Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1910.01137 · Physical review. B./Physical review. B

Localization from Hilbert space shattering: From theory to physical realizations

Vedika Khemani, Michael Hermele, Rahul Nandkishore

Published May 15, 2020• 480 citations
arXivHTMLPDF

Abstract

Known routes to ergodicity breaking -- integrability and many-body localization -- require extensively many explicit or emergent conservation laws. Here we show that two conservation laws are sufficient to provably and robustly break ergodicity by ``shattering'' Hilbert space into exponentially many dynamically disconnected sectors. This represents a new paradigm for localization, which does not rely on disorder or energy conservation, which works in any number of spatial dimensions, and which should be realizable in near-term ultracold atom experiments.

DOI
10.1103/physrevb.101.174204

Related papers

Ergodicity Breaking Provably Robust to Arbitrary Perturbations

David T. Stephen, Oliver Hart, Rahul Nandkishore

We present a new route to ergodicity breaking via Hilbert space fragmentation that displays an unprecedented level of robustness. Our construction relies on a single emergent (prethermal) conservation law. In the limit when the conservation law is exact, we prove the emergence of Hilbert space fragmentation with an exponential number of frozen configurations. These configurations are low-entanglement states in the middle of the energy spectrum and therefore constitute examples of quantum many-body scars. We further prove that every frozen configuration is absolutely stable to arbitrary perturbations, to all finite orders in perturbation theory. In contrast to previous constructions, our proof is not limited to symmetric perturbations, or to perturbations with compact support, but also applies to perturbations with long-range tails, and even to arbitrary geometrically nonlocal k-body perturbations, as long as k/L→0 in the thermodynamic limit, where L is linear system size. Additionally, we identify one-form U(1) charges characterizing some nonfrozen sectors, and discuss the dynamics starting from typical initial conditions, which we argue is best interpreted in terms of the magnetohydrodynamics of the emergent one-form symmetry.

arXiv:2209.03966•Physical Review Letters•Jan 23, 2024•32 citations

Hilbert space shattering and dynamical freezing in the quantum Ising model

Oliver Hart, Rahul Nandkishore

We discuss quantum dynamics in the transverse field Ising model in two spatial dimensions. We show that, up to a prethermal timescale, which we quantify, the Hilbert space ``shatters'' into dynamically disconnected subsectors. We identify this shattering as originating from the interplay of a $\text{U}(1)$ conservation law and a one-form ${\mathbb{Z}}_{2}$ constraint. We show that the number of dynamically disconnected sectors is exponential in system volume, and includes a subspace exponential in system volume within which the dynamics is exactly localized, even in the absence of quenched disorder. Depending on the emergent sector in which we work, the shattering can be weak (such that typical initial conditions thermalize with respect to their emergent symmetry sector), or strong (such that typical initial conditions exhibit localized dynamics). We present analytical and numerical evidence that a first-order-like ``freezing transition'' between weak and strong shattering occurs as a function of the symmetry sector, in a nonstandard thermodynamic limit. We further numerically show that on the ``weak'' (melted) side of the transition domain wall dynamics follows ordinary diffusion.

Physical review. B./Physical review. B•Dec 21, 2022•39 citations

Hilbert space shattering and dynamical freezing in the quantum Ising model

Hart, Oliver, Rahul Nandkishore

We discuss quantum dynamics in the transverse field Ising model in two spatial dimensions. We show that, up to a prethermal timescale, which we quantify, the Hilbert space 'shatters' into dynamically disconnected subsectors. We identify this shattering as originating from the interplay of a $\mathrm{U}(1)$ conservation law and a one-form $\mathbb{Z}_2$ constraint. We show that the number of dynamically disconnected sectors is exponential in system volume, and includes a subspace exponential in system volume within which the dynamics is exactly localized, even in the absence of quenched disorder. Depending on the emergent sector in which we work, the shattering can be weak (such that typical initial conditions thermalize with respect to their emergent symmetry sector), or strong (such that typical initial conditions exhibit localized dynamics). We present analytical and numerical evidence that a first order-like 'freezing transition' between weak and strong shattering occurs as a function of the symmetry sector, in a non-standard thermodynamic limit. We further numerically show that on the 'weak' (melted) side of the transition domain wall dynamics follows ordinary diffusion.

arXiv:2203.06188•Mar 11, 2022•3 citations

Stark many-body localization: Evidence for Hilbert-space shattering

Elmer V. H. Doggen, I. V. Gornyi, Dmitry G. Polyakov

The ergodic hypothesis lies at the heart of classical statistical physics. A crucial question, therefore, is how this idea translates into the quantum world. Many-body localization -- the analog of Anderson localization to the many-body case -- has emerged as a key example of nonergodicity. Here, the authors analyze a disorder-free Heisenberg spin chain under the influence of a constant field gradient (Stark many-body localization). Surprisingly, they find that nonergodicity results, even for a vanishingly small gradient.

arXiv:2012.13722•Physical review. B./Physical review. B•Mar 4, 2021•60 citations

Ergodicity breaking in a model showing many-body localization

Andrea De Luca, Antonello Scardicchio

We study the breaking of ergodicity measured in terms of return probability in the evolution of a quantum state of a spin chain. In the non-ergodic phase a quantum state evolves in a much smaller fraction of the Hilbert space than would be allowed by the conservation of extensive observables. By the anomalous scaling of the participation ratios with system size we are led to consider the distribution of the wave function coefficients, a standard observable in modern studies of Anderson localization. We finally present a criterion for the identification of the ergodicity-breaking (many-body localization) transition based on these distributions which is quite robust and well suited for numerical investigations of a broad class of problems.

arXiv:1206.2342•Europhysics Letters (EPL)•Feb 1, 2013•295 citations

Strong ergodicity breaking due to local constraints in a quantum system

Sthitadhi Roy, Achilleas Lazarides

This paper shows that local kinetic constraints in random many-body Hamiltonians can lead to ergodicity breaking, despite the Hilbert space not fragmenting. This is purely due to the constraints, in that the unconstrained system is ergodic. The mechanism underlying the emergence of the localized phase is qualitatively different from many-body localization in locally disordered Hamiltonians.

arXiv:1912.06660•Physical Review Research•May 12, 2020•50 citations
Known routes to ergodicity breaking -- integrability and many-body localization -- require extensively many explicit or emergent conservation laws. Here we show that two conservation laws are sufficient to provably and robustly break ergodicity by ``shattering'' Hilbert space into exponentially many dynamically disconnected sectors. This represents a new paradigm for localization, which does not rely on disorder or energy conservation, which works in any number of spatial dimensions, and which should be realizable in near-term ultracold atom experiments.
We present a new route to ergodicity breaking via Hilbert space fragmentation that displays an unprecedented level of robustness. Our construction relies on a single emergent (prethermal) conservation law. In the limit when the conservation law is exact, we prove the emergence of Hilbert space fragmentation with an exponential number of frozen configurations. These configurations are low-entanglement states in the middle of the energy spectrum and therefore constitute examples of quantum many-body scars. We further prove that every frozen configuration is absolutely stable to arbitrary perturbations, to all finite orders in perturbation theory. In contrast to previous constructions, our proof is not limited to symmetric perturbations, or to perturbations with compact support, but also applies to perturbations with long-range tails, and even to arbitrary geometrically nonlocal k-body perturbations, as long as k/L→0 in the thermodynamic limit, where L is linear system size. Additionally, we identify one-form U(1) charges characterizing some nonfrozen sectors, and discuss the dynamics starting from typical initial conditions, which we argue is best interpreted in terms of the magnetohydrodynamics of the emergent one-form symmetry.
We discuss quantum dynamics in the transverse field Ising model in two spatial dimensions. We show that, up to a prethermal timescale, which we quantify, the Hilbert space ``shatters'' into dynamically disconnected subsectors. We identify this shattering as originating from the interplay of a U(1)\text{U}(1)U(1) conservation law and a one-form Z2{\mathbb{Z}}_{2}Z2​ constraint. We show that the number of dynamically disconnected sectors is exponential in system volume, and includes a subspace exponential in system volume within which the dynamics is exactly localized, even in the absence of quenched disorder. Depending on the emergent sector in which we work, the shattering can be weak (such that typical initial conditions thermalize with respect to their emergent symmetry sector), or strong (such that typical initial conditions exhibit localized dynamics). We present analytical and numerical evidence that a first-order-like ``freezing transition'' between weak and strong shattering occurs as a function of the symmetry sector, in a nonstandard thermodynamic limit. We further numerically show that on the ``weak'' (melted) side of the transition domain wall dynamics follows ordinary diffusion.
We discuss quantum dynamics in the transverse field Ising model in two spatial dimensions. We show that, up to a prethermal timescale, which we quantify, the Hilbert space 'shatters' into dynamically disconnected subsectors. We identify this shattering as originating from the interplay of a U(1)\mathrm{U}(1)U(1) conservation law and a one-form Z2\mathbb{Z}_2Z2​ constraint. We show that the number of dynamically disconnected sectors is exponential in system volume, and includes a subspace exponential in system volume within which the dynamics is exactly localized, even in the absence of quenched disorder. Depending on the emergent sector in which we work, the shattering can be weak (such that typical initial conditions thermalize with respect to their emergent symmetry sector), or strong (such that typical initial conditions exhibit localized dynamics). We present analytical and numerical evidence that a first order-like 'freezing transition' between weak and strong shattering occurs as a function of the symmetry sector, in a non-standard thermodynamic limit. We further numerically show that on the 'weak' (melted) side of the transition domain wall dynamics follows ordinary diffusion.
The ergodic hypothesis lies at the heart of classical statistical physics. A crucial question, therefore, is how this idea translates into the quantum world. Many-body localization -- the analog of Anderson localization to the many-body case -- has emerged as a key example of nonergodicity. Here, the authors analyze a disorder-free Heisenberg spin chain under the influence of a constant field gradient (Stark many-body localization). Surprisingly, they find that nonergodicity results, even for a vanishingly small gradient.
We study the breaking of ergodicity measured in terms of return probability in the evolution of a quantum state of a spin chain. In the non-ergodic phase a quantum state evolves in a much smaller fraction of the Hilbert space than would be allowed by the conservation of extensive observables. By the anomalous scaling of the participation ratios with system size we are led to consider the distribution of the wave function coefficients, a standard observable in modern studies of Anderson localization. We finally present a criterion for the identification of the ergodicity-breaking (many-body localization) transition based on these distributions which is quite robust and well suited for numerical investigations of a broad class of problems.