars.
We discover a variety of exact quantum many-body scars in higher-spin kinetically constrained models through the recently developed DMRG-S algorithm [Zhang et al., Phys. Rev. Lett. 131, 020402 (2023)]. Specifically, for the higher-spin PXP model on arbitrary bipartite lattices of any spatial dimension, we find exact many-body scars that are equidistantly spaced in the energy spectrum and exhibit similar structures to the ground state of the Affleck-Kennedy-Lieb-Tasaki model. For the one-dimensional Fermi-Hubbard model with a tilted potential in a certain parameter regime, whose effective model is equivalent to a kinetically constrained spin model with four degrees of freedom on each site, we find several many-body scars at energy E=0E=0E=0 and $E=\ifmmode\pm\else\textpm\fi{}\sqrt{2}$ that can be exactly represented as matrix product states with finite bond dimensions. Our results demonstrate that larger local degrees of freedom in the kinetically constrained models provide a much broader space for the emergence of quantum many-body scars and weak ergodicity breaking.
Eigenstate thermalization in quantum many-body systems implies that eigenstates at high energy are similar to random vectors. Identifying systems where at least some eigenstates are nonthermal is an outstanding question. In this Letter we show that interacting quantum models that have a nullspace-a degenerate subspace of eigenstates at zero energy (zero modes), which corresponds to infinite temperature, provide a route to nonthermal eigenstates. We analytically show the existence of a zero mode which can be represented as a matrix product state for a certain class of local Hamiltonians. In the more general case we use a subspace disentangling algorithm to generate an orthogonal basis of zero modes characterized by increasing entanglement entropy. We show evidence for an area-law entanglement scaling of the least-entangled zero mode in the broad parameter regime, leading to a conjecture that all local Hamiltonians with the nullspace feature zero modes with area-law entanglement scaling and, as such, break the strong thermalization hypothesis. Finally, we find zero modes in constrained models and propose a setup for observing their experimental signatures.