arXiv:1904.04815

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• Link
• Title: Local constraints can globally shatter Hilbert space: a new route to quantum information protection
• Keywords (optional): fracton, quantum scar
• Authors: Vedika Khemani and Rahul M. Nandkishore

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Problems

• Dipole conservation on ED, count the reduced Hilbert space dimension.
• ED on floquet system: $U=\mathcal{T}\int_0^\tau e^{iH(t)t} dt$, eigenstates of evolution operator.

• This ‘shattering’ may be straightforwardly resolved by numerically extracting the ‘connectivity’ of the Floquet operator

  What does connectivity of Floquet op. mean?

Summary

• The authors first show that there are lots of low entanglement eigenstate in different Q and P sector of a general 3 gates fracton circuit setup.
• The authors then found that there are lots of product state in charge basis are localized due the restriction of conservation on both charge and dipoles. And by recursively induction, such 'inert' states are shown to exponential many with the system size, approxiamated as $2.2^L$.
• Such exponential large inert Hilbert space is a general feature, and also exist in 4/5 gates fracton circuit.
• Non degenerate states (OBC) is critical to obtain such conclusion, otherwise, only polynomial large inert Hilbert space exists. (arXiv:1903.06173)
• They further show that higher dimension of dynamical sector can be constructed, namely the structure of sector in Hilbert space is much richer than expected (P,Q sector).
• Namely, the Hilbert space emerge shattering into various subsectors instead of simple (P,Q) sector.

• There even exist states with strictly zero entanglement in subsectors with dimension greater than one. Thus, the ‘shattering’ of Hilbert space is part of the explanation for the broad distribution of entanglement entropies, but it is not the whole picture.

• Hilbert space fracture is basis dependent, maybe more sectors could be revealed by other basis than naive z basis which could explain the broad distribution of entanglement entropy even after including the consideration on subsectors.
• Note inert states are still measure zero compared to all states like other scar systems. The dynamics on random initial states show that the saturate entropy is well below the Page value.
• PXP model show no shattering by the similar argument for finding inert states in fracton circuits.
• Possible relation with automata. (John Von Neumann et al., “The general and logical theory of automata,” 1951 , 1–41 (1951).)

Comments

• Though the analysis of number of inert states is in general valid and cool, the shattering Hilbert space picture seems not so useful. As mentioned by the author, such block diagonal picture is basis dependent , and actually every Hamiltonian can shatter into all one dimension subsectors in a non trivial way (i.e. diagonalization to eigenstates).

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Similar work

• Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians, Pablo Sala, Tibor Rakovszky, Ruben Verresen, Michael Knap, Frank Pollmann, arXiv:1904.04266

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Follow Ups

• Localization from shattering: higher dimensions and physical realizations
• Experimental probes of Stark many-body localization
• Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian